CIRCUMSTELLAR HYDRODYNAMICS AND
SPECTRAL RADIATION IN ALGOLS
DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
In memory of Jean Stillwell
I do not believe it is possible to express the depth of my gratitude to my committee
Wilson. I thank him for the tremendous education I have received from
him as both a teacher and a supervisor. On countless occasions I have walked into his
immediately put down his own work to help me with mine. This disseration simply could
not have been done without him.
I would like to thank the faculty of the Astronomy Department for giving me the
opportunity to pursue this work. In particular, I would like to thank H.K. Eichhorn for his
I also thank R.J. Leacock for much help early on in my graduate career and
Dermott for his advice and assistance on my behalf.
Life during the last seven years would have been much less enjoyable without the
times I shared with my friends Billy Cooke, Dan Durda, and Dave Kaufiann. Lunchtime
discussions on just about every subject conceivable and trips to see Shuttle launches at all
hours of the day with these guys are things I will always cherish. But most of all, I am
very grateful to know people with such an infectious love of science. I have also been
very lucky to have Russ Poole as a friend. I thank him for taking the time to teach me a
few of the martial arts skills he has so well mastered.
Brooke. During the last two years of my undergraduate career at Clemson, they treated
me as if I were a member of their family. I simply cannot weave the words to express my
love and gratitude to them.
G. Albright, G. Peters, M. Plavec, and D.
Vesper kindly provided their observations
adapting it for use with our hydro program. I am greatly indebted to
Ko who made
great efforts in modifying her radiative transfer code to use the viscous heating from the
NASA deserves credit for both direct and indirect support of my achievements.
me to explore
Indirectly, NASA is partly responsible for my becoming a scientist. I was very young
when the Apollo program put humans on the Moon, and the sight of people walking on
the Moon and rockets flying into the sky made me want to learn about nature.
Finally, I could never have achieved the lofty goals I set for myself without my
Walter and Ayenda. From them I learned the need for hard work, discipline, and
patience. I thank my father for always leading by example.
The times he drove several
hours from his job sites to see my football games, only to have to drive back that night to
go to work,
will always remind me of how much he has given me. And I thank my
mother for never tiring of my questions as a child. She never seemed bothered by my
answer. By encouraging curiosity and learning, she molded me into a scientist.
TABLE OF CONTENTS
AC K.NOW LEDGM ENTS ... ..... .................. ............ ......................................... .. ..... ..
A B STR A CTER S.............. ........................................................... ..................................
HYDRODYNAM ICS .... .. ........................................... ......... ...... t............ ......
Previous W ork............ .. ..................................................................................
The Roche M odel......... ................................................................................
Our Scheme................... .......... e............... ..... ................ .... ....... ................ .........
Num erical M odels............................................................................................
RADIATIVE TRANSFER ............................................. ....... ......................
The Escape Probability
Technique ....... .... ... ..... ... ....... ... ....................... .. ..... ..
H eating and Cooling M mechanism s ........................................ ....... ......................
RESU LTS ............................ ....................... ..... ......................... ... ....... ..........
Spectroscopy ..... .............. ...... .e i..................................... ............. .. ............i.
BIOGRAPHICAL SKETCH ........................................
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
CIRCUMSTELLAR HYDRODYNAMICS AND
SPECTRAL RADIATION IN ALGOLS
Major Department: Astronomy
Algols are the remnants of binary systems that have undergone large scale mass
transfer. This dissertation presents the results of the coupling of a hydrodynamical model
is a fully
three-dimensional scheme with a novel
treatment of viscosity and an implementation of the smoothed particle hydrodynamics
method to compute pressure gradients.
Viscosity is implemented by allowing particles
within a specified interaction length to share momentum.
The hydrodynamical model
includes a provision for computing the self-gravity of the disk material, although not it is
not used in the present application to Algols.
Hydrogen line profiles and equivalent widths computed with a code by Drake and
Ulrich are compared
observations of both short and
sophisticated radiative transfer computations are done with the escape probability code of
Ko and Kallman which includes the spectral lines of thirteen elements. The locations and
velocities of the
supplied to the radiative transfer program,
which computes the equilibrium temperature
of the gas and generates its emission spectrum. Intrinsic line profiles are assumed to be
delta functions and are properly Doppler shifted and summed for gas particles that are not
eclipsed by either star.
Wilson-Liou polarization program. Although the results are preliminary, they show that
polarization observations show great promise for studying circumstellar matter.
The binary star Algol, also known as 3 Persei, is the prototype of a large class of
observationally by the large difference in depth between primary and secondary eclipses,
which is a result of the large difference in temperature between the two components.
Algols are also characterized by emission lines and light curve distortions arising from
The subject of this dissertation is the computation of both the fluid
flow and the resulting emission of radiation by the gas. By employing the method of
smoothed-particle hydrodynamics (SPH), we compute the flow of the gas from the inner
radiative transfer code of Ko and Kallman (1994).
In the early 1940's, spectra of the Algol-type binary RW Tauri taken by Joy (1942)
showed emission line features that he interpreted as arising from a ring of circumstellar
gas around the primary star. Struve observed similar features in other Algols
1944) and his immense volume of high quality spectroscopic observations demonstrated
characteristic features of Algols and not anomalies in just a few systems. Around the
same time, Kuiper published a theoretical treatment of 3 Lyrae and pointed out, for the
importance of the
binaries (Kuiper, 1941).
Spectroscopic and photometric observations of Algols showed that the hotter and
more massive primary stars have radii that placed them on the main sequence in the H-R
diagram, but the cooler secondaries were already evolved from the main sequence. A well
known deduction from stellar evolution theory is that the main sequence lifetime of a star
decreases with increasing mass.
Thus, Algols presented an apparent problem, the so-
called Algol Paradox- how could the lower mass secondary have evolved off of the
main sequence before the higher mass primary did so? A major step in the resolution of
the Algol paradox was achieved by Crawford (19
5). He recognized that Algols are the
remnants of binary systems that have experienced large scale mass transfer. The now less
massive secondary was once the more massive star and evolved to fill its Roche lobe,
spilling matter toward the other star through the L1 point. Morton (1960) showed that the
configuration in which the more massive star fills the Roche lobe is unstable and results
in mass transfer on a thermal time scale.
