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Magneto-Hydrodynamics of Mass Accretion in Close Binary Systems andProtostars

Stehle, R.

Publication date1997

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Citation for published version (APA):Stehle, R. (1997). Magneto-Hydrodynamics of Mass Accretion in Close Binary Systems andProtostars.

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Download date:22 May 2021

Hydrodynami cc simulations of CV accretion disks in outburst t

R.. Stehle1'2

11 Max-Planck-IntUtut fur Astrophysik, Karl-Schwarzschild-Str. 1, D-85740 Garching, Germany 22 Astronomy Group, University of Leicester, Leicester, LEl 1RH, UK

MNRA SS submitted

ABSTRACT T Wee study the outburst phase of CV accretion disks in the (r, 4) plane with full hydro-dynamics.. Vertically the disk is treated using the one-zone model of Stehle & Spruit (1997)) which allows us to follow the fundamental mode of vertical disk oscillations cor-rectly,, a shear-viscosity (Shakura & Sunyaev 1973) and energy loss at the surface of the accretionn disk by radiation are included. We run accretion disk models with different massess M\ and Mi for the primary and the secondary star and different values of a. In alll calculations we observe strong spiral shock arms, which dominate the disk evolution onn a hydrodynamical time scale. In cases where the surface mass density E(r) decreases withh radius r the disk pattern is stable. In cases where E(r) increases with r we observe aa quasi periodic cycle of the disk size. During the phase where the disk is large, the spirall arm close to the L\ point connects to the secondary and about 1% of the total diskk mass is lost to the secondary per cycle. However, in none of our calculations are wee able to find an eccentric and precessing accretion disk even though the disk edge is outsidee the 3:1 tidal resonance radius R$\. Our calculations therefore can not confirm thee tidal instability model, presently the only model for superhumps in SU UMa CVs duringg superoutbursts.

Keyy words: accretion, accretion disks - shock waves - novae, cataclysmic variables —— binaries: close

11 INTRODUCTION

Cataclysmicc Variables (CVs) are close binary systems with orbitall periods P«>rb of a few hours. They consist of a white dwarff i WD) primary with mass 0.3 M© £ M\ ^ 1.4 A/ö

andd a low mass secondary 0.07 MQ & M% ^ 1 MQ, The secondaryy loses mass through the inner Lagrangian point L\L\ and feeds an accretion disk around the primary (see Frank.. King k Raine 1992 for a general introduction). In systemss where the mass transfer rate is below a critical valuee Mct\\ ~ fewlO~9A/0/yr (Meyer k Meyer-Hofmeister 1982)) the accretion disk alternates in a semi periodic way be-tweenn phases of high and low accretion rates Mace (Warner 1995).. During the quiescence phase (low M4CC) the disk is coldd and faint and mass accretion is inefficient. Mass accu-mulatess in the disk and is accreted to the WD during the outburstt phase, which is 2-5 m* s brighter than the quies-centt phase. The time evolution of CV outbursts is success-fullyy explained by the thermal instability model where the smalll scale viscosity is correlated with the local disk pres-suree (Shakura k Sunyaev 1973, Meyer k Meyer-Hofmeister 1981).. In SU UMa type of CVs so-called superoutbursts are observedd which last longer and are l-2m* s brighter than the

normall outbursts. During a superoutburst the light curve is modulatedd in brightness by ~~ 0.15 - 0.3m** with a super-humpp period PSB about 3-6% longer than the orbital period Porbb (Stolz k Schömbs 1984).

Ann explanation of the superhump phenomenon was sug-gestedd by the results from particle simulations (Whitehurst 1988,, Hirose k Osaki 1990) and from Smooth Particle Hy-drodynamicc calculations, in short SPH calculations (White-hurstt 1988, Hirose k Osaki 1990, Lubow 1991, Murray 1996).. These show eccentric disks, precessing with a period slightlyy longer than the orbit. This is understood theoreti-callyy (Hirose k Osaki 1990, Whitehurst k King 1991, Lubow 1991)) as the result of a 3:1 tidal resonance instability. These precessingg disks have been found only in SPH simulations, however,, whereas simulations with grid-based hydro-codes havee had difficulties reproducing the effect. In the simula-tionss by Heemskerk (1994), for example, the tidal torques causedd the disk radius to shrink rapidly inside the 3:1 reso-nancee radius K31, preventing the instability from developing. Onlyy when Heemskerk replaced the tidal force by its m = A Fourierr component did the disk remain large enough for the instabilityy to develop. Evidently, the tidal stresses of the secondaryy remove angular momentum very efficiently from

388 II: Accretion Disks in Close Binary Systems, Chapter 4

aa disk which is large enough to reach the 3:1 radius. A very largee viscosity is needed for the disk to be spread far enough, inn spite of the strong tidal angular momentum loss at this radius.. This may explain why the phenomenon is seen in SPHH simulations, which typically are extremely viscous by thee nature of the method used. In real systems, such high a viscosityy apparently exists only during superoutbursts.

Thee aim of this paper is to study the development of the eccentricc instability by numerical simulations with a grid-basedd hydro-code. In later studies the models introduced heree wil l be compared to observations of accretion disks in closee binary systems in more detail.

Thee method used is described in Stehle & Spruit (1997). AA short note on the calculations is also found in Stehle (1997).. The calculations are two-dimensional like most other accretionn disk simulations, but differ from these in the way in whichh the disk thickness is treated. Instead of fixing the disk thickness,, we use a one-zone approximation for the equation off motion in the direction perpendicular to the disk. This al-lowss us to follow the time dependence of the disk thickness byy two additional equations. The method correctly repro-ducess the fundamental-mode oscillations of the thickness of geometricallyy thin disks.

Importantt processes in the disk are viscous spreading andd (tidally excited) shock waves. As long as there is no goodd theory for the viscosity of disks, it is quite possible thatt the processes determining the effective bulk viscosity (whichh affects shock waves) differs from those determining thee shear viscosity. For this reason, we implemented only ann a - t ype of shear viscosity as given by Shakura & Sun-yaevv (1973). The energy loss rate from the disk surface by radiationn is related to the mid-plane temperature Tc by a standardd radiative diffusion model. The bolometric and vi-suall disk luminosities are calculated at each time step for comparisonn with observed CV light curves.

