Two dimensional numerical model of gas flux in close binaries

0273—1177/88 $0.00 + .50 Copyright © COSPAR

Adv. Space Res. Vol. 8. No. 2—3. pp. (2)183-42)188. 1988 Printed in Great Britain. AU rights reserved.

TWO DIMENSIONAL NUMERICAL MODEL OF GAS FLUX IN CLOSE BINARIES M. Dimitrova and L. Filipov Space Research Institute, Sofia, Bulgaria

ABSTRACT Two dimensional numerical model of gas flux trough inner Lagrangian point is built up for close binary system with compact object. Parametric variation in dependence on substance inflow angle is examined for the region around the compact object.Computatiorss are based on hydrodinamic approach considering pressure gracizent in the disc. INTRODUCTION

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After the identification of large number discret X—ray courses as narrow binaries in UHURUCataIog the study of gas fluxes into such systems become very actual.Hihg luminosity (L j~31— 1O~ erg/c) and smallcharacteristic variation (t’O.Ol •~. 1 s) provide evidense on accretion process onto compact object (neutron star,white dwart or black hole). Prendergast and Barbidge [1] show that the gas owtf lowing throwgh the inner Lagrangian point to a compact object has angular momentum larger that to provide radial falling to the obiect.A disc is therefore formed into the system plane of almost Keplerian rotation. Due to the complex type of equations discribing the disc physical characteristic they may not be analytically expressed.Theref ore the efforts was concentrated on building numerical models of accretion discs arownd compact objects.Majority of them adopt the standart model,assuming geometrically thin and axially symetric disc. The pressure gradient of the plane is neglected. Lightman t2,3) and later Okuda (4] show that considering pressure gradient in the motion equations the disc characteristics sigrsif i— cantly change.The standart model is an initial approximation in their works. This paper examines the disc formation under assumption of axial insymmetry and considering pressure gradient. PROBLEM The model was built up under the foliwirig assumptions: entering substance is totaly ionised hidrogen plasmaa — medium is optically thick — flat flux is formed — with small thickness compared to the — distance to the compact object is small. Under these aassumptions the gas phisical characteristics are defined in equations — continuum equation —

1.+

S.~iv (2)183

(2)184

M. Dimitrova and L. Filipov —motion equation

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P The above formulas for O~and are given in £5] .This paper provides as well the value of constans C CC., Oi, ~ f or different dissipation mechanismus and scatering processes.

NUMERICAL METHOD

The discribed system of nonlinear differential equations was solved with the large pa rticle method.The method provides possibility to model any type of shock waves with out necessarily define their nature and the corresponding rupture surf aces.Point to point variation of phisical parameters is smooth and in agreement with hydrodinamic considerations.Artifical viscosity or initial substaans density introduction for the examined region is non required. Stability conditionrequires time step smaller thenthe time needed for substance transition throughout one domain (6]. This method is originaly used in an astrophisical model..The adoption implies centered cheme of finite differences of second accuracy order. The numerical model provides free selectionof large number of parameters such as: — angle,velocity,temp and location of substans inflow — mass,radius of the compaact object and the outer region boundaries — selection of dissipaation mechanism Where ,

V~

~VL

(— ax~i----~ ?x1

~ii.~O,5 C

state equation

P=~—f.T —

energybudiet equation

F

a

(r)

where

,aSz~t

1~

Fso,5~?~ 4

~

•

velocity is transport is the the energy irradiation flux

T

—

This scheme reflects the assumption that the total energy dissipating withviscous forses is transported by irradiation. Opacity ~ is formed by: — ff transitions — Tompson scatt:ering — Compton effect — cyclotron irradiation It is given with expression:

• ~ where

S C r,f,t)

J

N

(r,f,z,t) da is the surface density and o H is the depth Viscosity is defined in equation

2

=

~..SQ.rb.

WC

Gas Flux in Close Binaries

—

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number of iterations and domains to which the region is subdivided RESULTS AND DISCUSSIONS

Computations refer to gas flux into neutron star surounding space. lOsm. The region ou to RoutlOsm is considered ~or substance inflow of temp M=1OM~year and angular moment h~h=G.M.R.atdifferent incidence angles around the object.

Themass is M=Meand the radius R

Figures 1,2 and 3 illustrate the results from the numerical modelling of functionsl S(r,f) ~ (r,f,t = const) S(r,t)a~JS Cr,f,t) df and

V(r.’c)=~.i(r,f,tmc.on$+) respectively at incidence angle 5~

...

-

~

~

—

~—.

—

-

E~?~

j1(~lI: I

--

Figi

-

,I~

S

Fig.2

c.,4

‘

—:.;‘~~.‘-—-‘~~

..

-

:-:

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M. Dimitrova and L. Filipov

—

—

~ ~

•

t’~~

N

.

:.

.•

~çj.

\ ~~(J •

..-~

,i

/‘

Fig.3 Figures 4 and 5 represent functions S(r,f) and S(r,t) at angle 30? Figure 6 shows the magnitude as a function of r,f.Its value provides evidence of viscouse instability occurence.

~

-

.4

Fig.4

..

.

S

Fig. S

/~/ ~vV\ /~~AA

Gas Flux in Close Binaries

(2)187

t’~

Fig.6

As itis seen on the figures a ring system of nonhomogenious density distribution by azimuthal angle is formedaround the compact object rather then compact disc.Computations show that this distribution is quasistaationary as seen in Fig.2 and 5. Such distribution maynot be obtained based on the standart model due to axial symmetry assumption. It conf rims results obtained by some authors in C3,4,5]but in contrast to their computations we obtain here ring separation of substanse at the very outer region.The asi mutal inhomogenity is due tointeraction of entering and ring substance and the quasistationary behavious results from pressure gradient.That is why existing models do not consider this inhomoge nity. Computations mode with this model under assumption of axial symmetry and ne glection of pressure gradient into the disc planecoincide with standart results.This confrims owr reliability. Viscous insta .bility occures when:

(s.~.) <0 where the disc should decompose into rings.The values obtained and schematicaly shown in Fig. 6 correspond to substance distribution in Fig.4 CONCLUSION The model discussed here may be applied f or large class of close binaries of rather differing parameters. It may discribe accretion both due to inner Lagrangian point and stellar wind. Gas density irregularities as time dependent decay and their orbital motion will result in respective irregularities in disc irraadiaation.This might be the reason for small amplitude non periodic fluctuations in luminosity. It is interest tomake futhure computations on variation in temp of accretion rate.Such variations and of periodic nature in parti cular sshould resul in specifics novel in nature.

REFERENCES 1. K.H.Prendergasr 6.R.Barbedge “On the nature of Galactic X—ray Sourses” Ap.J. 151,1968 2. A.P.Lightmam “Time—dependment Accretion Discs Around Compact Obiects”.Theory and Basic Equations” Ap.J. 194,419,1974 3. A.p•Lightmam “Titne—dependment Accretion Discs Around Compact Objects.NumericalModels and Instability of Inneer Region” Ap.J. 194,419,1974

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M. Dimitrova and L. Filipov

4. T.Okuda

“On Accretion fliscs Around a Black Holes and Neutron Stars” Publ.Astr.Soc.Japan ,32, 1980 5. L.Filipov, Nonstationary disc accretion and cataclizmic variables, Adv.Spase Res. l984,v.3N 6. 0.4 Beloretckovskii ~Numerical modeling in the mechanics of fluids Moscow,Nauka, 1984