This meant that the mass transfer would be
relatively rapid and of such magnitude as to reverse the mass ratio of the system, making
the gainer the more massive star. At some point after the mass ratio reversal, the mass
transfer would stabilize and take place at the much longer nuclear time scale. A key point
that Morton failed to take into account, however, was the changing separation of the stars
as the mass transfer occurred. Later work by several researchers allowed for the changing
separation and more accurately described the evolution of the system (see the reviews by
Plavec, 1968 and Paczynski, 1971).
satellite led Plavec (1980) to identify a class of binaries whose ultraviolet spectra show
emission lines of highly ionized species at all phases, and he called these systems the W
stars are binaries in the rapid phase of mass transfer,
Lyrae (Wilson, 1974;
Wilson and Terrell, 1992). Structurally, the W Serpentis stars are
quantitatively. One model of these systems (Wilson, 1981) has the gainer spun up to the
centrifugal limit by the accretion stream. Being at the centrifugal limit, the gainer cannot
accept any more of the incoming gas, and a thick accretion disk is formed. Eventually the
gas is able to settle onto the star, and the system will then consist of a rapidly rotating
main sequence star and a lobe-filling subgiant, making it a rapidly rotating Algol. If the
morphological type (Wilson, 1979). Tidal forces will cause the rotation of the primary to
slow, eventually to synchronism with the orbit, and the system will be in the Algol stage.'
According to the foregoing theory, Algols represent a fairly long-lived state of certain
binaries that have undergone large scale mass transfer, and are therefore important in
understanding the evolution of close binaries. The rapidly rotating Algols are particularly
interesting because they tend to be more active than the ordinary Algols, but not quite so
active as the W Serpentis systems, which are more difficult to model.
The subject of this dissertation is the synthesis of observable quantities, in particular
emission line profiles and spectral energy distributions, that arise from mass transfer in
We also present some results from the merger of the hydro program with the
Wilson-Liou polarization program (Wilson and Liou,
1993). Although hydrodynamical
been done together so as
comparison of theory
The reason for this is
1 This is true assuming that the mass-gaining star that results after the rapid phase of mass
ftrnofnr 4c mon cnnna etor hl ;m n,, hi*<- i lt iirnViilAt k^ fnr1 theT frv Tt PlAnnPh nfT
simplifying assumptions has made the treatment of the problem feasible.
Our eventual goal is to have a computer model that is general in applicability to
matter-transferring binaries such as Algols, cataclysmic variables, and X-ray binaries.
Therefore, the implementation of the model as a computer program has been done with
generality and expandability in mind. The program is written so that future modifications
(such as the inclusion of magnetic forces) will be straightforward. However, a generalized
program should allow the user to "turn off'
calculations that may be unnecessary in a
particular application. An example occurs in our present application to Algols where the
The calculation of the emission of a moving gas in the presence of the radiation
fields of two stars is complicated.
The solution of that problem will lie in work well
gradually increasing the sophistication of our model, we believe that it will be possible to
model the observations of many types of interacting binaries accurately.
to begin our modelling efforts with the Algols
We have chosen
because they are relatively bright (and
easily observed) and because the mass ejection mechanism is relatively well understood.
Many high-quality observations of these systems exist. This dissertation is an attempt to
bring theory into comparison with observations.
integration of the restricted
for particles ejected
Lagrangian (L1) point at arbitrary angles and speeds. Kopal (1956) computed trajectories
for particles ejected at various speeds from L
plane. Gould (1957
at tangents to the Roche lobe in the orbital
1959) considered ejections from L, as well as radial ejections from
other locations along the equator of the star, and considered motion only in the orbital
plane. Both authors computed trajectories for a wide variety of mass ratios and ejection
speeds so as to illuminate the gross properties of these flows.
Plavec, Sehnal, and MikulBA (1964) computed ejections from L1
in a system with
mass ratio of
in an attempt to model the circumstellar gas in RW Tauri.
results showed that thermal ejection at L
was not sufficient to form a ring of material
around the pri
Only ejections at higher speeds could avoid impacting the primary
They also computed trajectories for ejections out of the orbital plane and
showed that such trajectories were short-lived and could not encircle the primary. In a
follow-up paper, Plavec and Kriz (1965) computed trajectories for a wider range of mass
0.3, and 0.5) and for ejections from
locations on the secondary's
surface, although mainly from L1. The new results showed that, at least over the range of
mass ratio considered, the trajectories were surprisingly independent of the mass ratio,
The ring-forming trajectories were also short-lived,
indicating that the disks
systems as RW Tauri and U Cephei must be transient phenomena, a conclusion supported
By assuming a conservatively low density of 1010
particles per cm and collision
cross-sections of 10"'5
cm2, Prendergast (1960) showed that the mean-free path of
the particles was much smaller than the orbital separation of the stars (1-10 km).
result showed that a particle trajectory treatment of the problem was insufficient, and that
a proper treatment should be
hydrodynamical. He derived approximate solutions to the
hydrodynamical equations, simplifying the problem by ignoring pressure gradient terms
in Euler's equation and assuming hydrostatic support perpendicular to the orbital plane.
Unfortunately, the computing power available was inadequate for a more general attack
equation, they simulated solutions of the Boltzmann equation rather than solving a set of
tendency to collide gave them a Maxwellian distribution. After dividing the circumstellar
region into a grid of volumes or boxes, particles were allowed to move in straight lines at
constant velocity between collisions. After moving for a certain interval of time (the
longest possible time that did not allow any particle to travel more than one box length),
The timestep assignment amounts to assigning a mean-free path for the
particles (of the order of the box size), so that viscosity and thermal conductivity are then
built-in viscosity is tied to the computational grid, and cannot be adjusted independently
of the grid resolution.
The results of their calculations for a system
similar to those of U Cephei showed that a stream was formed that impacted the primary.
However, some material did encircle the primary and some material was ejected from the
system as a whole,
with more material being ejected if the primary were rotating five
times faster than synchronously rather than synchronously.
