accretion flows: aspects of gas dynamic modeling

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ACCRETION FLOWS: ASPECTS OF GAS DYNAMIC MODELING NIKOLAI V. POGORELOV and IGOR A. KRYUKOV Institute for Problems in Mechanics, Russian Academy of Sciences 101-1 Vernadskii Avenue, Moscow 117526, Russia; E-mail: [email protected] Abstract. We present our view on the application of numerical models to accretion flows in as- trophysics. Special attention is paid to the problem of existence of steady-state solutions in time- dependent calculations and to origin of numerically induced instabilities. The problem is considered of the supersonic wind accretion onto gravitating objects. We also present the results of the gas dynamic simulation of accretion on a body imitating the shape of the star magnetosphere with holes in its polar regions. This shape can occur as a result of the cusp disintegration owing to the Rayleigh–Taylor instability in the equatorial region of the magnetosphere. 1. Introduction Accretion onto neutron stars and black holes produces the main energy supply in galactic X-ray sources. The problem of the wind accretion onto a moving gravit- ating center has long been the subject of interest in view of possible observations of black holes moving through interstellar gas. There is also a belief that solu- tions of these problems can explain the observations of the X-ray pulsars in which magnetic neutron stars accrete from massive O/B companions. Due to complexity of the problem, starting from the work of Hunt (1971), numerical methods have been intensively applied to its solution. Summarizing all these results, we can state that solutions of the axisymmetric problem generally have steady states for all parameters of the uniform wind. Non-axisymmetric calculations can be sub- divided into plane and genuinely three-dimensional ones. Plane accretion is a used to resolve certain methodological problems. Most of plane calculations show that accretion is very unsteady, exhibiting oscillations of the accretion cone together with its tip attached to the absorbing boundary. If the size of the accreting circle is small enough, the amplitude of spatial oscillations exceeds the diameter of the boundary and the so-called flip-flop instability originates. It is accompanied by a violent motion of the tip of the accretion column and the angular momentum sporadically increases so much that an accretion disk is formed and absorption of matter temporarily stops. Note that these instabilities can originate spontaneously from the initially symmetric flow. The flip-flop instability cannot account, however, for the periodic behavior observable in some transient Be-type X-ray binaries, since the amplitude of fluctuations observed in the plane case is not large enough (Shima Astrophysics and Space Science 274: 275–284, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands.

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ACCRETION FLOWS: ASPECTS OF GAS DYNAMIC MODELING

NIKOLAI V. POGORELOV and IGOR A. KRYUKOVInstitute for Problems in Mechanics, Russian Academy of Sciences

101-1 Vernadskii Avenue, Moscow 117526, Russia;E-mail: [email protected]

Abstract. We present our view on the application of numerical models to accretion flows in as-trophysics. Special attention is paid to the problem of existence of steady-state solutions in time-dependent calculations and to origin of numerically induced instabilities. The problem is consideredof the supersonic wind accretion onto gravitating objects. We also present the results of the gasdynamic simulation of accretion on a body imitating the shape of the star magnetosphere withholes in its polar regions. This shape can occur as a result of the cusp disintegration owing to theRayleigh–Taylor instability in the equatorial region of the magnetosphere.

1. Introduction

Accretion onto neutron stars and black holes produces the main energy supply ingalactic X-ray sources. The problem of the wind accretion onto a moving gravit-ating center has long been the subject of interest in view of possible observationsof black holes moving through interstellar gas. There is also a belief that solu-tions of these problems can explain the observations of the X-ray pulsars in whichmagnetic neutron stars accrete from massive O/B companions. Due to complexityof the problem, starting from the work of Hunt (1971), numerical methods havebeen intensively applied to its solution. Summarizing all these results, we canstate that solutions of the axisymmetric problem generally have steady states forall parameters of the uniform wind. Non-axisymmetric calculations can be sub-divided into plane and genuinely three-dimensional ones. Plane accretion is a usedto resolve certain methodological problems. Most of plane calculations show thataccretion is very unsteady, exhibiting oscillations of the accretion cone togetherwith its tip attached to the absorbing boundary. If the size of the accreting circleis small enough, the amplitude of spatial oscillations exceeds the diameter of theboundary and the so-called flip-flop instability originates. It is accompanied bya violent motion of the tip of the accretion column and the angular momentumsporadically increases so much that an accretion disk is formed and absorption ofmatter temporarily stops. Note that these instabilities can originate spontaneouslyfrom the initially symmetric flow. The flip-flop instability cannot account, however,for the periodic behavior observable in some transient Be-type X-ray binaries, sincethe amplitude of fluctuations observed in the plane case is not large enough (Shima

