Circumstellar hydrodynamics and spectral radiation in algols

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Circumstellar hydrodynamics and spectral radiation in algols
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Thesis (Ph. D.)--University of Florida, 1994.
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CIRCUMSTELLAR HYDRODYNAMICS AND
SPECTRAL RADIATION IN ALGOLS









By

DIRK TERRELL


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

1994





























In memory of Jean Stillwell









ACKNOWLEDGMENTS


I do not believe it is possible to express the depth of my gratitude to my committee


chairman, R.E.


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


office


see him


absorbed


thought


on some


problem,


never


did he


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


advice


many


stimulating


discussions


on matters


both


astronomical


non-


astronomical


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.


forever be


indebted to


Stillwell


family--


, Jean,


Elizabeth,


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

hydro program.

NASA deserves credit for both direct and indirect support of my achievements.


During


three


years


have


been


supported


a NASA


Graduate


Student


Researcher


Fellowship


allowed


me to explore


projects


addition


to this


one.


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


parents,


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


constant


barrage


curiosity


about


everything


imaginable,


always


gave


me an


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.............. ........................................................... ..................................

CHAPTERS


INTRODUCTION


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.

Polarization ........................................

Conclusions .....................................................................................................


REFERENCES..........


................................................................................................... 5


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

By

DIRK TERRELL


August, 1994


Chairman: R.E.


Wilson


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


a radiative


model


flow


from


inner


Lagrangian


point.


hydrodynamical model


is a fully


Lagrangian,


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


long


period Algols.


More


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


gas particles,


and the


viscous


heating


from


hydro


program


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.


Polarization


curves


are computed


combining


hydro


program


with


Wilson-Liou polarization program. Although the results are preliminary, they show that

polarization observations show great promise for studying circumstellar matter.














CHAPTER 1
INTRODUCTION


The binary star Algol, also known as 3 Persei, is the prototype of a large class of


eclipsing


binaries


referred


Algols.


Algols


eclipse


distinguished


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


circumstellar material.


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


Lagrangian


point


(LI).


radiative


properties


are computed


with


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


(e.g., Struve


1944) and his immense volume of high quality spectroscopic observations demonstrated


these


streams


rings,


now


more


commonly


referred


as disks,


were


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


first time,


importance of the


point in


understanding


flows


in interacting


binaries (Kuiper, 1941).





2


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).


Spectroscopic


observations


with


International


Ultraviolet


Explorer


(IUE)


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


Serpentis stars.


W Serpentis


stars are binaries in the rapid phase of mass transfer,









Lyrae (Wilson, 1974;


Wilson and Terrell, 1992). Structurally, the W Serpentis stars are


much


more


complicated


Algols


therefore,


much


more


difficult


to model


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


gainer


rotating


centrifugal


limit,


system


is of


double-contact


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


Algols.


We also present some results from the merger of the hydro program with the


Wilson-Liou polarization program (Wilson and Liou,


1993). Although hydrodynamical


treatments


very


simplified


radiative


treatments


problem


have


been


done


previously,


never before


have


been done together so as


to allow


a meaningful


comparison of theory


and observation.


The reason for this is


the complexity


of the


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









problem


increases


which


taxes even


m computing


most


power


powerful


combined


with


computers available.


judicious


However,


programming


recent


some


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


self-gravity


negligible,


time-consuming


self-gravity


calculations


omitted.

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


beyond


the material


presented in


this dissertation.


However,


starting


simply, and


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.














CHAPTER


HYDRODYNAMICS


Previous Work


earliest


attempts


to model


flows


in semidetached


binaries


were


integration of the restricted


three-body problem


for particles ejected


from


inner


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


0.2151


in an attempt to model the circumstellar gas in RW Tauri.


Their


results showed that thermal ejection at L


was not sufficient to form a ring of material


around the pri

during infall.


irmary


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


ratios (0.2,


0.3, and 0.5) and for ejections from


various


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


in such


systems as RW Tauri and U Cephei must be transient phenomena, a conclusion supported

by observations.


By assuming a conservatively low density of 1010


particles per cm and collision


cross-sections of 10"'5


to 10"16


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).


This


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


on the


problem,


Prendergast's


solutions


were


severely


limited


required


simplifying assumptions.


more


generalized


treatment


was


given


Prendergast


Taam


(1974).


Recognizing


equations


of hydrodynamics


are moments


Boltzmann


equation, they simulated solutions of the Boltzmann equation rather than solving a set of


difference


equations.


was treated


as a


collection


of particles


whose


strong


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),


particles


were


thermalized


a new


Maxwellian


velocity


distribution


was


constructed.