Wee restrict our study to CV disks in outburst, since in quiescencee the Mach numbers are too high to resolve the hydrodynamicss appropriately (Rózyczka & Spruit 1993).

AA full hydrodynamic calculation of a complete outburst cyclee is still beyond the possibilities of present computer technology. .

Ourr paper is organised as follows. In Section (2) we for-mulatee our model equations. They are the same as given in Stehlee fc Spruit (1997) except for the addition of the viscous andd radiative loss terms, which were not included there. In Sectionn (3) we show how the dynamics of the disk depends onn the binary mass ratio and the viscosity parameters. Sec-tionn (4) concludes the paper and discusses the results.

22 T H E E Q U A T I O N S

Wee model the dynamical evolution of viscous accretion disks byy solving the time dependent hydrodynamic equations in twoo dimensions. We use a cylindrical coordinate system (r,, <j>) which is centered on the primary and corotates with thee binary angular velocity I i 0 = fioez. We follow the evo-lutionn of the local disk thickness in the one-zone model off Stehle k. Spruit (1997). Viscous forces, as described by Shakuraa & Sunyaev (1973), are included in the momentum andd energy equations. We take care of energy losses by ra-diationn at the disk surface.

Inn terms of the surface density E, the radial and az-imuthall momenta pr, p# and the vertical momentum and potentiall energy variables pt and Ep<tXtS the equations are

0) )

(2) )

(3) )

a,EE + V h ( L i ; h ) = 0

dlPrdlPr + V h (prVh) = - a r n - Edr<I>|z=0

++ E— + 2fto£v* - -d*Tx4, rr r

&P** + vh (P+vh) = -d0n - zd*$\z=0

—— 2ftorEvr dT (r'Trt)

ft£ft£PP.t,..t,. + V_ (Ep.,,,t.b) = v'ïïg'.Hpï + £pol,, HtZ*£* (5) Sh Sh

y/2^pt y/2^pt ^e^e + V h (evh) = - n V h u h - II -

£ƒƒ ƒ ++ Q+-Q- (6) )

wheree a subscript h denotes components parallel to the disk plane,, 0, e are the vertically integrated gas pressure and internall energy, fio the orbital frequency of the binary, $ thee Roche potential, rr0 the implemented small scale vis-cosity,, H* the pressure scale height c / f i n t where QK is the Kepleriann frequency and g'. = —d~-$ the vertical derivative off the gravitational acceleration. The (time dependent) disk thicknesss H is related to Epot,- by Epoi,3 = ^g'zHH2. Q~ standss for the loss of energy by radiation (see section 2.2) andd Q+ for viscous heating:

*+=-[-H?) +¥)] ] wheree the shear viscosity is given by

TV** = -n i(*W W do do

(7) )

(8) )

Thee equation of state is that of an ideal gas of constant ratioo of specific heats 7. In our variables, this is

nn = ( 7 - l ) e (9) )

wheree the value f = 1.4 is used in all our calculations. The diskk temperature is given by

TT = AH (10) )

wheree % is the molar gas constant and /i the mean molecular weight. .

Too close the equations we have to find a formulation for thee small scale viscosity /J which we will introduce in the nextt section.

2.11 Viscosity

Thee formulation for the shear-viscosity (eq. 8) assumes physicss on smaller scales which is not resolved with our grid resolutionn or which is 3-dimensional in nature. The viscous transportt of angular momentum and heat generation is thus parametrised.. Such a parameterization should in principle bee based on local, 3-dimensional hydrodynamic disk calcu-lationss as done, e.g., by Brandenburg t;t al. (1095) or Stone

R.R. Stekte: Hydrodynamic simulations of CV accretion disks in outburst 39

ett al. (1996). These calculations, however, are not yet in the statee where a meaningful parameterization of their results is possible.. We therefore rely our calculations on the parame-terizationn suggested by thermal instability models which ex-plainn dwarf nova outbursts successfully (see Cannizzo 1993b forr a review). They follow the viscous evolution time scale byy setting

HH = uZ = -arc ,#£, (" ) )

wheree a & 1 is a constant. We adopt eq. (11) for all our investigationss and take the formulation for the viscosity also onn the much smaller hydrodynamical time scale literally. We assumee that the local value for the viscosity adjusts itself instantlyy to the local thermodynamics of the disk gas.

Thee viscosity is implemented in the code explicitly, but thee time step for the viscous integration is chosen indepen-dentlyy to the hydrodynamical time step. Usually for large valuess of a we perform about 5 time steps for the viscous integration,, while updating the hydrodynamical part once.

2.22 Energy loss by radiat ion : Q~

Finallyy we determine the loss of energy by radiation at the surfacee of the accretion disk. For CV disks in outburst the energyy is transported mainly by radiation. Convective trans-portt of energy is unimportant. The equation of radiative transferr in the diffusion approximation reads (Warner 1995, p.44) )

F(z)F(z) « -16*7** 8T

3KR/JJ dz

Thee optical depth r is defined by

drr = KRpdz,

(12) )

(13) )

wheree KR. is the Rosseland mean opacity. Eq.. (12) is solved for T(z) if the local heat generation

qq++ (z)(z) in the accretion disk

^ - ' M M (14) )

iss known. Att least two different processes will contribute to the lo-

call disk heating: small scale viscous heating ^ „ ( z ) ~ ac,Hp andd tidal heating q*ld{z) «- p. The vertical temperature distributionn will depend on the respective fraction of the twoo heating mechanisms. Additionally, on a hydrodynam-icall time scale, we also expect the disk to be heated and cooledd by gravo-thermal processes. To follow the different heatingg processes properly the disk has to be resolved in the verticall direction. This requires full 3-dimensional calcula-tions,, however, which are yet to come. Eq. (12) and (14) aree thus only solvable if we specify q+(z), which we do, to simplifyy matters, in the following way:

qq++ = FoS(z), (15) )

wheree FQ = const. = <77?ff and S(z) Dirac's delta function. Thuss from (15) we derive

T(T)T(T)44 = fair + 2/3),

whichh yields, using an opacity law KR = KopaTb and a Gaus-siann density distribution in the vertical direction in eq. (12),

(TT + 2 / 3 ) < l - 6 / 4 )) =

Q*T*Q*T* (Zyf* 1-6/4 . l+ 22 UJ (Wa)t/>J8o73,E H

(17) )

( I t a ) ' ' ass long as 6 < 4.