Astrophysics and Space Science274: 275–284, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

276 N.V. POGORELOV AND I.A. KRYUKOV

et al., 1998). Besides, three-dimensional computations (Ruffert, 1996) give theamplitude 3 to 10 times smaller than that observed in the plane case. There stillremains a doubt, however, that existence of nonstationary solutions is parameter-dependent or that some drawbacks of the computational methods can lead in certaincases to numerical instabilities.

Accreting matter can have an angular momentum. If it is sufficiently large, cent-rifugal forces push matter from the axis of rotation towards the equatorial plane.In this case, accretion can occur only through the accretion disk. If the angularmomentum is insufficient for formation of the accretion disk, the matter either fallsquasi-radially or the disk is very thick.

Development of computational fluid dynamics provided a variety of efficientnumerical tools for resolving multishocked high-speed gasdynamic flows. A num-ber of astrophysical phenomena can be modeled using the continuum mechanicsapproach. They possess, however, to an extreme extent difficulties which one canencounter in traditional gas dynamics. Among the methods which we apply tosolution of such flows are high-resolution numerical schemes, irregular adaptiveand overlapping grids (Ivanovet al., 1999), and efficient absorbing (nonreflecting)boundary conditions (Pogorelov, 1995). It becomes widely known that even applic-ation of well-validated codes for insufficiently understood problems can result inwrong results, excessive slow convergence, or even nonconvergent solutions (Yeeand Sweby, 1998).

We present calculations of the supersonic wind accretion obtained on the basisof a high-resolution numerical scheme with the proper entropy correction proced-ure necessary to avoid ‘weak’ nonmonotonicity intrinsic in these schemes. Godunovshowed that linear schemes can be monotone if their order of accuracy is not higherthan one. Modern high-resolution numerical schemes (Hirsch, 1990) are nonlinearand the above restriction can be avoided without adding any special semi-empiricalviscosity. Ostapenko (1998) showed, however, that nonlinear schemes, though pre-serving monotonicity of functions, can produce high-amplitude oscillations of dif-ference derivatives if the amount of the numerical viscosity is insufficient. We avoidthe lack of numerical viscosity by adding it in the vicinity of zero eigenvalues, thatis, near sonic points and stagnation points of the flow. Note that the amount ofviscosity is much smaller in this case than that in conventional linear numericalschemes. With application of these approach, we found steady-state solutions forthe plane accretion in certain cases even for sinusoidally perturbed and stronglynonuniform winds. It must be noted that nonstationary phenomena do exist, butthey are not very violent and can sometimes as well be attributed to numerical asto physical reasons (detailed results will be published elsewhere).

We also present some results on numerical modeling of accretion on the mag-netosphere of the star possessing magnetic field. Both the cases without prescribedangular momentum of the accreting matter at large distances from the star and thecases with slow rotation are considered.

ACCRETION FLOWS 277

2. Supersonic Wind Accretion onto Gravitating Objects

2.1. STATEMENT OF THE PROBLEM

The gas dynamic behaviour of accreting matter is governed by the two-dimensionalEuler equations, which in the Cartesian coordinate systemx, z (see Figure 1) read

∂U∂t+ ∂E∂x+ ∂G∂z+ H = 0, (1)

where

U = [ρ, ρu, ρw, e]T, (2)

E = [ρu, ρu2+ p, ρuw, (e + p)u]T, (3)

G = [ρw, ρuw, ρw2+ p, (e + p)w]T, (4)

H = H1+ H2, (5)