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





7


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


with parameters


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


binaries


used


matched


asymptotic


expansions.


Their work


showed


particle


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


another.


Their method makes use of continuum mechanics and is semi-analytic,


meamnng


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,


which is


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


binary.

wind


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


to 280.4


In a later paper (Lubow and


Shu,


1976),


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





8


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.


Their method


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


to the


local


sound speed and r


is the separation of the


particles.


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










simulating


flows.


Whitehurst


(1988a)


applied


model


to the


UMa


star Z


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,


which


based


on several


assumptions


about


stars.


These


assumptions


standard Roche model are as follows:


Gravitationally,


the stars act as point masses.


This assumption is


justified


because


high


central


condensation


stars.


In Algols,


more


distorted secondary has evolved beyond the main sequence and is even more

centrally condensed.

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.


stars


rotate


synchronously.


For the distorted


secondary,


synchronous


rotation is invariably observed because of the large tidal torques produced on


However,


primary


components


many


Algols


not rotate


synchronously,


Roche


model


must


extended


to treat


them


properly.

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


with


surfaces


constant


potential


energy,


equipotentials.


There


exists


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


one






10


point. This potential defines the Roche lobe of each star, beyond which it cannot expand.


Figure


1 shows the Roche lobes for the SX


Cas system. At the Li point,


which lies


along the


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


its Roche


lobe and dumps material toward the detached primary


intermittently while


expanding on a nuclear timescale.




1.0 I




0.5-




> 0.0 0




-0.5



-1.0o i ii I


Figure 2-1


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

(Wilson, 1979).


Our Scheme


large


number


of techniques


have


been


developed


to solve


equations of


hydrodynamics,


can be


classified


three


groups:


Lagrangian


(particle


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.


They require


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


new


effects (e.g.,


disk self-gravity


or magnetic


fields)


possible with minimal coding effort.


Therefore,


we have adopted a three-dimensional Lagrangian scheme similar to the


two-dimensional Lin and Pringle (1976) and Whitehurst (1988b) approaches. An early


version


program


was


described


previously


(Terrell


Wilson,


1993).


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


separation.


The x-axis of the system lies along the line of centers and is positive in the


direction


of the


lobe-filling


component.


The z-axis


is parallel


to the


orbital


angular


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


q
l+q


and write the acceleration components due to gravity of a particle located at (x,y,z) as


(2-2)


= 2jy+ x (x-x1)-


+y y-
rf


3 (X X2)
r2


(2-3)


ri
'r2


(2-4)


1-
ri


zr-
r 2


(2-5)


where


(2-6)









with x


1 and x2being the distances from the center of mass of the primary and secondary,


respectively.

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.


The physical


mechanism


of the viscosity


in accretion


disks


is not known,


although


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


following


quantities:


+ 2 )(1- )+(1
("T


rr
CT ,-~ n


r,,
A









12 13 +...+
Sa


(2-9)


n

The new x-component of the velocity for particle 1 is then computed by


kayg


+ Keep


and the y and z-components are computed in the same manner.


(2-10)


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

(1993).

Inefficiency in the determination of nearest neighbors can cause rapid escalation of


computing time.


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


this process,


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


dimensional array,


GRID(x,y),


corresponding to the grid.


Then


then grid location of


particle two is determined. If particle


happens to be in the same box as particle 1, then


an auxiliary


1-dimensional


array


POINT


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


n memory


locations.


simpler,


but much


less efficient


scheme would be to make GRID a 3-dimensional array, with dimensions 1 and


being


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





15


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.


Once al


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


gradient


forces


non-negligible.


code


uses


smoothed


particle


hydrodynamics


(SPH)


scheme developed


Lucy


(1977) and


Gingold and


Monaghan


(1977


1978,


1979a, and


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),





16


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


the geometry


of the system or the volume that the


fluid occupies.


SPH also has the


advantage


physical


laws


are applied


their most


fundamental


form


since


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


means of


an appropriate


smoothing function.


This procedure


the advantage that


spatial


derivatives


continuous


quantities


can be


replaced


analytically


calculated


derivatives of the smoothing function.


The smoothing


function,


must satisfy


(2-11)


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


P (r)


With


(2-12)


ml w(r -
j=1


and the acceleration acting on particle i due to pressure gradients is


dv, P,
d =-mI +
dt J pI


- *7)


(2-13)


Schiissler


Schmitt


(1981)


discuss


pitfalls


using


certain


smoothing


functions, and recommend the following smoothing function for modelling fluids where

self-gravity is negligible:


Iwcr


r\ 13-
-- ri~a r


Vw(~










forr


(2-14)


(2-15)


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


such as


W Serpentis stars,


self-gravity


must


considered (See


Wilson (1981) and


Wilson (1974) for a discussion of the disk


of 3


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


not use


it in


our present application to


Algols.


user can


enable


the self-gravity


calculations


setting


a control


integer


input


reduce


number of


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.