Lubow and Shu (1975) developed a hydrodynamical treatment of semi-detached
trajectory calculations could be useful over a large region that they called called the "orbit
region" since the mean-free path argument is valid only if particle trajectories cross one
Their method makes use of continuum mechanics and is semi-analytic,
that most of the integration could be reduced to a numerical integration of a set of
ordinary differential equations.
They exploited the existence of a parameter, e,
given by e = a/(2d),
where a is the isothermal sound speed (a (kT/m) /2), (2 is the
orbital angular speed (( = 2 /P),
and d is the separation of the two stars. Assuming
synchronous rotation of the secondary and isothermal flow under steady conditions, they
reduced the parameter space of the equations to a single quantity- the mass ratio of the
They also presented a mechanism for the flow near Li- a non-isotropic stellar
which reaches sonic speeds in the neighborhood of Li, and is throttled into a
narrow stream of gas making an angle with the line of centers ranging from 190
for the entire range of possible mass ratios (cf. their Table
In a later paper (Lubow and
they considered the dynamics of the gas flow perpendicular to the orbital
plane, and found that the scale height of the stream often exceeded the corresponding
hydrostatic value because the inertia of the gas prevented it from adjusting to the rapidly
About the same time, Lin and Pringle (1976) outlined a 2-dimensional, many-body
approach to the situation where the size of the primary star was very small compared to
the orbital separation, such as in cataclysmic variables and X-ray binaries.
was the first fully Lagrangian method for simulating gas flows in interacting binaries, and
included a viscous interaction for particles, but did not include pressure gradients. The
viscosity was effected by allocating the particles to the boxes of a Cartesian grid centered
on the primary star, and then allowing the particles within a box to interact with one
another so that the particles achieved solid-body rotation about their center of mass.
Unfortunately, this scheme has the undesirable consequence of allowing particles that are
close to one another, but not in the same box, to avoid interaction while more distant
particles in the same box will interact. Lin and Pringle concluded that the accretion disk
was quite well defined and comparable in size to the Roche lobe,
with all but a few
percent of the transferred mass being accreted by the primary.
Whitehurst (1988b) presented an extension of the
Lin and Pringle scheme that
included pressure gradients computed according to the scheme of Larson (1978), which is
a simplistic approach necessary to keep computational requirements reasonable, but it is
not as drastic as might first appear since the disk is dominated by angular momentum
transport. Larson's method assumes that the particles are extended, deformable gas clouds
in continual contact.
The repulsive acceleration of the particles is given as C2/r, where C
is a parameter proportional
sound speed and r
is the separation of the
Whitehurst's major simplifying assumption was that the energy dissipated by
particle interactions was radiated away instantaneously through the disk surfaces. This, of
course, made the pressure calculations somewhat crude, but it was the next obvious step
Chamaeleontis and was able to model the superhumps1 of the system successfully.
The Roche Model
The basic framework for our modelling of Algols is the extended Roche model,
standard Roche model are as follows:
the stars act as point masses.
This assumption is
distorted secondary has evolved beyond the main sequence and is even more
The stars orbits are circular. Light curves of Algols show that this assumption
is valid since the eclipses are equally spaced and of equal duration.
For the distorted
rotation is invariably observed because of the large tidal torques produced on
With the above assumptions, one can use a frame of reference that rotates with the
binary, and thus has the stars at fixed locations.
The surfaces of the stars will coincide
equipotential of critical importance for close binaries. That is the one that includes the L1
I Vl,.nnrbim nu n.ra -v. n n-r an oA;...-n n n; k.abna'cn, aMlAt i 1'_A n/rrTnt thlt nrnrirl
point. This potential defines the Roche lobe of each star, beyond which it cannot expand.
1 shows the Roche lobes for the SX
Cas system. At the Li point,
ine of centers closer to the less massive secondary component, the sum of the
gravitational and centrifugal forces vanishes. If one of the stars tries to expand beyond its
Roche lobe, the material at the L1 point,
with gas pressure on one side and essentially a
vacuum on the other, will flow toward the other star. In normal Algols,
the secondary fills
lobe and dumps material toward the detached primary
expanding on a nuclear timescale.
> 0.0 0
-1.0o i ii I
The Roche lobes for the components of SX Cas.
The surface of
Si I I I I---- i I I I
The extended Roche model allows for a more general treatment of close binaries. In
this model, the stars may have eccentric orbits and rotate non-synchronously (see Wilson,
1979). Although eccentric orbit generalization is not necessary for Algols, non-
synchronous rotation has proven to be reasonably common for the primary components.
In the rapidly rotating Algols, the primaries do not have a Roche lobe per se, but they do
have a critical surface. The critical surface is defined by the equipotential that, for a given
rotation rate, contains a point at the centrifugal limit. Double-contact binaries have the
secondary filling its Roche lobe while the primary fills its rotational critical surface
schemes), Eulerian (grid or mesh schemes), and a combination of the two. Each technique
has advantages and disadvantages that must be carefully considered for applications to
different situations. Non-adaptive grid codes are straightforward to program and analyze,
but can be very inefficient when the density of the fluid changes rapidly.
large amounts of memory when applied to three-dimensional flows. Particle methods, on
the other hand, benefit from the fact that the number density of the particles maps directly
onto the mass density of the fluid, giving increased resolution in high density areas.
Other advantages of a particle method make it attractive for computing gas flows in
binary star systems. One attractive feature is that there is no grid to impose an artificial
geometry on the system. Another is the fact that a article scheme can be "low level,"
which makes the addition
possible with minimal coding effort.
we have adopted a three-dimensional Lagrangian scheme similar to the
two-dimensional Lin and Pringle (1976) and Whitehurst (1988b) approaches. An early
Assumptions about the binary system appropriate to Algols are made, namely that the
stars act gravitationally as centrally condensed mass points and move in circular orbits.
Particle positions and velocities are specified in a rectangular coordinate system with the
origin at the center of mass of the
binary, and the unit of
distance being the stellar
The x-axis of the system lies along the line of centers and is positive in the
momentum vector of the binary, and the y-axis is defined such that the coordinate system
is right-handed. The unit of time is P/27, where P is the orbital period.