H1 = α ρux

[1, u, w,

e + pρ

]T

, (6)

H2 = ρ

2R2[0, sinθ, cosθ, u sinθ + w cosθ ]T (7)

with α = 1 for the axisymmetric andα = 0 for the plane flow. Hereρ, p, u, andw are density, pressure, and the components of the velocity vectorv, respectively;e is the total energy per unit volume. In the gravitational termH2, R andθ are thepolar coordinates (the angleθ is counted off thez-axis). The quantities of density,pressure, and velocity are normalized byρ∞, ρ∞V 2∞, andV∞, respectively, wherethe subscript∞ refers to the characteristic values of the stellar wind at infinity.Time and the linear dimensions are normalized byRHL/V∞ andRHL = 2GM/V 2∞(M is the mass of the star andG is the gravitational constant). System (1) can thenbe written out for each computational cell (Rl, θn) giving

Rl1Rl1θ∂Uk

l,n

∂t+ (Rl+1/2Fkl+1/2,n + Rl−1/2Fkl−1/2,n

)1θ

+(Fkl,n+1/2+ Fkl,n−1/2

)1Rl + Rl1Rl1θ Hk+1

l,n = 0, (8)

whereF is the flux normal to the boundary. We determine this flux by solving theRiemann problem at the cell interface.

The boundary conditions are fixed at the supersonic entrance. Extrapolation isused at the segments of supersonic exit. At the points of subsonic outflow, theabsorbing boundary conditions (Pogorelov and Semenov, 1997) are used.

278 N.V. POGORELOV AND I.A. KRYUKOV

0 1 2-1-2-3

0

-1

-2

-3

1

2

3

ax

z

0 1 2 3-1-2-3

-3

-2

-1

0

1

2

3

b

Figure 1. Streamlines for the plane accretion withM∞ = 5 andγ = 1.01 (a) andM∞ = 3 andγ = 5/3 (b).

2.2. NUMERICAL RESULTS

We present here only some numerical results for plane two-dimensional accretion.Axisymmetric accretion was found out to be steady for a wide range of the gov-erning parametersM∞ = 2–10 andγ = 1.01–5/3, whereM∞ andγ are the Machnumber of the uniform wind at infinity and the polytropic index, respectively. Theflow is directed from the right to the left in Figure 1, where we show the streamlinedistribution forM∞ = 5 andγ = 1.01 (a) andM∞ = 3 andγ = 5/3 (b). Thecomputational region lies betweenRmin = 0.05 andRmax = 10. No unsteady flip-flop behavior was obtained. We see two shocks attached to the inner boundary.

ACCRETION FLOWS 279

In principle, the accretion pattern for the first presented case is very much similarto the classical Hoyle–Littleton scheme, since the accretion column in the rearside of the accretor is rather thin. It is apparent that a stagnation point of the flowexists behind the accretor. It looks like a saddle singular point. It would not besurprising if it turned out to be unstable. We do not see this instability in oursolution. One can attribute this to addition of numerical viscosity in the vicinityof vanishing eigenvalues (sonic and stagnation points). On the other hand, thenumerical scheme becomes insufficiently stable without this viscosity. One cannotmake a final decision on the nature of instability of the accretion problem only onthe basis of numerical calculations. Further analysis must be made with the use ofthe dynamical systems theory (Yee and Sweby, 1998).

As noted by Pogorelovet al. (2000) the boundary conditions atR = Rmin canalso lead to numerical instabilities. Note in this connection that the relativistic ac-cretion calculations performed by Font and Ibañez (1998) are free of the boundarycondition problem (the inner boundary is always a supersonic exit boundary) andgive only steady solutions. On the other hand, if we want to analyze accretion ontothe star magnetosphere, it is more suitable to state the boundary conditions on itssurface.

