
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00004831/00001
Material Information
 Title:
 An improved algorithm for the determination of the system parameters of a visual binary by least squares
 Creator:
 Xu, YuLin, 1945
 Publication Date:
 1988
 Language:
 English
 Physical Description:
 xi, 138 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Approximation ( jstor )
Covariance ( jstor ) Error rates ( jstor ) Least squares ( jstor ) Matrices ( jstor ) Newtons method ( jstor ) Observational astronomy ( jstor ) Orbitals ( jstor ) Orbits ( jstor ) Space observatories ( jstor ) Double stars  Orbits ( lcsh ) Least squares ( lcsh )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1988.
 Bibliography:
 Includes bibliographical references.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by YuLin Xu.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 001098622 ( ALEPH )
AFJ4467 ( NOTIS ) 19508136 ( OCLC ) AA00004831_00001 ( sobekcm )

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Full Text 
AN IMPROVED ALGORITHM FOR
THE DETERMINATION OF THE SYSTEM PARAMETERS
OF A VISUAL BINARY BY LEAST SQUARES
By
YULIN XU
L_
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988
UNIVERSITY OF FLORIDA LIBRARIES
AN IMPROVED ALGORITHM FOR
THE DETERMINATION OF THE SYSTEM PARAMETERS
OF A VISUAL BINARY BY LEAST SQUARES
By
YULIN XU
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988
UNIVERSITY OF FLORIDA LIBRARIES
To my mother and late father
ACKNOWLEDGEMENTS
The author acknowledges his heartfelt gratitude to
Dr. Heinrich K. Eichhorn, his research advisor, for
proposing this subject, the considerate guidance and
encouragement throughout his research, and for the patience
in reading and correcting the manuscript. The author has
benefited in many ways as Dr. Eichhorn's student.
The author is also grateful to Drs. KwanYu Chen,
Haywood C. Smith, Frank Bradshaw Wood and Philip Bacon for
having served as members of his supervisory committee and
for helpful discussions, timely suggestions and the careful
review of this dissertation. Likewise, his deep apprecia
tion goes to Drs. W. D. Heintz and H. A. Macalister for
having provided data used in his dissertation.
The author is especially grateful to Drs. Jerry L.
Weinberg and RuTsan Wang in the Space Astronomy Laboratory
for their considerate encouragement and support. Without
their support, the fulfillment of this research could not be
possible.
It is a great pleasure to acknowledge that all the
calculations were performed on the Vax in the Space
Astronomy Laboratory.
in
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS in
LIST OF TABLES vi
LIST OF FIGURES viii
ABSTRACT X
CHAPTERS
I INTRODUCTION 1
II REVIEW AND REMARKS 4
Review of the Methods for OrbitComputation... 4
Definitions of the Orbital Parameters.... 7
The Method of M. Kowalsky and
S. Glasenapp 8
The Method of T. N. Thiele, R. T. A.
Innes and W. H. van den Bos 13
The Method of Least Squares 16
Remarks 23
III GENERAL CONSIDERATIONS 26
The Condition Equations 27
The General Statement of the Least Squares
Orbit Problem 33
About the Initial Approximate Solution 36
IV THE SOLUTION BY NEWTONS METHOD 39
The Solution by Newton's Method 39
Weighting of Observations 62
The Orthogonal Set of Adjustment Parameters
and the Efficiency of a Set of Orbital
Parameters 65
A Practical Example 70
Remarks 84
IV
page
V THE MODIFIED NEWTON SCHEME 85
The Method of Steepest Descent 86
The Combination of Newton's Method with the
Method of Steepest DescentThe Modified
Newton Scheme 92
The Application of Marquardt's Algorithm 98
Two Practical Examples 101
VI DISCUSSION 134
REFERENCES 136
BIOGRAPHICAL SKETCH 138
v
LIST OF TABLES
Table Page
41 Expressions for All the Partial Derivatives
in 48
42 Expressions for All the Partial Derivatives
in f 53
43 The Observation Data for 51 Tau 73
44 The Reduced Initial Data for 51 Tau 74
45 The Initial Approximate Solution g for 51 Tau... 77
46 The Final Solution for 51 Tau 78
47 The Residuals of the Observations for 51 Tau
in (p,6) and (x,y) 80
51 The Observation Data for (3738 105
52 The Reduced Initial Data for 3738 106
53 The Initial Approximate Solution g for 3738 109
54 The Solution #1 for 3738 110
55 The Residuals of the Observations for 3738
in (p, 6) and (x,y) in Solution #1 112
56 Heintz' Result for 3738 116
57 The Solution #2 for 3738 117
58 The Residuals of the Observations for 3738
in (p, 6) and (x,y) in Solution #2 119
59 The Observation Data for BD+195116 123
510 The Reduced Initial Data for BD+195116 124
511 The Initial Approximate Solution g for
BD+19 5116 127
vi
Table page
512 The Final Solution for BD+195116 by the
MQ Method 128
513 The Residuals of the Observations for BD+195116
in 6,p,x and y 130
vii
LIST OF FIGURES
Figure Page
41 Plot of the observation data for 51 Tau in
the xgy0 plane 75
42 Plot of a) the Xg b) ygcoordinates against
the observing epochs of the observation data
for 51 Tau 76
43 The residuals of the observation for 51 Tau
in ( p, 6 ) 81
44 The residuals of the observations for 51 Tau
in (x,y) 82
45 The original observations for 51 Tau compared
with the observations after correction 83
51 Plot of the observation data for (3738 in the
xgyg plane 107
52 Plot of a) the Xg b) ygcoordinates against
the observing epochs of the observation data
for [3738 108
53 The residuals of the observations for 3738
in (p,e) according to the solution #1 113
54 The residuals of the observations for 3738
in (x,y) according to the solution #1 114
55 The original observations of 3738 compared
with the observations after correction
according to the solution #1 115
56 The residuals of the observations for 3738
in (p,9) according to the solution #2 120
57 The residuals of the observations for 3738
in (x,y) according to the solution #2 121
viii
page
Figure
58 The original observations for 3738 compared
with the observations after correction
according to the solution #2 122
59 Plot of the observation data for BD+195116
in the XqYo plane 125
510 Plot of a) the xg b) ygcoordinates against
the observing epochs of the observations for
BD+195116 126
511 The residuals of the observations for
BD+195116 in (p,6) 131
512 The residuals of the observations for
BD+195116 in (x,y) 132
513 The original observations for BD+195116
compared with the observations after
correction 133
xx
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
AN IMPROVED ALGORITHM FOR
THE DETERMINATION OF THE SYSTEM PARAMETERS
OF A VISUAL BINARY BY LEAST SQUARES
By
YULIN XU
April 1988
Chairman: Dr. Heinrich K. Eichhorn
CoChairman: Dr. KwanYu Chen
Major Department: Astronomy
The problem of computing the orbit of a visual binary
from a set of observed positions is reconsidered. It is a
least squares adjustment problem, if the observational
errors follow a biasfree multivariate Gaussian distribution
and the covariance matrix of the observations is assumed to
be known.
The condition equations are constructed to satisfy both
the conic section equation and the area theorem, which are
nonlinear in both the observations and the adjustment
parameters. The traditional least squares algorithm, which
employs condition equations that are solved with respect to
the uncorrelated observations and either linear in the
adjustment parameters or linearized by developing them in
Taylor series by firstorder approximation, is inadequate in
x
our orbit problem. D. C. Brown proposed an algorithm
solving a more general least squares adjustment problem in
which the scalar residual function, however, is still
constructed by firstorder approximation. Not long ago, a
completely general solution was published by W. H. Jefferys,
who proposed a rigorous adjustment algorithm for models in
which the observations appear nonlinearly in the condition
equations and may be correlated, and in which construction
of the normal equations and the residual function involves
no approximation. This method was successfully applied in
our problem.
The normal equations were first solved by Newton's
scheme. Practical examples show that this converges fast if
the observational errors are sufficiently small and the
initial approximate solution is sufficiently accurate, and
that it fails otherwise. Newton's method was modified to
yield a definitive solution in the case the normal approach
fails, by combination with the method of steepest descent
and other sophisticated algorithms. Practical examples show
that the modified Newton scheme can always lead to a final
solution.
The weighting of observations, the orthogonal para
meters and the "efficiency" of a set of adjustment
parameters are also considered. The definition of
"efficiency" is revised.
xi
CHAPTER I
INTRODUCTION
The problem of computing the orbit of a visual binary
from a set of observed positions is by no means new.
A great variety of methods has been proposed. As is well
known, only a few of these suffice to cover the practical
contingencies, and the majority fails to handle the input
data efficiently and properly.
For the visual binary case, the determination of an
orbit normally requires a large number of observations. All
measures of position angles and separations are, as are all
observations, affected by observational errors. For the
purpose of our work, these errors are assumed to follow a
biasfree multivariate Gaussian distribution. Under this
assumption, orbitcomputing is a least squares adjustment
problem, in which the condition equations are nonlinear in
both the observations and the adjustment parameters. The
condition equations must incorporate all relationships that
exist between observations and the orbital parameters,*
*Usually, the term "orbital elements" is used. We prefer
"orbital parameters" instead. Strictly speaking, the
orbital elements are the constants of integration in the
twobody problem and, therefore, do not include the masses
of the components.
1
2
that is, must state both the area theorem, which follows
from Kepler's equation, and the condition that the projected
orbit is a conic section. H. Eichhorn (1985) has suggested
a new form for the construction of a complete set of
condition equations for this very problem.
The traditional least squares algorithm, which is based
on condition equations linearized with respect to the
observational errors, will not lead to those orbital
parameters which minimize the sum of the squares of observa
tional errors, because linearization, in this case, is too
crude an approximation. In some earlier papers, H. Eichhorn
and W. G. Clary (1974) proposed a least squares solution
algorithm, which takes into account the second (in addition
to the first) order derivatives in the adjustment residuals
(observational errors) and the corrections to the initially
available approximation to the adjustment parameters. A
completely general solution was published by W. H. Jefferys
(1980, 1981), who proposed a rigorous adjustment algorithm
for models in which the observations appear nonlinearly in
the condition equations. In addition, there may be non
linear constraints** among model parameters, and the
observations may be correlated. In practice, the method is
nearly as simple to apply as the classical method of least
**We use the term "constraints" for condition equations
which do not contain any observations explicitly.
squares, for it does not require calculation of any deriva
tives of higher than the first order.
In this paper, we present an approach to solve the
orbit problem by Jefferys' method, in which both the area
theorem and the conic section equation assume the function
of the condition equations.
CHAPTER II
REVIEW AND REMARKS
Review of the Methods for OrbitComputation
Every complete observation of a double star supplies us
with three data: the time of observation, the position
angle of the secondary with respect to the primary, and the
angular distance (separation) between the two stars. The
problem of computing the socalled orbital elements (in this
paper, called orbital parameters) of a visual binary from a
set of observations superficially appears analogous to the
case of orbits in the planetary system, yet in practice
there is little resemblance between the problems. The
problem of determining the orbit of a body in the solar
system is complicated by the motion of the observer who
shares the motion of the Earth, so that, unlike in the case
of a binary, the apparent path is not merely a projection of
the orbit in space onto the celestial sphere.
In the case of a binary, the path observed is the
projection of the motion of the secondary round the primary
onto a plane perpendicular to the line of sight. The
apparent orbit (i.e., the observed path of the secondary
about the primary) is therefore not a mere scale drawing of
the true orbit in space. The primary may be situated at any
4
point within the ellipse described by the secondary and, of
course, does not necessarily occupy either a focus or the
center.
5
The problem of deriving "an orbit" (meaning a set of
estimates of the orbital parameters) from the observations
was first solved by F. Savary in 1827. In 1829, J. F. Encke
quickly followed with a different solution method which was
somewhat better adapted to what were then the needs of the
practical astronomer. Theoretically, the methods of Savary
and Encke are excellent. But their methods utilize only
four complete pairs of measures (angle and distance) instead
of all the available data and closely emulate the treatment
of planetary orbits. They are therefore inadequate in the
case of binary stars (W. D. Heintz, 1971; R. G. Aitken,
1935) .
Later, Sir John Herschel (1832) communicated a
basically geometric method to the Royal Astronomical
Society. Herschel's method was designed to utilize all the
available data, so far as he considered them reliable.
Since then, the contributions to the subject have been many.
Some consist of entirely new methods of attack, others of
modifications of those already proposed. Among the more
notable investigators are Yvon Villarceau, H. H. Madler, E.
F. W. Klinkerfues, T. N. Thiele, M. Kowalsky, S. Glasenapp,
H. J. Zwiers, K. Schwarzchild, T. J. J. See, H. N. Russell,
R. T. A. Innes, W. H. van den Bos and others.
One may classify the various methods published so far
into "geometric" methods, which are those that enforce only
6
the constraint that the orbit is elliptical, and the
"dynamical" ones which enforce, in addition, the area
theorem.
The geometric treatment initiated by J. Herschel peaked
in Zwiers' method (1896) and its modifications, e.g. those
of H. M. Russell (1898) and of G. C. Comstock (1918). Every
geometric method has the shortcoming that it must assume the
location of the center of the ellipse to be known while it
ignores the area theorem and thus fails to enforce one of
the constraints to which the observations are subject. The
growing quantity and quality of observations called for
suitable computing precepts, and the successful return to
dynamical methods began with van den Bos (1926).
Of the many methods for orbitcomputation formulated,
some are very useful and applicable to a wide range of
problems, e.g. those by Zwiers, Russell and those by Innes
and van den Bos.
Zwiers' method (1896) is essentially graphical and
assumes that the apparent orbit has been drawn. This method
is therefore useless unless the apparent ellipse gives a
good geometrical representation of the observations and
satisfies the law of areas, and thus will not be further
described here since we are primarily concerned with the
analytical methods.
7
In the following, we will briefly review Kowalsky's
method and that by Thiele and Innes.
Definition of the Orbital Parameters
Seven parameters define the orbit and the space
orientation of its plane. The first three of these (P,T,e)
are dynamical and define the motion in the orbit; the last
four (a,i,i2,w) are geometrical and give the size and
orientation of the orbit. The parameters are defined
somewhat differently from those for the orbits of planets
and comets.
The first dynamical parameter P is the period of
revolution, usually in units of mean sidereal years; n is
the mean (usually annual) angular motion; since n=2n/P, P
and n are equivalent. The second, T, is the epoch of
periastron passage (usually expressed in terms of years and
fractions thereof). The third, e, is the eccentricity of
the orbital ellipse.
The geometrical parameter a is the angle subtended by
the semimajor axis of the orbital ellipse (usually
expressed in units of arcseconds). The angle i is the
inclination of the orbital plane to the plane normal to the
line of sight, that is, the angle between the plane of
projection and that of the orbit in space. It ranges from 0
to 180. When the position angle increases with time, that
is, for direct motion, i is between 0 and 90; for retro
grade motion, i is counted between 90 and 180;
8
and i is 90 when the orbit appears projected entirely onto
the line of nodes. The "node", ft, is the position angle of
the line of intersection between the tangential plane of
projection and the orbital plane. There are two nodes whose
corresponding values of ft differ by 180. That node in
which the orbital motion is directed away from the sun is
called the ascending node. We understand ft, which ranges
from 0 to 360, to refer to the ascending node. Because it
is, however, one of the peculiarities of the orbit
determination of a visual binary that it is in principle
impossiblefrom positional data aloneto decide whether
the node is the ascending or descending one ft may be
restricted to 0
the periastron in the plane of the orbit, counted positive
in the direction of the orbital motion and starting at the
ascending node. It ranges from 0 to 360.
These definitions are adhered to throughout our work.
Some of them may somewhat differ a little from those given
by previous authors. But any way in which one defines them
does not affect the principles of the method we describe.
The Method of M. Kowalsky and S. Glasenapp
This old method was first introduced by J. Herschel in
a rather cumbersome form and is better known in its more
direct formulation by M. Kowalsky in 1873 and by
S. Glasenapp in 1889. R. G. Aitken (1935) gives the
9
detailed derivation of the formulae in his textbook The
Binary Stars.
Kowalsky's method is essentially analytical. It
derives the orbit parameters from the coefficients of the
general equation of the ellipse which is the orthogonal
projection of the orbit in space, the origin of coordinates
being taken at the primary. The projected orbit can be
expressed by a quadratic in x and y, whose five coefficients
are related to those five orbital parameters which do not
involve time.
In rectangular coordinates, the equation of an ellipse,
and thus of a conic section, takes the form
c^x2 + 2c2xy + cjy2 + 2C4X + 2c5y +1=0 (21)
where the rectangular coordinates (x,y) are related to the
more commonly directly observed polar coordinates (p,9) by
the equations
x = pcosG ,
y = psine ,
(22a)
(2 2b)
where p is the measured angular distance and 0 the position
angle.
10
The five coefficients of equation (21) can be deter
mined by selecting five points on the ellipse or by all the
available observations in the sense of least squares.
There is an unambiguous relationship between the five
coefficients (0^,02,03,04,05) and the five orbital para
meters (a,e,i,oj,Q) We can find the detailed derivation of
the formulae in Aitken's The Binary Stars. Here, we will
therefore state only the final formulae without derivation.
The five orbital parameters (a,e,i,a>,Â£2) can be calcu
lated from the known coefficients (C1/C2/C3,04,05) by the
following procedure.
1) The parameter Q can be found from the equation
2(c^c.Cr)
2 4 5
(23)
tan2fi
To determine in which quadrant Q is located, we can use
two other equations:
c'sin2iJ = 2(C2~C4C5) ,
c'cos2Q = C52C42+C1C3
(23'b)
(23'a)
where
2.
tan 1
c
(23c)
2
P
which is always positive.
11
More elegantly, we write in Eichhorn's notation
= plg[2(c2C4C5) ,C52C42+c1C3]
(23d)
H. Eichhorn (1985), in his "Kinematic Astronomy"
(unpublished lecture notes), defines the plg(x,y) function
as follows:
plg(x,y) = arctan(x/y) + 90[2sgnx(1+sgny)] ,
where arctan is the principal value of the arctangent.
2)The inclination i is found from
tan^i = 2 +
(24)
Whether the i is in the first or second quadrant is deter
mined by whether the motion is direct or retrograde. If the
motion is direct, the position angle increases with time,
i<90; otherwise, i>90.
3) The equation
u = plg[l(c4sinQC5COsQ)cosi,C4COsQ+C5sinfi) (25)
gives the value of w.
4) With i,o),n known, two more parameters (a and e) can be
calculated from
12
2cos2Â£2(c4sinc,cosQ)cosi
e
(26)
2cos22
a
(27)
[cos2Q( c^^+c^^clc3 ) (Ce2c^+clc3 ) ] (ie^)
5) To complete the solution analytically, the mean motion n
(or the period P) and the time of periastron passage T, must
be found from the mean anomaly M, computed from the observa
tions by Kepler's equation:
(28)
M = n(tT) = EesinE
where E is the eccentric anomaly. Every M will give an
equation of the form
(29)
M = nt + T
where T=nT.
From these equations the values of n and T are computed
by the method of least squares.
This is essentially the socalled Kowalsky method. It
looks mathematically elegant. But because it is not only
severely affected by uncertainties of observations but also
ignores the area theorem, it has a very poor reputation
among seasoned practitioners. However, in our work, we use
13
it for getting the initial approximation to the solution.
It serves this purpose very well.
The Method of T. N. Thiele, R. T. A. Innes and
W. H. van den Bos
T. N. Thiele (1883) published a method of orbit
computation depending upon three observed positions and the
constant of areal velocity.
The radii vectors to two positions in an orbit subtend
an elliptical sector and a triangle, the sector being
related to the time interval through the law of areas.
Gauss introduced the use of the ratio: "sector to triangle"
between different positions into the orbit computation of
planets, and Thiele applied the idea to binary stars.
Although the method could have been applied in a wide range
of circumstances, it became widely used only after Innes and
van den Bos revived it.
In 1926, R. T. A. Innes (Aitken, 1935), seeking a
method simpler than those in common use for correcting the
preliminary parameters of an orbit differentially, indepen
dently developed a method of orbit computation which differs
from Thiele's in that he used rectangular instead of polar
coordinates. W. H. van den Bos's (1926, 1932) merit is not
merely a modification of the method (transcribing it for the
use with Innes constants) but chiefly in its pioneeringly
successful application. The device became most widely
applied. Briefly, the method of computation is as follows.
14
The computation utilizes three positions (p^e^) or the
corresponding rectangular coordinates (xj_,y) at the times
(i=l,2,3). The area constant C is the seventh quantity
required. Thiele employs numerical integration to find the
value of C.
First, from the observed data, find the quantities L by
l12 = t2tlD12/c '
(210a)
l23 = t3t2D23^c '
(210b)
L13 = t3t1D13/C ,
(210c)
where Dj_j = ppjsin( 6j6) are the areas
of the corre
sponding triangles.
Then, from the equations
nL]_2 = psinp ,
(2lla)
nL23 = qsinq ,
(2llb)
nL13 = (p+q)sin(p+q) ,
(2llc)
the quantities n, p and q can be found by
trials, for in the
three equations above there are only three unknowns, i.e.,
n, p and q. The eccentric anomaly E2 and the eccentricity e
can thus be computed from
(D23sinpD12sin(3)
esinEp =
(D23+D12D13>
(212a)
(212a)
(212b)
ecosE^
(D23cosp+D12COS(3D13
D23+D12"D13
After E2 and e are obtained, the two other eccentric
anomalies E]_ and E3 can be found from
E1 = E2 P (213a)
E3 = E2 + q (213b)
The E are used first to compute the mean anomalies Mj_ from
equations (28), which lead to three identical results for
T as a check, and second to compute the coordinates XÂ¡_ and
from
Xj_ = cos E e (214a)
Yj_ = sinE^V le^) (2 14b)
By writing
X
H
II
+ FYi ,
(215a)
Yi = BXi
+ gy ,
(215b)
the constants A, B, F, G are obtained from two positions,
the third again serving as a check. These four coeffi
cients, A, B, F, G, are the now socalled ThieleInnes
constants. In addition to n, T, e, which have already been
16
determined, the other four orbital parameters a, i, fi, w can
be calculated from A, B, F, G through
Q+u) = plg(BF,A+G) (216a)
nd) = pig (B+F, AG) (216b)
_i (AG)cos(u)+Q) (B+F)sin(w+Q)
tan = = (216c)
2 ( A+G)cos(jQ) (BF) sin( wfi) ,
A+G
2acos(u)+Q)cos
2
(216d)
In addition to the brief introduction to this method given
above, a detailed description of it can be found in many
books, e.g., Aitken's book The Binary Stars (1935) and W. D.
Heintz's book Double Stars (1971).
A more accurate solution is obtained by correcting
differentially the preliminary orbit which was somehow
obtained by using whatever method. This correction can be
achieved by a least squares solution.
The Method of Least Squares
The method of least squares was invented by Gauss and
first used by him to calculate orbits of solar system bodies
from overdetermined system of equations. It is the most
important tool for the reduction and adjustment of observa
tions in all fields, not only in astronomy. However, the
traditional standard algorithm, which employs condition
equations that are solved with respect to the (uncorrelated)
17
observations and either linear in the adjustment parameters
or linearized by developing them in Taylor series which are
broken off after the first order terms, is inadequate for
treating the problem at hand. An algorithm for finding the
solution of a more general least squares adjustment problem
was given by D. C. Brown (1955). This situation may briefly
be described as follows.
Let {x} be a set of quantities for which an approxima
tion set {xq} was obtained by direct observation. By
ordering the elements of the sets {x> and {xQ} and regarding
them as vectors, x and xq, respectively, the vector v=xxq
is the vector of the observational errors which are
initially unknown. Assume that they follow a multivariate
normal distribution and that their covariance matrix a is
regarded as known. Further assume that a set of parameters
{a} is ordered to form the vector a. The solution of the
least squares problem (or the adjustment) consists in
finding those values of the elements of {x} and {a} which
minimize the quadratic form vTa1v while at the same time
rigorously satisfying the condition equations
fi({x}i,{a}i)=0 (217)
This is a problem of finding a minimum of the function
vTa1v subject to condition equations. A general rigorous
and noniterative algorithm for the solution exists only for
18
the case that the elements of {x} and {a} occur linearly in
the functions f j_. When fj_ are nonlinear in the elements of
either {x} or {a}, or both, equations which are practically
equivalent to the f j_ and which are linear in the pertinent
variables can be derived in the following way.
Define a vector 6 by a=aQ+6. The condition equations
can then be written
f(x0+v, a0+6) = 0
(218)
where the vector of functions f=(f]_,f2 / fro)T, m being
the number of equations for which observations are
available. Now assume that all elements of {v} and {6} are
sufficiently small so that the condition equations can be
developed as a Taylor series and written
f0 + fxv + ^a +0(2) ... 0
(218'a)
If the small quantities of order higher than 1 can be
neglected, we can write the linearized condition equations
as
f0 + fxv + fa6 = 0
(218'b)
19
These are linear in the relevant variables, which are the
components of v and of 6.
In order to satisfy the conditions (218'b), we define
a vector 2y. of Lagrangian multipliers and minimize the
scalar function
S(v,6) = vTo1v 2n(f0 + fxv + fa6) (219)
in which the components of v and 6 are the variables.
Setting the derivatives (3S/3v) and (3S/36) equal to zero
and considering equations (218'b), we obtain
6 = (faT(fxafxT)fa]1faT(fxafxT)1f0 (220a)
U = (fxafxTr^faS+fo) (220b)
v = cjfxTu (220c)
where we have assumed that fxafxT is nonsingular. This is
the case only if all equations fj_=0 contain at least one
component of x; i.e., if there are no pure constraints of
the form gj_(a)=0. This case (which we shall not encounter
in our investigations) is discussed below in the description
of Jefferys' method.
Note that in constructing the scalar function S in
expression (219), first order approximations have
20
been used. In some cases, the linearized representation of
Eg. (218) by Eg. (218'b) is not accurate enough. In some
of these cases, the convergence toward the definitive
solution may be accelerated and sometimes be brought about
by a method suggested by Eichhorn and Clary (1974) when a
strictly linear approach would be divergent. Their solution
algorithm takes into account the second (as well as the
first) order derivatives in the adjustment residuals
(observational errors) and the corrections to the initially
available approximations to the adjustment parameters. They
pointed out that the inclusion of second order terms in the
adjustment residuals is necessary whenever the adjustment
residuals themselves cannot be regarded as negligible as
compared to the adjustment parameters, in which cases the
conventional solution technigues would not lead to the
"best" approximations for the adjustment parameters in the
sense of least sguares. The authors modified the condition
eguations as
M6
f0 + fxv + fa^ + Vv + D6 + or = 0 (218")
Nv
Correspondingly, the scalar function to be minimized becomes
M6
S"(v,6) = vTa1v 2u(fo+fxv+fa6+Vv+D6+or) (219")
Nv
21
Here, the ith line of the matrix D is
t
O' E. ,
2 1
and
E.
1
r
32fi
3a^ 3a
k'
(221)
the matrix of the Hessean determinant of fj_ with respect to
the adjustment parameters. Similarly, the ith line of
1
vector V is vTWj_, and
2
W.
l
' 32fi ^
,3Xj3xk>
the ith line of M is vTHj_, and
(222)
H.
1
r 32fi ^
. 3x 3a, .
v j
(223)
and that of N is 6thÂ£, so evidently M6=Nv.