Combinedd with eq. (16) this matches Te« to Tc. For the hott branch of accretion disks (T £ 104'2K) typical values forr Kramer's opacity are «o = 2.81024, a — 1, 6 — —3.5 (Cannizzo,, Shafter & Wheeler 1988) and thus we find

Tefff = 7 . 3 1 0 -7 T c1 5 / s£ -1 / 2/ / l / \ (18) )

Inn the one-zone model used we calculate the disk thickness HH = H(t,r) jt / /eq explicitly and H is also a free param-eter.. For disks in thermal equilibrium (H — //«), eq. (18) reducess to well known formulations of the hot branch of accretionn disks (Cannizzo 1993a, Ichikawa & Osaki 1992, Ludwigg 1996).

Thee energy loss by radiation for the hot, optically thick partt (T > 1) reads

Q~Q~>t>t = 2<r7?ff (19)

wheree the factor of 2 takes account of the two surfaces of thee accretion disk.

Forr optically thin parts (r < 1) we set T*tt = Tc and apply y

Q 7 < i = 2 < r 7 ? ( l - e x p ( - T ) ). . (20) )

(16) )

Thee energy loss by radiation for a given central tem-peraturee Tc is most efficient near T = 1. For r > l we find Tefff <C Tc'. the surface temperature is much smaller than the centrall temperature. For r <C 1 the gas is much too dilute too contribute significantly to the radiation.

Wee compute the total accretion disk luminosity by as-sumingg that each disk element radiates as a black body. Too derive the luminosity in the visual part of the spectrum wee fold the Planck spectrum B\(Tes) of each grid cell with ann optical filter function (Allen 1973). The transparency of thee filter peaks around 5500 A (see also Schandl, Meyer-Hofmeisterr & Meyer 1997).

33 MODE L C A L C U L A T I O N S

3.11 General i n t roduc t i on to the ca lcu la t ions

Wee study CV accretion disks in outburst. The primary mass MiMi ,, the secondary mass M-2 and the efficiency of the a-shear viscosityy differs from model to model. Guided by thermal instabilityy models we study cases where S(r) increases or decreasess with disk radius r. An outline of the initial pa-rameterss is given in Tab. (1).

Thee computations are done on an equidistant Eulerian gridd with 256x128 grid cells in the (r,<f>) direction respec-tively.. The grid is centred on the primary and the frame of referencee corotates with the binary orbital frequency. The computationall domain in model 1 & 2 extends up to the Lagrangiann point L\t i.e. up to r\,x and in model 3 fz 4 up too 1.25 TL,. In all 2-dimensional plots the Lagrangian L\ pointt and the secondary star are to the right.

400 II: Accretion Disks in Close Binary Systems, Chapter 4

T a b lee 1. Some initial values of our model calculations. The sym-bolss have the following meaning: nr: model number; M\, M2 the masss of the primary and secondary; Po r b the binary orbital pe-riodd and a the viscosity parameter. The initial model is either axisymmetr icc or started from a previous calculation with a dif-ferentt value of a.

nr. .

l a a l b b

2a a 2b b

3 3

4 4

initiall model

axisymmetric,, E, ~ r — 3 ' 4

l aa after 5.4 binary orbi ts

axisymmetric,, E; ~ r— 3 / 4

2aa after 5.6 binary orbi ts

axisymmetric,, E; ~ r

axisymmetric,, Ej ~ exp(r)

Mi Mi

Me Me 1.0 0 1.0 0

1.0 0 1.0 0

1.0 0

0.685 5

M 2 2

M 0 0

0.15 5 0.15 5

0.3 3 0.3 3

0.3 3

0.07 7

^orb b h h

1.88 8 1.88 8

2.3 3 2.3 3

2.3 3

1.51 1

a a

0.01 1 0.3 3

0.01 1 0.3 3

0.3 3

1.0 0

T hee init ial model is s ta t i onary and ax isymmetr ic. It re-sembless a disk a round a single W D. Thus only £ ( r) can be chosenn freely. Fixing E ( r ), t he internal energy is given by Q ++ = Q~ and the az imu thal velocity by the radial force ba lance. .

T hee init ial ax i symmet r ic models are subsequent ly ex-posedd to t he t idal force of t he secondary star. After ~ 2-3 b inaryy orb i ts the surface dens i ty of the disk evolves into a p a t t e rnn which is i ndependent of the ra ther special initial cond i t ion.. In models 1 & 2 we resume a calculat ion from a prev iouss model with a different value of a. We thus study thee effect of a on the disk evolut ion.

Al ll our calculat ions a re run without any mass t rans-ferr f rom t he secondary s ta r. Th is neglect has two different effectss on the disk evolut ion.

F i rs t,, as the specific angu lar momentum of the accre-t ionn s t r e am is small compared to the disk gas, the accret ion s t r e amm ac ts very much lik e an angular momentum sink. We expectt t he ra te at which the disk increases in size by viscous spread ingg to be reduced when the mass accret ion s t ream fromm the secondary s tar is included in the calculat ions. We thereforee overest imate the disk size in our calculat ions.

A ss a second effect we also miss the hydrodynamical in-te rac t ionn of the accret ion s t r e am with the disk gas. Accret ion diskss in ou tbu rs t, however, a re huge and the disk mass near too the ou ter edge is high compared to the mass supplied byy the accret ion s t ream dur ing the outburst phase (Ludwig &:: Meyer 1997). T he neglect of the accretion s t ream might thereforee be of only minor impor tance.

Subsequent lyy we discuss or model calculations.