Some


Algols


have


primary


components


rotate


faster


synchronously


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


allowing stream


particles to interact with surface particles moving at


velocities specified by an input parameter F, which is defined as the angular rotation rate


,-C *L,


Cetr AhiiAxAc


---I---


1 An\


IU* **fl-. *~~I 1flnnrllrnrlll trrni'J runp l~a Lfafrt Il~l Lr tA)l III Il4I ** l- I


forr


where the normalization constant is given by


47th3


I


f\1 | n l f i


W(r)


TI,,,,


,,,,,1


Tf 1*: 1,,,









strength


stream-surface


interaction.


radial


component


stream


velocity at the star surface is reversed and multiplied by the bounce parameter.


With the


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.)

Numerical Models


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


Table


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






































Mass Ratio


Figure


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.


Figure


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.


Computed


Ha profiles


for both


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


orbital


periods


an elliptical


formed.


comparison,


a simulation


with


c = 0.1 was also run. Figure 2-6 shows the SX


Figure


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


simulations


of Whitehurst (1988a,1988b) and Lubow


(1992)


gives a


detailed


discussion of the instability mechanisms.

























* ( #*


-1.0 -0.5 0.0 0.5 1.0 1.5
X


Figure 2-3. A simulation of SX Cas, showing the stream trajectory 0.38 orbital periods
after the beginning of the transfer event.


- ,:: ( (,
'z. '


I i I -I


Figure 2-4. The early stages of disk formation in SX Cas at 0.8 orbital periods after the


"















































X



Figure 2-5. The elliptical disk that forms 1.6 orbital periods after the onset of mass

transfer in SX Cas.


1.5 -




1.0




0.5




0.0




-0.5 --




-1.0 -
-1.0


.I


-


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



0.5



0.0 ~-.



-0.5 -



- 1 .0 -- --------- --0- -.--- -
-1.0 -0.5 0.0


1.0 1.5


X


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.













CHAPTER 3
RADIATIVE TRANSFER


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






25


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


Kallman (1994).


3.5e-10


3.0e-10


2.5e-10


2.0e-10


1.5e-10


1,0e-10


50e-1 1


O.Oe+O


1200


1300


1400


1500


1800


Al Ill


Fe III


2000


Wavelength (Angstroms)


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
material.




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


I





26


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

achieved.

For line radiation, there are three probabilities to consider in computing the escape

probability:


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


Pesci.,(i)


= 1 for t,


(3-1)


S- -


esci,(xi)


for 10-5


(3-2)


tc,,1 (t,


S, 1i 12 +


O.5Jiog(r,
1+T,/T,


for


(3-3)


where -, is the optical depth where the line wings become optically thick.


<10-5






27


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


P
escl,c


I KT
+ K,I


(3-4)


where KT is


the Thomson scattering opacity.


The probability that the photon is scattered out of the line core by Thomson

scattering given by


Jutl~


+ Kc,


The escape probability for a line photon is then given by


el =
escI


Pescl


+ ( escl.c


t+ P^escp)(1 Pesc, ).


(3-6)


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 -


(3-8)


where x


- s/s, with


c being the energy of the photons and ei the ionizing threshold of the


level.


The normalizing constant c is given by

J,(x)dxp


(3-9)


which upon integration yields

c=y-1.

At the absorption edge at optical depth ta, the escape probability is


(P ,~(x)dx


(3-10)


where xe is defined such that


(3-11)


which yields


cP i~(x, )z.








and escape probabilities


escc (' a)


=(Y-1)t0]


for x


>1/(y-1)


(3-13)


for t


-1)


(3-14)


For recombining continuum the profile function is


cP,=x) =


r~


(3-15)


(x)/x dx


given the level photoionization cross section ai c For or(x) equal to a constant times

X-


(3-16)


which yields


)-p/(p+I)


2>1/p


(3-17)


Pesc(oa)


=1 for


ra < 1/P


(3-18)


with 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.


Therefore,


certain


heating


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


Pescc,p (


< lily


Pso (Z,


ico (")lx


(P





29


(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.,


where the


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.


The radiation


field of the primary star is taken to be that of a blackbody of temperature T and is given

by the Planck function


F(s)= y
h c2


which leads to the photoionization heating rate


tH)~t


I:


(3-19)


ci,


(3-20)


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
Ixx


nk ni
n -
t00


8x~.-'
X J*k F +t32 Pec, e
\he


dx f,


(3-21)


and the line cooling rate is


element ion

k m j>i i


nI


(3-22)


element ion level
= T Y


rad=









where


is the line energy and the total escape probability


Pesc, j is given by


escl,j esc,!