Minimizing S", also 6, v can be calculated.
For this algorithm in detail, one can refer to the
original papers of Eichhorn and Clary. The notation used
here is slightly different from the original one used by the
authors.
Jefferys (1980, 1981) proposed an even more accurate
algorithm which also improves the convergence of the
22
conventional least squares method. Furthermore, his method
is nearly as simple to apply in practice as the classical
method of least squares, because it does not require any
second order derivatives. Jefferys defines the scalar
function to be minimized as
(224)
where c=Xq+v; x,a are the current "best" approximations to x
and a; and g is another vector function consisting of the
constraints on the parameters; i, y are vectors of
Langrangian multipliers and evaluated at (x, ). Minimizing
S with respect to x and a, he arrives at the normal equa
tions
a_1v + fT(x,) = 0 ,
fxT(x,) + gT() y = 0
f(x,) = 0 ,
g() = 0 ,
(225c)
(225d)
(225a)
(225b)
where
x, a
These equations are exact and involve no approximations.
This is the significant difference between Jefferys' method
and those of Brown and of Eichhorn and Clary.
23
Remarks
As mentioned before, Kowalsky's method will most likely
not produce the best obtainable results, because the
relative observed coordinates (x, y, t) are subjected only
to the condition (21), which involves only five of the
seven necessary orbital parameters as adjustment parameters
and does not enforce the area theorem. It can therefore
never be used for a definitive orbit determination since it
completely ignores the observation epochs.
Yet, Eg. (21) has the advantage that it appears to be
simple, in particular it is linear in the adjustment
parameters albeit not in the observations. When it is used
for the determination of orbits, the righthand sides of the
equations which result from inserting a pair (x, y) of
observed rectangular coordinates into Eg. (21) are regarded
as errors with a univariate Gaussian distribution (i.e., as
normally distributed errors). One may then perform a least
squares adjustment which is linear in the adjustment
parameters. As Eichhorn (1985) pointed out, this approach,
while it has the advantage that approximation values for the
adjustment parameters need not be available at the outset,
fails to take into account two facts.
1) It is not the righthand sides of the condition equations
which are to be considered as normally distributed errors,
but rather the observations (x,y) or (p,6). The condition
equations (21) thus contain more than one observation each.
24
Since the observations occur in the condition equations
nonlinearly, the matrix
rdf(x,a)>
, 3x
must be found. This requires knowledge of approximate
values ag for a. Approximate values xg for x are
availablethey are the observations themselves.
Approximate values ag for a may sometimes indeed be obtained
in the classical way by regarding the righthand sides of
the condition equations as normally distributed errors. In
addition, it should also be taken into account that the
covariance matrix a of the observations is not necessarily
diagonal.
2) In some cases, especially when the binary under study is
very narrow, the errors of the observations are not
negligibly small compared with the adjustment parameters.
This requires either that secondorder terms in the observa
tional errors v be carried in the equations or, as Jefferys
has pointed out, that iterations be performed using in the
evaluation of the matrices fx and fa not only improved
approximations for a but also improved values for the
observed quantities as they become available.
If Kowalsky's methods were so modified, the algorithm
would yield better values for the adjustment parameters a
25
than the traditional approach. Either way, one can usually
find an initial approximation by Kowalsky's method.
With respect to both the theoretical clarity and the
practical applicability, as far as it is concerned, the
ThieleInnesvan den Bos method leaves nothing to be
desired. However, the three places selected, even when
smoothed graphically or by some computation, may not suffice
to describe the motion with sufficient accuracy, so that
large and systematic residuals may remain, particularly near
periastron. The method is seriously inadequate even if one
of the ratios sector to triangle is very close to 1 and thus
strongly affected by the measurement errors or if the area
constant C is not initially known to the required accuracy.
The computation may then produce an erroneous orbit with
spuriously high eccentricity, perhaps a hyperbolic one, or
no solution at all. And obviously, different combinations
of the three positions selected from a set of observations
will not likely give the same result. This method therefore
fails to use the information contained in the observations
in the best possible way.
In our work we try to present a fairly general least
squares algorithm to solve the orbit problem. We shall
adopt Jefferys' least squares method as our basic approach.
CHAPTER III
GENERAL CONSIDERATIONS
This chapter contains a general discussion of the least
squares orbit problem. We shall set up condition equations
which simultaneously satisfy the ellipse equation and the
area theorem.
We have seen that it is not sufficient to use Eq. (21)
as the only type of condition equation because this would
ignore the observing epochs, cf. last chapter. Completely
appropriate condition equations must explicitly contain the
complete set of the seven independent orbital parameters as
the adjustment parameters. Also, to be useful in practice,
they must impose both the geometric and dynamical condi
tions, and must lead to a convergent sequence of iterations.
After the condition equations are established, we
present the general outline of the algorithm which solves
the orbit problem by Jefferys' method of least squares.
We also discuss some further suggestions for obtaining
the initial approximate solution required for the least
squares algorithm.
26
27
The Condition Equations
Assume that a set of observations {xq} was obtained
consisting of complete data triples (t, p, 6), which
measure the positions of the fainter component (secondary)
with respect to the brighter one (primary): the position
angle 0 is counted counterclockwise from North and ranges
from 0 to 360; the angular separation p (also called
distance) is usually given in seconds of arc, and t is
the observing epoch. The conversion of (p, 0) to
rectangular coordinates (x, y) in seconds of arc is as
following:
Declination difference 6c6p = x = pcos0, (3la)
Right ascension difference (acOp)cos6 = y = psin0, (3lb)
where 6C, ac are the declination and right ascension,
respectively, of the secondary; 6p, Op those of
primary.
Equivalently, the observations can also be regarded
as relative coordinates (t, x, y) of the secondary with
respect to the primary.
It might be worthwhile to point out that 1) even though
the formulae (31) are approximations valid only for small
values of p, they may be regarded as practically rigorous
for binaries; 2) we are following the custom in double star
28
astronomy by having the xaxis along the colure* and the y
axis tangential to the parallel of declination.
All observations in {xg} are affected by observational
errors. Let {x} be the set of the true values of the
observations, that is, those values the observations would
have had if there were no observational errors. By ordering
the elements of the sets {xg} and {x} and regarding them as
vectors, xg and x respectively, we have seen that we may
write the vector of observational errors as v = x Xg.
These errors are of course unknown, but as mentioned already
in the last chapter, we assume that they follow a multi
variate normal distribution with known covariance matrix.
For visual binaries, the relative orbit must be an ellipse
(strictly speaking, a conic section) in space as well as in
projection. All pairs (x, y) must therefore satisfy the
condition equations (21):
C^x2 + 2C2xy + C3y2 + 2C4x + 2C5y +1=0 ,
which implicitly involve five of the seven orbital para
meters but do not enforce the area theorem.
The wellknown relationships between the five coeffi
cients (Ci, C2, C3, C4, C5) in Eq. (21) and the five
*Following Eichhorn's terminology who uses the term "colure"
generally for any locus of constant right ascension.
29
orbital parameters (e, a, i, a>, n) by way of the Thiele
Innes constants, have been discussed in the last chapter.
Consider a righthanded astrocentric coordinate system
K whose XY plane is the true orbital plane such that the
positive Xaxis points toward the periastron (of the
secondary with respect to the primary). The positive Yaxis
is obtained by rotating the Xaxis by 90 on the Zaxis in
the direction of the orbital motion. The axes of a second
astrocentric, righthanded coordinate system k are parallel
to those of the equator system Q. The two systems K and k
are related by the transformation
xK = R3(w)R1(i)R3(n)Xk (32)
From the theory of the twobody problem we know that,
in the system K, the coordinates of the secondary with
respect to the primary are given by
K
x =
f X 1
Y
= a
'cosE e ^
sinEZle^
l 2 J
l o J
where E is the eccentric anomaly, which is the solution of
Kepler's equation
n(tT) = EesinE
(34)
30
Here, n and T, the mean motion and the periastron epoch,
the two orbital parameters not involved in Eq. (21).
From Eq. (32) we obtain
'A B
"x"
XK =
F G H
y
,K L
where
Kx + Ly
z =
and
r A B
F G
. K L
or, in detail,
A = cosicosa) sinfsinwcosi ;
B = sinQcoso) + cosQsinojcosi ;
C = sinojsini ;
F = cosfisino) sinQcostocosi ;
G = sinfisinoj + cosQcosuicosi ;
H = cosuisini ;
M
C ^
H
M j
= R3(a))R1(i)R3(n) ,
(
(
(
(3
(3
(3
(3
(3
(3
(3
are
35)
36)
37)
8a)
8b)
8c)
8d)
8e)
8f)
8g)
K = sinQsini
31
L = cosfisini ;
M = cos ,
where in this notation, the traditional ThieleInnes
constants would be aA, aB, aF and aG.
From Eq. (37) and (35) we can get
X = Ax + By + Cz =
Gx Fy
M
(3
Y = Fx + Gy + Hz = 
Bx Ay
M
(3
Thus, we see that X and Y can be expressed in terms of
and the observations (x, y).
From Eq. (33) we obtain
X
cosE = + e ,
a
(3
sinE =
f. 2
avle
(3
Combining (310) with Kepler's equation (34), we get
X
+ e =
eY
n(tT) +
aVle2
38h)
38i)
9a)
9b)
i co Q
10a)
10a)
a
cos
(3lla)
32
a^e2
= sin
eY
n(tT) +
a
More succinctly, we have
(3llb)
u =
V =
or
u =
V =
with
U =
cost n(tT)
sin[ n(tT)
cost n(tT)
sin[ n(tT)
X
 + e ,
a
+ eV ] ,
+ eV ] ,
+ e7lU2 ] ,
+ eV ] ,
Y
V =
L 2
avle
(312a)
(312b)
(312'a)
(312'b)
(312c)
After X and Y are expressed in terms of i, w, Q and (x, y)
as in Eqs. (39), Eqs. (311) or (312) involve exactly the
seven orbital parameters (n, T, a, e, i, to, 2) and the
observations (t, x, y). The observing epoch t now appears
explicitly, as it must if the area theorem is to be
enforced.
Now, we see that if both equations (311) are satis
fied, Kepler's equation which enforces area theorem would be
satisfied and furthermore, the ellipse equation would also
be automatically satisfied as can be seen if t is eliminated
from Eqs. (311) so that these equations are reduced to one
equation in X and Y. If we select the two equations (311)
33
as the condition equations, we need no longer carry
Eq. (21) separately. Of course, we can use any one of
Eqs. (311) as well as Eq. (21) as the condition equations.
However, using Eq. (21) is not convenient, for it contains
the five coefficients directly, but not the five orbital
parameters themselves, even though there are unique rela
tionships between them. Ideally, the condition equations
should have the adjustment parameters explicitly as
variables.
In our work, we use the Eqs. (311) as the complete set
of condition equations.
The General Statement of the Least
Squares Orbit Problem
As seen in the last section, the vector of "true
observations" x (presumably having been adjusted from the
observation xq by the residuals v) and the vector of "true
orbital parameters" a, a=(n, T, a, e, i, w, Q)T, must
satisfy the condition equations
X
eY
f!(x0+v,a)
+ e cos n(tT)+
0
(313a)
a
Y
eY
f2(xQ+v,a)
sin n(tT)+
0
(3 13b)
where
34
' X
r A B C >
r X "
Y
=
F G H
y
, o
, K L M ,
, z >
with
Kx + Ly
z = 
and
f A B C ^
F G H
v K L M j
= R3(w)R1(i)R3(Q)
In our problem, there are no constraints between the
parameters which involve no observations so that Jefferys' g
function does not occur. The problem can therefore be
stated as follows.
Assume that the residuals {v} (regarded as vector v) of
a set of observations {xg} (regarded as vector xg) follow a
multivariate normal distribution, whose covariance matrix a
is regarded as known; we are to find the best approximations
of v (for v, the residuals) and (for a, the parameters)
such that
f^xg+v^a) = 0
and
f2(x0+v,a) = 0
35
are both satisfied while at the same time the quadratic form
1 T 1
S 0= V v ( 3 i 4 )
U 2
is minimized.
Following the wellknown procedure introduced by
Lagrange, the solution is obtained by minimizing the scalar
function
S = vTa_1v + fT(x,)ji (315)
2
where x = Xg + v, and ii is the vector of Lagrangian multi
pliers, together with satisfying the equations f]_=0=f2
Denoting the matrix of partial derivatives with respect to a
variable by a subscript, this is equivalent to solving the
following normal equations:
a 'v + fT(ic,) = 0 (316a)
fT = 0 (316b)
f(x,a) = 0 (316c)
We have stated before that these equations are exact and
therefore involve no approximations. Before Jefferys, all
authors used first order or second order approximations in
forming S in equation (315). This is the significant
36
difference that distinguishes Jefferys' method from those
employed by previous authors.
It is evident that the solution of the equations (316)
would solve the posed problem.
About the Initial Solution
The least squares algorithm requires an initial
solution as starting point. Any approach which leads to
approximate values of the orbital parameters serves this
purpose, because our algorithm does not require a very
accurate initial approximation. As long as the initial
approximation is not too different from the final result,
convergence can always be achieved. To obtain an initial
solution, the following procedures may lead to an initial
solution.
1) As mentioned in Chapter II, Kowalsky's method can produce
a preliminary solution. Inserting the pairs (x, y) of
observed rectangular coordinates into Eq. (21), we have a
set of linear equations in which the five coefficients (c^,
C2, C3, C4, C5) are the unknowns. By making a classical
least squares solution based on these linear equations, the
five coefficients can be computed. An estimate of the five
parameters (a, e, i, ai, Q) can be obtained in turn from the
unique relationships between them and the five coefficients.
The remaining two parameters (n, T) also can then be
calculated from the known quantities simply by classical
least squares, as described in Chapter II.
37
As Eichhorn (1985) has pointed out, it is of course
better to use the modified Kowalsky method. Using (21) as
condition equations and the five coefficients as the
adjustment parameters, one may iterate by Jefferys'
algorithm toward the best fitting adjustment parameters and
the best corrections to the observations, v, that is, to
arrive at the values of and v which minimize the scalar
function Sq (see equation 314) while simultaneously
satisfying the condition equations.
This method is simple to apply in practice. But
unfortunately, especially when the observations are not very
precise, the five coefficients (c^, C2, C3, C4, C5) in some
cases do not always satisfy the conditions for an ellipse;
i.e., they do not meet the requirements
C^>0, C3>0 and C^C3_C2^>0.
However, in these cases, it does not mean that there is no
elliptic solution at all and other approaches can be tried.
When this happens, one may for instance take the approach
outlined in Chapter 23 of Lawson and Harsson (1974).
2) By using some selected points among the observation data
instead of using all the data points, sometimes a solution
can be found by Kowalsky's method. Such a solution is
usually also good enough to be a starting approximation.
38
3) Or, carefully selecting three points among the observa
tion data, one may use the ThieleInnesvan den Bos method
to calculate an initial approximation. The method has been
described in Chapter II.
CHAPTER IV
THE SOLUTION BY NEWTON'S METHOD
The Solution by Newton's Method
In our problem, the normal equations (316) are
nonlinear. They must be solved by linearization and
successive approximations. Assume that approximate initial
estimates of the unknowns in the normal equations have
somehow been obtained (using whatever methods). This
initial approximation may be improved by Newton's method,
which consists of linearizing the normal equations about the
available solution by a first order development and
obtaining an improved solution by solving the linearized
equations. This process is iterated with the hope that
convergence to a definite solution would eventually be
obtained. This expectation is reasonable if the initial
approximation is sufficiently close to the final solution
and if the observational errors are not too large.
Following Jefferys' notation (which is also the
notation we have used in Chapter III), let the initial
approximate solution (and also the current approximate
solution during iteration) be given by (x,a), where
x = xg + v, xg being the vector of observation, v the vector
of observational errors (for which we adopt the nullvector
39
40
as initial approximation); a is the initial approximation of
the vector of the seven orbital parameters (adjustment
parameters); also let the corrections to both x and be
denoted by and 6, respectively.
The normal equations in our problem now become
(4la)
(4lb)
(4lc)
Here we have ignored (as Jefferys did) products of e and 6
with Lagrangian Multipliers. This does not affect the final
result, as Jefferys also pointed out. A caret in equations
(41) above means evaluation at current values of x and a.
Similar to Jefferys' procedure, we solve the equations
(41) as follows.
Solving Eq. (4la) for e we have
= v afu
(42)
Substituting (42) into (4lc) for , Eq. (4lc) becomes
f fv fafxTl + f6 = 0
(43)
Solving Eq. (43) for we obtain
41
= w(f fv + f6) (44)
where the "weight matrix" w is given by
w = (f^af^T)1 (45)
Inserting this solution for into Eg. ( 4lfe), we have
fTw(f fv + f5) = 0 (46)
If we now define
$ = Â£ fv (47)
and rearrange Eq. (46), the equation for 5 will have the
form
(fTwf)6 = fTw$ (48)
This set of linear equations is easy to solve for the
corrections S by general methods. With this solution for 6,
the improved residuals vn are obtained from the equation
Vn = CTfTw($+f6) (49)
42
which follows from Egs. (4la), (44) and (47). We then
get the new vectors of an and xn as
n = + 6 ,
^n = x0 +
which constitute the improved solution.
After each iteration, we check the relative magnitude
of each component in v and 5 against the corresponding
component in x and a and get the maximum value among all of
these and test if this value is smaller than some specified
number, say 10_. If the improved solution is still not
sufficiently accurate (i.e. if the above found maximum value
is still not smaller than the specified value), the process
of iterations has to be continued until convergence has been
attained, that is, until subsequent iterations no longer
give significant corrections.
At the outset, the obvious starting point for this
scheme is to adopt x=xg as the initial approximation for the
"true observation" vector x (in this case, v=0), and to use
as first approximation of a for a a vector g, an initial
solution of the seven orbital parameters, which has been
obtained somehow.
It is important for convergence that the initial
solution of (x,a) is not too different from the final
solution which is obtained by the process given by Eq. (41)
(410a)
(410b)
43
through (410). In Chapter III, we discussed how to find a
good approximation as an initial solution.
According to the scheme outlined above, the application
of Newton's method in our problem would consist of the
following steps.
Step 1.
Calculate f, f and f from the current values of x
and ;
Step 2.
Calculate $ from $ = f fv ;
Step 3.
Calculate the "weight matrix" w from w = (fafT)l ;
Step 4.
Solve the corrections to the parameters, 6, from the
linear equations (fTwf)6 = fTw$ ;
Step 5.
Calculate the improved residuals vn from
vn = afxTw($+fS) ;
Step 6.
Find the new approximate solution from
n = + 6 ,
n = xo + vn ;
Step 7.
Test the relative magnitude of each component of 6 and
v against and x, and decide if a further iteration is
needed, in which case all the steps above must be repeated.
44
In detail, the steps are as follows.
Step 1.
Calculate the vector of functions in condition equa
tions f, and the vectors of partial derivatives f and f
from the current values of x and a, where x=xq+v.
The dimension of the vectors xq, v, x all are 2m, m
being the number of observed positions.
The observation vector is
x0 (x10'Yl0'* *'xiO'YiO'*xm0'YmO ^ (411)
The current vector of corrections to observations is
V ( Vjr}_ / Vyl . Vj^j_ Vyj_ , VjQ^j Vyj^ ) ^ (4~12)
From Xq and v, x can be easily found by x=Xq+v,
x = (x10+vxl,y10+vyl,...,xi0+vxi,yi0+vy,...
'xmO+vxm'YmO+vym)T (413)
The current approximation for the seven parameters, , is
= (n,T,a,e,i,o),Q)T (414)
at current values. Insert the known (x,a) into the
45
condition equations to get the function 2mvector f. It has
a form
Â£ [ f]_i 21 *' ^li' ^2i' * ^lm* ^2m^r^' 1^)
where
f
li
+ e
cosE.
i
(416a)
f
2i
Yi
b
sinE^
(416b)
with
b = a/le2
(416c)
and
E. = n(t.T) +  (416d)
11 b
The coordinates X,Y are calculated from Eqs. (39),
A,B,F,G,M from Eqs. (38).
The first derivatives f, f are calculated at current
values of x and .
The partial derivatives of Â£ with respect to the
observations x, f, is a blockdiagonal 2mx2m square matrix
of the form
46
fx =
' 8fll 3fll
3x,
3x,
3Yi
9f21 3f21
3Yi
afii 9fu
3x.
3x.
9f2i 9f2i
9yi
3f
lm
3flm
0
3x
m
9ym
3f2m
3Â£2m
3x
m
9ym
i.e.,
(417)
with
^tag(g^,g2/ /9i* /9m)
(418)
3Â£u
3fu 1
9xi
9yi
gi =
3f2i
3Â£2i
3x.
1 i
3yi J
i=l ,m
(419)
From Eqs. (416), (38) and (39), we obtain
47
g. =
f 1
e ^
f 3X.
3X. ^
sinE.
1
1
a
b 1
3xi
3Yi
0
1
(1ecosE.)
3Yi
3Yi
v.
b 1
J
9xi
SYi J
(420)
particular,
we have
ax.
i
ax
G
aXj.
ax
ax.
i
3x
M ,
3yi
ay
3Yi _
ay
B
3Yi
3Y
3xi
3x
M ,
3yi
ay
therefore
r i
e

a
sinE.
b 1
gi =
1
0
(1ecosE.)
b 1
F
M
G
M
B
M
F
M
A
M
(421)
(422)
The expressions for all the partial derivatives in are
listed in Table 41.
The dimension of f, the vector of the partial deriva
tives of f with respect to the seven parameters, is 2mx7.
It has the form
fa (q1'G2
,Gi
(423)
48
Table 41.
Expressions for All the Partial Derivatives in f
fli
f2i
3
3x
1fG
M^a
B
esinE^
1 B
 (1ecosE.)
Mb 1
3
3y
1(F A
 esinE.
M^a b 1>
1 A
(1ecosE.)
Mb 1
49
where
G.
i
3f
li
2i
3n
3f
li
3fu 3fu
3n 3T 3 a 3e
3f 3f. 3f. 3f
2i
2i
3T 3a
2i
3e
3fli 3fu 3fu '
3i
3f
2i
3i
3o)
3f
2i
3w
32
3f
2i
3Q
The expressions for all the elements in Gj_ are listed below.
3Â£li
3fu 1
r \
sinE.
3n
3T
1
3f 2i
3Â£2i
cosE.
^ 3n
3T J
1
C J
( vT n )
(424a)
3f
li
Xi eYi .
~ sinE,
3a a ab
(424b)
3f2i
38.
(1ecosE.) ;
ab 1
(424c)
3f
li
3e
Y^sinE^
b(1e2)
+ 1
(424d)
3f?. e cosE.
2, i
3e
b(1e )
(424e)
50
3fli
3fii
3fu 'i
r 1
e '
sinE.
r sxi
SX.
1
SX. 'J
l
Si
S oo
SQ
a
b 1
Si
Soo
SQ
Sf_.
2i
3Â£2i
Sf_.
2i
0
1
(1ecosE.)
!!i
SY,
l Si
S (o
SQ J
b 1 J
l Si
Soo
SQ j
( 4 2 4 f )
From Eqs. (38) and (39) we can find the expressions for
the following six partial derivatives.
sx
M = z sinoo
Si
M
9Y,
Si
= Z,COSOO
SX, SY.
M  = Ay, Bx, M  = Fy, Gxi
S oo S oo
SX, SY,
M = Gy. + FX. M = By. Ax.
SQ 1 1 SQ 1 1
(424g)
In terms of these derivatives, we have
PÂ£ii
3Â£ii
3fii3
' 1
e '
sinE.
r
z,sinoo
Ayi_Bxi
Gyi+Fxi
Si
Soo
SQ
Ma
Mb 1
3f2i
l Si
Sf2i
Soo
3Â£2i
SQ J
0
1 e
cosE.
Mb Mb \
z,cosoo
Fyi_Gxi
By,Ax,
y
(424h)
All expressions in f are listed in Table 42
51
Table 42.
Expressions for All the Partial Derivatives in f
fli
*2i
3
3n
(t^T)sinE^
(t^T)cosE^
3
3T
nsinE^
ncosE^
3
3a
Yi .
esinE.
ab ]
(1ecosE^)
3
3e
1 +
Yi
b(1e )
27 sinEi
b(1e )
(ecosE^)
3
3i
"sinu
M a
e
+ sinE.coscj
b 1
1
(1ecosE. )z.cosu>
Mb 1
3
3(i)
1 Ay.Bx, e
 + sinE.(Fy.Gx.)
M a b 1 1 1
1
(1ecosE.)(Fy.Gx.)
Mb ill
3 1 Gy^+Fx^
3fi M a
e
sinE.(By.+Ax.)
b ill
1
(ecosE.1)(By.+Ax. )
Mb 1 11
52
Step 2.
Calculate $ from f, v and f by The dimension
of the vector $ is also 2m. It has the form
4> (i / 2 / i / / ^m (425)
where
r 8fii
3fli ]
r >
.
11 8x 'Xi
V .
3Yi Y1
il
3f2i
3f 2i
$i2
< >
f  v .
21 ax,
 v .
ay, J
(425')
Step 3.
Calculate the weight matrix w from and a.
The matrix has been calculated in step 1. The
covariance matrix a is assumed to be known. The dimension
of a is 2mx2m.
An example for computing a is shown below.
The relationship between the covariance matrix of
rectangular coordinates (x, y) and that of polar coordinates
(p, 0) is
3(x,y)
3(x,y)
axy
. 3 ( p 6 )
CTP0
. 9 ( p / 9 )
(426)
53
where x=pcos6, y=psin9 and
3(x,y)
s(p,e)
r cose psine >
^ sine pcose
(426')
Thus, we have
xy
cose
^ sine
psine '
pcose
r
P
0 a
0 ^
e '
cose sine "
^ psine pcose
and therefore
xy
2 2 2
r a cos 6 + aQp sin 6
P 0
2
(a aQp )cos6sin9
P o
(a aQp)cos6sin0 ^
p e
2 2 1
apsin 6 + aQp cos'e j
(426")
In these expressions, cp=A^p, Gq=A^6 and Ap, A6 are the
observational errors in p and 6, respectively. For the
observations, the random errors are of similar order of
magnitude as the systematic ones, larger in separation than
in position angle. The average errors pA0 and Ap vary
somewhat with the separation p and can be assumed, for many
series of observations, to follow the form Cp*/3, where C
varies with different observers. For a single good observa
tion C will not exceed 0V03 in position angle (pA9) and
0V08 in separation (Ap) (Heintz, 1971). Errors will be
somewhat larger and difficult to measure if one or both
components are faint. If the errors are expressed in the
54
dimensionless (relative) forms A0 and Ap/p, it is seen that
they increase as pairs become closer.
Based on the considerations above, we can, for example,
put Ap=0V08pl/3 an pA0=O'.'O3pl/3, i.e. A0=O'.'O3p2/3. if we
allow each pair of numbers (x^,yj_) in an observation to be
correlated, but no correlations between different observa
tions, a, would be blockdiagonal. In this case, we have
o diag(0^,02, , /) /
where
( Q (T ^
uxixi uxiyi
' ail
ai3 "
i =
^ ayixi ayiyi J
ai4
ai2 >
with xiyi_ayixi> i, i3 ^i4
The form of the weight matrix is simpler in this case;
it is also blockdiagonal.