3 .22 M o d el 1: M , = 1.0 A/c,, M2 = 0 . 1 5 M ,; , a nd Sjj ~ r - 3 / 4

Inn model 1 we follow the evolut ion of a disk in a binary with MiMi = 1.0M@, M2 = 0.15 A/,:, and an orbital period of Po r b = 1.88h.. We first choose a r a t h er small value of a — 0.01. T hee init ial d is t r ibut ion of t he surface density and the disk t e m p e r a t u ree decreases w i t h radius, as given by s ta t ionary S h a k u raa fc Sunyaev disk models:

// r \ - 3 / 4

£(t/T-- 5.407)

£(t/T«« S.IB9)

Figuree 1. The disk pattern of model 1 where a = 0.01 (above) respectivelyy a = 0.3 (below). The secondary is to the right of the plot.. The spiral shock arms are tightly wound. The size of the diskk barely extends beyond the 2:1 resonance radius where the fundamentall mode of vertical disk oscillation is commensurate withh its synodic orbital period (inner dashed curve). The disk exceedss the 3:1 tidal resonance radius (outer dashed curve) only inn the case where a — 0.3 (lower figure) , and then mainly in the spirall shock arms.

and d

(( r \_3/4

Wee choose E0 = 450 g / cm2 and thus find T0 ~ 1.35 10 'K. Thee initial size of the disk is equal to the tidal resonance

radiuss Ri3 (outer dashed line in Fig. 1). The tidal forces of thee secondary star are appl ied and a stat ionary spiral shock pat te rnn with tightly wound spiral arms evolves. After about twoo binary orbits the disk has shrunk below the Rï3 radius.

Ass seen in Fig. (1 above) the outer edge of the disk iss after 5.4 binary orbi ts close to the 2:1 resonance radius #211 < /Ï31 (inner dashed r ing). At that radius the synodic frequencyy |QK — fio| is commensura te with the fundamental verticall disk oscillation frequency w ~ y/\ + y OK (Stehle fc Spruitt 1997).

Thee disk is stable in size: angular momentum t rans-portedd outward by viscous forces is efficiently removed by tidall stresses at the outer part of the disk.

Too enlarge the disk beyond the R2\ radius and to force thee disk gas into the t idal potent ial field of the secondary

R.R. Stehle: Hydrodynamic simulations of CV accretion disks in outburst 41

wee increase the value of a from a = 0.01 to a = 0.3 after 5.44 binary orbits. This artificial change in a is made for thee whole disk simultaneously. The change in the total disk luminosityy to the new equilibrium value takes only ~ 1/4 of aa binary orbit.

Wit hh the new value of a = 0.3 we additionally run ~15 binaryy orbits. After a short transition phase the disk is again stationaryy in a frame of reference corotating with the binary. Becausee of the higher value of a, viscous spreading operates withh higher efficiency and thus the disk is ~ 10% larger in radius.. The spiral shock arms are further extended into the potentiall field of the secondary (Fig. 1 below) and are well beyondd the R31 resonance radius.

Forr each time step we follow the surface density of the diskk by recording its second moments (see Appendix A). Thee deformation of the disk from azimuthal symmetry is describedd by the ratio of the minor semi axis 6 to its major semii axis a. The angle between the major axis a of the disk andd the line drawn from the WD to the secondary star is calledd o».

Whenn we increase a we observe that b/a changes from b/ab/a ~ 0.80 in model l a to b/a ~ 0.71 in model lb, whereas <£aa decreases from </>,. ~ 85° to <£a ~ 75°. In model lb the masss centre of the disk coincides with the gravitational cen-tree of the WD by about 0.015 n,, . The Mach number M decreasess by a factor of two and, in the case of a = 0.3, has aa value of M ~ 12. The slightly elliptically deformed disk is, however,, not becoming eccentric and is not precessing. We regardd this calculation as representing the evolution of an accretionn disk during a normal outburst.

3.33 M o d el 2: Mi = 1.0 M©, M2 = 0.3 M© and

Inn model 2 we choose a higher mass for the secondary star: M2M2 = 0.3 MQ and Porb = 2.3 h. Except for the higher sec-ondaryy mass, the initial structure of the accretion disk is similarr to that of model 1. The initial axisymmetric dis-tributionn is again given by eqs. (21) and (22). We choose E00 = 500g/cm2 and thus derive To ~ 1.03105oK. We ex-posee the disk to the tidal field of the secondary star, first withh a = 0.01 for At ~ 5.6 binary orbits and subsequently too a = 0.3 for another 15 binary orbits.

Inn Fig. (2) we plot the distribution of the disk thickness H{r,4>)H{r,4>) for a = 0.01 at t = 5.6 T (Fig. 2 upper panel) and forr a = 0.3 at t = 12.7 T (Fig. 2 lower panel). We only plot regionss where E > lg/cm2. The vertical axis is enlarged by aa factor of 5.

Wee find that the local disk thickness H can differ from itss corresponding thermal equilibrium value Hc by about a factorr of 2.5. This is especially true for the spiral shock arms closee to the outer edge of the accretion disk.

Inn the calculations of model 2. E(r) decreases with r. Againn we find a stationary spiral shock pattern. The disk evolvess in a very similar way to that of model 1, although thee secondary mass is by a factor of 2 larger: the size of thee disk hardly exceeds the Ü21 radius and is largest in the prominentt spiral shock arms. The disk is stable in size. The transportt of angular momentum by small scale viscosity is inn equilibrium with tidal removal of angular momentum at thee outer part of the accretion disk. This calculation shows

Figuree 2. The stationary disk pattern of model 2a (upper plot: aa = 0.01, t/T = 5.5) and of model 2b (lower plot: a = 0.3, t/Tt/T = 12.3). We show the disk semi-thickness H as given by the one-zonee model of Stehle & Spruit (1997). The vertical axis is enlargedd by a factor of 5. Plotted are only those regions of the diskk where £ > lg/cm2. The secondary star is to the right.

800 0

6000 -

uu 400

2000 -.... r-0.50 r„

00 1 2 3 4 5 6

Figuree 3. The surface density D as a function of 0 for three dif-ferentt disk radii (model 2b at t/T = 12.3). Phase zero is towards thee secondary star.

thatt disks, which basically differ only in the secondary mass, evolvee very similar.