The mass ratio is
defined in the sense of loser to gainer,
We define an auxiliary quantity
and write the acceleration components due to gravity of a particle located at (x,y,z) as
= 2jy+ x (x-x1)-
3 (X X2)
1 and x2being the distances from the center of mass of the primary and secondary,
Particles are ejected from Lwith initial velocities specified in the program input file.
The user can specify an angle that is the projection of the initial velocity in the orbital
plane, measured from the positive x-axis. (Thus |)=1800 means the particle is ejected
toward the center of the primary star.) The angle 0 is measured from the positive z-axis to
the ejection vector, so 0=900 means that the ejection is in the orbital plane and 0=0
means that ejection is parallel to the orbital angular momentum. Angles A( and A9 are
user-specified ranges over which the particles will be randomly ejected, centered on ) and
That is, the particles will be ejected randomly with initial angles 1 + AA /2 and 0
A9/2. The initial velocity specification is completed by the initial dimensionless speed. In
our experiments the initial ejection velocities were consistent with the calculations of
Lubow and Shu (1975).
Of critical importance in simulating astrophysical disks is the handling of viscosity.
of the viscosity
is not known,
turbulence and magnetic fields have been suggested (see Shakura and Sunyaev (1973)
and Pringle (1981)).
We have chosen to implement the viscosity by specifying a viscous
interaction length, a, for the particles. Particles within a distance a of one another are
allowed to share momentum if they are approaching one another. At each timestep all
particles within a of a particular particle are found. Label the particle whose new velocity
we wish to compute by the number 1, and the interacting particles by 2,3,4
... n+1 (where
n is the number of particles interacting with particle
), then compute the
+ 2 )(1- )+(1
CT ,-~ n
12 13 +...+
The new x-component of the velocity for particle 1 is then computed by
and the y and z-components are computed in the same manner.
The particles retain their
positions while having their velocities modified in a way that conserves momentum and
does not depend on the order of calculation as in the method used in Terrell and Wilson
Inefficiency in the determination of nearest neighbors can cause rapid escalation of
The simplest, and most inefficient method would be to compare each of
n particles with the others, requiring
Z n(n- 1)comparisons.
To improve the efficiency of
we have implemented a grid location scheme to eliminate particles too
distant from a given particle to interact with it. A grid centered on the primary star with
box sizes of length a is used to generate a linked list of interacting particles. Starting with
the first particle, its grid location is determined and the particle number is placed in a
corresponding to the grid.
then grid location of
particle two is determined. If particle
happens to be in the same box as particle 1, then
is updated so
that POINT(2) contains a
indicating that particle
2 "points" to particle 1.
Now suppose particle 3 is also in the same
box. The value in GRID is changed to a 3, and POINT(3) is set to
This scheme has the
advantage that it uses only
scheme would be to make GRID a 3-dimensional array, with dimensions 1 and
the x and y locations of the grid box and dimension 3 being the labels of the particles in
that grid box. Although very easy to implement, this latter method is very inefficient
in order to avoid possible overflow errors. And if the search grid must be extended to
three dimensions (for geometrically thick disks), GRID would have to be a 4-dimensional
array, further increasing the memory requirements.
of the particles have been assigned to the search grid, interacting particles
can be quickly identified.
To find all particles interacting with particle i, one need only
search the box containing particle i and the eight boxes that surround it (assuming the box
containing particle i is not on the edge of the grid).
The viscous interactions between the particles cause their total kinetic energy to
decrease, and this loss of kinetic energy shows up in the form of thermal energy.
hydro program computes the total viscous heating at each timestep by comparing the new
kinetic energy of each particle with its previous value.
The acquired thermal energy is
then distributed to each particle based on the strength of its viscous interaction. The
radiative transfer program uses the viscous heating rate (in ergs/cm3/sec) to compute the
equilibrium temperature of the gas.
The heating rate is computed by assuming that the
viscously deposited energy is added to the gas evenly over the timestep, and evenly over
the volume of the gas particle.
Forces arising from pressure gradients in the disk are typically small compared to
gravitational and centrifugal forces, but can be important nonetheless since the latter two
forces can be anti-parallel, thus may largely cancel one another and make the pressure
1979b) to avoid certain limitations of grid-based finite difference schemes.
Since its introduction, SPH has become a standard tool in computational astrophysics,
with applications to diverse areas such as the formation of planetoids (Benz, et al., 1989),
star formation (Prongracic, et al., 1993), galactic collisions (Hernquist and Katz, 1989),
and the formation of galactic clusters (Thomas and Couchman, 1992).
Most of the advantages of SPH arise from the fact that it treats fluid elements as
extended clouds of material whose centers of mass move according to the conservation
laws of hydrodynamics. Unlike grid-based methods, there are no artificial constraints on
of the system or the volume that the
SPH also has the
interaction between particles is specified.
To compute values of continuum variables such as pressure and density from the
discrete distribution of
particles, an average over the particle distribution is taken by
the advantage that
derivatives of the smoothing function.
where V represents the volume of interest and r. is the position vector of particles.
n particles of mass m, the smoothed density is given by
ml w(r -
and the acceleration acting on particle i due to pressure gradients is
d =-mI +
dt J pI
functions, and recommend the following smoothing function for modelling fluids where
self-gravity is negligible:
-- ri~a r
and h is the user-specified smoothing length.
This smoothing function avoids artificial
clustering that occurs in situations where self-gravity is negligible (such as Algol disks),
and we employ it in our program.
some disk models
W Serpentis stars,
Wilson (1981) and
Wilson (1974) for a discussion of the disk
Lyrae.). Because our program was designed to be extended easily,
we were able to add
the computation of forces due to disk self-gravity with very little effort, although we do
our present application to
computations, the gravitational interaction between particles is computed with a hybrid
grid and n-body scheme. Particles close to one another are treated as individual sources of
gravity and the forces are computed in the usual softened n-body manner. More distant
particles are treated as groups and the force is computed based on the sum of the masses
and their common center of mass.
because of spin-up during mass transfer.