3. Accretion onto Magnetosphere of a Star

The model of accretion implemented in this paper corresponds to the scenario ofArons and Lea (1976). According to it, the plasma flow is, at first, decelerated bythe star magnetosphere with the cusps formed in the polar regions. Later on, owingto the Rayleigh–Taylor instabilities, the clusters of plasma penetrate beneath themagnetopause and fall freely along the magnetic field lines onto the poles underaction of gravitation. In our paper we choose a simplified approach which allowsus to stay within purely gas dynamic statement of the problem. We assume that, onentering inside the magnetopause, the clusters of plasma are homogenized within alayer adjacent to the inner side of the initial magnetospheric surface and soon afterthat we can again consider the gas flow in the continuum approximation. This flowoccurs, however, around another shape of magnetosphere which is characterizedby presence of the polar holes. As a result, the homogenized plasma moves alongthe solid surface corresponding to the modified magnetopause and accretes nearthe poles. To attain larger accretion rates, we state the free penetration conditionsat the holes (see Section 2.1).

3.1. NUMERICAL METHOD

For numerical solution of the above problem we use the numerical method sim-ilar to that applied in the previous section. The similarity is in application of thesolution to the Riemann problem to find numerical fluxes at the computational cell

280 N.V. POGORELOV AND I.A. KRYUKOV

interfaces. There are, however, certain differences. In the current calculation, weuse an irregular structured numerical grid adapted to the shape of the inner bound-ary. It possesses a higher order of accuracy in regions of smooth functions and ismonotonicity preserving on discontinuities. It is a modification of the Godunovscheme and can be attributed either to the TVD (total variation diminishing) orENO (essentially nonoscillatory) classes, depending on the implementation of itsconstituent elements.

LetSij be the decomposition of the planex−z into convex quadranglesABCD(cells) and denotePij = (xij , zij ) the center of mass of each quadrangleSij .

Let us integrate system (1) over the cellABCD

∂Uij (t)

∂t= − 1

aij

[FAB(t)+ FBC(t)+ FCD(t)+ FDA(t)

], (9)

whereaij andUij (t) are the area of the cell and the value ofU averaged over thecell at timet , respectively. Then

Uij (t) = 1

aij

∫ABCD

U(x, z, t)dxdz. (10)

The fluxF through the cell interface can be determined as

FAB(t) =∮AB

(Edx −Gdy). (11)

We shall use the Gauss quadrature formula to evaluate the integral in (11). If thefunctionu(ξ) belongs toC2K , theK-point Gauss formula reads∫ b

a

u(ξ)dξ = b − a2

K∑k=1

cku(ξk) = (K!)4(b − a)2K+1

(2K + 2)[(2K)!]3u(2K)(ξ). (12)

Hence, for the integral (11) we can write out

FAB(t) ≈K∑k=1

ck

(E(U(xk, zk, t))

1zAB

2−G(U(xk, zk, t))

1xAB

2

). (13)

ForK = 1 we obtain the most frequently used form of numerical flux

FAB(t) ≈ E (U(x1, z1, t)) 1zAB −G (U(x1, z1, t)) 1xAB, (14)

wherex1 = (xA+xB)/2 and,y1 = (yA+yB)/2. ForK = 2 numerical flux acquiresthe form

FAB(t) ≈ 12 [E (U(x1, z1, t))1zAB −G (U(x1, z1, t)) 1xAB ]

+12 [E (U(x2, z2, t)) 1zAB −G (U(x2, z2, t)) 1xAB ] , (15)

ACCRETION FLOWS 281

where

x1/2 = xA + xB2

∓ ξ21xAB, z1/2 = zA + zB

2∓ ξ

21zAB, ξ = 1√

3.

For determination of numerical fluxes through the cell sides in the Gauss pointswe use either exact or some of the approximate solutions of the Riemann problemwith initial data referring to the quantities on the left- and right-hand sides of thecell surfaces. In order to obtain the value of the vectorU in the Gauss points viathe averaged values ofU at the cell centers, we use a substantially two-dimensionalreconstruction procedure (Ivanovet al., 1999).

Time integration is performed using the Runge–Kutta procedure up to the thirdorder of accuracy.