Inn Fig. (3) we show the surface density Er(0) as a func-tionn of azimuth 6 at the disk radii r = 0.22 r\.l, r = 0.36 rt,, andd r = 0.50 TL, . At the spiral shock arms the surface den-sityy is about a factor of 2-5 larger than outside the shock regionn and by a factor 1.5-4 larger compared to the az-imuthallyy averaged surface density < E >«. According to ourr prescription for the shear viscosity, angular momentum iss mainly transported near to the spiral shock arms and thus

422 / / : Accretion Disks in Close Binary Systems, Chapter 4

mostt heat is generated there. As long as this description of thee small scale viscosity in terms of an «-model is appli-cable,, we expect most energy to be radiated from a region nearr to the spiral shock arms.

Modelss 1 & 2 showed that disks where E(r) decreases withh r are stationary on a hydrodynamical timescale. The spirall shock pattern was essentially independent of Mi and iss basically a function of the disk temperature. Only when aa is large does the disk exceed the R\z resonance radius, mainlyy in the spiral shock arms. However, the relative mass storedd near to the tidal resonance radius is small in these models.. This might therefore explain why the tidal reso-nancee instability was unable to operate and to drive the diskk into an elliptically deformed, eccentric and precessing state.. To get over this shortcoming models 3 & 4 start with aa surface density which increases with r: drE(r) > 0. This initiall set up is justified by thermal instability calculations. Ludwigg &: Meyer (1997) show, that at the time where the superoutburstt starts, the surface density increases rapidly withh radius. Most disk mass is close to the Rz\ resonance radiuss during the superoutburst and thus sensitive to the tidall resonance instability.

3.44 M o d el 3: Mi = 1.0 MQ, M2 = 0.3 M 0 and £- ~ r

Inn model 3 we follow the disk in a binary system of M\ = 1.00 MQ and M2 = 0.3 MQ. According to various SPH calcu-lationss (see, e.g., Murray 1996) we do not expect the tidal resonancee instability to operate as q = M2/M\ is slightly tooo large and R3i close to the Roche radius RR1 of the WD. Thee disk is tidally truncated before its outer edge can reach thee tidal resonance radius. This calculation is done as a com-parisonn to model 4, where q is small and the tidal instability expectedd to operate.

Wee show in Fig. (4) the initial density and temperature distributionn of model 3. £ ( r) increases linearly with r up to rr = 1010 3 8cm and then drops rapidly. The value of a = 0.3 iss constant throughout the whole calculation. We run about 322 binary orbits. This corresponds to a total evolution time off 3.1 days.

Againn we find prominent spiral shock arms. Due to the highh value of a they are efficiently driven into the poten-tiall field of the secondary star. We observe the disk to be unstablee in size. The disk oscillates between two phases:

Inn the phase where the disk is large, the rate at which angularr momentum is extracted from the disk by tidal forces exceedss the rate at which angular momentum is transported outwardd by shear viscosity. At that time one of the massive spirall shock arms extends up to the secondary star near too the L\ point. The disk loses mass to the secondary and contractss again.

Inn the other phase, where the size of the disk is small, tidall forces are inefficient in removing angular momentum fromm the disk. Thus, driven by small scale viscosity, the disk iss forced to grow in size.

Inn Fig. (5) we show the evolution of the disk surface densityy in four snapshots around t/T ~ 16.5 for ~ half a bi-naryy period. At t/T = 16.2 the shock arms are small and the diskk starts to rotate in a retrograde fashion (Fig. 5a, b, c). Att t/T ~ 16.55 (Fig. 5c), the shock arm has already grown inn size and forms a bridge from the disk to the secondary. Masss is lost from the disk with a high specific angular mo-

10.00 10.2 10.4 10.6 10.8 logg (r /cm)

9.88 10.0 10.2 10.4 10.6 logg ( r /cm)

Figuree 4. The initial axisymmetric setup (E(r), Tc{r) and ^eff(r))) f° r model 3. E(r) increases linearly up to r andd then drops rapidly.

10' '

mentum.. Additionally angular momentum is extracted from thee disk by tidal forces, which operate very efficiently close too the secondary star. The shock arm falls back onto the disk.. Its impact at the outer disk edge prevents the disk evolvingg into precession (Fig. 5d). We observe the disk to precesss only around t/T ~ 16.5 for a rather short time. The cyclicc variation of the disk radius, however, is an ongoing processs throughout the whole integration time. Models 2 andd 3 differ in the existence of this cycle. In model 2 the diskk was stable in size, probably because the surface density decreasedd rapidly with disk radius.

3.55 Model 4: Mi = 0.685 MQ, M2 = 0.07 MQ and Hii ~ exp(r)

Wee finally model a SU UMa type of CV: OY Car with Mii = O.685M0, M2 = 0.07.V/,., and Porb = l.Slh (Ritter kk Kolb 1995). We choose a = 1. E(r) increases exponen-tiallyy with disk radius. The initial conditions for £, Tc and Teftt are shown in Fig. (6). According to our initial setup mostt mass is stored at the outer part of the disk close to R31R31 and should be sensitive to the tidal resonance instabil-ity.. A rather high total disk mass was chosen to give the tidal resonancee instability enough time to develop. Due to or = 1 viscouss spreading drives the disk beyond the /fei radius.

Inn model 4 we cover ~ 85 binary orbits. This is 5.3 days inn the life of a CV disk. Observations of various OVs show thatt superhumps are visible within two days after the onset off a superoutburst (Warner 1995). We thus expect the total integrationn time of our simulation to be long enough to show superhumpss if our initial setup was chosen appropriately.