According to Eq. (45), w=(faf)^. The matrices f,
a, fT now are all blockdiagonal. Therefore
w = diag(w1,W2,.. ,Wj_, ,wm) (429)
with
55
wi =
wil
wi3 '
wi4
wi2 >
(430)
The computation of w is straightforward. We first find w"l.
If we denote u=w1=fafÂ£T, it is obvious that u has the same
form
u = diag(u1,u2,,Ui,...,um) ,
(431)
where
r u
ui3 1
ui =
l ui4
ui2 J
9fu
3fii 1
f
OXj.
3yi
CTil
3f2i
3f2i
Ox.
1 i
3yi J
i4
s.
i3
i2
Of
li
Ox.
Of
li
Oy.
9Â£2i ^
Ox.
i
8f2i
3yi 2
i. e.,
(432)
rof, .y of,. of,, rof,.^
ii ii ii ii
Uxi J 11 0X;L 0Y;L loYi
i2
(433a)
56
f3f,0
2i
z
3f2i
3f2i
rsfO
2i
u. =
i2
il+ 2
3x.
a i3 +
3Yi
,3Yi >
i2 '*
(433b)
u
i3
!Â£li 2i
3x^ 3x^
a. +
i
8f2i + !Â£u !Â£2'1
3x. 3y. 3y. 3x.
ai3 +
3f,. 3f.
li 2i
3Yi 3Yi
ai2
(433c)
ui4 = ui3
(433d)
After computing u, we can find its inverse w very easily.
If we denote ^=UiiUi2ui3ui4=:uiiU'i2''ui3 / we have
wii
Wi3
Ui2
. Ui4 1
=
u
u
Wi2 ,
Ui3
, u
uil
U ,
As seen above, we can calculate the components of w directly
from the components of u.
If the quantities which constitute different observa
tions were also correlated, we would have to find the
product of the matrices f, a, fT and calculate its inverse
to get w. This would be more complicated by far.
Step 4.
57
Find fTwf and fT$ from f, $ and w, then solve the
linear equations
(fTwf)6 = fTw$
to get the correction vector 6.
The dimension of the matrix fTwf is 7x7 and fTW(j>, 5
are 7vectors. Denoting A=fTwf and B=fTw, we have
f 3fli 3fli .. 3f2i 3fli ..
A(j,k) = > wn + wi3 +
i=l
3aj 3ak
9aj 3ak
3fli 9f2i 9f2i 9f2i ..
+ w. n + w. n
i3 i2
3aj 3ak
3aj 3ak
and
j=l,7;
k=l,7; (43 5a)
m
( 3f
B(j) = T
2=1
33. .
V 3
 wn +n + r21 wi3 *11+
3a.
3fli , 3f2i
+ w13 *12 + wi2 *12
3 3.
8aj
j=l,7; (435b)
where
58
3 3..,
9f
li
9n ,
3a~
9T ,
9f
li
9f
li
9f . 9f1.
li li
3a.
3 a
3a,
9e ,
..., and so forth.
To solve the linear equations a6=B for 6, we can apply any
linear equation solution method, e.g. Gaussian Elimination,
GaussJordan Elimination, the Jacobi Iterative method, etc.
In a private communication, Eichhorn (1985) proposed a
special solution method for this particular problem, because
the covariance matrix, i.e., the inverse of the coefficient
matrix of 6, is fortunately a positive definite matrix. The
method will be described in another section of this chapter.
In our work, we use this special method for solving the
linear equations above for 5.
Step 5.
Calculate the new vector vn, which now is the improved
correction vector to the observations, from
vn = afTw($+f6)
The dimension of the vector v is 2m. Denote Q=$f5 and
R=fTwQ, Q and R both are also 2mvectors. We can see that
Q (Ql/Q2' 'Qi' ' Qm ( 4 36)
where
59
Qi
3f. 3f. 7 3f..
f. v liv.+J 6 .
11 3Xi X1 3Yi yi fer 3a j 3
f 9Â£2i !Â£
21 8xi xi 3Yj
2i
v +
yi
3Â£2i
i=l ia.
i.e. ,
Qi
r
7
afii
3aj
Â£u
+ S
6i
Qn
f2i
7
i=l
3f2i
3a.
5j
2i2
V.
3
J
V 7
(436' )
In the case of w being blockdiagonal,
R ( R^_ R2 / r 1
, Rm)T ,
(437)
where
r Ri! '
Ri =
Ri2
60
3Â£U (QilWil+Qi2Wi3) + ISilWi4+2i2Wi2l
3x.
3yi
3x.
l
3f2i
(QilMil+Qi2wi3) + (enwi4+Q12wi2)
3yi
(437')
Therefore, in this case,
vn = (Vx, V2, .Vif ..., Vm)T ,
(438)
where
V. =
i
fv. )
xi
V .
v yi j
r a. R. + a. .R. ^
ll ll i3 i2
i4Eil + ai2Ri2
(438')
Step 6.
Find the new approximate solutions an and xn from
n = + 6
xn = x0 + vn
Step 7.
Determine if another iteration is needed.
Calculate all quantities of
xi(new) xi(old)
Yi(new) Yi(old)
and
6 .
3
xi(old)
/
Yi(old)
aj
(altogether 2ra+7 quantities) and pick up the maximum value
among them. If this maximum exceeds a preset specified
61
value, say 10^, return to the very beginning and take all
of the steps once again. Only when the maximum is less than
the specified value, we assume that good convergence has
been achieved.
Above is the scheme of Newton's method in our orbit
problem. In real programming, some additional considera
tions have been taken into account. For example, in the
actual program, all variables are dimensionless. For
rectangular coordinate pairs (x, y), two new quantities are
defined,
x y
x' = y' = (439)
x0 *0 '
where (xg, yg) are the "observations" in the initial data,
so that x = Xgx1, y = ygy'. For the seven parameters, seven
new variables are defined as well. They are
n T a e i
al = a2 = a3 = a4 = a5 = ~
n0 T0 a0 e0 ^O '
(439 )
where (ng,Tg,ag,eg,ig,ojg,ftg) in g is the initial
approximate solution and e is defined as e=sintj) and eg=sin(J)g.
00 ft
62
In terms of these new variables, all formulae for
computing the partial derivatives need only slight modifica
tions, and the whole process remains in principle unchanged.
In addition, we have seen that f, a, fT are square
2mx2m matrices. If the number of observations is, e.g.
m=100, the covariance matrix a has 40,000 components, and
these three matrices alone occupy a huge storage, 120,000,
in the computer. In practical programming, we should keep
the required computer storage to a possible minimum. A
blockdiagonal covariance matrix a can be stored in a 2mx2
matrix, and the same applies to all other related matrices.
This greatly reduces the need for storage.
Weighting of Observations
As we know, the measures of any binary star will be
affected by the accidental and systematic observational
errors and, occasionally, from blunders, i.e., actual
mistakes. The points, when plotted, will not lie exactly on
an ellipse but will occupy a region which is clustered
around an ellipse only in a general way.
Observations are frequently combined into normal
places. Among the observed quantities, the time is the one
observed most precisely. To investigate the measurements
for discordance before using them to calculate an orbit, the
simplest method is to plot the distance p in seconds of arc
and the position angles 6, separately, as ordinates, against
the times of observation as abscissae. Smooth curves are
63
drawn to represent the general run of the measures and in
drawing these curves, more consideration will naturally be
given to those observation points which are relatively more
precise (for example, a point based upon several well
agreeing measures by a skilled observer and supported by the
preceding and following observations) than to the others.
The deviation of the observation points is in the ordinate
direction only and gives a good idea of the accuracy of both
observed quantities. The curves will show whether or not
the measures as a whole are sufficiently good to warrant the
determination of a reasonably reliable orbit. Observations
which are seriously in error will be clearly revealed and
should be rejected or given very low weights. The curves
will also give a general idea of how the observations are to
be weighted. The points which show larger deviations should
be given lower weights and the welldetermined points should
be given higher weights.
It is hard to recommend a general rule for the
weighting of measurements and normal positions. Some
precept could be considered (W. D. Heintz, 1971): compute a
weight p_ according to the number of observations, and P2
according to the "weight" assigned to the observers, then
the normal place receives the weight pVpiP2 (if computations
are made in the quantities d9 and dp/p). This precept could
avoid unduly high weights for single measurements as well as
for very many observations by few observers, and would
64
reduce the influence of residual systematic errors. Its
implicit assumption is a proportionality of the errors pde
and dp with Jp. This holds better for the multiobserver
average, as the share by less accurate data usually
increases at larger separations.
In our work, we given the points on or nearly on the
smooth curves equal unit weights, and assign lower weights
to those farther from the lines.
When we take into account the weights for the observa
tions, some slight and very simple modification is needed in
the solution process described above.
Let G be the matrix of weights for the observational
errors v. The dimension of G is 2mx2m, and G is diagonal.
Taking into account G, the residual function Sq will be
modified as
1 1
T T 1 ~T 1 
Sn = v Go Gv = v Go Gv ,
(440)
because GT = G. If we define Â£~1 = Gct ^G, then
T 1
Sn = vXE v ,
(440')
which has the exact form as before except a is replaced by
Â£. Therefore we need only to replace a by E, by E"l
wherever ct of appears.
65
Because G is diagonal, the calculations of E and E 1
from G and a are also very simple.
The Orthogonal Set of Adjustment Parameters
and the Efficiency of a Set of Orbital Parameters
As mentioned in the first section of this chapter,
Eichhorn (1985) has proposed a special method in a private
communication for solving the linear normal equations a6=B
of our problem. This method is as follows.
According to its definition, A=fTwf is obviously a
symmetrical positive definite matrix.
Given a symmetrical positive definite matrix Q, we look
for its eigenvectors X which have the property that
QX = XX (441)
Any value X which satisfies Eg. (441) is an eigenvalue of
Q, and since Q is positive definite we know that all X>0.
Obviously X must satisfy the equation QIX = 0, the
"characteristic equation" of Q. This is a polynomial in X
of the same order n as that of Q. The solutions are
Xq_, X2,...,Xn. The solution X^ of the homogeneous system
(441) with X=XK is the kth eigenvector. Since this is
determined only with the uncertainty of an arbitrary scale
factor, we may always achieve XK = 1 for all k. Let the
matrix P=(X]_,X2,... ,Xn) be the matrix of the normalized
eigenvectors of Q. It can be shown that P is therefore
66
orthogonal, so that PT=p"l. Equation (441) can be written
as
QP = PdiagUi, Xn) (442)
Writing the diagonal matrix diag(X]_, X2, Xn)=D, we
rewrite Eq. (442) as
QP = PD ,
whence
PTQP = D Q = PDPT
(443)
and
PtQ_1P = D1 Q1 = PD1Pt (444)
The equation (444) shows incidentally that the eigenvectors
of a matrix are identical to those of its inverse, but that
the eigenvalues of a matrix are the inverses of the eigen
values of the inverse.
Let Q be the covariance matrix of a set of statistical
variates X. We look for another set Y as function of X,
such that the covariance matrix of Y is diagonal.
67
Putting dX=a, the quadratic form which is minimized is
aTQ1a, where Q is the covariance matrix of X. If we define
Y = PTX (445)
then we have
X = PY
and
dX = PdY or a = PfJ aT = pTPT .
and in terms of 3 = dY, the quadratic form minimized which
led to the values X becomes PTPTQ_1PP whence the covariance
matrix of Y is seen to be (PTQ~*P)*=PTQP=D by Eq. (444).
It is then shown that Y=PTX is the vector of linear
transforms of X whose components are uncorrelated. A vector
of such components is called a vector of statistical
variates with efficiency 1 (Eichhorn and Cole, 1985). If
any of its components are changed (e.g. in the process of
finding their values in a course of iteration), none of the
other components of Y would thereby be changed in a tradeoff
between changes of correlated unknowns.
68
Now, back to the normal equations AS=B. The problem
of solving normal equations above can therefore be attacked
as follows.
1) Find the eigenvalues and eigenvectors of A, i.e., D1 and
P. (A=Q1, according to the notation above.)
2) The covariance matrix of 6, Q, can then be calculated
from Eq. (443), Q=A1=PDPT.
3) Let 6'=PT6. This gives APS'=B, whence S'=PTQB, or from
Eq. (443),
S' = DPtB (446)
Finding D* from D is very easy because D is diagonal. The
vector 6 is then easily calculated directly from 6=PS'.
One of the advantages of this method is that we can
calculate both 5 and S', the correlated and uncorrelated
elements of the corrections to the adjustment parameters, at
the same time. The vector S' is the set of the corrections
to the orthogonal adjustment parameters.
Furthermore, using this method, a measure for the
"efficiency" of a set of adjustment parameters can be easily
calculated, cf. Eichhorn and Cole (1985). They point out
that the information carried by the vector 5 of the
correlated estimates (whose covariance matrix is Q) is
partly redundant because of the nonvanishing correlations
between the estimates. What is the information content
69
carried by these estimates? We see that the matrix PTQP is
diagonal if P is the (orthogonal) matrix of the normalized
eigenvectors of Q and that it is also the (therefore
uncorrelated) linear transforms S'=P6 of 6. We might regard
the number
I Q I
= (447)
qll**qnn
that is, the product of the variances of the components of
vector S' (the product of the variances of the uncorrelated
parameters) divided by the product of the variances of the
components of vector 6 (the product of the variances of the
correlated parameters) as a measure for the "efficiency" of
the information carried by the estimates 5. But we note
that according to the definition (447), the value of is
severely affected by the number of variables. In order to
eliminate the effect of n, we redefine as
e =
I Q I
1
>
lll ,ynn
(447' )
For any set of uncorrelated estimates we would have e=l,
which is evidently the largest value this number may assume.
In our work, for every model calculated, the efficiency
of the set of adjustment parameters and their covariance
matrix are calculated.
70
A Practical Example
Table 43 lists the observation data for 51 Tau. These
data are provided by H. A. Macalister. The author would
like to thank Macalister and Heintz for their data which he
has used in this dissertation.
Theoretically, the first step in our computation should
be the reduction of the measured coordinates to a common
epoch by the application to the position angles of correc
tions for precession and for the proper motion of the
system. The distance measures need no corrections.
Practically, both corrections are negligibly small unless
the star is near the Pole, its proper motion unusually
large, and the time covered by the observations long. The
precession correction, when required, can be found with
sufficient accuracy from the approximate formula
A0 = 0 6q = +0?00557sinasec6(ttg) (448)
which is derived by differentiating Eq. (31) with respect
to 0, and introducing the precessional change A(ncosa) for
d6. The position angles are thus reduced to a common
equinox tg (for which 2000.0 is currently used), and the
resulting node Qg also refers to tg, because
A0=00g=QQg=Afi. Computing ephemerids, the equinox is
reduced back from tg to t.
The change of position angle by proper motion,
71
0 60 = y.asina( ttg) (449)
where the proper motion component in right ascension ua is
in degrees, can be neglected in most cases.
The two formulae above can be found either in Heintz'
book "Double Stars" or in Aitken's book "The Binary Stars".
In table 44, all position angles have been reduced to
the common equinox 2000.0 and the converted rectangular
coordinates (xg,yg) are also listed.
In figure 41, all pairs of rectangular coordinates
(xQ,yo) are plotted in the xgyg plane. We can see at a
glance that all observation points are distributed closely
to an ellipse. Furthermore, in Figures 2a and 2b, we plot
the distance p in seconds of arc and the position angles
against the observing epoch, respectively. From Figure 2,
we see that the observations fall upon nearly sinelike
curves. Thus, we get the impression that these data are
rather precise and therefore give all observations equal
weights.
For these data, an initial approximate solution is
easily obtained by Kowalsky's method. The initial
approximation g is listed in Table 45.
Starting from this, the final solution for is
obtained after only three iterations and shown in Table 46.
The calculation required only 47?62 CPU time on a VAX.
72
The residuals (observational errors) in (p,6) and (x,y)
are shown in Table 47. The residuals in (p,0) and (x,y)
are plotted against the observing epoch in Figures 43 and
44, respectively. They show a random distribution as
expected. Also, Figure 45 shows the comparison of the
observation points with the corresponding points after
correction in the apparent plane.
The "efficiency," the covariance matrix, the correla
tion matrix and transformation matrix (which transforms the
correlated parameters to the uncorrelated ones) of the
adjusted parameters in the final solution are calculated and
listed in Table 46. In addition, Table 46 also lists the
standard deviations of a) the original and b) the
uncorrelated parameters in the final solution.
73
Table 43.
The Observation Data for 51 Tau.
HR1331
51 Tau
HD 27176
SAO
76541
04185+2135 n
1975.7160
106.0
2.0
0.080
0.003
A1
1975.9591
91.9
1.7
0.074
0.003
A1
1976.8574
34.9
1.5
0.069
0.005
A2
1976.8602
33.5
1.5
0.073
0.009
A2
1976.9229
22.9
1.0
0.072
0.008
A3
1977.0868
26.7
1.0
0.083
0.008
A3
1977.6398
8.8
0.101
A6
1977.7420
3.1
0.8
0.110
0.008
A5
1978.1490
352.2
0.5
0.113
0.010
A5 n
1978.6183
340.7
0.8
0.108
0.008
A5
1978.8756
333.3
2.0
0.086
0.013
B4
1978.7735
304.3
0.090
A7
1980.1532
285.9
0.075
A8
1980.7182
259.0
0.079
A8
1980.7263
255.8
0.085
A8
1980.7291
259.1
0.087
A8
1982.7550
191.80
0.1343
C2
1982.7579
192.65
0.1362
C2
1982.7605
190.36
0.1315
C2
1982.7633
192.90
0.1381
C2
1982.7661
193.39
0.1308
C2
1983.0472
186.18
0.1333
C2
1983.0637
187.21
0.1499
C2
1983.7108
182.05
0.1456
C2
1983.7135
179.56
0.1480
C2
1983.9337
181.0
1.9
0.149
0.010
FA
1983.9579
176.7
0.157
RB
1984.0522
175.01
0.1446
C2
1984.0576
174.79
0.1445
C2
1984.0603
172.73
0.1355
C2
1984.779
157.3
0.146
RC
1984.9308
164.5
2.6
0.141
0.013
FB
1985.1063
161.0
2.7
0.137
0.013
FB
1985.2048
158.0
3.0
0.125
0.013
FB
1985.8378
145.69
0.1141
##
1985.8406
144.53
0.1202
##
1985.8541
145.71
0.1200
##
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
74
Table 44.
The Reduced Initial Data for 51 Tau
t
e0
P0
x0
Y0
1975.7160
106.00
0.0800
0.0222
0.0769
1975.9591
91.90
0.0740
0.0026
0.0740
1976.8574
34.90
0.0690
0.0565
0.0396
1976.8602
33.50
0.0730
0.0608
0.0404
1976.9229
32.90
0.0720
0.0604
0.0392
1977.0868
26.70
0.0830
0.0741
0.0374
1977.6398
8.80
0.1010
0.0998
0.0156
1977.7420
3.10
0.1100
0.1098
0.0061
1978.1490
352.20
0.1130
0.1120
0.0151
1978.6183
340.70
0.1080
0.1020
0.0355
1978.8756
333.30
0.0860
0.0769
0.0385
1979.7735
304.30
0.0900
0.0508
0.0743
1980.1532
285.90
0.0750
0.0207
0.0721
1980.7182
259.00
0.0790
0.0150
0.0776
1980.7263
255.80
0.0850
0.0207
0.0824
1980.7291
259.10
0.0870
0.0163
0.0855
1982.7550
191.80
0.1343
0.1314
0.0176
1982.7579
192.65
0.1362
0.1329
0.0300
1982.7605
190.36
0.1315
0.1293
0.0238
1982.7633
192.90
0.1381
0.1346
0.0310
1982.7661
193.39
0.1308
0.1272
0.0305
1983.0472
186.18
0.1333
0.1325
0.0145
1983.0637
187.21
0.1499
0.1487
0.0190
1983.7108
182.05
0.1456
0.1455
0.0054
1983.7135
179.56
0.1480
0.1480
0.0010
1983.9337
181.00
0.1490
0.1490
0.0028
1983.9579
176.70
0.1570
0.1568
0.0088
1984.0522
175.01
0.1446
0.1441
0.0124
1984.0576
174.79
0.1445
0.1439
0.0129
1984.0603
172.73
0.1355
0.1344
0.0170
1984.7790
157.30
0.1460
0.1348
0.0562
1984.9308
164.50
0.1410
0.1359
0.0375
1985.1063
161.00
0.1370
0.1296
0.0445
1985.2048
158.00
0.1250
0.1160
0.0467
1985.8378
145.69
0.1141
0.0943
0.0642
1985.8406
144.53
0.1202
0.0980
0.0696
1985.8541
145.71
0.1200
0.0992
0.0675
y0 (seconds of arc)
xQ (seconds of arc)
Figure 41. Plot of the observation data for 51 Tau in the xQy0 plane.
(seconds of arc)
Time of observation
Figure 42. Plot of a) the Xg b) y^ coordinates against the observing
epoch of the observation data for 51 Tau.
CTl
y0 (seconds of arc)
77
Table 45.
Initial Approximate Solution g for 51 Tau.
P0(Yr) T0 ag e0 i0 w0 Q0
11.18 1966.4 0.128 0.181 127.3 152.9 170.2
Table 46
The Final Solution for 51 Tau
P(yrs)
T
a"
e
i
(i)
Q
the final solution
11.22
1966.5
0.128
0.173
125.5
157.3
171.2
standard deviations
0.039
0.031
0.0004
0.002
0.32
1.51
0.33
standard deviations
of the uncorrelated
parameters
0.00003
2.091
0.0002
0.0005
0.28
1.39
2.32
the efficiency
0.466
.36E06
.77E09
.17E06
.23E06
.12E06
.75E06
.66E08
. 77E09
.74E11
.47E09
.12E08
.17E09
.98E09
.38E09
. 17E06
.47E06
.30E06
.34E06
.14E06
.34E06
.66E08
the covariance matrix .23E06
.12E08
.34E06
.47E05
.11E06
.14E05
.27E06
.12E06
.17E09
.14E05
.11E06
.19E06
.29E06
.16E08
.7 5E06
.98E09
.34E06
.14E05
.29E06
.29E05
.30E06
.66E08
.38E09
.66E08
.27E06
.16E08
.30E06
.11E06
00
Table 46 (continued)
P(yrs)
T
a"
e
i
0)
Q
1.0000
0.4709
0.5131
0.1762
0.4604
0.7339
0.0327
0.4709
1.0000
0.3119
0.2075
0.1481
0.2099
0.4115
0.5131
0.3119
1.0000
0.2859
0.6074
0.3661
0.0355
the correlation
0.1762
0.2075
0.2859
1.0000
0.1127
0.3811
0.3718
matrix
0.4604
0.1481
0.6074
0.1127
1.0000
0.3992
0.0112
0.7339
0.2099
0.3661
0.3811
0.3992
1.0000
0.5273
0.0327
0.4115
0.0355
0.3718
0.0112
0.5273
1.0000
0.0074
0.9999
0.0002
0.0000
0.0002
0.0026
0.0033
0.4926
0.0068
0.2021
0.0225
0.1789
0.1955
0.8036
0.1416
0.0018
0.5317
0.0168
0.8053
0.0501
0.2145
the transformation
0.4483
0.0034
0.6345
0.1522
0.4443
0.3079
0.2848
matrix
0.6780
0.0034
0.4681
0.0428
0.3145
0.0186
0.4691
0.2546
0.0001
0.2331
0.4862
0.1520
0.7869
0.0466
0.1086
0.0003
0.0195
0.8590
0.0121
0.4949
0.0703
j
vo
80
Table 47.
Residuals of the Observations for 51 Tau in (p,6) and (x,y)
t
v6
VP
vx
VY
1
1975.7160
1.0090
0.0042
0.0000
0.0044
2
1975.9591
1.1702
0.0039
0.0012
0.0039
3
1976.8574
2.5044
0.0082
0.0048
0.0073
4
1976.8602
3.7620
0.0042
0.0007
0.0064
5
1976.9229
1.2477
0.0070
0.0050
0.0051
6
1977.0868
0.0465
0.0009
0.0009
0.0002
7
1977.6398
2.0624
0.0015
0.0010
0.0039
8
1977.7420
0.5795
0.0083
0.0083
0.0004
9
1978.1290
0.2656
0.0058
0.0057
0.0011
10
1978.6183
0.5497
0.0015
0.0018
0.0006
11
1978.8756
0.2196
0.0173
0.0152
0.0082
12
1979.7735
2.1815
0.0054
0.0059
0.0027
13
1980.1632
1.2302
0.0036
0.0008
0.0039
14
1980.7182
2.8268
0.0005
0.0038
0.0014
15
1980.7263
0.0249
0.0064
0.0014
0.0063
16
1980.7291
3.4623
0.0084
0.0032
0.0093
17
1982.7550
1.9542
0.0022
0.0031
0.0038
18
1982.7579
1.0538
0.0041
0.0045
0.0013
19
1982.7605
3.2987
0.0007
0.0009
0.0074
20
1982.7633
0.7102
0.0058
0.0060
0.0001
21
1982.7661
0.1717
0.0015
0.0014
0.0006
22
1983.0472
2.7391
0.0052
0.0043
0.0069
23
1983.0637
1.4488
0.0111
0.0115
0.0019
24
1983.7108
2.8848
0.0019
0.0020
0.0075
25
1983.7135
0.4324
0.0005
0.0005
0.0013
26
1983.9337
4.9126
0.0004
0.0007
0.0129
27
1983.9579
0.9444
0.0083
0.0085
0.0022
28
1984.0522
0.5459
0.0042
0.0040
0.0020
29
1984.0576
0.3999
0.0043
0.0042
0.0016
30
1984.0603
1.6232
0.0133
0.0136
0.0023
31
1984.7790
6.9626
0.0028
0.0031
0.0173
32
1984.9308
2.5201
0.0004
0.0022
0.0060
33
1985.1063
1.7784
0.0000
0.0015
0.0042
34
1985.2048
0.3956
0.0097
0.0086
0.0046
35
1985.8378
0.2949
0.0015
0.0008
0.0014
36
1985.8406
0.8016
0.0047
0.0037
0.0039
37
1985.8541
0.6865
0.0050
0.0050
0.0016
8.0
0.020
av
V
e
p
5.2
CD
CD
Q)
CD
CD
TD
CD
>
2.4
0.4
3.2
A
X
X
x
A
0.013
0.006
0.001
0.008
Q.
>
6.0
1975
i i i
1977 1979 1981
1983
i
1985
1 0.015
1987
Time of observation
Figure 43. The residuals of the observations for 51 Tau in (p,0).
00
(seconds of arc)
(seconds of arc)
Time of observation
Figure 44. The residuals of the observations for 51 Tau in (x,y).
00
ro
(seconds of arc)
y (seconds of arc)
0.13
0.08
0.03
0.02
0.07
0.12
0.20 0.13 0.06 0.01 0.08 0.15
x (seconds of arc)
Figure 45. The original observations for 51 Tau compared with the
observations after correction.
o original observations
observations after correction
J   L
84
Remarks
Newton's method would converge well if the observa
tional errors are small enough and the initial approximate
solution is sufficiently accurate. But, when the errors in
the observed distances and position angles are not suffi
ciently small, or the initial approximation is not close
enough to the final solution, two things will happen:
1) In some iterations, the corrections to the adjust
ment parameters are too big so that some parameters go to
unreasonable values; e.g., the semimajor axis a becomes
negative or the mean motion n, the eccentricity e becomes
negative, which are unacceptable; and further calculation
becomes pointless.