3.2. NUMERICAL RESULTS

With no rotation, the determining dimensionless parameters of the problem arethe Mach numberM0 = U0/a0 at the outer boundaryR = R0 = 100, andS = GM/U2

0R∗ = U2K/U

20 (UK is the Keplerian velocity), and the polytropic

indexγ . HereR∗ is the reference length related with the size of the magnetosphere,see Figure 2a, where we present the numerical results for the following determiningparameters:S = 100,M0 = 8, γ = 1.1, andd = 0.2 (the last parameter specifiesthe size of the polar hole). The value of the polytropic index was chosen sufficientlysmall to account for the radiative cooling effects. In the first quadrant, we usearrows to show the velocity vectors. The size of each arrow is proportional to themagnitude of the velocity vector. In order to avoid misinterpreting, we limited thesize of the vectors shown. The regions with small velocity manifest themselvesas white zones free of arrows. The second and third quadrants contain the chartsof constant pressure and density logarithms, respectively. In these charts we candistinguish the discontinuities which are likely to occur in the progress of flowdeceleration by the impermeable surface of the magnetosphere. In the fourth quad-rant, we show the chart of the constant Mach number lines. In addition, dotted linesrepresent the levelM = 1.

In the adopted model, the flow is essentially the combination of the blunt bodyand nozzle flows. It is therefore expectable that a bow shock will appear aheadof an impermeable portion of the magnetosphere. In addition, the surface of themagnetosphere has such a shape that the radially oriented gravitation force alwayshas a component which makes the matter move downward along this surface to-wards the poles. The bow shock is finely resolved by the numerical scheme on thegrid adapted to the magnetosphere surface. The supersonic stream of matter caninitially be freely accreted through the polar holes. In the course of time, however,as we approach the steady state, the freely falling column of matter surroundingthe polar axis meets the jet formed in the shock layer around the impermeable partof the magnetosphere. Collision of these two streams decelerates the former one,

282 N.V. POGORELOV AND I.A. KRYUKOV

Figure 2.Accretion onto magnetosphere of the star: no rotation (a), slow rotation (b).

thus resulting in the origin of another shock. These two shocks form a combinedbow shock around the magnetosphere. The resulting shock cannot generically besmooth and the derivativeRsθ of the shock surfaceR = Rs(θ) is discontinuous. Itis clear that there must appear a third shock at this point. Thus, the jet is becomessupersonic. There exists another sonic line in the vicinity of the polar holes. Nearthis line the jet decelerates to subsonic velocities again. This implies existence of ashock wave. In Figure 2a, the slip line between the two portions of the flow passingthrough the different two parts of the bow shock is also very well seen.

If accreting matter has an angular momentum at the outer boundary, one moredetermining parameter exists, namely the ratioSr = GM/ω2∗R3∗, whereω∗ is the

ACCRETION FLOWS 283

angular velocity which could be attained at a distanceR∗ from the axis of rotationin assumption of constant angular momentum distribution. The larger isSr , thesmaller is rotation in comparison with the Keplerian one. LetSr = 250,S = 100,M0 = 2, γ = 1.1, andd = 0.2 (Figure 2b). Though rotation is rather slow in thiscase, we can see a noticeable difference with the similar results for a quasisphericalaccretion shown in Figure 2a. Namely, the centrifugal force acts to decelerate thejet formed in the shock layer and there appears a large chance for accretion straightalong the rotation axis (see the velocity vector distribution). Note that the shapeof the bow shock becomes more complicated due to faster rotation about the polaraxis.

3.3. CONCLUSION

We obtained steady-state numerical solutions both for the accretion onto a super-sonically moving black hole and for the quasispherical accretion of matter onto themodel magnetosphere of a star. High-resolution numerical methods of the Godunovtype with proper entropy fix procedures designed to eliminate inadmissible shockwaves were used. Analysis shows that application of high-resolution numericalmethods for simulation of discontinuous flows can give either steady or unsteadysolutions, depending not only on the statement of the physical problem, but alsodue to numerical reasons. The main difficulty in this case is related with nonlinear-ity of numerical schemes which can give spuriously chaotic solutions.

Acknowledgements

This work was supported, in part, by the Russian Foundation for Basic Researchgrant 98-01-00352. The authors are grateful to G.S. Bisnovatyi-Kogan and T. Mat-suda for stimulating discussions.

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