To simulate these systems, it is necessary to to
model the interaction of the impacting stream and the surface of the star. In our model
this is done by
particles to interact with surface particles moving at
velocities specified by an input parameter F, which is defined as the angular rotation rate
IU* **fl-. *~~I 1flnnrllrnrlll trrni'J runp l~a Lfafrt Il~l Lr tA)l III Il4I ** l- I
where the normalization constant is given by
f\1 | n l f i
Tf 1*: 1,,,
velocity at the star surface is reversed and multiplied by the bounce parameter.
bounce parameter set to 0, the stream plows into the star and is engulfed by it. A value of
1 for the bounce parameter causes a perfectly elastic collision.
Finally, the mass transfer rate (in Me yr
) is specified in the input file along with
the number of timesteps between the ejection of particles.
Once these two parameters
have been chosen, the mass of each gas particle is determined. Particles will be emitted
from L1 until the user-specified limit has been reached and the program will run until all
particles have either been accreted by either star or lost from the binary system. (Note that
the length of the mass transfer event is determined by the maximum number of particles
and the number of timesteps between ejections.)
Algol systems can be split into two groups in terms of the type of disk formed
during mass transfer. The difference between the two is determined by whether or not the
matter stream strikes the primary star before encircling it. Lubow and Shu (1975) in their
2 list a parameter, 'amin,
which is the minimum distance of the stream from the
center of the primary star (in units of the orbital separation). Figure 2-1 shows a plot of
relative primary radius versus mass ratio (r-q diagram) with 'min indicated by the smooth
curve and the values for the primaries of a number of Algols and W Serpentis stars. If the
radius of the primary star is larger than mmin, then the stream will impact the star, and a
transient disk will form. If the radius of the star is less than Smin, then the stream flows
around the star and impacts itself, as seen in a simulation of SX Cassiopeiae shown in
Relative radius versus mass ratio for several Algols and W Serpentis
stars. The smooth curve indicates the Lubow-Shu stream impact
condition. Systems observed to have permanent disks are indicated by
triangles and those having transient disks by circles.
There are some systems in Figure
that seem not to obey this logic, in
particular V356 Sgr,
S Cnc, and RW Per. In the case of V356 Sgr, the primary is most
likely rotating at the centrifugal limit (Wilson and Caldwell, 1978), thus inhibiting the
accretion of the circumstellar material. Although its location in the r-q diagram indicates
Hamme and Wilson,
1993). RW Per was long thought to have a thick accretion disk
(Hall, 1969; Hall and Stuhlinger, 1978), but Wilson and Plavec (1988) found no evidence
of such a disk in their light curve modelling. They conclude that RW Per is an "ordinary
Algol system" with the rapidly rotating primary filling about 94% of its limiting lobe. So,
although there are some anomalies in the r-q diagram, the Lubow-Shu criterion does seem
to be a generally good indicator of the type of disk that will form in a binary.
For a given absolute radius, the relative radius of the primary will scale as P-2/3
where P is the orbital period.
Thus we expect long-period systems generally to have
permanent disks and short-period systems to have transient disks.
Peters (1989) finds that
systems with P
< 4.5 days rarely show permanent H, emission,
while those with P
days generally exhibit permanent Ha emission. Highly variable emission is exhibited by
systems whose periods lie in
range of 5-6 days.
transient and permanent disk systems will be presented in Chapter 4.
Figure 2-4 and Figure
show the formation of the SX Cas disk as a mass transfer
event continues. In these simulations, the viscous interaction parameter was 0.06. By 1.6
c = 0.1 was also run. Figure 2-6 shows the SX
Cas disk at the same time as shown in
With stronger viscous interaction, the disk shows greater radial spreading, as
expected, and is also elliptical. The elliptical shape is maintained to at least 8 orbits, when
the simulation was stopped. Figure
-7 shows the disk after 8 orbits and the gravitational
influence of the secondary is seen in spiral arms that have developed. The spiral arms are
caused by the 3:1 resonance which leads to stresses in the disk. This phenomenon is seen
of Whitehurst (1988a,1988b) and Lubow
discussion of the instability mechanisms.
* ( #*
-1.0 -0.5 0.0 0.5 1.0 1.5
Figure 2-3. A simulation of SX Cas, showing the stream trajectory 0.38 orbital periods
after the beginning of the transfer event.
- ,:: ( (,
I i I -I
Figure 2-4. The early stages of disk formation in SX Cas at 0.8 orbital periods after the
Figure 2-5. The elliptical disk that forms 1.6 orbital periods after the onset of mass
transfer in SX Cas.
Figure 2-6. A simulation at the same phase and with the same parameters used for Figure
1~~~ -- A I. I 1.
1.5 -.?-- r* -*--
- 1 .0 -- --------- --0- -.--- -
-1.0 -0.5 0.0
Figure 2-7. The same simulation as in Figure 2-6 at 8 orbits after the onset of mass
transfer. Note the spiral arms that have formed due to the 3:1 resonance.
The Escape Probability Technique
An early version of our program (Terrell and Wilson
,1992) used the hydrogen-only
radiative transfer code of Drake and Ulrich (1980; hereafter DU) to compute Ha profiles.
This code is useful for computing line profiles because its computational demands are
fairly modest. Many ground-based observations of Algols are available (mainly He and
Hp) and the DU code provides a means of rapidly computing hydrogen line profiles. DU
include all relevant collisional processes, and explicitly treat the first six angular
momentum sublevels for the first six energy levels. The angular momentum sublevels for
higher energy levels are computed by assuming statistical equilibrium. Justifications for
these assumptions are given by DU, and the assumptions are valid for the conditions
found in Algols.
Unfortunately, the only hydrogen line observable by the International Ultraviolet
Explorer (IUE) is Lyman a, although there are many lines of other species in the
wavelength range covered by the IUE short wavelength camera (see Figure 3-1).