2) The iterations do not converge; i.e., the residual
function S becomes larger in the next iteration, although
all parameters remain in the ranges of reasonable values.
In these cases, Newton's method will fail to yield a
solution. In the next chapter, two other approaches (the
modified Newton methods) are proposed to deal with these
cases.
CHAPTER V
THE MODIFIED NEWTON SCHEME
The scheme for solving the nonlinear condition equa
tions by Newton's method has been discussed in the last
chapter. Although rapid convergence can be expected if the
initial approximate solution is sufficiently accurate and
the residuals are small enough, Newton's method often fails
to yield a solution, particularly if the initial approxima
tion g of the vector is greatly in error or the residuals
are very large. As mentioned in the last chapter, two
problems arise in these cases:
1) Some of the corrections 6 to the adjustment parameters
are too big, which is unacceptable. For example, eg=0.15,
but 6e=0.16, so that the new value e=0.01; in this case,
the residual function calculated is no longer meaningful and
any further calculation becomes pointless.
2) Divergence occurs directly; i.e., the value of residual
function Sg is larger after the iteration than it was
before.
For both of these cases, the major problem is that the
step size could be too large. If we decrease the step size,
the situation might be improved to some extent. We still
rely on Newton's method indicating the right direction, but
85
86
we no longer apply the full corrections which would follow
from the original formalism. In this chapter, we will
discuss the combination of Newton's method with the method
of steepest descent and other sophisticated methods.
The Method of Steepest Descent
We retain A=fTwf and introduce a numerical factor 1/f
into the normal equation (48), which thus becomes
1 T
 AS = fw$ (51)
f a
By choosing an appropriate value of f, we would reduce the
step size and make the residual function Sq gradually
smaller and smaller. This is analogous to the basic idea of
"the method of steepest descent" which is to step to the
next in a sequence of better approximations to the solution
by moving in the direction of the negative of the gradient
of Sq. If the step is not too large, the value of Sq must
necessarily decrease from one iteration to the next. Our
goal is to arrive at the stable absolute minimum value of Sq
and to the corresponding set of parameters which is the best
solution.
At this point, two new problems arise:
1) How does one calculate the residual function Sq and its
gradient?
2) How does one choose the best value of fl
87
Suppose we were to pretend that the current values of
at any iteration are the true values of a. This assumption
is of course not true. But it makes it in general possible
to estimate the vector v as a function of . This is so
because if the present value of were the true value of a,
there would be a definite set of residuals v which causes
the corrected observations to satisfy the conditions
equations rigorously. It is not difficult to find those
components of v which correspond to certain value of .
Suppose we have an approximation and a and vg of v, we
wish to obtain the "best" approximation of v, assuming to
be correct. We then need to minimize
T 1 T
S = V CT V + f L
relative to the remaining variables v and which yields
f(Xq + v,) = 0 (52a)
o1v + fT = 0 (52b)
Setting v=vg+e, and expanding (52a) into powers of e, we
have
f(x0,) + f0 = o
(53)
88
and Eq. (52) becomes
a1(vo+e) + fxOTÂ£ = 0 (54)
where xq = xq + vq. Solving Eq. (54) for yields
= Vq afQTl (55)
Substituting now (55) into Eq. (53) for , we get
f(x0,) fovQ f0afx0T =0. (56)
From Eq. (56) we obtain
= w[f(Xq,) fxov0] (57)
From Eqs. (55) and (57) we arrive therefore at the
expression
v = af0w[f(xQ,) fxov0] (58)
where w=fQTafQ and xq = xq + vq still. This equation may
be iterated, if needed, to arrive at a definite v; i.e.,
until the value of v on the left hand side and the value of
vq on the right hand side are the same. There is thus a

Full Text 
.t^!Xi*S.TY OF FLORIDA
HIM 1111
1262 08556 8003
AN IMPROVED ALGORITHM FOR
THE DETERMINATION OF THE SYSTEM PARAMETERS
OF A VISUAL BINARY BY LEAST SQUARES
By
YULIN XU
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988
UNIVERSITY OF FLORIDA LIBRARIES
To my mother and late father
ACKNOWLEDGEMENTS
The author acknowledges his heartfelt gratitude to
Dr. Heinrich K. Eichhorn, his research advisor, for
proposing this subject, the considerate guidance and
encouragement throughout his research, and for the patience
in reading and correcting the manuscript. The author has
benefited in many ways as Dr. Eichhorn's student.
The author is also grateful to Drs. KwanYu Chen,
Haywood C. Smith, Frank Bradshaw Wood and Philip Bacon for
having served as members of his supervisory committee and
for helpful discussions, timely suggestions and the careful
review of this dissertation. Likewise, his deep appreciaÂ¬
tion goes to Drs. W. D. Heintz and H. A. Macalister for
having provided data used in his dissertation.
The author is especially grateful to Drs. Jerry L.
Weinberg and RuTsan Wang in the Space Astronomy Laboratory
for their considerate encouragement and support. Without
their support, the fulfillment of this research could not be
possible.
It is a great pleasure to acknowledge that all the
calculations were performed on the Vax in the Space
Astronomy Laboratory.
in
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS in
LIST OF TABLES vi
LIST OF FIGURES viii
ABSTRACT X
CHAPTERS
I INTRODUCTION 1
II REVIEW AND REMARKS 4
Review of the Methods for OrbitComputation... 4
Definitions of the Orbital Parameters.... 7
The Method of M. Kowalsky and
S. Glasenapp 8
The Method of T. N. Thiele, R. T. A.
Innes and W. H. van den Bos 13
The Method of Least Squares 16
Remarks 23
III GENERAL CONSIDERATIONS 26
The Condition Equations 27
The General Statement of the Least Squares
Orbit Problem 33
About the Initial Approximate Solution 36
IV THE SOLUTION BY NEWTONâ€™S METHOD 39
The Solution by Newton's Method 39
Weighting of Observations 62
The Orthogonal Set of Adjustment Parameters
and the Efficiency of a Set of Orbital
Parameters 65
A Practical Example 70
Remarks 84
IV
page
V THE MODIFIED NEWTON SCHEME 85
The Method of Steepest Descent 86
The Combination of Newton's Method with the
Method of Steepest DescentThe Modified
Newton Scheme 92
The Application of Marquardt's Algorithm 98
Two Practical Examples 101
VI DISCUSSION 134
REFERENCES 136
BIOGRAPHICAL SKETCH 138
v
LIST OF TABLES
Table Page
41 Expressions for All the Partial Derivatives
in 48
42 Expressions for All the Partial Derivatives
in f 53
43 The Observation Data for 51 Tau 73
44 The Reduced Initial Data for 51 Tau 74
45 The Initial Approximate Solution Ã¡g for 51 Tau... 77
46 The Final Solution for 51 Tau 78
47 The Residuals of the Observations for 51 Tau
in (p,6) and (x,y) 80
51 The Observation Data for (3738 105
52 The Reduced Initial Data for 3738 106
53 The Initial Approximate Solution Ã¡g for 3738 109
54 The Solution #1 for 3738 110
55 The Residuals of the Observations for 3738
in (p, 6) and (x,y) in Solution #1 112
56 Heintz' Result for 3738 116
57 The Solution #2 for 3738 117
58 The Residuals of the Observations for 3738
in (p, 6) and (x,y) in Solution #2 119
59 The Observation Data for BD+19Â°5116 123
510 The Reduced Initial Data for BD+19Â°5116 124
511 The Initial Approximate Solution Ã¡g for
BD+19 Â° 5116 127
vi
Table page
512 The Final Solution for BD+19Â°5116 by the
MQ Method 128
513 The Residuals of the Observations for BD+19Â°5116
in 6,p,x and y 130
vii
LIST OF FIGURES
Figure Page
41 Plot of the observation data for 51 Tau in
the xgy0 plane 75
42 Plot of a) the Xg b) ygcoordinates against
the observing epochs of the observation data
for 51 Tau 76
43 The residuals of the observation for 51 Tau
in ( p, 6 ) 81
44 The residuals of the observations for 51 Tau
in (x,y) 82
45 The original observations for 51 Tau compared
with the observations after correction 83
51 Plot of the observation data for (3738 in the
xgyg plane 107
52 Plot of a) the Xg b) ygcoordinates against
the observing epochs of the observation data
for [3738 108
53 The residuals of the observations for (3738
in ( p, 0 ) according to the solution #1 113
54 The residuals of the observations for 3738
in (x,y) according to the solution #1 114
55 The original observations of 3738 compared
with the observations after correction
according to the solution #1 115
56 The residuals of the observations for 3738
in (p, 6) according to the solution #2 120
57 The residuals of the observations for 3738
in (x,y) according to the solution #2 121
viii
Figure page
58 The original observations for 3738 compared
with the observations after correction
according to the solution #2 122
59 Plot of the observation data for BD+19Â°5116
in the XqYo plane 125
510 Plot of a) the xg b) ygcoordinates against
the observing epochs of the observations for
BD+19Â°5116 126
511 The residuals of the observations for
BD+19Â°5116 in (p,0) 131
512 The residuals of the observations for
BD+19Â°5116 in (x,y) 132
513 The original observations for BD+19Â°5116
compared with the observations after
correction 133
xx
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
AN IMPROVED ALGORITHM FOR
THE DETERMINATION OF THE SYSTEM PARAMETERS
OF A VISUAL BINARY BY LEAST SQUARES
By
YULIN XU
April 1988
Chairman: Dr. Heinrich K. Eichhorn
CoChairman: Dr. KwanYu Chen
Major Department: Astronomy
The problem of computing the orbit of a visual binary
from a set of observed positions is reconsidered. It is a
least squares adjustment problem, if the observational
errors follow a biasfree multivariate Gaussian distribution
and the covariance matrix of the observations is assumed to
be known.
The condition equations are constructed to satisfy both
the conic section equation and the area theorem, which are
nonlinear in both the observations and the adjustment
parameters. The traditional least squares algorithm, which
employs condition equations that are solved with respect to
the uncorrelated observations and either linear in the
adjustment parameters or linearized by developing them in
Taylor series by firstorder approximation, is inadequate in
x
our orbit problem. D. C. Brown proposed an algorithm
solving a more general least squares adjustment problem in
which the scalar residual function, however, is still
constructed by firstorder approximation. Not long ago, a
completely general solution was published by W. H. Jefferys,
who proposed a rigorous adjustment algorithm for models in
which the observations appear nonlinearly in the condition
equations and may be correlated, and in which construction
of the normal equations and the residual function involves
no approximation. This method was successfully applied in
our problem.
The normal equations were first solved by Newton's
scheme. Practical examples show that this converges fast if
the observational errors are sufficiently small and the
initial approximate solution is sufficiently accurate, and
that it fails otherwise. Newton's method was modified to
yield a definitive solution in the case the normal approach
fails, by combination with the method of steepest descent
and other sophisticated algorithms. Practical examples show
that the modified Newton scheme can always lead to a final
solution.
The weighting of observations, the orthogonal paraÂ¬
meters and the "efficiency" of a set of adjustment
parameters are also considered. The definition of
"efficiency" is revised.
xi
CHAPTER I
INTRODUCTION
The problem of computing the orbit of a visual binary
from a set of observed positions is by no means new.
A great variety of methods has been proposed. As is well
known, only a few of these suffice to cover the practical
contingencies, and the majority fails to handle the input
data efficiently and properly.
For the visual binary case, the determination of an
orbit normally requires a large number of observations. All
measures of position angles and separations are, as are all
observations, affected by observational errors. For the
purpose of our work, these errors are assumed to follow a
biasfree multivariate Gaussian distribution. Under this
assumption, orbitcomputing is a least squares adjustment
problem, in which the condition equations are nonlinear in
both the observations and the adjustment parameters. The
condition equations must incorporate all relationships that
exist between observations and the orbital parameters,*
*Usually, the term "orbital elements" is used. We prefer
"orbital parameters" instead. Strictly speaking, the
orbital elements are the constants of integration in the
twobody problem and, therefore, do not include the masses
of the components.
1
2
that is, must state both the area theorem, which follows
from Kepler's equation, and the condition that the projected
orbit is a conic section. H. Eichhorn (1985) has suggested
a new form for the construction of a complete set of
condition equations for this very problem.
The traditional least squares algorithm, which is based
on condition equations linearized with respect to the
observational errors, will not lead to those orbital
parameters which minimize the sum of the squares of observaÂ¬
tional errors, because linearization, in this case, is too
crude an approximation. In some earlier papers, H. Eichhorn
and W. G. Clary (1974) proposed a least squares solution
algorithm, which takes into account the second (in addition
to the first) order derivatives in the adjustment residuals
(observational errors) and the corrections to the initially
available approximation to the adjustment parameters. A
completely general solution was published by W. H. Jefferys
(1980, 1981), who proposed a rigorous adjustment algorithm
for models in which the observations appear nonlinearly in
the condition equations. In addition, there may be nonÂ¬
linear constraints** among model parameters, and the
observations may be correlated. In practice, the method is
nearly as simple to apply as the classical method of least
**We use the term "constraints" for condition equations
which do not contain any observations explicitly.
squares, for it does not require calculation of any derivaÂ¬
tives of higher than the first order.
In this paper, we present an approach to solve the
orbit problem by Jefferys' method, in which both the area
theorem and the conic section equation assume the function
of the condition equations.
CHAPTER II
REVIEW AND REMARKS
Review of the Methods for OrbitComputation
Every complete observation of a double star supplies us
with three data: the time of observation, the position
angle of the secondary with respect to the primary, and the
angular distance (separation) between the two stars. The
problem of computing the socalled orbital elements (in this
paper, called orbital parameters) of a visual binary from a
set of observations superficially appears analogous to the
case of orbits in the planetary system, yet in practice
there is little resemblance between the problems. The
problem of determining the orbit of a body in the solar
system is complicated by the motion of the observer who
shares the motion of the Earth, so that, unlike in the case
of a binary, the apparent path is not merely a projection of
the orbit in space onto the celestial sphere.
In the case of a binary, the path observed is the
projection of the motion of the secondary round the primary
onto a plane perpendicular to the line of sight. The
apparent orbit (i.e., the observed path of the secondary
about the primary) is therefore not a mere scale drawing of
the true orbit in space. The primary may be situated at any
4
point within the ellipse described by the secondary and, of
course, does not necessarily occupy either a focus or the
center.
5
The problem of deriving "an orbit" (meaning a set of
estimates of the orbital parameters) from the observations
was first solved by F. Savary in 1827. In 1829, J. F. Encke
quickly followed with a different solution method which was
somewhat better adapted to what were then the needs of the
practical astronomer. Theoretically, the methods of Savary
and Encke are excellent. But their methods utilize only
four complete pairs of measures (angle and distance) instead
of all the available data and closely emulate the treatment
of planetary orbits. They are therefore inadequate in the
case of binary stars (W. D. Heintz, 1971; R. G. Aitken,
1935) .
Later, Sir John Herschel (1832) communicated a
basically geometric method to the Royal Astronomical
Society. Herschel's method was designed to utilize all the
available data, so far as he considered them reliable.
Since then, the contributions to the subject have been many.
Some consist of entirely new methods of attack, others of
modifications of those already proposed. Among the more
notable investigators are Yvon Villarceau, H. H. Madler, E.
F. W. Klinkerfues, T. N. Thiele, M. Kowalsky, S. Glasenapp,
H. J. Zwiers, K. Schwarzchild, T. J. J. See, H. N. Russell,
R. T. A. Innes, W. H. van den Bos and others.
One may classify the various methods published so far
into "geometric" methods, which are those that enforce only
6
the constraint that the orbit is elliptical, and the
"dynamical" ones which enforce, in addition, the area
theorem.
The geometric treatment initiated by J. Herschel peaked
in Zwiers' method (1896) and its modifications, e.g. those
of H. M. Russell (1898) and of G. C. Comstock (1918). Every
geometric method has the shortcoming that it must assume the
location of the center of the ellipse to be known while it
ignores the area theorem and thus fails to enforce one of
the constraints to which the observations are subject. The
growing quantity and quality of observations called for
suitable computing precepts, and the successful return to
dynamical methods began with van den Bos (1926).
Of the many methods for orbitcomputation formulated,
some are very useful and applicable to a wide range of
problems, e.g. those by Zwiers, Russell and those by Innes
and van den Bos.
Zwiers' method (1896) is essentially graphical and
assumes that the apparent orbit has been drawn. This method
is therefore useless unless the apparent ellipse gives a
good geometrical representation of the observations and
satisfies the law of areas, and thus will not be further
described here since we are primarily concerned with the
analytical methods.
7
In the following, we will briefly review Kowalsky's
method and that by Thiele and Innes.
Definition of the Orbital Parameters
Seven parameters define the orbit and the space
orientation of its plane. The first three of these (P,T,e)
are dynamical and define the motion in the orbit; the last
four (a,i,i2,w) are geometrical and give the size and
orientation of the orbit. The parameters are defined
somewhat differently from those for the orbits of planets
and comets.
The first dynamical parameter P is the period of
revolution, usually in units of mean sidereal years; n is
the mean (usually annual) angular motion; since n=2n/P, P
and n are equivalent. The second, T, is the epoch of
periastron passage (usually expressed in terms of years and
fractions thereof). The third, e, is the eccentricity of
the orbital ellipse.
The geometrical parameter a is the angle subtended by
the semimajor axis of the orbital ellipse (usually
expressed in units of arcseconds). The angle i is the
inclination of the orbital plane to the plane normal to the
line of sight, that is, the angle between the plane of
projection and that of the orbit in space. It ranges from 0Â°
to 180Â°. When the position angle increases with time, that
is, for direct motion, i is between 0Â° and 90Â°; for retroÂ¬
grade motion, i is counted between 90Â° and 180Â°;
8
and i is 90Â° when the orbit appears projected entirely onto
the line of nodes. The "node", ft, is the position angle of
the line of intersection between the tangential plane of
projection and the orbital plane. There are two nodes whose
corresponding values of ft differ by 180Â°. That node in
which the orbital motion is directed away from the sun is
called the ascending node. We understand ft, which ranges
from 0Â° to 360Â°, to refer to the ascending node. Because it
is, however, one of the peculiarities of the orbit
determination of a visual binary that it is in principle
impossiblefrom positional data aloneto decide whether
the node is the ascending or descending one ft may be
restricted to 0Â°
the periastron in the plane of the orbit, counted positive
in the direction of the orbital motion and starting at the
ascending node. It ranges from 0Â° to 360Â°.
These definitions are adhered to throughout our work.
Some of them may somewhat differ a little from those given
by previous authors. But any way in which one defines them
does not affect the principles of the method we describe.
The Method of M. Kowalsky and S. Glasenapp
This old method was first introduced by J. Herschel in
a rather cumbersome form and is better known in its more
direct formulation by M. Kowalsky in 1873 and by
S. Glasenapp in 1889. R. G. Aitken (1935) gives the
9
detailed derivation of the formulae in his textbook The
Binary Stars.
Kowalsky's method is essentially analytical. It
derives the orbit parameters from the coefficients of the
general equation of the ellipse which is the orthogonal
projection of the orbit in space, the origin of coordinates
being taken at the primary. The projected orbit can be
expressed by a quadratic in x and y, whose five coefficients
are related to those five orbital parameters which do not
involve time.
In rectangular coordinates, the equation of an ellipse,
and thus of a conic section, takes the form
c^x2 + 2c2xy + cjy2 + 2C4X + 2c5y +1=0 , (21)
where the rectangular coordinates (x,y) are related to the
more commonly directly observed polar coordinates (p,9) by
the equations
x = pcosG ,
y = psine ,
(22a)
(2 â€” 2b)
where p is the measured angular distance and 0 the position
angle.
10
The five coefficients of equation (21) can be deterÂ¬
mined by selecting five points on the ellipse or by all the
available observations in the sense of least squares.
There is an unambiguous relationship between the five
coefficients (0^,02,03,04,05) and the five orbital paraÂ¬
meters (a,e,i,oj,Q) . We can find the detailed derivation of
the formulae in Aitken's The Binary Stars. Here, we will
therefore state only the final formulae without derivation.
The five orbital parameters (a,e,i,,Â£2) can be calcuÂ¬
lated from the known coefficients (C1/C2,Â£3,04,05) by the
following procedure.
1) The parameter Q can be found from the equation
2(c^c.Cr)
2 4 5
(23)
tan2Q
To determine in which quadrant Q is located, we can use
two other equations:
c'sin2i2 = 2(C2~C4C5) ,
c'cos2Q = C52C42+C1C3
(23'b)
(23'a)
where
2.
tan 1
c
(23â€™c)
2
P
which is always positive.
11
More elegantly, we write in Eichhorn's notation
2Q = plg[2(c2C4C5),C52C42+c1C3]
(23â€™d)
H. Eichhorn (1985), in his "Kinematic Astronomy"
(unpublished lecture notes), defines the plg(x,y) function
as follows:
plg(x,y) = arctan(x/y) + 90Â°[2sgnx(1+sgny)] ,
where arctan is the principal value of the arctangent.
2)The inclination i is found from
tan^i = 2 +
(24)
Whether the i is in the first or second quadrant is deterÂ¬
mined by whether the motion is direct or retrograde. If the
motion is direct, the position angle increases with time,
i<90Â°; otherwise, i>90Â°.
3) The equation
u = plg[l(c4sinQC5COsQ)cosi,C4COsQ+C5sinfi) (25)
gives the value of w.
4) With i,o),Q known, two more parameters (a and e) can be
calculated from
12
2cos2Â£2(c4sinÃic,cosQ)cosi
e
(26)
2cos2Ã2
a
(27)
[cos2Q(c^^+c^^clc3)(c^^c^^+clc3)](1e^)
5) To complete the solution analytically, the mean motion n
(or the period P) and the time of periastron passage T, must
be found from the mean anomaly M, computed from the observaÂ¬
tions by Kepler's equation:
(28)
M = n(tT) = EesinE
where E is the eccentric anomaly. Every M will give an
equation of the form
(29)
M = nt + T
where T=nT.
From these equations the values of n and T are computed
by the method of least squares.
This is essentially the socalled Kowalsky method. It
looks mathematically elegant. But because it is not only
severely affected by uncertainties of observations but also
ignores the area theorem, it has a very poor reputation
among seasoned practitioners. However, in our work, we use
13
it for getting the initial approximation to the solution.
It serves this purpose very well.
The Method of T. N. Thiele, R. T. A. Innes and
W. H. van den Bos
T. N. Thiele (1883) published a method of orbit
computation depending upon three observed positions and the
constant of areal velocity.
The radii vectors to two positions in an orbit subtend
an elliptical sector and a triangle, the sector being
related to the time interval through the law of areas.
Gauss introduced the use of the ratio: "sector to triangle"
between different positions into the orbit computation of
planets, and Thiele applied the idea to binary stars.
Although the method could have been applied in a wide range
of circumstances, it became widely used only after Innes and
van den Bos revived it.
In 1926, R. T. A. Innes (Aitken, 1935), seeking a
method simpler than those in common use for correcting the
preliminary parameters of an orbit differentially, indepenÂ¬
dently developed a method of orbit computation which differs
from Thiele's in that he used rectangular instead of polar
coordinates. W. H. van den Bos's (1926, 1932) merit is not
merely a modification of the method (transcribing it for the
use with Innes constants) but chiefly in its pioneeringly
successful application. The device became most widely
applied. Briefly, the method of computation is as follows.
14
The computation utilizes three positions (^ ) or the
corresponding rectangular coordinates (xj_,yÂ¿) at the times
tÂ¿ (i=l,2,3). The area constant C is the seventh quantity
required. Thiele employs numerical integration to find the
value of C.
First, from the observed data, find the quantities L by
l12 = t2_tl_D12/c '
(210a)
l23 = t3t2D23^c '
(210b)
L13 = t3t1D13/C ,
(210c)
where Dj_j = pj_pjsin( 6j6Â¡_) , are the areas
of the corre
sponding triangles.
Then, from the equations
nL]_2 = psinp ,
(2lla)
nL23 = qsinq ,
(2llb)
nL13 = (p+q)sin(p+q) ,
(2llc)
the quantities n, p and q can be found by
trials, for in the
three equations above there are only three unknowns, i.e.,
n, p and q. The eccentric anomaly E2 and the eccentricity e
can thus be computed from
(D23sinpD12sin(3)
esinEp =
(D23+D12D13>
(212a)
(212a)
(212b)
ecosE^
(D23cosp+D12COS(3D]_3
D23+D12"D13
After E2 and e are obtained, the two other eccentric
anomalies E]_ and E3 can be found from
E1 = E2  P , (213a)
E3 = E2 + q â€¢ (213b)
The EÂ¿ are used first to compute the mean anomalies Mj_ from
equations (28), which lead to three identical results for
T as a check, and second to compute the coordinates Xj_ and
from
Xj_ = cos E e , (214a)
Yj_ = sinEj_y( le^) . (2 â€” 14b)
By writing
X
H
II
+ FYi ,
(215a)
Yi = BXi
+ gyÂ± ,
(215b)
the constants A, B, F, G are obtained from two positions,
the third again serving as a check. These four coeffiÂ¬
cients, A, B, F, G, are the now socalled ThieleInnes
constants. In addition to n, T, e, which have already been
16
determined, the other four orbital parameters a, i, fi, w can
be calculated from A, B, F, G through
Q+u) = plg(BF,A+G) ,
Ã20) = plg(B+F,AG) ,
(216a)
(216b)
_i ( AG)cos(go+Â£3) (B+F) sin( w+Q)
tan â€” = = 
2 ( A+G)cos(ojQ) (BF) sin( o)Q) ,
(216c)
A+G
2acos(u)+Q)cos
2
(216d)
In addition to the brief introduction to this method given
above, a detailed description of it can be found in many
books, e.g., Aitken's book The Binary Stars (1935) and W. D.
Heintz's book Double Stars (1971).
A more accurate solution is obtained by correcting
differentially the preliminary orbit which was somehow
obtained by using whatever method. This correction can be
achieved by a least squares solution.
The Method of Least Squares
The method of least squares was invented by Gauss and
first used by him to calculate orbits of solar system bodies
from overdetermined system of equations. It is the most
important tool for the reduction and adjustment of observaÂ¬
tions in all fields, not only in astronomy. However, the
traditional standard algorithm, which employs condition
equations that are solved with respect to the (uncorrelated)
17
observations and either linear in the adjustment parameters
or linearized by developing them in Taylor series which are
broken off after the first order terms, is inadequate for
treating the problem at hand. An algorithm for finding the
solution of a more general least squares adjustment problem
was given by D. C. Brown (1955). This situation may briefly
be described as follows.