+ pc, p (1


(3-23)













CHAPTER 4
RESULTS


Spectroscopy


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


Keplerian


showing


that the


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


subtracted


from


observed


spectrum.


subtraction


was


done


interpolating


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,


1984;


Hamme


Wilson,


1990).


simulations


system


assume


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 -



1.0



0.5



0.0



-0.5



-tO --
-1.0
-1.0


X


Figure 4-1. The stream in RW Tauri impacts the primary star at phase 0.22.


*t -


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


Honeycutt.


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


1.00


0.95
6520


65---40---
6540


4-- -;.... ----..


6580


Wavelength (Angstroms)


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


Vesper)


6600


.222.












- -,-


6560


Wavelength (Angstroms)


Figure 4-4 Same as Figure 4-3,


6520


except the observation was made at phase 0.463.


6540


6560


6600


Wavelength (Angstroms)


Figure 4-5. Same as Figure 4-3 with the observation made at phase 0.685.


6520


6580


-j- _- 1.- T, ^


1






35
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.





0.9
0.8


0.6 I-


z 04 t
0.3 I
0.2 |
0. I--t I II


6520 6540 6560 6580 6600
Wavelength (Angstroms)

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.


6520


6560


6580


6600


Wavelength (Angstroms)


Figure 4-7


Same as Figure 4-6 except that twice and many gas particles were used in the


simulation.


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













1.20 -



1.15 -



1.10



1.05 -


1.00 -

na I
I


6540


Wavelength (Angstroms)


Figure 4-8


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 . -- .... .. .. -


1.15
1.15 -i


1.10 -



1.05


0.95 --
6540


6560


6580


6600


Wavelength (Angstroms)


Figure 4-9.


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).


Figure 4-10


shows the distribution of circumstellar matter 0.35 orbits after the onset of mass transfer.


Figure 4-10.


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


phase 0.631


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.













-r -
-1


6540


US
-4 I-."-- a-!--"-~ .- -- -----


6560


6580


Wavelength (Angstroms)


Figure 4-11. The computed He profile of TX UMa at phase 0.631 for the disk shown in
Figure 4-9.


0.3 --

0.2 -i i


0.0 4- -----


6520


6540


6560


6580


,i 1 4


6600


Wavelength (Angstroms)


Figure 4-12. The computed H, profile at phase 0.306 for the disk in TX UMa shown in


6600


1






40
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


Wilson,


1986).


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


A striking


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.


6520


6540


6560


6580


6600


Wavelength (Angstroms)

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).













1.20 --

1.15

1.10 -

1.05




0.95 -

0.90


0.80 -i -----
6520


4....f -- -----i t


6540


6560


Wavelength (Angstroms)


Figure 4-14.


Observed HI profile of U Sge at phase 0.219.


1.15 F
1-


0.95


6520


6560


6580


Wavelength (Angstroms)


Figure 4-15


Observed H, profile of U Sge at phase 0.499.


-I---


6580


i


?. .1.~


"^~1~~


"~;"~"~































1-,~~~.-~_~._t------2-


Wavelength (Angstroms)


Figure 4-


Observed Ha profile of U Sge at phase 0.672.


L,_~I i


6520


Wavelength (Angstroms)


Figure 4-17. Observed Ha profile ofU Sge at phase 0.888.


6520


6540


6580











1.0
0.9

0.8
0.7
0.6

0.5
0.4 1
0.3

0.2
0.1
o o -0 .......4 ...... 4- 4


6520


6540


656


I |
ii

I-

1~




-4





0 6580 66i


Wavelength (Angstroms)


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


Monocerotis


a long-period


(11.1


days)


system


close


to the


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


6600


Wavelength (Angstroms)


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) .


6520


6540


6580


6600


Wavelength (Angstroms)


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





























6580


Wavelength (Angstroms)


Figure 4-21.


The H, profile of AU Mon at phase 0.33,


taken 1 night later than the profile


in Figure 4-19


'I I
hj
-I

ii1


0.0


6520


6540


6560


I






6580


Wavelength (Angstroms)


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


I2


I
34


6560


"~SS"~v
















I I 1 I I 1 i i i

a)


s... *~
C>
* s-at,
., ,
,.


II11


II I1


b) 2








I I ~ i II


0.0 0.5 1.0


ITT TT












I Ir l-
I
-1.0 -0.5


II 4


I II II


II1


'liii, liii I1)


',III


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.