Therefore, we require the capability to include other atomic species such as helium,
carbon, and nitrogen. Fortunately the necessary program was completed recently in the
1994). This program includes the lines of the thirteen cosmically abundant elements- H,
He, C, N, O, Ne, Mg, Si, S, Ca, Fe, Ni and Ar. The following discussion covers the
general techniques employed in the program. For details the reader is referred to Ko and
Figure 3-1. An IUE spectrum of the Algol-type binary U Cephei during its total primary
eclipse showing the rich set of emission lines arising from the circumstellar
The Ko program is based on non-local thermodynamic equilibrium calculations of
ion and level populations employing a large number of atomic processes. The radiative
transfer for both line and continuum radiation is treated by the escape probability
formalism (e.g., Krolick and McKee, 1978; Drake and Ulrich, 1980). This method works
given optical depth of a transition, one estimates the fraction of photons that escape from
the gas. Then the spontaneous radiative transition probability for each transition is set
equal to the Einstein A value multiplied by the escape probability and the rate equations
for the level populations are solved. If the optical depths from the newly calculated level
populations do not agree with those originally assumed, then the calculation is redone
with the new optical depth. This iterative procedure is continued until self-consistency is
For line radiation, there are three probabilities to consider in computing the escape
The probability that the photon can escape without taking part in any other
atomic processes, PJ',t (t), which depends on the optical depth of the line and
is written, using the notation of Ko and Kallman (1994), as
= 1 for t,
S, 1i 12 +
where -, is the optical depth where the line wings become optically thick.
The probability that the line photon is destroyed by photo-absorption, Pes ,
which depends on the line opacity, K,
, and the continuum opacity at the line
energy, Kce,, is written as
where KT is
the Thomson scattering opacity.
The probability that the photon is scattered out of the line core by Thomson
scattering given by
The escape probability for a line photon is then given by
+ ( escl.c
t+ P^escp)(1 Pesc, ).
For the continuum, taking the line analogy for the escape probability of the
recombining photons, the assumed profile function of the recombining photon level i is
(po(x) = cx -
- s/s, with
c being the energy of the photons and ei the ionizing threshold of the
The normalizing constant c is given by
which upon integration yields
At the absorption edge at optical depth ta, the escape probability is
where xe is defined such that
cP i~(x, )z.
and escape probabilities
escc (' a)
For recombining continuum the profile function is
given the level photoionization cross section ai c For or(x) equal to a constant times
ra < 1/P
= 3 for hydrogenic photoionization cross section.
Heating and Cooling Mechanisms
program was designed to compute the emission of a disk in an X-ray
binary. In these systems the disk surrounds a neutron star, and the accretion of
material onto the neutron star releases X-radiation which is then absorbed and processed
by the disk. In Algols, the gravitational potential well of the primary star is not as deep,
and X-radiation is not produced.
and cooling mechanisms
employed by the Ko program are negligible when applied to Algol disks, but our eventual
goal is to build a program that will handle a wide variety of matter-transferring binaries
(including X-ray binaries and cataclysmic variables) and the Ko program is well-suited
for our work.
The viscous interaction in the circumstellar gas is a major contributor to its heating,
and the method of computing the heating was described in Chapter
The Ko program
was modified to read the viscous heating from the hydro program, add it to the other
heating rates, and compute the gas temperature for thermal equilibrium (i.e.,
heating and cooling rates are equal). Additional heating mechanisms included in the Ko
program are photoionization (due to radiation from the primary star) from all levels and
Compton heating, which is significant in X-ray binaries but not in Algols.
field of the primary star is taken to be that of a blackbody of temperature T and is given
by the Planck function
which leads to the photoionization heating rate
The cooling mechanisms that are computed are radiative recombination cooling,
line cooling, Compton cooling, and bremsstrahlung cooling, the latter two of which are
very small in the Algol regime. The radiative recombination cooling rate is
element ion level
X J*k F +t32 Pec, e
and the line cooling rate is
k m j>i i
element ion level
= T Y
is the line energy and the total escape probability
Pesc, j is given by
+ pc, p (1
Transient Disk Systems
RW Tau. Kaitchuck and Honeycutt (1982) used time resolved spectroscopy during
the total eclipse of the short period Algol RW Tauri and found that the disk was highly
variable, on the time scale of one orbital period.
These observations dispelled the idea
that Algol-type accretion
disks are axisymmetric and
radius of the disk varied from
widths were at least a factor of
the rotation of a Keplerian disk.
to 1.7 times the radius of the primary and that line
2 greater than would be expected for broadening due to
Vesper and Honeycutt (1993) made high resolution (-1
A) observations of the Ha line over the entire orbit and discussed two distinct emission
features in the line profiles.
One emission feature ("Type A") is
strong enough to be seen
in raw spectra while the other ("Type B") shows up only when the light of the two stars is
Kurucz (1979) atmospheres for the primary and using the spectrum of HR 88
7 for the
secondary, and then convolving them with limb darkening and rotation profiles.
The primary of RW Tau rotates at greater than twice the synchronous rate (Olson,
parameters given in
Terrell, et al. (1992), and a mass transfer rate of 108 M yr
Figure 4-1 shows, the stream impacts the primary star and rebounds to a distance of about
two primary radii from the center of the star. As mass transfer continues, a thin ring forms
1 .5 -
Figure 4-1. The stream in RW Tauri impacts the primary star at phase 0.22.
Figure 4-2. By phase 0.32, a ring of material has formed around the primary star as matter
continues to flow from the secondary.
Figure 4-3 shows the He line in RW Tau at phase 0.222 as observed by Vesper and
The strongest emission feature is red-shifted while a weaker feature is blue-
shifted, and spectra taken at the same phase during other epochs show similar features.
Near phase 0.75 the spectra are similar, except that the Doppler shifts are reversed (see
Figure 3 of Vesper and Honeycutt (1993)). At phase 0.463 the strongest feature is blue-
shifted and a weaker component lies near the rest wavelength' as shown in Figure 4-4.
Figure 4-5 shows the line at phase 0.685.
For the computed H, profiles, we chose the disk as shown in Figure 4-2 and used
the Drake program to compute the profiles as seen from different orbital phases. It must
4-- -;.... ----..
Figure 4-3 The He difference profile for RW Tauri at phase 0
line indicates the rest wavelength of the line. (Data courtesy D.
The dashed vertical
Figure 4-4 Same as Figure 4-3,
except the observation was made at phase 0.463.
Figure 4-5. Same as Figure 4-3 with the observation made at phase 0.685.
-j- _- 1.- T, ^
be stated that we have not attempted to formally fit the observed profiles. At this point we
only seek to show that the computed spectra are generally similar to the observed ones.