Let {x} be a set of quantities for which an approximaÂ¬
tion set {xq} was obtained by direct observation. By
ordering the elements of the sets {x> and {xQ} and regarding
them as vectors, x and xq, respectively, the vector v=xxq
is the vector of the observational errors which are
initially unknown. Assume that they follow a multivariate
normal distribution and that their covariance matrix a is
regarded as known. Further assume that a set of parameters
{a} is ordered to form the vector a. The solution of the
least squares problem (or the adjustment) consists in
finding those values of the elements of {x} and {a} which
minimize the quadratic form vTa1v while at the same time
rigorously satisfying the condition equations
fi({x}i,{a}i)=0 . (217)
This is a problem of finding a minimum of the function
vTa1v subject to condition equations. A general rigorous
and noniterative algorithm for the solution exists only for
18
the case that the elements of {x} and {a} occur linearly in
the functions f j_. When fj_ are nonlinear in the elements of
either {x} or {a}, or both, equations which are practically
equivalent to the f j_ and which are linear in the pertinent
variables can be derived in the following way.
Define a vector 6 by a=aQ+6. The condition equations
can then be written
f(x0+v, a0+6) = 0
(218)
where the vector of functions f=(fj_,f2 ..., fro)T, m being
the number of equations for which observations are
available. Now assume that all elements of {v} and {6} are
sufficiently small so that the condition equations can be
developed as a Taylor series and written
f0 + fxv + ^aÂ® +0(2) ...  0
(218'a)
If the small quantities of order higher than 1 can be
neglected, we can write the linearized condition equations
as
f0 + fxv + fa6 = 0
(218'b)
19
These are linear in the relevant variables, which are the
components of v and of 6.
In order to satisfy the conditions (218'b), we define
a vector 2y. of Lagrangian multipliers and minimize the
scalar function
S(v,6) = vTo1v  2n(f0 + fxv + fa6) (219)
in which the components of v and 6 are the variables.
Setting the derivatives (3S/3v) and (3S/36) equal to zero
and considering equations (218'b), we obtain
6 =  (faT(fxafxT)fa]1faT(fxafxT)1f0 , (220a)
U =  (fxafxTjiif^+fo) , (220b)
v = cjfxTu , (220c)
where we have assumed that fxafxT is nonsingular. This is
the case only if all equations fj_=0 contain at least one
component of x; i.e., if there are no pure constraints of
the form gj_(a)=0. This case (which we shall not encounter
in our investigations) is discussed below in the description
of Jefferys' method.
Note that in constructing the scalar function S in
expression (219), first order approximations have
20
been used. In some cases, the linearized representation of
Eg. (218) by Eg. (218'b) is not accurate enough. In some
of these cases, the convergence toward the definitive
solution may be accelerated and sometimes be brought about
by a method suggested by Eichhorn and Clary (1974) when a
strictly linear approach would be divergent. Their solution
algorithm takes into account the second (as well as the
first) order derivatives in the adjustment residuals
(observational errors) and the corrections to the initially
available approximations to the adjustment parameters. They
pointed out that the inclusion of second order terms in the
adjustment residuals is necessary whenever the adjustment
residuals themselves cannot be regarded as negligible as
compared to the adjustment parameters, in which cases the
conventional solution technigues would not lead to the
"best" approximations for the adjustment parameters in the
sense of least sguares. The authors modified the condition
eguations as
M6
f0 + fxv + fa6 + Vv + D6 + or = 0 . (218")
Nv
Correspondingly, the scalar function to be minimized becomes
M6
S"(v,6) = vTa1v  2u(fo+fxv+fa6+Vv+D6+or) â€¢ (219")
Nv
21
Here, the ith line of the matrix D is
t
O' E. ,
2 1
and
E.
1
r
32fi
3a^ 3a
k'
(221)
the matrix of the Hessean determinant of fj_ with respect to
the adjustment parameters. Similarly, the ith line of
1
vector V is â€”vTWj_, and
2
W.
l
' 32fÂ± ^
,3Xj3xk>
the ith line of M is vTHj_, and
(222)
H.
1
r 32fi ^
. 3x . 3a, .
v j
(223)
and that of N is 6thÂ£, so evidently M6=Nv.
Minimizing S", also 6, v can be calculated.
For this algorithm in detail, one can refer to the
original papers of Eichhorn and Clary. The notation used
here is slightly different from the original one used by the
authors.
Jefferys (1980, 1981) proposed an even more accurate
algorithm which also improves the convergence of the
22
conventional least squares method. Furthermore, his method
is nearly as simple to apply in practice as the classical
method of least squares, because it does not require any
second order derivatives. Jefferys defines the scalar
function to be minimized as
(224)
where Ãc=Xq+v; x,a are the current "best" approximations to x
and a; and g is another vector function consisting of the
constraints on the parameters; Â¿i, y are vectors of
Langrangian multipliers and evaluated at (x, Ã¡). Minimizing
S with respect to x and a, he arrives at the normal equaÂ¬
tions
a_1v + fÂ¿T(x,Ã¡)Ãœ = 0 ,
fxT(x,Ã¡)Ã¼ + gÂ¿T(Ã¡) y = 0
f(x,Ã¡) = 0 ,
g(Ã¡) = 0 ,
(225c)
(225d)
(225a)
(225b)
where
x, a
These equations are exact and involve no approximations.
This is the significant difference between Jefferys' method
and those of Brown and of Eichhorn and Clary.
23
Remarks
As mentioned before, Kowalsky's method will most likely
not produce the best obtainable results, because the
relative observed coordinates (x, y, t) are subjected only
to the condition (21), which involves only five of the
seven necessary orbital parameters as adjustment parameters
and does not enforce the area theorem. It can therefore
never be used for a definitive orbit determination since it
completely ignores the observation epochs.
Yet, Eg. (21) has the advantage that it appears to be
simple, in particular it is linear in the adjustment
parameters albeit not in the observations. When it is used
for the determination of orbits, the righthand sides of the
equations which result from inserting a pair (x, y) of
observed rectangular coordinates into Eg. (21) are regarded
as errors with a univariate Gaussian distribution (i.e., as
normally distributed errors). One may then perform a least
squares adjustment which is linear in the adjustment
parameters. As Eichhorn (1985) pointed out, this approach,
while it has the advantage that approximation values for the
adjustment parameters need not be available at the outset,
fails to take into account two facts.
1) It is not the righthand sides of the condition equations
which are to be considered as normally distributed errors,
but rather the observations (x,y) or (p,6). The condition
equations (21) thus contain more than one observation each.
24
Since the observations occur in the condition equations
nonlinearly, the matrix
rdf(x,a)>
, 3x
must be found. This requires knowledge of approximate
values ag for a. Approximate values xg for x are
availablethey are the observations themselves.
Approximate values ag for a may sometimes indeed be obtained
in the classical way by regarding the righthand sides of
the condition equations as normally distributed errors. In
addition, it should also be taken into account that the
covariance matrix a of the observations is not necessarily
diagonal.
2) In some cases, especially when the binary under study is
very narrow, the errors of the observations are not
negligibly small compared with the adjustment parameters.
This requires either that secondorder terms in the observaÂ¬
tional errors v be carried in the equations or, as Jefferys
has pointed out, that iterations be performed using in the
evaluation of the matrices fx and fa not only improved
approximations for a but also improved values for the
observed quantities as they become available.
If Kowalsky's methods were so modified, the algorithm
would yield better values for the adjustment parameters a
25
than the traditional approach. Either way, one can usually
find an initial approximation by Kowalsky's method.
With respect to both the theoretical clarity and the
practical applicability, as far as it is concerned, the
ThieleInnesvan den Bos method leaves nothing to be
desired. However, the three places selected, even when
smoothed graphically or by some computation, may not suffice
to describe the motion with sufficient accuracy, so that
large and systematic residuals may remain, particularly near
periastron. The method is seriously inadequate even if one
of the ratios sector to triangle is very close to 1 and thus
strongly affected by the measurement errors or if the area
constant C is not initially known to the required accuracy.
The computation may then produce an erroneous orbit with
spuriously high eccentricity, perhaps a hyperbolic one, or
no solution at all. And obviously, different combinations
of the three positions selected from a set of observations
will not likely give the same result. This method therefore
fails to use the information contained in the observations
in the best possible way.
In our work we try to present a fairly general least
squares algorithm to solve the orbit problem. We shall
adopt Jefferys' least squares method as our basic approach.
CHAPTER III
GENERAL CONSIDERATIONS
This chapter contains a general discussion of the least
squares orbit problem. We shall set up condition equations
which simultaneously satisfy the ellipse equation and the
area theorem.
We have seen that it is not sufficient to use Eq. (21)
as the only type of condition equation because this would
ignore the observing epochs, cf. last chapter. Completely
appropriate condition equations must explicitly contain the
complete set of the seven independent orbital parameters as
the adjustment parameters. Also, to be useful in practice,
they must impose both the geometric and dynamical condiÂ¬
tions, and must lead to a convergent sequence of iterations.
After the condition equations are established, we
present the general outline of the algorithm which solves
the orbit problem by Jefferys' method of least squares.
We also discuss some further suggestions for obtaining
the initial approximate solution required for the least
squares algorithm.
26
27
The Condition Equations
Assume that a set of observations {xq} was obtained
consisting of complete data triples (t, p, 6), which
measure the positions of the fainter component (secondary)
with respect to the brighter one (primary): the position
angle 0 is counted counterclockwise from North and ranges
from 0Â° to 360Â°; the angular separation p (also called
distance) is usually given in seconds of arc, and t is
the observing epoch. The conversion of (p, 0) to
rectangular coordinates (x, y) in seconds of arc is as
following:
Declination difference 6c6p = x = pcos0, (3la)
Right ascension difference (acOp)cos6 = y = psin0, (3lb)
where 6C, ac are the declination and right ascension,
respectively, of the secondary; 6p, Op those of
primary.
Equivalently, the observations can also be regarded
as relative coordinates (t, x, y) of the secondary with
respect to the primary.
It might be worthwhile to point out that 1) even though
the formulae (31) are approximations valid only for small
values of p, they may be regarded as practically rigorous
for binaries; 2) we are following the custom in double star
28
astronomy by having the xaxis along the colure* and the y
axis tangential to the parallel of declination.
All observations in {xg} are affected by observational
errors. Let {x} be the set of the true values of the
observations, that is, those values the observations would
have had if there were no observational errors. By ordering
the elements of the sets {xg} and {x} and regarding them as
vectors, xg and x respectively, we have seen that we may
write the vector of observational errors as v = x  Xg.
These errors are of course unknown, but as mentioned already
in the last chapter, we assume that they follow a multiÂ¬
variate normal distribution with known covariance matrix.
For visual binaries, the relative orbit must be an ellipse
(strictly speaking, a conic section) in space as well as in
projection. All pairs (x, y) must therefore satisfy the
condition equations (21):
C^x2 + 2C2xy + C3y2 + 2C4x + 2C5y +1=0 ,
which implicitly involve five of the seven orbital paraÂ¬
meters but do not enforce the area theorem.
The wellknown relationships between the five coeffiÂ¬
cients (Ci, C2, C3, C4, C5) in Eq. (21) and the five
*Following Eichhorn's terminology who uses the term "colure"
generally for any locus of constant right ascension.
29
orbital parameters (e, a, i, co, n) by way of the Thiele
Innes constants, have been discussed in the last chapter.
Consider a righthanded astrocentric coordinate system
K whose XY plane is the true orbital plane such that the
positive Xaxis points toward the periastron (of the
secondary with respect to the primary). The positive Yaxis
is obtained by rotating the Xaxis by 90Â° on the Zaxis in
the direction of the orbital motion. The axes of a second
astrocentric, righthanded coordinate system k are parallel
to those of the equator system Q. The two systems K and k
are related by the transformation
xK = R3(u))R1(i)R3(n)Xk . (32)
From the theory of the twobody problem we know that,
in the system K, the coordinates of the secondary with
respect to the primary are given by
K
x =
f X 1
Y
= a
'cosE  e ^
sinEZle^
l 2 J
l o J
where E is the eccentric anomaly, which is the solution of
Kepler's equation
n(tT) = EesinE
(34)
30
Here, n and T, the mean motion and the periastron epoch,
the two orbital parameters not involved in Eq. (21).
From Eq. (32) we obtain
'A B
"x"
XK =
F G H
y
,K L
where
Kx + Ly
2 =  â–
and
r A B
F G
. K L
or, in detail,
A = cosicosa)  sinfÃsinwcosi ;
B = sinQcoso) + cosQsinojcosi ;
C = sinojsini ;
F = cosfisino)  sinQcostocosi ;
G = sinfisinoj + cosQcosuicosi ;
H = cosuisini ;
M
C ^
H
M j
= R3(a))R1(i)R3(n) ,
(
(
(
(3
(3
(3
(3
(3
(3
(3
are
35)
36)
37)
8a)
8b)
8c)
8d)
8e)
8f)
8g)
K = sinQsini
31
L = cosfisini ;
M = cosÃ ,
where in this notation, the traditional ThieleInnes
constants would be aA, aB, aF and aG.
From Eq. (37) and (35) we can get
X = Ax + By + Cz =
Gx  Fy
M
(3
Y = Fx + Gy + Hz = 
Bx  Ay
M
(3
Thus, we see that X and Y can be expressed in terms of
and the observations (x, y).
From Eq. (33) we obtain
X
cosE = â€” + e ,
a
(3
sinE =
f. 2
avle
(3
Combining (310) with Kepler's equation (34), we get
X
â€” + e =
eY
n(tT) +
aVle2
38h)
38i)
9a)
9b)
i , co , Q
10a)
10a)
a
cos
(3lla)
32
a Jle2
= sin
eY
n(tT) +
a
More succinctly, we have
(3llb)
u =
V =
or
u =
V =
with
U =
cost n(tT)
sin[ n(tT)
cost n(tT)
sin[ n(tT)
X
 + e ,
a
+ eV ] ,
+ eV ] ,
+ eJlU2 ] ,
+ eV ] ,
Y
V = â€”
L 2
avle
(312a)
(312b)
(312'a)
(312'b)
(312c)
After X and Y are expressed in terms of i, w, Q and (x, y)
as in Eqs. (39), Eqs. (311) or (312) involve exactly the
seven orbital parameters (n, T, a, e, i, to, Â£}) and the
observations (t, x, y). The observing epoch t now appears
explicitly, as it must if the area theorem is to be
enforced.
Now, we see that if both equations (311) are satisÂ¬
fied, Kepler's equation which enforces area theorem would be
satisfied and furthermore, the ellipse equation would also
be automatically satisfied as can be seen if t is eliminated
from Eqs. (311) so that these equations are reduced to one
equation in X and Y. If we select the two equations (311)
33
as the condition equations, we need no longer carry
Eq. (21) separately. Of course, we can use any one of
Eqs. (311) as well as Eq. (21) as the condition equations.
However, using Eq. (21) is not convenient, for it contains
the five coefficients directly, but not the five orbital
parameters themselves, even though there are unique relaÂ¬
tionships between them. Ideally, the condition equations
should have the adjustment parameters explicitly as
variables.
In our work, we use the Eqs. (311) as the complete set
of condition equations.
The General Statement of the Least
Squares Orbit Problem
As seen in the last section, the vector of "true
observations" x (presumably having been adjusted from the
observation xq by the residuals v) and the vector of "true
orbital parameters" a, a=(n, T, a, e, i, w, Q)T, must
satisfy the condition equations
X
eY
f!(x0+v,a)
+ e  cos n(tT)+
0
(313a)
a
Y
eY
f2(xQ+v,a)
sin n(tT)+
0
(3 â€” 13b)
where
34
' X
r A B C >
' X "
Y
=
F G H
y
, o â€ž
, K L M ,
, z >
with
Kx + Ly
z = 
and
f A B C ^
F G H
v K L M j
= R3(u)R1(i)R3(n)
In our problem, there are no constraints between the
parameters which involve no observations so that Jefferys' g
function does not occur. The problem can therefore be
stated as follows.
Assume that the residuals {v} (regarded as vector v) of
a set of observations {xg} (regarded as vector xg) follow a
multivariate normal distribution, whose covariance matrix a
is regarded as known; we are to find the best approximations
of v (for v, the residuals) and Ã¡ (for a, the parameters)
such that
f^xg+v^a) = 0
and
f2(x0+v,a)
0
35
are both satisfied while at the same time the quadratic form
1 T 1
S 0= " VÃ¼ v ( 3 â€” i 4 )
U 2
is minimized.
Following the wellknown procedure introduced by
Lagrange, the solution is obtained by minimizing the scalar
function
S =  vTa_1v + fT(x,Ã¡)y. , (315)
2
where x = Xq + v, and is the vector of Lagrangian multiÂ¬
pliers, together with satisfying the equations f]_=0=f2
Denoting the matrix of partial derivatives with respect to a
variable by a subscript, this is equivalent to solving the
following normal equations:
a â– 'â– v + fÂ¿T(x,Ã¡)Ãº = 0 (316a)
fÃ¡TÃš = 0 , (316b)
f(x,a) = 0 . (316c)
We have stated before that these equations are exact and
therefore involve no approximations. Before Jefferys, all
authors used first order or second order approximations in
forming S in equation (315). This is the significant
36
difference that distinguishes Jefferys' method from those
employed by previous authors.
It is evident that the solution of the equations (316)
would solve the posed problem.
About the Initial Solution
The least squares algorithm requires an initial
solution as starting point. Any approach which leads to
approximate values of the orbital parameters serves this
purpose, because our algorithm does not require a very
accurate initial approximation. As long as the initial
approximation is not too different from the final result,
convergence can always be achieved. To obtain an initial
solution, the following procedures may lead to an initial
solution.
1) As mentioned in Chapter II, Kowalsky's method can produce
a preliminary solution. Inserting the pairs (x, y) of
observed rectangular coordinates into Eq. (21), we have a
set of linear equations in which the five coefficients (c^,
C2, C3, C4, C5) are the unknowns. By making a classical
least squares solution based on these linear equations, the
five coefficients can be computed. An estimate of the five
parameters (a, e, i, ai, Q) can be obtained in turn from the
unique relationships between them and the five coefficients.
The remaining two parameters (n, T) also can then be
calculated from the known quantities simply by classical
least squares, as described in Chapter II.
37
As Eichhorn (1985) has pointed out, it is of course
better to use the modified Kowalsky method. Using (21) as
condition equations and the five coefficients as the
adjustment parameters, one may iterate by Jefferys'
algorithm toward the best fitting adjustment parameters and
the best corrections to the observations, v, that is, to
arrive at the values of Ã¡ and v which minimize the scalar
function Sq (see equation 314) while simultaneously
satisfying the condition equations.
This method is simple to apply in practice. But
unfortunately, especially when the observations are not very
precise, the five coefficients (c^, C2, C3, C4, C5) in some
cases do not always satisfy the conditions for an ellipse;
i.e., they do not meet the requirements
C^>0, C3>0 and C^C3_C2^>0.
However, in these cases, it does not mean that there is no
elliptic solution at all and other approaches can be tried.
When this happens, one may for instance take the approach
outlined in Chapter 23 of Lawson and Harsson (1974).
2) By using some selected points among the observation data
instead of using all the data points, sometimes a solution
can be found by Kowalsky's method. Such a solution is
usually also good enough to be a starting approximation.
38
3) Or, carefully selecting three points among the observaÂ¬
tion data, one may use the ThieleInnesvan den Bos method
to calculate an initial approximation. The method has been
described in Chapter II.
CHAPTER IV
THE SOLUTION BY NEWTONâ€™S METHOD
The Solution by Newton's Method
In our problem, the normal equations (316) are
nonlinear. They must be solved by linearization and
successive approximations. Assume that approximate initial
estimates of the unknowns in the normal equations have
somehow been obtained (using whatever methods). This
initial approximation may be improved by Newton's method,
which consists of linearizing the normal equations about the
available solution by a first order development and
obtaining an improved solution by solving the linearized
equations. This process is iterated with the hope that
convergence to a definite solution would eventually be
obtained. This expectation is reasonable if the initial
approximation is sufficiently close to the final solution
and if the observational errors are not too large.
Following Jefferys' notation (which is also the
notation we have used in Chapter III), let the initial
approximate solution (and also the current approximate
solution during iteration) be given by (x,a), where
x = xg + v, xg being the vector of observation, v the vector
of observational errors (for which we adopt the nullvector
39
40
as initial approximation); Ã¡ is the initial approximation of
the vector of the seven orbital parameters (adjustment
parameters); also let the corrections to both x and Ã¡ be
denoted by Ã© and 6, respectively.
The normal equations in our problem now become
(4la)
(4lb)
(4lc)
Here we have ignored (as Jefferys did) products of e and 6
with Lagrangian Multipliers. This does not affect the final
result, as Jefferys also pointed out. A caret in equations
(41) above means evaluation at current values of x and a.
Similar to Jefferys' procedure, we solve the equations
(41) as follows.
Solving Eq. (4la) for e we have
Ã© =  v  afÂ¿u
(42)
Substituting (42) into (4lc) for Ã©, Eq. (4lc) becomes
f  fÂ¿v  fÂ¿afxTÃl + fÂ¿6 = 0
(43)
Solving Eq. (43) for Ãš, we obtain
41
Ãœ = w(f  fÂ¿v + fÂ¿6) , (44)
where the "weight matrix" w is given by
w = (f^af^T)1 . (45)
Inserting this solution for u into Eg. ( 4lfe), we have
fgTw(f  fÂ¿v + fÂ¿5) = 0 . (46)
If we now define
$ = Â£  fÂ¿v , (47)
and rearrange Eq. (46), the equation for 5 will have the
form
(fÂ¿TwfÂ¿)6 =  fÂ¿Tw$ . (48)
This set of linear equations is easy to solve for the
corrections S by general methods. With this solution for 6,
the improved residuals vn are obtained from the equation
Vn =  CTfÂ¿Tw($+fÂ¿6) , (49)
42
which follows from Egs. (4la), (44) and (47). We then
get the new vectors of an and xn as
Ã¡n = Ã¡ + 6 ,
^n = x0 + â€¢
which constitute the improved solution.
After each iteration, we check the relative magnitude
of each component in v and 5 against the corresponding
component in x and a and get the maximum value among all of
these and test if this value is smaller than some specified
number, say 10"Â®. If the improved solution is still not
sufficiently accurate (i.e. if the above found maximum value
is still not smaller than the specified value), the process
of iterations has to be continued until convergence has been
attained, that is, until subsequent iterations no longer
give significant corrections.
At the outset, the obvious starting point for this
scheme is to adopt x=xg as the initial approximation for the
"true observation" vector x (in this case, v=0), and to use
as first approximation of a for a a vector Ã¡g, an initial
solution of the seven orbital parameters, which has been
obtained somehow.
It is important for convergence that the initial
solution of (x,a) is not too different from the final
solution which is obtained by the process given by Eq. (41)
(410a)
(410b)
43
through (410). In Chapter III, we discussed how to find a
good approximation as an initial solution.
According to the scheme outlined above, the application
of Newton's method in our problem would consist of the
following steps.
Step 1.
Calculate f, fÂ¿ and fÂ¿ from the current values of x
and Ã¡ ;
Step 2.
Calculate $ from $ = f  fÂ¿v ;
Step 3.
Calculate the "weight matrix" w from w = (fÂ¿afÂ¿T)â€œl ;
Step 4.
Solve the corrections to the parameters, 6, from the
linear equations (fÂ¿TwfÂ¿)6 =  fÂ¿Tw$ ;
Step 5.
Calculate the improved residuals vn from
vn =  afxTw($+fÂ¿S) ;
Step 6.
Find the new approximate solution from
Ã¡n = Ã¡ + 6 ,
Â¿n = xo + vn ;
Step 7.
Test the relative magnitude of each component of 6 and
v against Ã¡ and x, and decide if a further iteration is
needed, in which case all the steps above must be repeated.
44
In detail, the steps are as follows.
Step 1.
Calculate the vector of functions in condition equaÂ¬
tions f, and the vectors of partial derivatives fÂ¿ and fÂ¿
from the current values of x and a, where x=xq+v.
The dimension of the vectors xq, v, x all are 2m, m
being the number of observed positions.
The observation vector is
x0 â€” (x10'Yl0'* *â€¢'xiO'YiO'â€¢*â€¢â€¢xm0'YmO ^ * (411)
The current vector of corrections to observations is
V â€” ( Vjr}_ / Vyl , . . . , Vj^j_ , Vyj_ , , VjQ^ , ) ^ . (4~12)
From Xq and v, x can be easily found by x=xg+v,
x = (x10+vxl,y10+vyl,...,xi0+vxi,yi0+vy,...
â€¢â€¢â€¢'xmO+vxm'YmO+vym)T * (413)
The current approximation for the seven parameters, Ã¡, is
Ã¡ = (n,T,a,e,i,w,Q)T , (414)
at current values. Insert the known (x,a) into the
45
condition equations to get the function 2mvector f. It has
a form
Â£ â€” [ f]_i , Â¿21 â€™ â€¢ â€¢ *' ^li' ^2i' * * * ' ^lm* ^2m^r^' * 1^)
where
f
li
ÃÃ + e
cosE.
i
(416a)
f
2i
Yi
b
sinE^
(416b)
with
b = a/le2
(416c)
and
E. = n(t.T) + â€” (416d)
11 b
The coordinates X,Y are calculated from Eqs. (39),
A,B,F,G,M from Eqs. (38).
The first derivatives fÂ¿, fÂ¿ are calculated at current
values of x and Ã¡.
The partial derivatives of Â£ with respect to the
observations x, fÂ¿, is a blockdiagonal 2mx2m square matrix
of the form
46
fx =
' 8fll 3fll
3x,
3x,
3Yi
9f21 3f21
3Yi
afii 9fu
3x.
3x.
9f2i 9f2i
9yi
3f
lm
3flm
0
3x
m
9ym
3f2m
3Â£2m
3x
m
9ym
i.e.,
(417)
with
â€” ^tag(g^,g2/ â€¢ â€¢ â€¢ /9i* â€¢ â€¢ â€¢ /9m)
(418)
3Â£u
3fu 1
9xi
9yi
gi =
3f2i
3Â£2i
3x.
1 i
3yi J
i=l ,m
(419)
From Eqs. (416), (38) and (39), we obtain
47
f 1
e ^
f 3X.
3X. ^
â€”
â€”sinE.
1
1
a
b 1
3xi
3yi
0
1
â€”(1ecosE.)
3Yi
3Yi
b x
J
9xi
SYi J
particular,
we
have
3X.
i
3X
G
3Xj. .
3X
F
3x.
i
3x
M
/
3yi
3y
M
3Yi _
3Y
B
3yi _
3Y
A
3xi
3x
M ,
3Yi
3y
M ,
and therefore
â€˜ 1 e
G F
â€” â€”sinE.
_  â€”
a b
M M
1
B A
0 â€”(1ecosE.)
 â€” _
b 1
M M
The expressions for all the partial derivatives in are
listed in Table 41.
The dimension of fÂ¿, the vector of the partial derivaÂ¬
tives of f with respect to the seven parameters, is 2mx7.
It has the form
(G1'G2'â€¢â€¢â€¢'Gi'â€¢â€¢â€¢'Gm)T
t
(423)
48
Table 41.
Expressions for All the Partial Derivatives in fÂ¿
fli
f2i
3
3x
1fG
M^a
B
â€” esinE^
1 B
 (1ecosE.)
Mb 1
3
3y
1(F A
  esinE.
M^a b 1>
1 A
(1ecosE.)
Mb 1
49
where
G.
i
3f
li
â– 2i
3n
3f
li
3fu 3fu
3n 3T 3 a 3e
3f . 3fÂ«. 3f. 3f
â– 2i
â– 2i
3T . 3a
2i
3e
3fli 3fu 3fu '
3i
3f
2i
3i
3o)
3f
2i
3w
3Ã2
3f
2i
3Q
The expressions for all the elements in Gj_ are listed below.