R.R. Stehle: Hydrodynamic simulations of CV accretion disks in outburst 43

£(t/T== '6.201) E(tA== 16.376)

7.50E+02 2

5.62E+02 2

175E+02 2

1.88E+02 2

0.00E+00 0

7.5OE+02 2

5.62E+02 2

3.75E+02 2

1.88E+02 2

O.OOE+00 0

I(tA == 16.552) I(t/T== 16.729)

7.50E+02 2

5.62E+02 2

3.75E+02 2

1.88E+02 2

O.OOE+00 0

7.5OE+02 2

5.62E+02 2

3.75E+02 2

1.88E+02 2

O.OOE+00 0

Figuree 5. The evolution of the surface density of model 3 in 4 snapshots around t/T ~ 16.5. The secondary star is to the right of the pictures.. Around t/T ~ 16.5 the disk becomes eccentric and begins to precess (Fig. 5a. b, c). At t/T = 16.55 one spiral arm, extended upp to the secondary star, falls back onto the disk and hits the disk at its outer edge (Fig. 5d). Any evolution into an elliptically deformed andd precessing disk is thus suppressed (see the text for further details).

10.00 10.2 10.4 10.6 logg (r /cm)

10.00 10.2 logg (r /cm)

10.44 10.6

Figuree 6. The initial axisymmetric distribution of £(r), Teff(r) andd Tc(r) for model 4. E(r) increases exponentially with disk radius.. Most mass is initially placed in the outer part of the disk andd thus expected to be sensitive to the tidal resonance instability.

Inn Fig. (7) we show the visual light curve of the disk for 255 < t/T < 55. The total luminosity decreases during these 300 binary orbits by a factor of 1.5. However no superhump modulationn is found.

Onee might ask for the geometrical evolution of the ac-cretionn disk, such as the evolution of the disk centre of mass, orr the angle d to the major semi axis. Following the geo-metricall evolution of the disk mass in its second moments (Appendixx A) for each time step, we find that the mass centree < r > of the disk revolves around the gravitational centree of the WD at about < r >< 0.02 rx, (see Fig. 9). Correspondinglyy the ratio of the semi axis b/a oscillates be-tweenn 0.65 and 0.75. The angle <£a to the major semi axis turnss between 45° and 65°. but no whole circle of 360° is observed. .

Thee disk pattern shows that the innermost part of the diskk is rather stationary. In the outer part of the disk the shockk arms go through a cycle, as already observed in model 3.. The different phases of the cycle are shown in Fig. (8). Thee corresponding points are marked in the light curve with dotss (Fig. 7).

Thee cycle starts with a rather small disk (Fig. 8a). Sub-sequentlyy the small scale shear viscosity drives the spiral armss into the potential field of the secondary (Fig. 8b). At t/Tt/T ~ 35.1 one of the spiral shock arms forms a bridge from thee disk to the secondary (Fig. 8c) and the disk loses mass withh a high specific angular momentum. Additionally the to-tall angular momentum of that shock arm decreases rapidly duee to tidal forces, which are very efficient close to the L\ point.. After a short mass loss phase, where about 1% of the totall disk mass is lost, the spiral shock arm truncates from

444 II: Accretion Disks in Close Binary Systems, Chapter 4

33. 11 O

3 3 . 0 5 5

3 3 . O O O

\\ ' '

^ ^ ^ 11 'J f F

•• ' ' - - - ' - - -

3 3 . 0 7 0 0

3 3 . 0 6 5 5

3 3 . 0 6 0 0

3 3 . 0 5 5 5

3 3 . 0 5 0 0

33.04- 5 5

2 66 2 8 3 0 3 2 3 4 t/ T T

34-.5 0 0

33 .1 0 0

33 .0 0 0

36.O O O

Figuree 7. The evolution of the total visual luminosity of the disk (model 4) for 25 < t/T < 55. Within about 30 orbits the luminosity dropss by a factor of 1.5 but no superhump modulation is observed.

thee secondary (Fig. 8d). It subsequently falls back onto the diskk (Fig. 8e,f) where it finally hits the disk at the outer edge.. The cycle starts again with a small disk and viscous spreadingg (Fig. 8g,h). The cycle is best seen in the evolution off the total disk mass (Fig. 9).

Followingg the evolution of the total disk mass M = ƒƒ rE(r. 4>) dr d<f> (Fig. 9) we find that about 1% of the total diskk mass is lost to the secondary within one cycle. Fig. (9) showss that the cycle is repetitive in a quasi periodic way on aa time scale of roughly half a binary orbit.

Wee find that the disk evolution is dominated by the cycle.. As the surface density is highest in the shock arms itt is there where most angular momentum is redistributed byy small scale viscosity. Any redistribution of angular mo-mentumm in the disk gas close to the secondary star has a largee effect in the disk geometry because the tidal potential fieldd close to the L\ point is flat. This is the reason why the diskk varies in size mostly at that shock arm which is close to thee L\ point. The disk shrinks again due to the loss of spe-cificc angular momentum, both by tidal forces and by mass loss.. This cycle is repetitive in a quasi periodic way and is observedd for the whole time of integration. No sign of an ellipticallyy deformed and precessing accretion disk is found.

44 SUMMAR Y AN D DISCUSSION

Wee introduced hydrodynamic calculations of accretion disks whichh are applicable for CVs in outburst. These disks are exposedd to the tidal force of the secondary star. In this paper wee focused on the global evolution of hot accretion disks. Thee goal of this paper was to test the current model to explainn superhumps as they are observed in SU UMa type off CVs during superoutbursts. Presently the model is only supportedd by Smooth Particle Hydrodynamic calculations (inn short: SPH calculations).

Ourr models are also designed as the basis for an ex-tensivee comparison of 2-dimensional hydrodynamic calcu-lationss with observations. This will be done in forthcoming papers. .

Wee run our simulations with the hydrodynamic code of Stelilee k. Spruit (1997) on an Eulerian grid of 256x128 grid cellss in (r, </>) respectively. In addition to Stehle fc Spruit

(1997),, an a-type of small scale shear viscosity is included inn the parameterization of Shakura & Sunyaev (1973). A radiativee diffusion model connects the disk surface temper-aturee with the disk mid-plane temperature. The radiative losss of energy at the disk surfaces was thus derived.

Wee calculated accretion disks in binaries with different massess M\ and Mi for the primary and secondary star and differentt values for a. We studied cases where the initial surfacee density E(r) was both increasing and decreasing.