We have not tried to fit the observations by varying the parameters of the hydro program.
In spite of this, the computed profiles do match the observations reasonably well. Figure
4-6 shows a computed spectrum for a phase 0.463 view of the disk 0.22 orbits after the
onset of mass transfer (see Figure 4-1). In the observed profile we see three peaks, the
strongest of which is blue-shifted to about 6554 A. A slightly weaker peak lies at the rest
wavelength and a much weaker peak lies at 6572 A. The computed profile, which has
been smoothed to the same resolution as the observed one, shows similar features with a
peak around 6552 A, but the second peak is slightly red-shifted.
z 04 t
0. I--t I II
6520 6540 6560 6580 6600
Figure 4-6. The computed profile for RW Tau 0.22 orbits after the onset of mass transfer
and as seen from phase 0.463. Figure 4-1 shows the disk that produces this
profile and Figure 4-4 shows an observed profile at this phase.
As a check on possibility that features in the computed spectrum might be spurious
and dependent on the number of gas particles, a simulation of RW Tau was run with the
same parameters, except that the number of particles was doubled. Figure 4-7 shows the
computed profile for the resulting mass distribution at the same phase as the profile
shown in Figure 4-6. Only minor differences exist between the two profiles.
Same as Figure 4-6 except that twice and many gas particles were used in the
TX Ursae Majoris. A comprehensive observational study of the circumstellar gas in
this system was done by Albright and Richards (1993) and they also discuss the extensive
observational history of the system. Inspection of their difference profiles shows that the
circumstellar emission is highly variable,
with rapid changes occurring over intervals as
short as one orbital cycle. Figure 4-8 shows an observed Ha profile at phase 0.306 and
An observed H
a profile of TX UMa at phase 0.306.
The dashed vertical line
indicates the rest wavelength (Data courtesy G. Albright).
1 2 0 . -- .... .. .. -
An observed H, profile of TX UMa at phase 0.631.
Simulations of TX UMa were done assuming a mass transfer rate of 10-9 M yr
and the parameters given in Terrell, et al. (1992), except that the rotation parameter for
the primary was set to 3.5 as suggested by
Albright and Richards (1993).
shows the distribution of circumstellar matter 0.35 orbits after the onset of mass transfer.
View of TX UMa 0.35 orbits after a mass transfer event has begun.
Ha profiles were computed for the disk shown in Figure 4-10. The computed profile for
is shown in Figure 4-11
and has some features in common with observed
profile in Figure 4-9 including a peak at 6560 A and another near the rest wavelength.
However, the strengths of the two peaks are reversed and the computed profile shows a
small peak around 6570 A that is not seen in the observed profile.
-4 I-."-- a-!--"-~ .- -- -----
Figure 4-11. The computed He profile of TX UMa at phase 0.631 for the disk shown in
0.2 -i i
0.0 4- -----
,i 1 4
Figure 4-12. The computed H, profile at phase 0.306 for the disk in TX UMa shown in
U Sagittae. U Sagittae is a short-period system that lies in the transient disk region
of the r-q diagram and slightly non-synchronous rotation with F=1.31 (Van Hamme and
The Ha difference profiles of the system are highly variable and show
strong absorption at some epochs as can be seen in Figures 4-13 through 4-17
feature of the observed profiles is the absorption core seen at some phases.
Simulations of the system were run with the parameters in
Terrell, et al. (1992).
Unfortunately, the absorption feature makes it difficult to compare the computed profile
with the observed ones. However, the profile computed at phase 0.499 does share the
locations of peaks and overall width with the observed profile in Figure 4-15.
Figure 4-13. An Ha difference profile of U Sge at phase 0.171 during a time of high
activity. (All U Sge data courtesy G. Albright).
0.80 -i -----
4....f -- -----i t
Observed HI profile of U Sge at phase 0.219.
Observed H, profile of U Sge at phase 0.499.
Observed Ha profile of U Sge at phase 0.672.
Figure 4-17. Observed Ha profile ofU Sge at phase 0.888.
o o -0 .......4 ...... 4- 4
0 6580 66i
Figure 4-18. The computed Ha profile of U Sge at phase 0.499 for comparison with the
observed profile in Figure 4-15.
Permanent Disk Systems
permanent/transient disk dividing line of the r-q diagram. An extensive discussion of the
system is given by Sahade and Ferrer (1982).
Peters and Polidan (1984) analyzed IUE
spectra of the system and found evidence for a high temperature accretion region with an
electron temperature of approximately 100,000 K and electron density of about 109 cm-3
Figure 4-19 shows an HE profile taken by G.
Peters at phase 0
In spectra taken about
seven months earlier, and shown in Figures 4-20 and 4-21, striking absorption features
are seen at the core of the profile
Simulations of AU Mon were done using the system parameters given in Terrell, et
al. (1992). This system is particularly interesting because it lies very close to the Lubow-
Shu curve in the r-q diagram. As can be seen in Figure 4-23, the stream barely impacts
6540 6560 6580
Figure 4-19. He profile ofAU Mon at phase 0.22 with the flux normalized to the
continuum (All AU Mon data courtesy G. Peters) .
Figure 4-20. The Ha profile ofAU Mon at the same phase (0.22) as the profile in Figure
4-18, but taken about seven months earlier.
profiles in Figures 4-19 and 4-19 show absorption cores and cannot be modeled by our
nraP, ,nm nlrrrn ynl n n. nn l ..nr.,m,.,1 + Ihkl i,-- ..T..rL.a. +- 4L ,r +-l1, a.-i nrn ,,rf-o-+Arl
The H, profile of AU Mon at phase 0.33,
taken 1 night later than the profile
in Figure 4-19
Figure 4-22. The computed H, profile of AU Mon at phase 0.22 for the disk formed 5.7
orbits after mass transfer began.
I I 1 I I 1 i i i
I I ~ i II
0.0 0.5 1.0
I Ir l-
I II II
'liii, liii I1)
Figure 4-23. The appearance of the disk of AU Mon at several phases.
a) 0.25 orbits; b) 0.64 orbits; c) 2.4 orbits; d) 5.7 orbits.
who showed that eclipses would break the disk symmetry and
.4 ~ ~ ~ I J ~ t1 165 r4 ** l f n.
formulae for computing the Stokes quantities.' Wilson and Liou (1993) combined the
Chandrasekhar and Brown et al. treatments with the Wilson-Devinney binary star light
curve model (Wilson and Devinney
1971; Wilson, 1979,
Wilson, 1990), and were able to
achieve good fits to the Kemp et al.