3Â£li
3fu "i
r \
sinE.
3n
3T
1
3f 2i
3Â£2i
cosE.
^ 3n
3T J
1
C J
( vT n )
(424a)
3f
li
Xi eYi .
~ sinE,
3a a ab
(424b)
3f2i
3s.
â€”(1ecosE.) ;
ab 1
(424c)
3f
li
3e
Y^sinE^
b(1e2)
+ 1
(424d)
3f?. e  cosE.
2, i
3e
b(1e )
(424e)
50
3fli
3fii
3fu 'i
r 1
e '
â€”sinE.
r sxi
SX.
1
SX. 'J
l
Si
S oo
SQ
a
b 1
Si
Soo
SQ
Sf_.
2i
3Â£2i
Sf_.
2i
0
1
â€”(1ecosE.)
!!i
l Si
S (o
SQ J
b 1 J
l Si
Soo
SQ j
( 4 â€” 2 4 f )
From Eqs. (38) and (39) we can find the expressions for
the following six partial derivatives.
sx
M = z . sinoo
Si
M
9Y,
Si
= Z^COSOO
SX, SY.
M â€” = Ayi  BxÂ± , M â€”= = FyL  Gxi
S oo S oo
SX, SY,
M = Gy. + FX. , M =  By.  Ax.
SQ 1 1 SQ 1 1
(424g)
In terms of these derivatives, we have
i3Â£u
3Â£ii
3fli3
' 1
e '
â€”sinE.
r
z^sinoo
Ayi_Bxi
Gyi+Fxi
Si
Soo
SQ
Ma
Mb 1
3f 2i
l Si
Sf2i
Soo
3Â£2i
SQ J
0
1 e
â€”  â€”cosE.
Mb Mb \
z^cosoo
Fyi_Gxi
ByiAxi
>
(424h)
All expressions in fÂ¿ are listed in Table 42
51
Table 42.
Expressions for All the Partial Derivatives in fÂ¿
fli
*2i
3
3n
(t^T)sinE^
(t^T)cosE^
3
3T
nsinE.
ncosE^
3
3a
Xi Yi .
â€”rr  â€” esinE.
a2 ab 1
 â€” (1ecosE.)
ab 1
3
3e
Yi
1 + =â€” sinE.
b(leÂ¿)
b(le )
â€” (ecosE^)
3
3i
"sinu
M a
e
+ â€” sinE.coscj
b 1
1
â€” (1ecosE. )z.cosu>
Mb 1
3
3(i)
1 Ay.Bx, e
+ sinE.(Fy.Gx.)
M a b 1 1 1
1
â€”(1ecosE.)(Fy.Gx.)
Mb ill
3 1 Gy^+Fx^
3fi M a
e
â€”sinE.(By.+Ax.)
b ill
1
â€”(ecosE.1)(By.+Ax. )
Mb 1 11
52
Step 2.
Calculate $ from f, v and fÂ¿ by The dimension
of the vector $ is also 2m. It has the form
4* â€” (i, 2 / â€¢ â€¢ â€¢ / 4>i, â€¢
(425)
where
â– li
3fn
3xi
v
xi
3f
fÂ«. â€” v .
21 3x. X1
9Â£li
yi
3Â£2i
3yi
yi
>
r
â– N
â™¦n
$i2
>
(425')
Step 3.
Calculate the weight matrix w from and a.
The matrix has been calculated in step 1. The
covariance matrix a is assumed to be known. The dimension
of a is 2mx2m.
An example for computing a is shown below.
The relationship between the covariance matrix of
rectangular coordinates (x, y) and that of polar coordinates
(p, 0) is
3(x,y)
3(x,y)
Â°xy
. 3(p,0)
CTP0
. 3 ( p / 9 )
(426)
53
where x=pcos6, y=psin9 and
3(x,y)
s(p,e)
r cose psine >
^ sine pcose
(426')
Thus, we have
xy
cose
^ sine
psine '
pcose
r
P
0 a
0 ^
e '
cose sine "
^ psine pcose
and therefore
xy
2 2 2
r a cos 6 + aQp sin 6
P 0
2
(a  aQp )cos6sin9
P o
(a  aQpÂ¿)cos6sin0 ^
p e
2 2 1
apsin 6 + aQp cosâ€œe j
(426")
In these expressions, cp=A^p, aq=A^6 and Ap, A6 are the
observational errors in p and 6, respectively. For the
observations, the random errors are of similar order of
magnitude as the systematic ones, larger in separation than
in position angle. The average errors pA0 and Ap vary
somewhat with the separation p and can be assumed, for many
series of observations, to follow the form Cp*/^, where C
varies with different observers. For a single good observaÂ¬
tion C will not exceed 0V03 in position angle (pA9) and
0V08 in separation (Ap) (Heintz, 1971). Errors will be
somewhat larger and difficult to measure if one or both
components are faint. If the errors are expressed in the
54
dimensionless (relative) forms A0 and Ap/p, it is seen that
they increase as pairs become closer.
Based on the considerations above, we can, for example,
put Ap=0V08pl/3 anÂ¿ pA0=O'.'O3pl/3, i.e. A0=O'.'O3p2/3. if we
allow each pair of numbers (x^,yj_) in an observation to be
correlated, but no correlations between different observaÂ¬
tions, a, would be blockdiagonal. In this case, we have
o â€” diag(0^,02, â€¢ â€¢ * , ^ , â€¢ â€¢ â€¢ /) /
where
( q rr . . ^
uxixi uxiyi
' ail
ai3 "
Â°i =
â€”
^ ayixi ayiyi J
ai4
ai2 >
with C7xiyiayixir i.s., 0^3 ^i4
â€¢
The form of the weight matrix is simpler in this case;
it is also blockdiagonal.
According to Eq. (45), w=(fÂ¿afÂ¿)^. The matrices fÂ¿,
a, fÂ¿T now are all blockdiagonal. Therefore
w = diag(w1,W2,.. . ,Wj_,â€” ,wm) , (429)
with
55
wi =
wil
wi3 '
wi4
wi2 >
(430)
The computation of w is straightforward. We first find w"l.
If we denote u=w1=fÂ¿afÂ£T, it is obvious that u has the same
form
u = diag(u1,u2,â€¢â€¢â€¢,Ui,...,um) ,
(431)
where
r uÃ¼
ui3 1
ui =
l ui4
ui2 J
9fu
3fii 1
f
3xi
3yi
CTil
3f2i
3f2i
3x.
1 i
3Yi J
Â°i4
s.
i3
/ â€¢ Â«*â– *
i2
3f
li
3x.
3f
li
3y.
9Â£2i ^
3x.
l
3f2i
3yi 2
i. e.,
(432)
rsf.r 3f. 3f,. raf,^
ii , _ ii ii . ii
Uxi J 11 3xÂ± 3Y;L L3Yi
i2
(433a)
56
f3f0
2i
z
3f2i
3f2i
rsfO
2i
u.  =
i2
Â°il+ 2
3x.
a Â°i3 +
3Yi
,3Yi >
Â°i2 ;
(433b)
u
i3
8fli 9Â£2i
3x^ 3x^
a.  +
iÂ±
8f2i + !Â£u !Â£2Ã'1
3x. 3y. 3y. 3x.
ai3 +
3f,. 3f.
li 2i
3Yi 3Yi
ai2
(433c)
ui4 = ui3
(433d)
After computing u, we can find its inverse w very easily.
If we denote 1i=:,iiiUL2ui3ui4=:uiiui2''ui3 / we have
wii
Wi3
Ui2
. Ui4 1
=
u
u
Wi2 ,
Ui3
, u
uil
U ,
As seen above, we can calculate the components of w directly
from the components of u.
If the quantities which constitute different observaÂ¬
tions were also correlated, we would have to find the
product of the matrices fÂ¿, a, fÂ¿T and calculate its inverse
to get w. This would be more complicated by far.
Step 4.
57
Find fÃ¡TwfÃ¡ and fÂ¿T$ from fÂ¿, $ and w, then solve the
linear equations
(fÂ¿TwfÂ¿)6 = fÃ¡Tw$
to get the correction vector 6.
The dimension of the matrix fÂ¿TwfÂ¿ is 7x7 and fÂ¿TW(j>, 5
are 7vectors. Denoting A=fÂ¿TwfÂ¿ and B=fÂ¿Tw, we have
f 3fli 3fli .. . 3f2i 3fli ..
A(j,k) = > wn + wi3 +
i=l
9aj 9ak
9aj 3ak
9fli 9f2i 9f2i 9f2i ..
+ w. n + w. n
i3 ^ _ i2
9aj 9ak
9aj 3ak
and
j=l,7;
k=l,7; (435a)
m
( 3f
B(j) = T
2=1
93. .
^ 3
 wii +ii+ r21 wi3 *11+
9a.
3fli , , 3f2i
+ w13 *12 + wi2 *12
9 a.
3aj
j=l,7; (435b)
where
58
ÃœÃ¼
3 3..,
9f
li
9n ,
ÃœÃ¼
3a~
ÃœÃ¼
9T ,
9f
li
9f
li
9f . 9f1.
li li
3a.
3 a
3a,
9e ,
..., and so forth.
To solve the linear equations a6=B for S, we can apply any
linear equation solution method, e.g. Gaussian Elimination,
GaussJordan Elimination, the Jacobi Iterative method, etc.
In a private communication, Eichhorn (1985) proposed a
special solution method for this particular problem, because
the covariance matrix, i.e., the inverse of the coefficient
matrix of 6, is fortunately a positive definite matrix. The
method will be described in another section of this chapter.
In our work, we use this special method for solving the
linear equations above for S.
Step 5.
Calculate the new vector vn, which now is the improved
correction vector to the observations, from
vn =  afÂ¿Tw($+fÂ¿6)
The dimension of the vector v is 2m. Denote Q=$fÂ¿5 and
R=fÂ¿TwQ, Q and R both are also 2mvectors. We can see that
Q â€” (Ql/Q2' â€¢ â€¢ â€¢ 'Qi' â€¢ â€¢ â€¢' Qm ( 4 36)
where
59
3f. 3f. 7 3f..
f. . â€” v . iÂ± v . + 5â€œ â€”â€” 6 .
11 3Xi X1 3Yi Y1 fel 3a j 3
f 9Â£2i _ !Â£
21 8xi xi 3Yj
2i
v . +
yi
3Â£2i
i=l ia.
i.e. ,
Qi
r
7
afli
3aj
r >
Â£u
+ S
6i
Qn
f2i
7
i=l
3f2i
3a.
5j
2i2
V.
3
J
V 7
(436' )
In the case of w being blockdiagonal,
R â€” ( R^_ , R2 / â€¢ â€¢ â€¢ r 1
, Rm)T ,
(437)
where
r Ri! '
Ri =
Ri2
60
3Â£U (QHWil+8i2Wi3) +â€” ISilWi4+2i2Wi2l
3x.
Mii
3Yl
3x.
l
3f2i
(QnMii+Qi2wi3) +â€” iQnWii^i 2"i2)
3yl
(437')
Therefore, in this case,
vn = (Vx, V2, â€¢Vif ..., Vm)T ,
(438)
where
V. =
i
r v . )
XI
V .
v yi j
r a. , R. + a. .R. ^
ll ll i3 i2
Â°i4Eil + ai2R12
(438')
Step 6.
Find the new approximate solutions an and xn from
Ã¡n = Ã¡ + 6
xn = x0 + vn
Step 7.
Determine if another iteration is needed.
Calculate all quantities of
xi(new) xi(old)
Yi(new) Yi(old)
and
6 .
3
xi(old)
/
Yi(old)
aj
(altogether 2ra+7 quantities) and pick up the maximum value
among them. If this maximum exceeds a preset specified
61
value, say 10^, return to the very beginning and take all
of the steps once again. Only when the maximum is less than
the specified value, we assume that good convergence has
been achieved.
Above is the scheme of Newton's method in our orbit
problem. In real programming, some additional consideraÂ¬
tions have been taken into account. For example, in the
actual program, all variables are dimensionless. For
rectangular coordinate pairs (x, y), two new quantities are
defined,
x y
x' = â€” y' = â€” (439)
x0 , Y0 '
where (xg, yg) are the "observations" in the initial data,
so that x = Xgx', y = ygy'. For the seven parameters, seven
new variables are defined as well. They are
n T a e i
al = a2 = a3 = a4 = a5 = T
n0 ' T0 ' a0 ' e0 ' ^0 '
(439â€™ )
where (ng,Tg,ag,eg,ig,ojg,ftg) in Ã¡g is the initial
approximate solution and e is defined as e=sincf) and eg=sin(J)g.
00 ft
62
In terms of these new variables, all formulae for
computing the partial derivatives need only slight modificaÂ¬
tions, and the whole process remains in principle unchanged.
In addition, we have seen that fÂ¿, a, fÂ¿T are square
2mx2m matrices. If the number of observations is, e.g.
m=100, the covariance matrix a has 40,000 components, and
these three matrices alone occupy a huge storage, 120,000,
in the computer. In practical programming, we should keep
the required computer storage to a possible minimum. A
blockdiagonal covariance matrix a can be stored in a 2mx2
matrix, and the same applies to all other related matrices.
This greatly reduces the need for storage.
Weighting of Observations
As we know, the measures of any binary star will be
affected by the accidental and systematic observational
errors and, occasionally, from blunders, i.e., actual
mistakes. The points, when plotted, will not lie exactly on
an ellipse but will occupy a region which is clustered
around an ellipse only in a general way.
Observations are frequently combined into normal
places. Among the observed quantities, the time is the one
observed most precisely. To investigate the measurements
for discordance before using them to calculate an orbit, the
simplest method is to plot the distance p in seconds of arc
and the position angles 6, separately, as ordinates, against
the times of observation as abscissae. Smooth curves are
63
drawn to represent the general run of the measures and in
drawing these curves, more consideration will naturally be
given to those observation points which are relatively more
precise (for example, a point based upon several well
agreeing measures by a skilled observer and supported by the
preceding and following observations) than to the others.
The deviation of the observation points is in the ordinate
direction only and gives a good idea of the accuracy of both
observed quantities. The curves will show whether or not
the measures as a whole are sufficiently good to warrant the
determination of a reasonably reliable orbit. Observations
which are seriously in error will be clearly revealed and
should be rejected or given very low weights. The curves
will also give a general idea of how the observations are to
be weighted. The points which show larger deviations should
be given lower weights and the welldetermined points should
be given higher weights.
It is hard to recommend a general rule for the
weighting of measurements and normal positions. Some
precept could be considered (W. D. Heintz, 1971): compute a
weight p_ according to the number of observations, and P2
according to the "weight" assigned to the observers, then
the normal place receives the weight pVpiP2 (if computations
are made in the quantities d9 and dp/p). This precept could
avoid unduly high weights for single measurements as well as
for very many observations by few observers, and would
64
reduce the influence of residual systematic errors. Its
implicit assumption is a proportionality of the errors pd9
and dp with Jp. This holds better for the multiobserver
average, as the share by less accurate data usually
increases at larger separations.
In our work, we given the points on or nearly on the
smooth curves equal unit weights, and assign lower weights
to those farther from the lines.
When we take into account the weights for the observaÂ¬
tions, some slight and very simple modification is needed in
the solution process described above.
Let G be the matrix of weights for the observational
errors v. The dimension of G is 2mx2m, and G is diagonal.
Taking into account G, the residual function Sq will be
modified as
1 1
T T 1  ~T 1 
Sn = â€” vGo Gv =  v Go Gv ,
(440)
because GT = G. If we define Â£~1 = Gct ^G, then
T 1
Sn = â€” vxE v ,
(440')
which has the exact form as before except a is replaced by
Â£. Therefore we need only to replace a by E, by E"l
wherever ct of appears.
65
Because G is diagonal, the calculations of E and E 1
from G and a are also very simple.
The Orthogonal Set of Adjustment Parameters
and the Efficiency of a Set of Orbital Parameters
As mentioned in the first section of this chapter,
Eichhorn (1985) has proposed a special method in a private
communication for solving the linear normal equations a6=B
of our problem. This method is as follows.
According to its definition, A=fÂ¿TwfÂ¿ is obviously a
symmetrical positive definite matrix.
Given a symmetrical positive definite matrix Q, we look
for its eigenvectors X which have the property that
QX = XX . (441)
Any value X which satisfies Eg. (441) is an eigenvalue of
Q, and since Q is positive definite we know that all X>0.
Obviously X must satisfy the equation Qâ€”IX = 0, the
"characteristic equation" of Q. This is a polynomial in X
of the same order n as that of Q. The solutions are
X]_, X2,...,Xn. The solution X^ of the homogeneous system
(441) with X=XK is the kth eigenvector. Since this is
determined only with the uncertainty of an arbitrary scale
factor, we may always achieve XK = 1 for all k. Let the
matrix P=(X]_,X2,... ,Xn) be the matrix of the normalized
eigenvectors of Q. It can be shown that P is therefore
66
orthogonal, so that PT=p"l. Equation (441) can be written
as
QP = PdiagUi, Xn) . (442)
Writing the diagonal matrix diag(X]_, X2, Xn)=D, we
rewrite Eq. (442) as
QP = PD ,
whence
PTQP = D , Q = PDPT
(443)
and
PtQ_1P = D1 , Q1 = PD1Pt . (444)
The equation (444) shows incidentally that the eigenvectors
of a matrix are identical to those of its inverse, but that
the eigenvalues of a matrix are the inverses of the eigenÂ¬
values of the inverse.
Let Q be the covariance matrix of a set of statistical
variates X. We look for another set Y as function of X,
such that the covariance matrix of Y is diagonal.
67
Putting dX=a, the quadratic form which is minimized is
aTQ1a, where Q is the covariance matrix of X. If we define
Y = PTX , (445)
then we have
X = PY
and
dX = PdY or a = PP , aT = pTPT .
and in terms of p = dY, the quadratic form minimized which
led to the values X becomes PTPTQ1PP , whence the covariance
matrix of Y is seen to be (PTQâ€*P)*=PTQP=D by Eq. (444).
It is then shown that Y=PTX is the vector of linear
transforms of X whose components are uncorrelated. A vector
of such components is called a vector of statistical
variates with efficiency 1 (Eichhorn and Cole, 1985). If
any of its components are changed (e.g. in the process of
finding their values in a course of iteration), none of the
other components of Y would thereby be changed in a tradeoff
between changes of correlated unknowns.
68
Now, back to the normal equations AS=B. The problem
of solving normal equations above can therefore be attacked
as follows.
1) Find the eigenvalues and eigenvectors of A, i.e., D1 and
P. (A=Q1, according to the notation above.)
2) The covariance matrix of 6, Q, can then be calculated
from Eq. (443), Q=A1=PDPT.
3) Let 6'=PT6. This gives APS'=B, whence S'=PTQB, or from
Eq. (443),
S' =  DPtB . (446)
Finding D* from D is very easy because D is diagonal. The
vector 6 is then easily calculated directly from 6=PS'.
One of the advantages of this method is that we can
calculate both 5 and S', the correlated and uncorrelated
elements of the corrections to the adjustment parameters, at
the same time. The vector S' is the set of the corrections
to the orthogonal adjustment parameters.
Furthermore, using this method, a measure for the
"efficiency" of a set of adjustment parameters can be easily
calculated, cf. Eichhorn and Cole (1985). They point out
that the information carried by the vector 5 of the
correlated estimates (whose covariance matrix is Q) is
partly redundant because of the nonvanishing correlations
between the estimates. What is the information content
69
carried by these estimates? We see that the matrix PTQP is
diagonal if P is the (orthogonal) matrix of the normalized
eigenvectors of Q and that it is also the (therefore
uncorrelated) linear transforms 6'=P6 of 6. We might regard
the number
I Q I
Ã© = (447)
qll*â€™*qnn
that is, the product of the variances of the components of
vector S' (the product of the variances of the uncorrelated
parameters) divided by the product of the variances of the
components of vector 6 (the product of the variances of the
correlated parameters) as a measure for the "efficiency" of
the information carried by the estimates 5. But we note
that according to the definition (447), the value of Ã© is
severely affected by the number of variables. In order to
eliminate the effect of n, we redefine Ã© as
e =
I Q I
1
> Ã±
lll * ' ,ynn
(447â€™ )
For any set of uncorrelated estimates we would have e=l,
which is evidently the largest value this number may assume.
In our work, for every model calculated, the efficiency
of the set of adjustment parameters and their covariance
matrix are calculated.
70
A Practical Example
Table 43 lists the observation data for 51 Tau. These
data are provided by H. A. Macalister. The author would
like to thank Macalister and Heintz for their data which he
has used in this dissertation.
Theoretically, the first step in our computation should
be the reduction of the measured coordinates to a common
epoch by the application to the position angles of correcÂ¬
tions for precession and for the proper motion of the
system. The distance measures need no corrections.
Practically, both corrections are negligibly small unless
the star is near the Pole, its proper motion unusually
large, and the time covered by the observations long. The
precession correction, when required, can be found with
sufficient accuracy from the approximate formula
A0 = 0  6q = +0?00557sinasec6(ttg) , (448)
which is derived by differentiating Eq. (31) with respect
to 0, and introducing the precessional change A(ncosa) for
d6. The position angles are thus reduced to a common
equinox tg (for which 2000.0 is currently used), and the
resulting node Qg also refers to tg, because
A0=00g=QQg=Afi. Computing ephemerids, the equinox is
reduced back from tg to t.
The change of position angle by proper motion,
71
6  6g = uasina(ttg) , (449)
where the proper motion component in right ascension ua is
in degrees, can be neglected in most cases.
The two formulae above can be found either in Heintz'
book "Double Stars" or in Aitken's book "The Binary Stars".
In table 44, all position angles have been reduced to
the common equinox 2000.0 and the converted rectangular
coordinates (xg,yg) are also listed.
In figure 41, all pairs of rectangular coordinates
(xO,Yo) are Plotted in the Xgyg plane. We can see at a
glance that all observation points are distributed closely
to an ellipse. Furthermore, in Figures 2a and 2b, we plot
the distance p in seconds of arc and the position angles
against the observing epoch, respectively. From Figure 2,
we see that the observations fall upon nearly sinelike
curves. Thus, we get the impression that these data are
rather precise and therefore give all observations equal
weights.
For these data, an initial approximate solution is
easily obtained by Kowalsky's method. The initial
approximation Ã¡g is listed in Table 45.
Starting from this, the final solution for Ã¡ is
obtained after only three iterations and shown in Table 46.
The calculation required only 47?62 CPU time on a VAX.
72
The residuals (observational errors) in (p,6) and (x,y)
are shown in Table 47. The residuals in (p,6) and (x,y)
are plotted against the observing epoch in Figures 43 and
44, respectively. They show a random distribution as
expected. Also, Figure 45 shows the comparison of the
observation points with the corresponding points after
correction in the apparent plane.
The "efficiency," the covariance matrix, the correlaÂ¬
tion matrix and transformation matrix (which transforms the
correlated parameters to the uncorrelated ones) of the
adjusted parameters in the final solution are calculated and
listed in Table 46. In addition, Table 46 also lists the
standard deviations of a) the original and b) the
uncorrelated parameters in the final solution.
73
Table 43.
The Observation Data for 51 Tau.
HR1331
51 Tau
HD 27176
SAO
76541
04185+2135 n
1975.7160
106.0Â°
2.0
0.080
0.003
A1
1975.9591
91.9
1.7
0.074
0.003
A1
1976.8574
34.9
1.5
0.069
0.005
A2
1976.8602
33.5
1.5
0.073
0.009
A2
1976.9229
22.9
1.0
0.072
0.008
A3
1977.0868
26.7
1.0
0.083
0.008
A3
1977.6398
8.8
0.101
A6
1977.7420
3.1
0.8
0.110
0.008
A5
1978.1490
352.2
0.5
0.113
0.010
A5 n
1978.6183
340.7
0.8
0.108
0.008
A5
1978.8756
333.3
2.0
0.086
0.013
B4
1978.7735
304.3
0.090
A7
1980.1532
285.9
0.075
A8
1980.7182
259.0
0.079
A8
1980.7263
255.8
0.085
A8
1980.7291
259.1
0.087
A8
1982.7550
191.80
0.1343
C2
1982.7579
192.65
0.1362
C2
1982.7605
190.36
0.1315
C2
1982.7633
192.90
0.1381
C2
1982.7661
193.39
0.1308
C2
1983.0472
186.18
0.1333
C2
1983.0637
187.21
0.1499
C2
1983.7108
182.05
0.1456
C2
1983.7135
179.56
0.1480
C2
1983.9337
181.0
1.9
0.149
0.010
FA
1983.9579
176.7
0.157
RB
1984.0522
175.01
0.1446
C2
1984.0576
174.79
0.1445
C2
1984.0603
172.73
0.1355
C2
1984.779
157.3
0.146
RC
1984.9308
164.5
2.6
0.141
0.013
FB
1985.1063
161.0
2.7
0.137
0.013
FB
1985.2048
158.0
3.0
0.125
0.013
FB
1985.8378
145.69
0.1141
##
1985.8406
144.53
0.1202
##
1985.8541
145.71
0.1200
##
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
74
Table 44.
The Reduced Initial Data for 51 Tau
t
e0
P0
x0
Y0
1975.7160
106.00
0.0800
0.0222
0.0769
1975.9591
91.90
0.0740
0.0026
0.0740
1976.8574
34.90
0.0690
0.0565
0.0396
1976.8602
33.50
0.0730
0.0608
0.0404
1976.9229
32.90
0.0720
0.0604
0.0392
1977.0868
26.70
0.0830
0.0741
0.0374
1977.6398
8.80
0.1010
0.0998
0.0156
1977.7420
3.10
0.1100
0.1098
0.0061
1978.1490
352.20
0.1130
0.1120
0.0151
1978.6183
340.70
0.1080
0.1020
0.0355
1978.8756
333.30
0.0860
0.0769
0.0385
1979.7735
304.30
0.0900
0.0508
0.0743
1980.1532
285.90
0.0750
0.0207
0.0721
1980.7182
259.00
0.0790
0.0150
0.0776
1980.7263
255.80
0.0850
0.0207
0.0824
1980.7291
259.10
0.0870
0.0163
0.0855
1982.7550
191.80
0.1343
0.1314
0.0176
1982.7579
192.65
0.1362
0.1329
0.0300
1982.7605
190.36
0.1315
0.1293
0.0238
1982.7633
192.90
0.1381
0.1346
0.0310
1982.7661
193.39
0.1308
0.1272
0.0305
1983.0472
186.18
0.1333
0.1325
0.0145
1983.0637
187.21
0.1499
0.1487
0.0190
1983.7108
182.05
0.1456
0.1455
0.0054
1983.7135
179.56
0.1480
0.1480
0.0010
1983.9337
181.00
0.1490
0.1490
0.0028
1983.9579
176.70
0.1570
0.1568
0.0088
1984.0522
175.01
0.1446
0.1441
0.0124
1984.0576
174.79
0.1445
0.1439
0.0129
1984.0603
172.73
0.1355
0.1344
0.0170
1984.7790
157.30
0.1460
0.1348
0.0562
1984.9308
164.50
0.1410
0.1359
0.0375
1985.1063
161.00
0.1370
0.1296
0.0445
1985.2048
158.00
0.1250
0.1160
0.0467
1985.8378
145.69
0.1141
0.0943
0.0642
1985.8406
144.53
0.1202
0.0980
0.0696
1985.8541
145.71
0.1200
0.0992
0.0675
y0 (seconds of arc)
xQ (seconds of arc)
Figure 41. Plot of the observation data for 51 Tau in the xQy0 plane.