Inn all cases we found very prominent spiral shock arms. Theyy are mainly driven by the dominant m = 2 component inn the azimuthal Fourier Transform of the secondary's po-tentiall field. Spiral shock arms dominate the disk evolution onn a hydrodynamical time scale. Including small scale vis-cosity,, most energy is released in these arms by small scale viscouss processes. Recently, a spiral structure has been found onn the basis of Doppler tomographs of the accretion disk in IPP Pegasi (Steeghs, Harlaftis k. Home 1997). The theoreti-call prediction of spiral shock waves in close binary accretion diskss are thus corroborated in at least this case.

Nonee of our calculations confirm the current explana-tionn for the superhumps found in SU UMa CVs during su-peroutbursts.. None of our calculations found the reaction off the disk upon the tidal resonance instability sufficient too drive the disk into eccentricity and precession. The tidal resonancee instability is sensitive only to the in = 3 compo-nentt of the azimuthal Fourier Transform of the secondary's potentiall field. It is much weaker than the m = 2 compo-nent.. In our calculations the m = 2 pattern in the density distributionn dominates over the m = 3 component, and in thee presence of the m = 2 component the tidal resonance instabilityy is unable to evolve.

Thee set of calculations done so far is sufficiently large too allow for a comparison of disk evolution with different initiall setups.

lowlow a versus high a: In models la and 2a we studied ac-cretionn disks with a — 0.01, whereas in models lb and 2b a wass 30 times higher. As expected, the rate of viscous spread-ingg in low a accretion disks is so small, that these disks barelyy exceed the Ri3 resonance radius. This was found al-readyy before by several authors, including Heemskerk (1991) orr Rózyczka feSpruit (1993). In the case where a is large.

R.R. Stehle: Hydrodynamic simulations of CV accretion disks in outburst 45

E(tA== 34.902) E(t/T== 35.000)

3.00E+02 2

2.25E+02 2

1.50E+02 2

7.50E+01 1

O.OOE+00 0

3.O0E+02 2

II 2.25E+02

1.50E+02 2

7.50E+01 1

0.00E+00 0

S(t/T== 35.096) S(tA== 35.195)

3.00E+02 2

2.25E+02 2

1.50E+02 2

7.50E+01 1

O.OOE+OO O

3.00E+02 2

^^ 2.25E+02

1.50E+02 2

7.50E+01 1

0.00E+00 0

E(tA== 35.296) S(t/T== 35.399)

3.00E+02 2

2.25E+02 2

1.50E+02 2

7.50E+01 1

O.OOE+00 0

3.00E+02 2

2.25E+02 2

1.50E+02 2

7.50E+01 1

0.00E+00 0

E(t/T== 35.502) E(tA== 35.602)

3.00E+02 2

|| 2.25E+02

1.50E+02 2

7.50E+01 1

O.OOE+00 0

3.OOE+02 2

11 2.25E+02

1.50E+02 2

7.50E+01 1

O.OOE+00 0

F i g u ree 8. A typical t ime evolution of the disk surface density in 8 snapshots (model 4). The first 6 plots cover a complete cycle. The cyclee starts with a small disk (a) which grows in size by viscous spreading. The spiral arm close to the L\ point forms a bridge from the diskk to the secondary (b). Mass is lost through this spiral arm from the disk to the companion star (c). In addition angular momentum iss efficiently extracted from the spiral arm by tidal forces. Thus the shock arm falls back onto the disk (d) and hits the disk at the outer edgee (e). The cycle starts all over again with viscous spreading (f, g).

466 II: Accretion Disks in Close Binary Systems, Chapter 4

0.040 0

.. 0.030 - I I

u u

A** 0.020

VV 0.010 0.0001 1 300 31 32 33 34 35

t/T T

0.65 5

0.60 0 300 31 32 33 34 35

t/T T

300 31 32 Z3 34 35 t/T T

23.29 9 23.28 8

^^ 23.27

?? 23.26

"" 23.25 23.24 4

; V --

—\ \ ^ \\

300 31 32 33 34 35 t/T T

Figuree 9. The geometrical evolution of the disk derived from the second moments of the surface density distribution. Shown is the evolutionn of the centre of mass < r > in units of ri,,, the ratio b/a of the minor semi axis to the major one, the angle <t>* and the total diskk mass M.

thee disk exceeds the R\$ resonance radius mainly in the spi-rall shock arms. It is there where E(</>) at a specific radius iss highest and where viscous spreading is most efficient. We thuss conclude that a sufficiently large disk viscosity is nec-essaryy to force the disk far enough into the potential field of thee secondary star to give superhumps.

d r E( r)) <; 0 versus drE(r) < 0: From now on we com-paree disk models with a high value of a.

Inn models 1 & 2 the initial distribution of the disk mass wass decreasing as a power law with index —3/4, whereas in modell 3 & 4 we chose drE(r) > 0. In the later case most masss is stored close to the R\3 resonance radius. We expect thiss mass to be sensitive to the tidal resonance instability.

Ass a result of our calculations we find that accretion diskss with a rapidly decreasing initial surface density are stationary.. Models where E increases with r or where E is nearlyy constant go through a cycle: Starting small, the disk growss in size by viscous spreading until tidal stresses of the secondaryy force the disk to shrink again. When the disk is large,, the spiral arm close to the L\ point reaches the sec-ondaryy star and mass is lost from the disk with high specific angularr momentum. When subsequently enough specific an-gularr momentum has been lost by tidal stresses and mass loss,, the spiral arm disconnects again from the secondary. Ass it falls back it hits the disk at its outer edge. Any on-sett of a 3:1 tidal resonance instability is thus destroyed. We conclude,, that the tidal resonance instability is unable to

evolvee in the presence of the dominant m = 2 spiral shock component. .

qq = M2/Mi > 0.25 versus q = M2/Mi < 0.25: Cal-culationss with different mass ratios have been performed to alloww for a comparison. When we compare model 1 with modell 2 and model 3 with model 4 we can hardly find any differencee in the global evolution of the accretion disk. Ac-cordingg to SPH calculations only models with q < 0.25 are sensitivee to the tidal resonance instability. It confirms that thee tidal resonance instability was only of minor importance forr our models.