Polarization


Interest


polarized


radiation


from


binary


stars


began


with


treatment


Chandrasekhar (1946),


who showed that eclipses would break the disk symmetry and


allow


limb


polarization


to be


observed.


Other workers


extended


Chandrasekhar


.4 ~ ~ ~ I J ~ t1 165 r4 ** l f n.


'-I--I--I--


11111








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.


Although


preliminary


work


stage,


on this


results


hydrodynamic


demonstrate


flow-polarization


polarization


(HFP)


program


observations


show


is at a


great


promise as a probe of circumstellar flows.


simplistic.


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


program


computed


electron


polarization


densities


to the


curves


polarization


a short


burst


program.


event


Figure


in SX


4-24


shows


assuming


Saha


ionization.


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,












0.10


0.05


0.00 --


0' -0.05 -


-0.105 -


-0.15


-0.20 ---------|---- \ -- -- --
0 1 2 3 4
Phase


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
curve.


1 2 3


Phase


Figure 4-25. Polarization curves for a medium-length (1.4 orbital periods) transfer event


in SX Cas.












0.4 -.---- ~


0,2 -


0.0


-0.2 4


-0.4


-0.6


-08 -


i<~A4~vAX


::9


1


Phase


Figure 4-26. Polarization curves for mass transfer event in SX Cas lasting 2.4 orbital

periods.


Vt


-0.2


-0.4 4


-0.6


-0.8 -
h---^


~I 4


--.. ~.....+


Phase


Figure 4-27. Polarization curves for a medium-length (1.4 orbital periods) burst event

assuming partial ionization of all circumstellar gas.


I-


."1Mni~ I































Phase


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


Phase

Figure 4-29. Same as Figure 4-4 except that ionization occurs within a radius of 0.1 of the


primary.


~----t --------~~---~







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


systems


are too dim


present


telescope/polarimeter


combinations.


However,


interest


polarimtery


greatly


increased


in recent


years,


combination of sensitive polarimeters and large aperture telescopes like the


Keck


10-meter


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


from them.


Conclusions


work


was


to compute


spectral


radiation


arising


from


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


serve


as a foundation


future


modelling


not only


Algols,


other types


interacting binaries as well.


The first step in using a new model is to become familiar


with


predicted


observables


a wide


range


input


parameters


so that


good


estimates of the input parameters can be made before beginning an impersonal adjustment

procedure.


Some parts of the model,


in particular, the radiative transfer program, push the






52

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


observed stars,


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


model,


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


telescope.


combining


our work with


Wilson-Devinney


binary


star model and


Mukherjee's (1993) absorption line profile model, we will be able to compute observables


that can


be directly


compared to observations.


This combination enables us to better


determine the parameters of the binary by simultaneously satisfying different kinds of


observations.


With such a combined model we could simultaneously fit light and radial


velocity


curves, line profiles, polarization curves, and X-ray pulse delay


curves.


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


X-ray


binary


77581


which


to non-radial


pulsations


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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BIOGRAPHICAL SKETCH


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


life. In


1978 the family moved to Greenville, South Carolina, and he graduated from


Wade Hampton High School in June,


1983,


where he was a letterman in football and


track as well as the 1983 U.S. Army Reserve National Scholar-Athlete. He was admitted


to the


physics


program


at Clemson


University


participated


Cooperative


Education


program


as a Research


Associate


at the


Department


Energy's


Morgantown Energy Technology Center in Morgantown,


West Virginia, in 1985. In May,


1987


he received


a Bachelor


of Science degree


physics.


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.







certify


have


study


in my opinion


it conforms


acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.


13 t2rw-


Robert Wilson, Chairman
Professor of Astronomy


certify


have


study


that irf


acceptable standards of scholarly presentation and is
as a dissertation for the degree of Doctor of Philosop


equal


opinion
ite, in}4


it conforms


:ope and quality,


Heinrich Eichhom


Professor of Astronomy


certify


have


study


in my


opinion


it conforms


acceptable standards of scholarly presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.


A'
K o-tiaC-


Janies Hunter, Jr.
Professor of Astronomy


cr


certify that


have


study


in my


opinion


it conforms


acceptable standards of scholarly presentation and is fully adequate, in
as a dissertation for the degree of Doctor of Philosophy. /


and quality,


Charles Ho6per
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


partial


fulfillment


of the requirements


for the degree


of Doctor of


Philosophy.

August, 1994


Dean, Graduate School


I-I. k~ ~US
krz


*





















N ~C,


UNIVERSITY OF FLORIDA
uI I I1 1 I iu 1114i
3 1262 08557 1114


1 8 i3