(seconds of arc)
Time of observation
Figure 42. Plot of a) the Xg b) y^ coordinates against the observing
epoch of the observation data for 51 Tau.
CTl
y0 (seconds of arc)
77
Table 45.
Initial Approximate Solution Ã¡g for 51 Tau.
P0(Yr) T0 ag e0 i0Â° w0Â° Q0Â°
11.18 1966.4 0.128 0.181 127.3 152.9 170.2
Table 46
The Final Solution for 51 Tau
P(yrs)
T
a"
e
i Â°
(i)Â°
QÂ°
the final solution
11.22
1966.5
0.128
0.173
125.5
157.3
171.2
standard deviations
0.039
0.031
0.0004
0.002
0.32
1.51
0.33
standard deviations
of the uncorrelated
parameters
0.00003
2.091
0.0002
0.0005
0.28
1.39
2.32
the efficiency
0.466
.36E06
.77E09
.17E06
.23E06
.12E06
.75E06
.66E08
. 77E09
.74E11
.47E09
.12E08
.17E09
.98E09
.38E09
. 17E06
.47E06
.30E06
.34E06
.14E06
.34E06
.66E08
the covariance matrix .23E06
.12E08
.34E06
.47E05
.11E06
.14E05
.27E06
.12E06
.17E09
.14E05
.11E06
.19E06
.29E06
.16E08
.7 5E06
.98E09
.34E06
.14E05
.29E06
.29E05
.30E06
.66E08
.38E09
.66E08
.27E06
.16E08
.30E06
.11E06
00
Table 46 (continued)
P(yrs)
T
a"
e
i Â°
U)Â°
1.0000
0.4709
0.5131
0.1762
0.4604
0.7339
0.0327
0.4709
1.0000
0.3119
0.2075
0.1481
0.2099
0.4115
0.5131
0.3119
1.0000
0.2859
0.6074
0.3661
0.0355
the correlation
0.1762
0.2075
0.2859
1.0000
0.1127
0.3811
0.3718
matrix
0.4604
0.1481
0.6074
0.1127
1.0000
0.3992
0.0112
0.7339
0.2099
0.3661
0.3811
0.3992
1.0000
0.5273
0.0327
0.4115
0.0355
0.3718
0.0112
0.5273
1.0000
0.0074
0.9999
0.0002
0.0000
0.0002
0.0026
0.0033
0.4926
0.0068
0.2021
0.0225
0.1789
0.1955
0.8036
0.1416
0.0018
0.5317
0.0168
0.8053
0.0501
0.2145
the transformation
0.4483
0.0034
0.6345
0.1522
0.4443
0.3079
0.2848
matrix
0.6780
0.0034
0.4681
0.0428
0.3145
0.0186
0.4691
0.2546
0.0001
0.2331
0.4862
0.1520
0.7869
0.0466
0.1086
0.0003
0.0195
0.8590
0.0121
0.4949
0.0703
j
vo
80
Table 47.
Residuals of the Observations for 51 Tau in (p,0) and (x,y)
t
v6
VP
vx
VY
1
1975.7160
1.0090
0.0042
0.0000
0.0044
2
1975.9591
1.1702
0.0039
0.0012
0.0039
3
1976.8574
2.5044
0.0082
0.0048
0.0073
4
1976.8602
3.7620
0.0042
0.0007
0.0064
5
1976.9229
1.2477
0.0070
0.0050
0.0051
6
1977.0868
0.0465
0.0009
0.0009
0.0002
7
1977.6398
2.0624
0.0015
0.0010
0.0039
8
1977.7420
0.5795
0.0083
0.0083
0.0004
9
1978.1290
0.2656
0.0058
0.0057
0.0011
10
1978.6183
0.5497
0.0015
0.0018
0.0006
11
1978.8756
0.2196
0.0173
0.0152
0.0082
12
1979.7735
2.1815
0.0054
0.0059
0.0027
13
1980.1632
1.2302
0.0036
0.0008
0.0039
14
1980.7182
2.8268
0.0005
0.0038
0.0014
15
1980.7263
0.0249
0.0064
0.0014
0.0063
16
1980.7291
3.4623
0.0084
0.0032
0.0093
17
1982.7550
1.9542
0.0022
0.0031
0.0038
18
1982.7579
1.0538
0.0041
0.0045
0.0013
19
1982.7605
3.2987
0.0007
0.0009
0.0074
20
1982.7633
0.7102
0.0058
0.0060
0.0001
21
1982.7661
0.1717
0.0015
0.0014
0.0006
22
1983.0472
2.7391
0.0052
0.0043
0.0069
23
1983.0637
1.4488
0.0111
0.0115
0.0019
24
1983.7108
2.8848
0.0019
0.0020
0.0075
25
1983.7135
0.4324
0.0005
0.0005
0.0013
26
1983.9337
4.9126
0.0004
0.0007
0.0129
27
1983.9579
0.9444
0.0083
0.0085
0.0022
28
1984.0522
0.5459
0.0042
0.0040
0.0020
29
1984.0576
0.3999
0.0043
0.0042
0.0016
30
1984.0603
1.6232
0.0133
0.0136
0.0023
31
1984.7790
6.9626
0.0028
0.0031
0.0173
32
1984.9308
2.5201
0.0004
0.0022
0.0060
33
1985.1063
1.7784
0.0000
0.0015
0.0042
34
1985.2048
0.3956
0.0097
0.0086
0.0046
35
1985.8378
0.2949
0.0015
0.0008
0.0014
36
1985.8406
0.8016
0.0047
0.0037
0.0039
37
1985.8541
0.6865
0.0050
0.0050
0.0016
8.0
0.020
aâ€”v
â–¼â€”V
e
p
â–¼
5.2
CD
CD
Q)
CD
CD
TD
CD
>
2.4
0.4
3.2
â–¼
â™¦
A
X
X
x
A
â–¼
0.013
0.006
0.001
0.008
Q.
>
6.0 â€”
1975
i i i
1977 1979 1981
1983
i
1985
â€”1 0.015
1987
Time of observation
Figure 43. The residuals of the observations for 51 Tau in (p,0).
00
(seconds of arc)
(seconds of arc)
Time of observation
Figure 44. The residuals of the observations for 51 Tau in (x,y).
00
ro
(seconds of arc)
y (seconds of arc)
0.13
0.08
0.03
0.02
0.07
0.12
0.20 0.13 0.06 0.01 0.08 0.15
x (seconds of arc)
Figure 45. The original observations for 51 Tau compared with the
observations after correction.
o â€”original observations
â€¢ â€” observations after correction
J   L
84
Remarks
Newton's method would converge well if the observaÂ¬
tional errors are small enough and the initial approximate
solution is sufficiently accurate. But, when the errors in
the observed distances and position angles are not suffiÂ¬
ciently small, or the initial approximation is not close
enough to the final solution, two things will happen:
1) In some iterations, the corrections to the adjustÂ¬
ment parameters are too big so that some parameters go to
unreasonable values; e.g., the semimajor axis a becomes
negative or the mean motion n, the eccentricity e becomes
negative, which are unacceptable; and further calculation
becomes pointless.
2) The iterations do not converge; i.e., the residual
function S becomes larger in the next iteration, although
all parameters remain in the ranges of reasonable values.
In these cases, Newton's method will fail to yield a
solution. In the next chapter, two other approaches (the
modified Newton methods) are proposed to deal with these
cases.
CHAPTER V
THE MODIFIED NEWTON SCHEME
The scheme for solving the nonlinear condition equaÂ¬
tions by Newton's method has been discussed in the last
chapter. Although rapid convergence can be expected if the
initial approximate solution is sufficiently accurate and
the residuals are small enough, Newton's method often fails
to yield a solution, particularly if the initial approximaÂ¬
tion Ã¡g of the vector Ã¡ is greatly in error or the residuals
are very large. As mentioned in the last chapter, two
problems arise in these cases:
1) Some of the corrections 6 to the adjustment parameters
are too big, which is unacceptable. For example, eg=0.15,
but 6e=0.16, so that the new value e=0.01; in this case,
the residual function calculated is no longer meaningful and
any further calculation becomes pointless.
2) Divergence occurs directly; i.e., the value of residual
function Sg is larger after the iteration than it was
before.
For both of these cases, the major problem is that the
step size could be too large. If we decrease the step size,
the situation might be improved to some extent. We still
rely on Newton's method indicating the right direction, but
85
86
we no longer apply the full corrections which would follow
from the original formalism. In this chapter, we will
discuss the combination of Newton's method with the method
of steepest descent and other sophisticated methods.
The Method of Steepest Descent
We retain A=fÂ¿TwfÃ¡ and introduce a numerical factor 1/f
into the normal equation (48), which thus becomes
1 T
 AS =  fw$ . (51)
f a
By choosing an appropriate value of f, we would reduce the
step size and make the residual function Sq gradually
smaller and smaller. This is analogous to the basic idea of
"the method of steepest descent" which is to step to the
next in a sequence of better approximations to the solution
by moving in the direction of the negative of the gradient
of Sq. If the step is not too large, the value of Sq must
necessarily decrease from one iteration to the next. Our
goal is to arrive at the stable absolute minimum value of Sq
and to the corresponding set of parameters which is the best
solution.
At this point, two new problems arise:
1) How does one calculate the residual function Sq and its
gradient?
2) How does one choose the best value of f?
87
Suppose we were to pretend that the current values of Ã¡
at any iteration are the true values of a. This assumption
is of course not true. But it makes it in general possible
to estimate the vector v as a function of Ã¡. This is so
because if the present value of Ã¡ were the true value of a,
there would be a definite set of residuals v which causes
the corrected observations to satisfy the conditions
equations rigorously. It is not difficult to find those
components of v which correspond to certain value of Ã¡.
Suppose we have an approximation Ã¡ and a and vg of v, we
wish to obtain the "best" approximation of v, assuming Ã¡ to
be correct. We then need to minimize
^ 1 
S = â€” V CT V + f (1
relative to the remaining variables v and y. which yields
f(Xq + v,Ã¡) = 0 (52a)
o1v + fÂ¿TÃº = 0 (52b)
Setting v=vq+0, and expanding (52a) into powers of e, we
have
f(x0,Ã¡) + fÂ¿0Ã© = 0
(53)
88
and Eq. (52) becomes
a1(vo+e) + fxOTÂ£ = 0 (54)
where xq = xq + vq. Solving Eq. (54) for Ã© yields
Ã© = Vq  afÂ¿QTy. . (55)
Substituting now (55) into Eq. (53) for Ã©, we get
f(x0,Ã¡)  fÂ¿ovQ  fÂ¿0afx0TÃº =0. (56)
From Eq. (56) we obtain
Ãœ = w[f(Xq,Ã¡)  fxov0] . (57)
From Eqs. (55) and (57) we arrive therefore at the
expression
v = afÂ¿0w[f(xQ,a)  fxov0] , (58)
where w=fÂ¿QTafÂ¿Q and xq = xq + vq still. This equation may
be iterated, if needed, to arrive at a definite v; i.e.,
until the value of v on the left hand side and the value of
vq on the right hand side are the same. There is thus a
89
corresponding v to each choice of parameters Ã¡ and we can
write v=v(a). Although Eg. (58) formally depends on v0 in
the first order, it is in fact rather insensitive to the
actual value of Vg used, since it is really quadratic in vg.
To appreciate this, note that
f(xg,Ã¡)  fÂ¿o(XgXg) = f(Xg,Ã¡) + 0(Vg2) . (59)
After we get the best value of v which corresponds to a
definite value of Ã¡, S is given by
^ ~T 1  T ^ _T 1 
S = â€” v a v + f Ãº = â€” v a v ;
i.e., S Sg,
because in this case, we already have f=0 for this value of
v. Fixing the value of Ã¡, we can therefore get the best
corresponding value of v, then calculate the value of the
residual function Sg which corresponds to this value of a.
In other words, we may also write Sg=Sg(Ã¡).
Furthermore, when we calculate the best value of v at Ã¡
from Eg. (58 ),
V =  afÂ¿gw$ ,
where again $ = f(x,Ã¡)fÂ¿v
90
Hence, the expression for Sq will be
1 _ 1 1 T  ^ T 
Sn = â€” v a v = â€” wfaa af~ w$ = â€” $ w$ . (510)
u 2 2 x x 2
This expression is exact if it is evaluated with the optimum
value of v. In fact, however,
$ = f(x,Ã¡)  fÂ¿v = f(x0,a)
is not sensitive to the value of v used.
From Eq. (510), we can estimate the negative gradient
6g of Sq with respect to Ã¡:
0So 3$T _ 1 3w
 6 = = w + â€”$ â€” $
^ 3Ã¡ 3a 2 3a
+
0(v)
w$ + O(v^)
= fTw$ , (511)
d
an expression which also depends only weakly on v. In
deriving Eq. (511), the symmetry of w has been used to
combine terms, and terms 0(v2) have been neglected. The
vector Sq points along the negative gradient of Sq, i.e.,
the direction of steepest descent. At this point, let us
91
delineate the method of steepest descent in our problem
first. It can be modified as follows.
At each step, use the current values of v and Ã¡ to
compute
$ = f(x,Ã¡)  fÂ¿v
(512a)
v =
(512b)
and iterate Egs. (512) above to get the best value v at Ã¡.
Since $ is quite insensitive to the values of v, this should
converge rapidly. Then calculate the residual function at
the best value of v, using
1 1
Sn = â€” vTa "'"v or â€” $TW(j> . (512c)
u 2 2
The corrections to the parameters are next computed from
6g =  ffÃ¡Tw$ , (512d)
Ã¡n = Ã¡ + 6g . (512e)
Now an is available. Using the procedure (512a) to (512c)
again, we can get the new value Sg. In Eq. (512d), the
proportionality constant f must be chosen judiciously so
that all components of an do not exceed reasonable ranges
and the new value of SQ is smaller than the old one.
92
This procedure can always be made to converge. But, as
we know, the method of steepest descent too has its drawÂ¬
backs, principally because typically the convergence may be
rather slow.
The Combination of Newton's Method with the Method
of Steepest DescentThe Modified Newton Scheme
Because of the drawbacks of the method of steepest
descent we would like to incorporate the results above into
alternative algorithms which combine the best of the
Newton's method with the method of steepest descent by
"interpolating" between them; i.e, combine the certain
improvement obtained by the steepest descent method when the
available approximation is far from the definitive solution,
with the rapid convergence of Newton's method when the
current approximation is already close to it.
In order to implement this idea, Eg. (512d) must be
slightly modified.
Consider the equation
1 T
 D6 =  fw$ (513)
f a
and compare it with Eqs. (512d) and (48). In Eq. (512d),
D=I, a unit matrix, and in Eq. (48), f=1 and D=A=fÂ¿Tw$.
The equation (512d) leads to the pure steepest descent
solution, which turns out to be undesirable in general.
Actually, we can retain the flexibility to choose D
93
more freely. As Jefferys remarked, this will allow us to
improve certain characteristics of the solution.
Set D=A=fgTw$ in Eq. (513) to obtain
1 T 
â€” AS =  f~ w$
f a
which is the same as Eq. (51).
If Eq. (51) is used instead of (512d), the only
difference between the normal equations of Newton's method
and Eq. (51) is the numerical factor f.
We adopt a modification of Newton's method which
introduces only the numerical factor f into the normal
equation (48) as in Eq. (51). In this algorithm, we
choose the value of f such that Sg (=$Tw$) , evaluated as a
function of i, is a minimum. Once 5 has been chosen in a
current iteration, that value of f is determined which
minimizesfor this iterationthe value of Sg. It is
plausible that this optimizes the gain of accuracy during
this iteration, and the next iteration proceeds from there.
In other words, we use the usual normal equations (/=1) to
estimate the direction of the vector at each step, and then
choose the value of f to get the optimum length.
For arriving at the optimum value of f, the socalled
"Golden section method" or "0.618 method" is used. This
method was originally conceived for searching for the
minimum of a function of one variable. The absolute minimum
94
may be found by this method if the function has only a
single peak or a strict vertex. Otherwise, for a function
with more than one peaks, the method will find a local
minimum. In any case, a point at which the function is
smaller than at the initial point can be always reached.
For every set of parameters a, there is a corresponding
best set of corrections v, and for every value of f, there
is a corresponding set of corrections 5. Therefore,
actually we can write Sg=Sg(f); the scalar residual function
Sq is a function of variable f. Anyway, we can thus
obviously search at least for that f which will produce a
local minimum of Sg.
The scheme for finding best value of f (i.e. the
minimum Sg) at each iteration consists of two parts. The
first of these is to delineate an interval in which the
minimum is located; then look for the minimum by the "0.618
method" within this interval. The searching procedure is as
follows.
1) Choose u as the length for the initial step as well as a
positive number F=(V5l)/2 (F=0.618) and a small quantity e.
2) Define f=10u; Sg=Sg(f) now is equivalent to Sg=Sg(u); put
u(0)=o. The case u=0, i.e., f=l, corresponds to Newton's
method. The superscript indicates the sequence number of
the current iteration.
If Sg(u(0) + u)
If Sg(u(Q) + U)>Sg(u(Â°)), put
95
u =  u ,
u(1) + u ,
and also go to step 3.
3) Calculate u(k+l)=u(k) + u and Sq(k+l))â€¢
4) If S0(u(k+1h
u=2u; K=K+1; go to step 3.
If S0(u(k+1)) > S(u(k)), then
u3 (0) = u(k+l), Ul(0) = U(kD ;
d(0) = u3(0)Ul(0) .
i = 0 ;
go to step 5.
5) Let y^i) = U3^)Fd^), y2^) = u^^+Fd^i).
6) If S0(y!(i)) < S0(y2(i)), then
d(i+D = y2^^  ux(i) , = Ul(i> ,
U3^+^^ = y2^) and
go to step 7.
If S0(yi(1)) > S0(y2(1)), then
Â¿(i+1) =  y1^i) , ^ itl) _ y^(i)
U3^+^) = U3^^ and
also go to step 7.
If S0(yi(i) = S0(y2(i)), then
Â¿(i+1) = y2(i)u1(i) = U3^)  yj^i) and
Ul(i+1) = u^(i) f u3 (1+1> = y2^ or
u^(i+l) = y^(i) / U3^+^) = U3^^
and also go to step 7.
7) If d(i+D
96
u â€¢ = ui (i+l) + Â¿(i+U/2
Otherwise i=i+l and return to step 5.
This process searches for the minimum of Sq at each iteraÂ¬
tion and accelerates the convergence.
In summation, Newton's modified method consists of the
following steps.
Step 1.
Using Eqs. (512a) and (512b) iteratively, get the
best value of v at Ã¡ (initially, set a=ag and since v is
unknown, set v0=0). This converges rather rapidly.
Step 2.
Compute Sg=So(a)=$Tw$ at Ã¡
Step 3.
Set the starting value for f equal to 1, i.e., u=0 and
use Eq. (51) (in this case, equivalent to Eq. (48)), to
compute 5.
Step 4.
Calculate an from Ã¡n = Ã¡ + 6
Step 5.
Check every element in an to see if they are all
located in the ranges of the allowed values (e.g., a>0,
00,...). If not, reduce u=u0.5 (i.e., f=10uâ€œÂ®*^)
and recompute the correction vector 6 using the equation
6' = /6 ,
(514)
97
and set 6=6', return to step 4.
This step may be iterated until every element in Ã¡ has
converged to a reasonable value.
Step 6.
Repeat the process of step 1 to get the best v at an
and compute
S0' = SQ(Ã¡n) = SgÃÃ¡+S).
If Sqâ€™Sq, set u=u0.5 and return to step 5.
Step 7.
If Sq1
If Sq1
Step 8.
Use the "0.618 method" to find the optimum value of f
to get the minimum of the residual function, Smj_n, at this
iteration, and the corresponding optimum improvement of an.
Step 9.
Test for convergence. That is, test again the size of
each component of 5 and v against the corresponding comÂ¬
ponent in Ã¡ and x. When the change is sufficiently small
for all components, the process may be said to have
converged. If the convergence has not yet been achieved,
return to step 3 for next iteration.
The scheme described above is efficient. In comparison
with Newton's method which was described in the last
chapter, we find two essential differences:
98
1) In this method, a corresponding best estimate v has to be
found for every estimate Ã¡ by using Eg. (512a) and (512b)
iteratively. In Newton's method, this is not taken into
account.
2) In the method proposed here, the numerical factor f is
applied to the normal equation for 6 and the best value of
f, which leads to a minimum Smj_n, must be found by the
"0.618 method".
We see, however, that if the an, which is computed from
that value of 5 which is the solution of the same normal
equations for 5 as in Newton's method (here, f=l), already
produces a smaller value of Sg, the scaler residual funcÂ¬
tion, one does not need to bother with searching for the
best value of f for the iteration. This iteration is then
similar to what one would do in Newton's method.
Jefferys (1981) points out that this method is somewhat
like a modified FletcherPowell algorithm (1963). We
therefore call this as "FP method". In addition, as a
slight further modification to scheme above, Marquardt's
algorithm (D. W. Marquardt, 1963) can also be incorporated
to suggest another approach for solving our orbit problem.
The Application of Marquardt's Algorithm
In a sense, the basic idea behind Marquardt's algorithm
is to interpolate between Newton's method and the method of
steepest descent such that initially, far from the solution,
steepest descent dominates, and that the more efficient
99
Newton method dominates the calculations as the solution is
approached. This is similar to what we have done above in
our improved Newton's method.
Comparison of Marquardt's paper (1963) with Eq. (51)
suggests that the natural generalization of this fundamental
equation is
1 T 
(A + â€” D) 5 =  f~ w$ , (515)
f a
where D is to be chosen appropriately. Because the gradient
methods are not scale invariant, it is desirable to choose D
so that it has the form
D = diag (Aq, A22/ A77), (516)
where Ajj are the elements of the matrix A. The effect of
this is to scale the gradient along each coordinate axis by
the factor
This is also exactly Jefferys' (1981) suggestion. In
our case, Marquardt's algorithm would take the same form as
described above for the improved Newton scheme, namely the
"FP method", except for step 3, where Eq. (51) would have
to be replaced by
1 T
(A +  D)6 = f w$ .
f a
100
We call this the "MQ method".
Using the two modifications of Newton's method just
described (FP and MQ method, respectively) for all observaÂ¬
tion data, the same result should be obtained. Our calculaÂ¬
tion confirmed this for all the practical examples in our
paper. We noticed, however, that Marguardt's method
converges much faster than the other one. The reason for
this might be as follows.
Comparing the two equations (51) and (515):
1 T
â€” A6 = f w$ and
f a
(A +  D) 5 = fTw$ .
f 3
where A=fÂ¿TwfÂ¿ and D=diag(An, A22> â€¢â€¢â€¢Â» A77), we see that
in equation (51), which is used in "FP Method", the only
difference from Afi=fÂ¿Tw$, the normal equation in the Newton
scheme, is the numerical factor f; all elements in A are
multiplied by the same factor 1/f. The solution 6 is
therefore only changed by the same numerical factor, i.e.,
the ratios between all elements in 5 remain unchanged.
However, in the "MQ method", as seen in Eq. (515), the
coefficient matrix of 6 is modified as
1
A +  D
f
101
in which only the diagonal elements are changed. The
coefficient matrix of 5 is gradually becoming diagonal in
the course of the "MQ" iterations. This means that the
corrections to the parameters (but not the parameters
themselves) become uncorrelated. However, we have to notice
that the inverse of the coefficient matrix of 6 and the
covariance matrix of the parameters are no longer the same
in this method, since the covariance matrix of the paraÂ¬
meters is intrinsically the inverse of A. Thus we calculate
only the "efficiency" of the adjusted parameters in final
solution.
Two Practical Examples
Table 51 lists the observations for (3738. These
observations were collected by Heintz. Table 52 contains
the initial data. All the position angles in the observaÂ¬
tions have been reduced to equator 2000.0. Figure 51 shows
all the observation points plotted in the Xgyg plane. In
Figure 52, the Xg, yg coordinates are respectively plotted
against the observing epoch. From Figure 52, we see that
the four earliest observations are scattered widely. The
initial approximate solution is listed in Table 53.
Initially, the same weights were assigned to all 26
observations. The same final solution is obtained using
both the "FP" and the "MQ" scheme as shown in Table 54;
this is called "solution #1" for (3738. "FP" requires 13
iterations and spent 12m06?18 CPU time on the VAX while "MQ"
102
requires only 10 iterations and 9mll?01 CPU time for
arriving at the same final result. The covariance, the
correlation and the transformation matrices as well as the
standard deviations of the parameters in this solution are
also listed in Table 54.
For evaluating a final result, the residuals are
important. They must be random and reasonably small. The
residuals in p, 0, x and y for this solution are listed in
Table 55, which shows that some are very big, especially
those for the four earliest points which were observed
before 1921. In Figures 53 and 54, the residuals in p, 0,
x and y are respectively plotted against the observing
epoch. Also, Figure 55 displays all the initial observaÂ¬
tion points compared with those after correction; short bars
connect every two corresponding points. From the relevant
tables and plots above, we see that the residuals seem to be
not very reasonable.
Heintz had already computed an orbit for (3738 from the
same observation data. His result is shown in Table 56
which differs greatly from the solution #1. This
discrepancy originates in earliest four uncertain observaÂ¬
tions. These weak observations affect the overall result.
Giving all the observations the same weights is obviously
inappropriate. We therefore assigned a much lower weight
(0.03) to the earliest four observations. Using these
weighted observation data, both the "FP" and "MQ" scheme
103
still yield exactly the same solution which agrees fairly
well with Heintz' result. This solution is listed in Table
57 and called "solution #2" for (3738. This time, both
methods run through only 6 iterations and used the same CPU
time, around 2m on the VAX, because in all iterations by
both methods Newton's method dominates, that is, neither
"FP" nor "MQ" has been really called into action. The
residuals in p, 6, x and y in this solution are generally
much smaller than in solution #1, which are listed in Table
58, and plotted in Figures 56 and 57. Figure 58 shows
again the comparison of initial observations with those
after correction. Table 57 shows also the final
covariance, correlation and transformation matrices as well
as the standard deviations.
In addition, we also removed the four observations
before 1921 from the input data. Using only the 22 observaÂ¬
tions that remained, the solution is essentially the same as
the weighted 26 observations by both the methods as we would
expect.
Another example is BD+19Â°5116. The observations (35
points) and the reduced initial data are shown in Tables 59
and 510. The observations are again plotted in the Xgyo
plane in Figure 59, and x0, yg coordinates are also plotted
against the time of observation in Figure 510. From Figure
57, we see that the observations cover only a very short
104
arc on the ellipse. The computation from these data will be
more difficult.
Using the "MQ" method, we arrived at a final solution
which is shown in Table 512. Only 5 iterations are needed
and around 9m CPU time is used. The residuals in p, 0, x
and y are listed in Table 513, and plotted in Figures 511
and 512. The initial observations are again compared with
those after correction in Figure 513. Also, the
covariance, the correlation and the transformation matrices
and the standard deviations are given in Table 512.
For these data, the "FP" scheme was also used.