Comparingg in particular models 3 & 4, the cyclic vari-ationn of the disk radius seems to be almost independent of thee mass ratio (which varies by a factor of ~ 3 from model 33 to 4) and of the value of a (which also varies by about a factorr of 3). The latter might only be true as long as a is largee enough to enforce viscous spreading at a sufficient rate. Basedd on our model calculations so far, we expect the ac-cretionn disk to grow in size mainly at the spiral shock arms. Wee are confident that the cyclic variation in the disk radius willl be found also for other combinations of n and q as long ass a is large enough and the surface density increases witl i diskk radius.

Grid-basedGrid-based calculations versus SPII calculations: SPH-calculationss confirm that superhumps are caused by thee tidal resonance instability. However, they fail to repro-ducee the spiral shock structure. This is surprising, as the spi-

R.R. Stehle: Hydrodynamic simulations of CV accretion disks in outburst 47

rall shock arms are mainly caused by the strongest azimuthal Fourierr component of the secondary's potential field, i.e. the mm = 2 component. The tidal resonance instability is driven, however,, by its much weaker m = 3 component.

Finitee differencing methods do show the opposite be-haviour:: namely, a strong response to the m = 2 component, butt only a weak or zero response to the m = 3 component. Thee reason why SPH calculations fail to form spiral shocks iss at present not clear. We suggest either a lack of spatial resolutionn due to an insufficient number of particles, or a super-efficientt bulk viscosity which smears the spiral shock armss out.

Thee first suggestion might be supported by some results off SPH calculations themselves. Osaki & Hirose (1990) find superhumpss in accretion disks up to binary mass ratios of qq = 0.5 if they use ~ 5000 particles. While increasing the numberr of particles by a factor of ~ 5, only disks with q < 0.255 are still unstable to the tidal resonance instability.

Comparingg disks based on SPH and finite differencing algorithmss suggests, however, that the tidal resonance insta-bilit yy can only evolve in the absence of spiral shock arms. AA strong bulk viscosity can smear the spiral shock arms out andd destroy the cyclic variation of the disk radius. Then the ratherr "gentle" tidal resonance instability might operate, as sketchedd by SPH calculations.

A C K N O W L E D G E M E N TS S

Wee thank A. King and J. Papaloizou for many stimulat-ingg and interesting discussions. I am especially thankful to H.. Spruit who introduced the whole subject to me.

R E F E R E N C ES S

Allen,, C.W., 1973, Astrophysical Quantities, 3r d ed., Univ. Lon-don.. The Athlone Press, p.205

Brandenburg,, A., Nordlund, A., Stein, R.F., Torkelsson, U., 1995, ApJJ 446, 741

Cannizzo,, J.K., 1993a, ApJ 419, 318 Cannizzoo J., 1993b, in Accretion disks in compact stellar systems,

eds.. J.C.Wheeler, World Scientific Publishing, Singapore, p. 6 6

Cannizzo,, J.K., Shafter, A.W., Wheeler, J.C., 1988, ApJ 333, 227 Frankk J., King A., Raine D., 1992 Accretion Power in Astro-

physics,physics, Cambridge Astrophysics Series 21, Cambridge Uni-versityy Press

Heemskerk,, MM 1994, A&A 288, 807 Hirose,, M , Osaki, Y., 1990, PASJ 42, 135 Ichikawa,, S., Osaki, Y„ 1992, PASJ 44, 15 Lubow,, S.H., 1991, ApJ, 381, 259 Ludwig,, K.: 1996, PhD thesis, Ludwig-Maxtmilians Universitat,

München n Ludwig,, K., Meyer, F., 1997, A%A, submitted Meyer,, F., Meyer-Hofmeister, E., 1981, A&A 104, L10 Meyer,, F., Meyer-Hofmeister, E., 1983, A&A 121, 29 Murray,, J.R., 1996, MNRAS 279, 402 Ritterr H., Kolb U., 1995, in: X-ray Binaries, eds. W.H.G. Lewin.

J.. van Paradijs & E.P.J, van den Heuvel, Cambridge Univer-sityy Press, p. 578

Rózyczka,, M., Spruit, H.C., 1993, ApJ 417, 677 Schandl,, S., Meyer-Hofmeister, E., Meyer, F., 1997, A&A 318, 73 Schneider,, P., Seitz, S., 1996, A&A 294, 411 Shakuraa N.I., Sunyaev R.A., 1973, A&A 24, 337

Spruit,, H.C., 1987, A&A 184, 173 Steeghs,, D., Harlaftis, E., Home, K., 1997, MNRAS, in press Stehle,, R., 1997, ASP conference in Wyoming: 13th North Amer-

icanican Workshop on Cataclysmic Variables Stehle.. R., Spruit, H.C., 1997, MNRAS, submitted Stolz,, R., Schömbs, R., 1984, A&A 132,187 Stone.. J.M., Hawley, J.F., Gammie. C.F., Balbus, S.A., 1996, ApJ

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A P P E N D I XX A: S E C O ND M O M E N T S OF T H E DISKK MAS S D ISTRIBUTIO N

Too describe the 2-dimensional distribution of the disk sur-facee density we introduce the matrix of its second moments (Schneiderr k. Seitz 1995). We find the mass centre of the diskk by

ƒƒ r £ ( r ) d2 r << r >= ^ —

/ £ ( r ) d * r r

andd its quadrupole matrix by

ƒƒ ( n - < n > ) ( n - < rj > ) 2 ( r ) d2 r Qiii =

/ £ ( r ) d * r r

(Al ) )

(A2) )

Thee shape of the mass distribution is described by the com-plexx eUipticity %

.,, ( Q n - Q 22 ) + 2tQi a XX~~ Q11+Q22 ' ( A 3 J

whichh reads for an elliptical mass distribution

,££ = TTÖe x p ( 2 ,' * * ) ' (A4)

wheree a and b are the major and minor semi axis and 0» is thee angle which is enclosed by the major semi axis and the linee between M\ and Mj .

Wee integrate eqs. (Al ) k (A2) which yields % (&1- A3) andd thus finally (b/a) and <£a (eq. A4).