(1983) observations of Algol by assuming fixed
locations of circumstellar material.
The hydro program has been interfaced to the
Wilson-Liou polarization program,
making it possible to compute polarization curves for time-dependent circumstellar flows.
is at a
promise as a probe of circumstellar flows.
The ionization treatment at present is rather
The ionization is computed either by the Saha formula or by assuming that a
constant fraction of the gas within a specified radius of the primary star is ionized. Future
improvements to the ionization calculations will include coupling the results of the Ko
A longer burst event results in a dramatically different set of polarization
curves, as seen in Figure 4-25.
An even longer burst event is shown in Figure 4-26.
computed curves are sensitive to the assumed ionization scheme.
Figures 4-27 through 4-
29 show the variation of the computed curves for the constant ionization method as the
radius of ionization is varied.
Polarization observations of Algol-type binaries are relatively rare owing to the fact
that necessary combination of sensitive polarimeters and large aperture telescopes has
not been developed. Because the polarization in Algols is very small,
large numbers of
1 The Stokes aunantities 0 and 1_ are freauentlv referred to as the Stokes parameters,
0' -0.05 -
-0.20 ---------|---- \ -- -- --
0 1 2 3 4
Figure 4-24. The polarization curves for a mass transfer event in SX Cas lasting 0.3
orbital periods. The upper curve is the U curve and the lower one is the Q
1 2 3
Figure 4-25. Polarization curves for a medium-length (1.4 orbital periods) transfer event
in SX Cas.
0.4 -.---- ~
Figure 4-26. Polarization curves for mass transfer event in SX Cas lasting 2.4 orbital
Figure 4-27. Polarization curves for a medium-length (1.4 orbital periods) burst event
assuming partial ionization of all circumstellar gas.
Figure 4-28. Same as Figure 4-4 except that the ionization occurs within a radius of 0.15
of the primary.
0 1 2 3
Figure 4-29. Same as Figure 4-4 except that ionization occurs within a radius of 0.1 of the
photons must be collected in order to achieve a reasonable signal-to-noise ratio. Although
it is relatively bright, Algol unfortunately has very low circumstellar activity. More active
are too dim
combination of sensitive polarimeters and large aperture telescopes like the
Telescope will be able to observe polarization in many of the more active Algols.
Because of the episodic nature of these gas flows, the ideal observing platform is one in
space where weather and day-night interruptions are eliminated and where observations
in the ultraviolet can be made.
the observations are made, w
We plan to further develop the HFP program so that when
re will have a tool for extracting quantitative information
circumstellar gas in Algol-type binaries. Although the agreement between computed and
observed spectra is not perfect,
we are encouraged by the similarities seen in some of
them. Of course, there is still much work left to be done, and the work presented here will
as a foundation
interacting binaries as well.
The first step in using a new model is to become familiar
estimates of the input parameters can be made before beginning an impersonal adjustment
Some parts of the model,
in particular, the radiative transfer program, push the
the simplifying assumptions necessary in this work. Enhancements to the present work
will be made for many years to come.
In this project, the emission lines of the gas were compared to difference profiles of
where the light of the two stars is removed from the observations.
state of affairs is similar to the way light curves were solved with the Russell-Merrill
where certain effects were removed from the observed curve before solving for
the system parameters. Unfortunately, this leads us in the case of our emission line work
to view the stellar light as a contaminant rather than a source of information on the
binary. As our experience with physical light curve models has taught us, it is better to
include all sources of light in the model and predict what is actually
observed at the
our work with
star model and
Mukherjee's (1993) absorption line profile model, we will be able to compute observables
compared to observations.
This combination enables us to better
determine the parameters of the binary by simultaneously satisfying different kinds of
With such a combined model we could simultaneously fit light and radial
curves, line profiles, polarization curves, and X-ray pulse delay
advantage of such a procedure is that all relevent parameters are determined in a logical
and self-consistent way. Another advantage is that simultaneous solutions may extract
information not readily apparent in separate solutions, such as the discovery of a tidal lag
supergiant as the neutron star spirals inward toward it (Wilson and Terrell, 1994). It is
hoped that the present work will spawn new efforts both theoretical and observational, so
that a coherent picture of binary star evolution may be achieved.
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Dirk Terrell was born on August 14, 1965,
in San Francisco, California. Six months
later his parents moved to Jacksonville, Florida, where he spent his early childhood. After
a divorce, his father married a woman who was to have a profound influence on Dirk's
1978 the family moved to Greenville, South Carolina, and he graduated from
Wade Hampton High School in June,
where he was a letterman in football and
track as well as the 1983 U.S. Army Reserve National Scholar-Athlete. He was admitted
as a Research
Morgantown Energy Technology Center in Morgantown,
West Virginia, in 1985. In May,
of Science degree
He was admitted to
graduate program in the Astronomy Department at the University of Florida in the fall of
1987 where he taught both laboratory and lecture courses on astronomy. He joined the
faculty of Santa Fe Community College as an adjunct professor of physical science in
1990 and he continues to teach there. In 1991, he was named a NASA Graduate Research
Fellow and awarded a three-year fellowship to complete his doctorate, which he expects
to receive in the summer of 1994. His spare time is spent cave diving, learning the martial
art Ninkido, riding his motorcycle, and writing computer software.
in my opinion
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Robert Wilson, Chairman
Professor of Astronomy
acceptable standards of scholarly presentation and is
as a dissertation for the degree of Doctor of Philosop
:ope and quality,
Professor of Astronomy
acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Janies Hunter, Jr.
Professor of Astronomy
acceptable standards of scholarly presentation and is fully adequate, in
as a dissertation for the degree of Doctor of Philosophy. /
Professor of Physics
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
of the requirements
for the degree
of Doctor of
Dean, Graduate School
I-I. k~ ~US
UNIVERSITY OF FLORIDA
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