Although it gave the same result, it ran through around 900
iterations and required 4 hours CPU time!
105
Table 51.
The Observation Data for [3738
(courtesy W. D. Heintz)
3 738
02232
s2952
(2000)
7.5
â€¢7.8 F8
1879.70
183.1
0.64
2
3
1891.80
174.7
0.55
3
13
1900.21
177.5
.75
3
Doo 2
Booth 1
05.98
193.8
.66
2
01
21.15
50.5
.44
10
6
26.67
43.5
.46
15
V 8 B
7
29.77
40.0
.50
8
B V 4
32.45
38.5
.55
15
4 6 3
34.76
37.3
.57
11
B V 4
6 3
38.16
35.0
.59
14
Sim 1C
1 B 4
42.52
32.3
.57
7
B 4 V
3
45.80
30.1
.62
12
Strom
8 V 4
47.98
30.4
.44
8
B
55.00
25.6
.22
4
B 2
int 2
56.45
13.2
.19
4
B 2 cp
int 2
59.03
352.1
.13
10
cp int
7 B 3
59.97
307.1
.10
6
cp int
3 B 3
61.03
249.2
.11
3
cp int
62.53
222.7
.14
2
cp int
64.58
223.5
.19
14
cp int
10 B 4
66.02
217.5
.26
4
cp int
2 B 2
66.94
220.9
.32
5
Kni
69.04
218.8
.56
3
cp int
75.55
217.4
.62
9
Wor 7
Hln 2
77.28
217.1
0.75
7
Hln 4
hz 3
1983.71
215.4
1.03
2
hz
Finsen published an
elliptical (P=110yr) and
a parabolic
solution (cf the 1969 orbit catalogue). Heintz' result is:
P 290 yr,
T 1952.0,
a 1.460,
e 0.63,
i 96.7,
co 38.0, node
29.4 (2000).
The position angles are oriented to equator 2000. The
observations before 1920 are from Northern instruments, and
are correspondingly uncertain.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
106
Table 52.
The Reduced Initial Data for (3738
t
e0
P0
x0
VO
1879.70
183.1
0.64
0.639
0.035
1891.80
174.7
0.55
0.548
0.051
1900.21
177.5
0.75
0.749
0.033
1905.98
193.8
0.66
0.641
0.157
1921.25
50.5
0.44
0.280
0.340
1926.67
43.5
0.46
0.334
0.317
1929.77
40.0
0.50
0.383
0.321
1932.45
38.5
0.55
0.430
0.342
1934.76
37.3
0.57
0.453
0.345
1938.16
35.0
0.59
0.483
0.338
1942.52
32.3
0.57
0.482
0.305
1945.80
30.1
0.62
0.536
0.311
1947.98
30.4
0.44
0.380
0.223
1955.00
25.6
0.22
0.198
0.095
1956.45
13.2
0.19
0.185
0.043
1959.03
352.1
0.13
0.129
0.018
1959.97
307.1
0.10
0.060
0.080
1961.03
249.2
0.11
0.039
0.103
1962.53
222.7
0.14
0.103
0.095
1964.58
223.5
0.19
0.138
0.131
1966.02
217.5
0.26
0.206
0.158
1966.94
220.9
0.32
0.242
0.210
1969.04
218.8
0.56
0.436
0.351
1975.55
217.4
0.62
0.493
0.377
1977.28
217.1
0.75
0.598
0.452
1983.71
215.4
1.03
0.840
0.597
y0 (seconds of arc)
xQ (seconds of arc)
Figure 51. Plot of the observation data for 3738 in the x^y^ plane.
107
(seconds of arc)
Figure 52. Plot of a) the xQ b) yQcoordinates against the observing epochs
of the observation data for 3738.
y0 (seconds of arc)
109
Table 53.
The Initial Approximate Solution Ã¡g for (3738
If
. O
O
_ o
p0
T0
a0
eo
!0
O30
n0
156.65
1955.2
0.958
0.523
105.2
57.4
31.2
Table 54
The Solution #1 for [3378
P(yrs)
T
a"
e
i Â°
coÂ°
Â£2Â°
solution #1
106.5
1952.9
7.426
0.241
101.6
48.6
26.1
standard deviations
4.73
2.09
0.038
0.033
1.55
8.78
2.37
standard deviations
of the uncorrelated
parameters
0.023
20.38
0.021
0.012
16.70
4.93
2.32
the efficiency
0.361
0.00105
0.00001
0.00025
0.00049
0.00004
0.00128
0.00009
0.00001
0.00000
0.00000
0.00001
0.00000
0.00004
0.00000
0.00025
0.00000
0.00035
0.00014
0.00008
0.00016
0.00046
the covariance
matrix
0.00049
0.00001
0.00014
0.00084
0.00001
0.00057
0.00006
0.00004
0.00000
0.00008
0.00001
0.00005
0.00009
0.00015
0.00128
0.00004
0.00016
0.00057
0.00009
0.00527
0.00015
o
Table 54 (continued)
P(yrs)
T
a"
e
i Â°
10Â°
QÂ°
1.00000
0.46336
0.41513
0.52158
0.18247
0.54280
0.07502
0.46336
1.00000
0.13857
0.38868
0.20111
0.97796
0.02163
0.41513
0.13857
1.00000
0.26090
0.60222
0.11928
0.67952
the correlation
matrix
0.52158
0.38868
0.26090
1.00000
0.05161
0.27167
0.05509
0.18247
0.20111
0.60222
0.05161
1.00000
0.17910
0.60616
0.54280
0.97796
0.11928
0.27167
0.17910
1.00000
0.05660
0.07502
0.02163
0.67952
0.05509
0.60616
0.05660
1.00000
0.00102
0.99997
0.00011
0.00320
0.00120
0.00629
0.00120
0.08818
0.00125
0.16364
0.06186
0.97745
0.04201
0.06672
0.59839
0.00132
0.59638
0.42977
0.17410
0.16814
0.20730
the transformation
matrix
0.37053
0.00059
0.68660
0.51859
0.02054
0.18073
0.29877
0.25804
0.00654
0.03721
0.08680
0.01713
0.96103
0.02395
0.43660
0.00065
0.36523
0.42745
0.09272
0.07716
0.69189
0.48954
0.00262
0.10657
0.59353
0.07052
0.08788
0.61969
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
112
Table 55.
Residuals of the Observations in 6,p
and x,y for Solution #1 of (3738
t
ve
VP
vx
VY
1879.70
22.281
0.201
0.121
0.326
1891.80
24.309
0.229
0.189
0.305
1900.21
15.235
0.138
0.153
0.168
1905.98
8.507
0.203
0.186
0.115
1921.25
30.522
0.239
0.248
0.141
1926.67
7.713
0.121
0.122
0.053
1929.77
3.396
0.076
0.075
0.030
1932.45
0.244
0.059
0.047
0.035
1934.76
1.703
0.030
0.014
0.031
1938.16
3.224
0.002
0.020
0.026
1942.52
4.821
0.041
0.060
0.023
1845.80
5.931
0.038
0.006
0.073
1947.98
8.737
0.100
0.122
0.023
1955.00
19.091
0.075
0.095
0.062
1956.45
345.659
0.043
0.048
0.048
1959.03
23.659
0.007
0.012
0.054
1959.97
0.265
0.020
0.013
0.016
1961.03
31.476
0.014
0.062
0.019
1962.53
30.273
0.024
0.055
0.062
1964.58
10.992
0.058
0.006
0.071
1966.02
10.225
0.051
0.003
0.072
1966.94
3.779
0.032
0.008
0.038
1969.04
0.972
0.120
0.098
0.069
1975.55
5.478
0.046
0.073
0.024
1977.28
6.502
0.039
0.014
0.091
1983.71
8.721
0.208
0.105
0.228
Time of observation
Figure 53. The residuals of the observations for 3738 in (p,0)
according to solution Ã1.
GJ
(seconds of arc)
(seconds of arc)
Figure 54. The residuals of the observations for 3738 in (x,y) according
to solution â€™ #1.
(seconds of arc)
y0 (seconds of arc)
Figure 55.
The original observations for 3738 compared with the
observations after correction according to solution #1.
115
116
Table 56.
Heintz' Result for (5738
P(yr) T a" e iÂ° ojÂ° QÂ°
290.0 1952.0 1.460 0.63 96.7 38.0 29.4
Table 57
The Solution #2 for 3378
P(yrs)
T
a"
e
i Â°
(j)Â°
QÂ°
solution #2
291.95
1953.5
1.281
0.630
96.59
47.05
31.37
standard deviations
90.90
1.31
0.377
0.109
0.78
7.02
1.07
standard deviations
of the uncorrelated
parameters
0.009
7.29
0.009
0.006
57.82
1.81
2.56
the efficiency
0.064
0.06440
0.00010
0.09938
0.06871
0.00031
0.02444
0.00482
0.00010
0.00000
0.00017
0.00010
0.00000
0.00008
0.00000
0.09938
0.00017
0.15552
0.10539
0.00072
0.04038
0.00676
the covariance
matrix
0.06871
0.00010
0.10539
0.07398
0.00023
0.02446
0.00529
0.00031
0.00000
0.00072
0.00023
0.00006
0.00047
0.00012
0.02444
0.00008
0.04038
0.02446
0.00047
0.01496
0.00141
0.00482
0.00000
0.00676
0.00529
0.00012
0.00141
0.00118
117
Table 57 (continued)
P(yrs)
T
a"
e
iÂ°
0)Â°
1.00000
0.59218
0.99306
0.99550
0.16324
0.78747
0.55320
0.59218
1.00000
0.13857
0.65664
0.52017
0.95382
0.17842
0.99306
0.65664
1.00000
0.98252
0.24660
0.83706
0.49879
the correlation
matrix
0.99550
0.52017
0.98252
1.00000
0.11618
0.73509
0.56656
0.16324
0.59504
0.24660
0.11618
1.00000
0.52217
0.45293
0.78747
0.95382
0.83706
0.73509
0.52217
1.00000
0.33507
0.55320
0.17842
0.49879
0.56656
0.45293
0.33507
1.00000
0.00602
0.99995
0.00014
0.00586
0.00163
0.00549
0.00071
0.09580
0.00107
0.00281
0.07470
0.97974
0.07131
0.14234
0.74444
0.00742
0.00185
0.62960
0.13253
0.17810
0.00842
the transformation
matrix
0.40571
0.00243
0.58166
0.37333
0.08224
0.34384
0.48239
0.46068
0.00077
0.71651
0.49012
0.00283
0.18190
0.03295
0.18105
0.00157
0.32895
0.14033
0.11015
0.30565
0.85660
0.16429
0.00623
0.20018
0.44577
0.06034
0.84760
0.11007
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
119
Table 58.
Residuals
in p,6,x and
y of the
Observations
for 3738 in
Solution
#2
t
ve
VP
vx
VY
1879.70
2.160
0.207
0.208
0.005
1891.80
10.692
0.309
0.316
0.016
1900.21
54.998
0.585
0.661
0.106
1905.98
107.227
0.473
0.652
0.344
1921.25
2.623
0.039
0.011
0.042
1926.67
0.680
0.020
0.018
0.009
1929.77
0.579
0.019
0.011
0.016
1932.45
0.384
0.002
0.004
0.002
1934.76
0.248
0.002
0.003
0.001
1938.16
0.715
0.004
0.007
0.004
1942.52
1.131
0.011
0.003
0.015
1845.80
1.485
0.073
0.070
0.024
1947.98
0.228
0.066
0.058
0.032
1955.00
4.324
0.043
0.046
0.000
1956.45
3.020
0.006
0.004
0.011
1959.03
5.384
0.047
0.048
0.001
1959.97
4.197
0.039
0.020
0.034
1961.03
16.641
0.034
0.034
0.027
1962.53
16.383
0.003
0.032
0.023
1964.58
4.004
0.044
0.020
0.041
1966.02
6.319
0.041
0.011
0.050
1966.94
1.320
0.024
0.013
0.022
1969.04
0.912
0.122
0.099
0.071
1975.55
1.468
0.079
0.073
0.033
1977.28
1.755
0.010
0.022
0.013
1983.71
1.629
0.068
0.040
0.062
Time of observation
Figure 56. The residuals of the observations for 6738 in (p,0) according to
solution D2.
to
o
(seconds of arc)
(seconds of arc)
Time of observation
Figure 57. The residuals of the observations for 8738 in (x,y) according to
solution #2.
(seconds of arc)
y (seconds of arc)
0.40
0.14
0.12
0.38
0.64
0.90
Â°â€”original observations
â€¢ â€” observations after correction
j 1 i i
1.00 0.66 0.32 0.02 0.36
0.70
x (seconds of arc)
Figure 58. The original observations for 8738 compared with the
observations after correction according to solution #2.
122
123
Table 59.
The Observation Data for BD+19Â°5116
(courtesy W. D. Heintz)
+19 0 5116
23319
N1956
(2000)
10.4
12.1 M4Ve
1945.473
174.47
3.563
2n
Ilex
S
49.809
165.72
3.430
3
21
51.964
161.08
3.435
3
14
54.804
154.45
3.583
2
10
55.971
152.54
3.565
2
10
58.482
147.18
3.616
3
14
60.152
143.91
3.648
6
50
61.682
141.51
3.708
4
46
62.835
139.46
3.759
8
70
63.757
138.49
3.770
4
31
64.860
136.50
3.810
6
56
65.766
134.67
3.840
3
23
68.289
129.65
3.967
2
21
69.838
127.34
4.101
6
76
70.792
126.12
4.098
2
28
71.939
124.54
4.134
5
36
73.903
121.48
4.236
7
32
75.872
118.90
4.297
2
16
77.049
117.65
4.361
4
25
78.904
115.01
4.431
2
12
80.529
112.47
4.621
3
12
81.856
111.72
4.602
4
18
1983.581
108.33
4.676
3
19
1941.75
182.17
3.678
9
M
42.65
179.70
3.643
6
43.79
177.20
3.750
5
44.72
174.82
3.679
4
46.03
172.59
3.605
5
49.75
165.33
3.644
4
1955.78
153.38
3.605
5
1952.19
161.8
3.52
7
VB
59.39
145.2
3.82
6
C Wor
3
62.07
141.1
3.76
11
B VB 4
C 3
65.53
137.0
3.97
7
VB
1981.64
112.3
4.24
2
hz
Pg. positions from parallax plates at Swarthmore (S)
refractor (meas. Heintz), from McCormick (M) refractor
(meas. Eichhorn); few micrometer observations. Position
angles equator 2000.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
124
Table 510.
The Reduced Initial Data for BD+19Â°5116
t
e0
P0
x0
Y0
1941.750
182.17
3.678
3.6754
0.1393
1942.650
179.70
3.643
3.6430
0.0191
1943.790
177.20
3.750
3.7455
0.1832
1944.720
174.82
3.679
3.6640
0.3322
1945.473
174.47
3.563
3.6454
0.3434
1946.030
172.59
3.605
3.5749
0.4649
1949.750
165.33
3.644
3.5252
0.9228
1949.809
165.72
3.430
3.3240
0.8460
1951.964
161.08
3.435
3.2494
1.1138
1952.190
161.80
3.520
3.3439
1.0994
1954.804
154.45
3.583
3.2326
1.5453
1955.780
153.38
3.605
3.2229
1.6153
1955.971
152.54
3.565
3.1633
1.6439
1958.482
147.18
3.616
3.0388
1.9599
1959.390
145.20
3.820
3.1368
2.1801
1960.152
143.91
3.648
2.9479
2.1489
1961.682
141.51
3.708
2.9023
2.3078
1962.070
141.10
3.760
2.9262
2.3611
1962.838
139.46
3.759
2.8567
2.4433
1963.757
138.49
3.770
2.8231
2.4986
1964.860
136.50
3.810
2.7637
2.6226
1965.530
137.00
3.970
2.9035
2.7075
1965.766
134.67
3.840
2.6996
2.7309
1968.289
129.65
3.967
2.5313
3.0544
1969.838
127.34
4.101
2.4874
3.2605
1970.792
126.12
4.098
2.4157
3.3103
1971.939
124.54
4.134
2.3439
3.4053
1973.903
121.48
4.236
2.2120
3.6126
1975.872
118.90
4.297
2.0767
3.7619
1977.049
117.65
4.361
2.0238
3.8630
1978.904
115.01
4.431
1.8733
4.0155
1980.529
112.47
4.621
1.7661
4.2707
1981.640
112.30
4.240
1.6089
3.9229
1981.856
111.72
4.602
1.7031
4.2753
1983.581
108.33
4.676
1.4706
4.4387
y0 (seconds of arc)
4.6
3.6
2.6
1.6
0.6
0.4
4.0 3.6 3.2 2.8 2.4 2.0 1.6 1.2
oo
o
Â° oÂ°
o o
o
cP
o o
o
o ^
o
o
J I I L
J L
xQ (seconds of arc)
Figure 59. Plot of the observation data for BEH19Â°5116 in the xnyn
plane.
125
(seconds of arc)
Figure 510. Plot of a) the xQ b) yQcoordinates against the observing
epochs of the observations for BD+19Â°5116.
K>
CTi
y0 (seconds of arc)
127
Table 511.
The Initial Approximate Solution Ã¡g fÂ°r BD+19Â°5116
P0(Yrs)
H
. O
O
_ o
T0
a0
e0
!0
o0
125.6
1891.1
5.31
0.61
115.2
94.0
76.0
Table 512
The Final
Solution
for BD+19
Â°5116 by MQ
Method
P(yrs)
T
a"
e
iÂ°
wÂ°
QÂ°
the final solution
130.88
1891.10
5.733
0.658
112.24
93.36
70.28
standard deviations
98.70
238.68
17.16
0.148
56.64
9.02
41.36
standard deviations
of the uncorrelated
parameters
0.0044
1.238
0.028
0.002
5.42
63.22
14.32
the efficiency
0.00018
12.86588
0.55060
14.11423
30.59438
4.41831
0.18458
2.30953
0.55060
0.02366
0.60254
1.30807
0.18866
0.00689
0.10037
14.11423
0.60254
15.51671
33.59297
4.85616
0.21927
2.51029
the covariance
matrix
30.59438
1.30807
33.59297
72.77996 
10.51481
0.45297
5.47306
4.41831
0.18866
4.85616
10.51481
1.51984
0.06812
0.78650
0.18458
0.00689
0.21927
0.45297
0.06812
0.01366
0.01665
2.30953
0.10037
2.51029
5.47306
0.78650
0.01665
0.43986
128
Table 512 (continued)
P(yrs)
T
a"
e
iÂ°
0)Â°
1.00000
0.99818
0.99907
0.99984
0.99928
0.45695
0.97320
0.99818
1.00000
0.99507
0.99718
0.99548
0.40265
0.98525
0.99907
0.99507
1.00000
0.99968
0.99999
0.49011
0.96431
the correlation
matrix
0.99983
0.99718
0.99968
1.00000
0.99979
0.46990
0.97003
0.99928
0.99548
0.99999
0.99979
1.00000
0.48690
0.96524
0.45695
0.40265
0.49011
0.46990
0.48690
1.00000
0.24247
0.97320
0.98525
0.96431
0.97003
0.96524
0.24247
1.00000
0.03975
0.99634
0.00037
0.00443
0.03125
0.06507
0.02247
0.01892
0.03091
0.21397
0.03665
0.97512
0.00385
0.02610
0.59874
0.02313
0.52455
0.52653
0.10091
0.16982
0.22264
the transformation
matrix
0.09483
0.05175
0.22637
0.11484
0.05393
0.85351
0.43877
0.65558
0.00198
0.55415
0.02590
0.09456
0.16166
0.47684
0.27365
0.05398
0.41417
0.03833
0.10707
0.46074
0.72488
0.35483
0.01514
0.38626
0.84028
0.12110
0.00528
0.06243
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
130
Table 513.
Residuals of the Observations for BD+19Â°5116
in 8,
p, x and y
t
ve
VP
vx
VY
1941.750
1.3854
0.0141
0.0119
0.0891
1942.650
0.6496
0.0041
0.0037
0.0414
1943.790
0.3677
0.1215
0.1226
0.0173
1944.720
0.1869
0.0636
0.0623
0.0175
1945.473
0.9591
0.0433
0.0368
0.0636
1946.030
0.1746
0.0046
0.0060
0.0103
1949.750
0.3556
0.0633
0.0670
0.0054
1949.809
0.8641
0.1506
0.1323
0.0894
1951.964
0.5481
0.1505
0.1311
0.0812
1952.190
1.7206
0.0667
0.0282
0.1226
1954.804
0.4275
0.0270
0.0359
0.0127
1955.780
0.4239
0.0180
0.0040
0.0320
1955.971
0.0417
0.0608
0.0552
0.0257
1958.482
0.5414
0.0549
0.0648
0.0005
1959.39
0.7957
0.1293
0.0772
0.1161
1960.152
0.6524
0.0607
0.0737
0.0015
1961.682
0.2183
0.0404
0.0405
0.0140
1962.070
0.0807
0.0008
0.0039
0.0036
1962.835
0.1737
0.0224
0.0244
0.0058
1963.757
0.5043
0.0394
0.0072
0.0511
1964.860
0.4534
0.0348
0.0042
0.0459
1965.530
2.1135
0.1028
0.1743
0.0324
1965.766
0.1890
0.0353
0.0157
0.0341
1968.289
0.6033
0.0009
0.0314
0.0275
1969.838
0.4151
0.0756
0.0227
0.0779
1970.792
0.1332
0.0350
0.0130
0.0339
1971.939
0.0557
0.0250
0.0174
0.0183
1973.903
0.0669
0.0465
0.0202
0.0422
1975.872
0.1861
0.0257
0.0246
0.0158
1977.049
0.5783
0.0409
0 0577
0.0161
1978.904
0.4527
0.0345
0.0461
0.0167
1980.529
0.0453
0.1591
0.0641
0.1457
1981.640
1.2982
0.2654
0.0058
0.2832
1981.856
0.9916
0.0883
0.1055
0.0537
1983.581
0.2496
0.0976
0.0118
0.0990
V0 (degrees)
Time of observation
u>
Figure 511.
The residuals of the observations for BEH19Â°5116 in (p,0).
(seconds of arc)
(seconds of arc)
Time of observation
Figure 512. The residuals of the observations for BD+19Â°5116 in (x,y).
co
NJ
(seconds of arc)
y (seconds of arc)
x (seconds of arc)
Figure 513. The original observations for BD+19Â°5116 compared with
the observations after correction.
133
CHAPTER VI
DISCUSSION
For solving our orbit problem, the Newton scheme and
its modifications ("FP" and "MQ" algorithms) have been
discussed in detail. As we have seen, the Newton scheme is
powerful and converges very fast if the observational errors
are sufficiently small and the initial approximations to the
parameters are sufficiently accurate. But in other cases,
Newton's scheme fails to converge. Some modifications of it
were carried out for these cases. Two other schemes, "FP"
and "MQ", are proposed. These two converge to a definitive
solution even when Newton's scheme diverges. As we have
seen, "FP" method only shortens, however, the length of the
step and retains all other features of Newton's scheme, so
that sometimes it converges still very slow. The "MQ"
method seems more powerful because it not only shortens the
length of the step but also makes the corrections to the
parameters gradually uncorrelated in subsequent iterations.
"MQ" method converges much faster than "FP" does.
In addition to which scheme is used, two things are
important about the observation data themselves.
134
135
1) The sufficiency of observation data itself is of
importance. Whether a set of observations is suitable for
computing an orbit depends on the amount, consistency and
distribution of the data. Some observation data define only
a short arc of small curvature variation like in our example
for BD+19 Â° 5116. The orbitcomputation from such data is
relatively more difficult.
2) The weighting of observation is also of importance.
The more a computation is based on more precise observations
the better the chances are of practical success. If using
weak data is unavoidable, low weights must be assigned to
them, in accordance with standard practice in the field.
REFERENCES
Aitken, Robert G. 1935. The Binary Stars, Dover
Publications, Inc., New York, 70.
Brown, D. C. 1955. Report No. 937, Ballistic Research
Laboratories, Aberdeen Proving Ground, Maryland.
Comstock, G. C. 1918. The Astronomical Journal, 31, 33.
Eichhorn, Heinrich. 1985. Astrophysics and Space Science,
110, 119.
Eichhorn, Heinrich, and Warren G. Clary. 1974. Mon. Not.
R. Astr. Soc., 166, 425.
Eichhorn, Heinrich, and Carl S. Cole. 1985. Celestial
Mechanics, 37, 263.
Fletcher, W., and M. J. D. Powell. 1963. Comput. J., 6,
163.
Glasenapp, S. 1889. Mon. Not. Roy. Astron. Soc., 49, 276.
Heintz, Wulff D. 1971. Double Stars, D. Reidel Publishing
Company, London, England, 38.
Herschel, John. 1833. Memoirs R. A. S. , 5, 171.
Jefferys, William H. 1980. The Astronomical Journal, 85,
177.
Jefferys, William H. 1981. The Astronomical Journal, 86,
149.
Lanson, C. L., and R. J. Harsson. 1974. Solving Least
Squares Problems, Englewood Cliffs, N.J. (Prentice
Hall) .
Marquardt, D. W. 1963. J. Soc. Ind. Appl. Math., 11, 431.
Russell, H. N. 1898. Astrophys. J., 19, 9.
136
137
Smart, W. M. 1930. Mon. Not. Roy. Astron. Soc., 90, 534.
Thiele, T. N. 1883. Astron. Nachr., 104, 245.
van den Bos, W. H. 1926. Union Obs. Circ., 2, 356.
van den Bos, W. H. 1932. Union Obs. Circ., 4^, 223.
Zwiers, J. H. 1896. Astron. Nachr., 139, 369.
BIOGRAPHICAL SKETCH
Yulin Xu was born on October 20, 1945, in Jinyun
County, Zhejiang, China. He received his elementary
education in Jinyun and was graduated from the Jinhua First
Middle School in 1962. He entered the Zhejiang University,
Hangzhou, in Septemoer 1962 and received his diploma from
the Department of Optical Instrument Engineering in July
1968.
He was an engineer in Hangzhou Camera Institute,
Hangzhou, China, from September 1968 through June 1983. In
July 1983, he entered the University of Florida to start
working toward a Ph.D. in astronomy.
In addition, he is working on the experimentation and
theory for the microwave scattering at the Space Astronomy
Laboratory, the University Florida.
He received the degree of Master of Science and the
degree of Doctor of Philosophy from the University of
Florida in April 1988.
He married Yuemei Li in January 1974, and they now
have two daughters, Yingxun and Huiyi.
138
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Heinrich K. Eichhorn, Chairman
Professor of Astronomy
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
 Ã¡vTfcvv Â¿I
KwanYu Chen,CoChairman
Professor of Astronomy
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
4
Frank B. Wood
Professor of Astronomy
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
c.
Haywood C. Smith
Associate Professor of Astronomy
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
P/JU A
PhiÃip Bacon
Associate Professor of
Mathematics
This dissertation was submitted to the Graduate Faculty of
the Department of Astronomy in the College of Liberal Arts
and Sciences and to the Graduate School and was accepted as
partial fulfillment of the requirements for the degree of
Doctor of Philosophy.
April 1988
Dean, Graduate School
.Uâ€žniyi^!TY 0F FLORIDA
VÂ»â€”1111,1111111 Bin
1262 08556 8003
PAGE 1
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