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Gas dynamic investigation of rotating gas accretion

We study axisymmetric accretion of slowly rotating matter onto a gravitation center. The process of the flow restructuring is presented as the angular velocity of accreting gas approaches the Keplerian velocity. For spherically-symmetric accretion we found conditions of existence of steady-state supersonic solutions for various external boundary conditions. These conditions are used for choosing proper initial and boundary conditions for accretion of the rotating matter. The study is performed on the basis of the numerical solution of the Euler gas dynamic equations using the second order extension of the Lax-Friedrichs scheme with the implicit approximation of the source term.

pulsars, whose origin is not yet clear (Nagase 1989;Lipunov 1992). Owing to highspeed stellar wind, the angular momentum of the falling matter is often not sufficient for the accretion disk formation at the level of the Alfvénic radius where the magnetic pressure of the neutron star is balanced by the dynamic pressure of the falling gas. The angular momentum acquired in this case by the neutron star from the falling gas was shown (Bisnovatyi-Kogan 1991) to be such that the equilibrium rotational period of the X-ray pulsar might be large. It is rather difficult to make estimations for the equilibrium rotational period because the formation is possible of outflowing streams which carry away the angular momentum (Illarionov & Kompaneetz 1990;Lovelace 1995). The accretion picture in this case is three-dimensional and its numerical modeling is very complicated.
It was performed only for the case of accretion onto a gravitating center, showing the presence of the Rayleigh-Taylor instabilities ). The nature of these instabilities is unclear and their numerical origin can not be excluded (Steinolfson et al. 1994).
To investigate the formation of long-periodic pulsars, the model is necessary for the accretion onto a magnetized neutron star from the stellar wind in the binary. To reduce a 3D problem to two dimensions, we can consider either the conical accretion of nonrotating gas or the accretion of slowly rotating gas onto a stationary star. In the first case (Koide et al. 1991), the average angular momentum acquired by the neutron star is zero due to the symmetry of the flow, so it can not be applied to long-periodic X-ray pulsars.
Here we use the second approach. Our aim is to consider accretion onto a magnetized star with the Alfvénic radius R A ≫ R * (star radius) taking into account the interaction between the gas and the magnetosphere and the possible outflow formation. As the first step, we consider accretion with the full penetration of the gas through the magnetosphere substituted by the inner boundary with the radius R * and neglect magnetogasdynamic effects. Beskin & Pidoprygora (1995) presented the approximate solution for the accretion of nonrotating gas onto a slowly rotating black hole in the framework of the general theory of relativity. In this work, we consider the accretion of rotating gas in the Newton approximation up to the rotation velocities close to the Keplerian velocity.
Three different modes are usually considered as constituent parts of the astrophysical accretion and are fairly well investigated (Lipunov 1992; Bisnovatyi-Kogan 1989).
1. Spherically-symmetric accretion occurs if the star velocity v ∞ is much smaller than the speed of sound a ∞ in an accreting matter and the angular momentum is negligible.
2. Cylindrical accretion is realized when v ∞ ≥ a ∞ with vanishing angular momentum.
3. Accretion disk is formed if the total angular momentum of the matter is sufficient for its formation.
Real accretion is, in fact, a combination of the above-mentioned modes.
From gasdynamic viewpoint, it is of interest to investigate the process of the transition from regime 1 to regime 3 for v ∞ ≪ a ∞ as the rotational velocity of the accreting matter approaches the Keplerian velocity. We consider the polytropic flow of ideal, perfect gas with the polytropic index γ = 1.4. Calculations are performed using the second order of accuracy in space high-resolution Lax-Friedrichs-type numerical scheme proposed by one of the authors (Barmin & Pogorelov 1995). Detailed description of the scheme is given by Barmin, Kulikovskiy & Pogorelov (1996). Recently, a three-dimensional Bondi-Hoyle accretion was investigated for the accretor (star) radius varying from 10 to 0.02 Bondi radii, so that both subsonic and supersonic regimes of accretion could be realized (Ruffert 1994).
In our study we are mainly interested in qualitative tracing the flow restructuring as its angular velocity increases. For this reason, the Euler equations are solved and the external flow is supposed to be supersonic. On the other hand, the size of the star is assumed to be sufficiently small; the inner, also supersonic, boundary is fixed at a finite distance from the gravitational center. The effect of the external boundary conditions is investigated on the existence and properties of a stationary supersonic accretion for the chosen computational region.
Let us consider the accretion gas flow between two spheres with the inner and outer radii equal to R * and R 0 respectively. If the flow at the distance R 0 is assumed to be supersonic, then all flow parameters should be fixed at this boundary. The question is what parameter values can be imposed at a given distance for the existence of a stationary solution. If the flow is shockless and polytropic, the following three conservation equations must be satisfied: Here ρ, p, U are, respectively, the density, the pressure and the radial velocity, R is the current distance from the gravitating center, M is the mass of the star. The values of the accretion rateṀ , the constant K, and the Bernoulli constant h t0 are to be fixed at R = R 0 . Introducing dimensionless variables with the units of velocity, pressure, density, and length equal to U 0 , ρ 0 v 2 0 , ρ 0 , and R * , respectively, and designating in the dimensionless form using the same notations for nondimensional values of U, R, p, ρ, and a = (γp/ρ) 1/2 (indices "0" and " * " correspond to the outer and to the inner boundary, respectively): Being subsonic at infinity and supersonic at R = R 0 , the flow becomes sonic U = In the sonic point Eqs. (4)-(7) reduce to If R B ≥ R 0 and the flow comes from infinity, the value h t0 must satisfy the inequality 3 Accretion with rotation.
In this section we consider the process of the axisymmetric rotating flow accretion onto a gravitating object. Analysis is performed on the basis of the numerical solution of the Euler gasdynamic equations. The gas is assumed to be ideal and perfect with the following caloric equation of state: ε = p/(γ − 1)ρ, where ε is the internal energy per unit mass and γ = 1.4 is the polytropic index. The system of governing equations in the Cartesian coordinate system shown in Fig. 1 reads: where Here e = p/(γ − 1) + ρ(u 2 + v 2 + w 2 )/2 is the total energy per unit volume. As far as the gravitational field is considered spherically symmetric, the polar computational domain (R, θ) is chosen with the inner and outer radii equal to R * and R 0 , see Fig. 1. On normalizing the quantities of density, pressure, and velocity by ρ 0 , ω * R * , ρ 0 ω 2 * R 2 * , where ρ 0 is the density at R = R 0 and ω * is the gas angular velocity at R = R * at the equator for the constant angular momentum distribution, the source term can be rewritten in the dimensionless form as Here S = GM/ω 2 * R 3 * . The form of the other vector components in Eq. (10) remains unchanged.
From now on we consider only nondimensional parameters.
The following procedure is used to construct the initial and boundary conditions.
1. Introducing dimensionless parameter α = U 0 /U K * , where U 0 is the radial velocity on the outer boundary and U K * = (GM/R * ) 1/2 is the Keplerian velocity on the inner boundary, we fix the values on the outer boundary as if the flow is spherically symmetric: where W 0 is the θ-component of the velocity and M 0 = U 0 /a 0 .
Corresponding parameter values inside the computational region are adopted equal to those at the boundary.
We can find the entropy and the total enthalpy dimensionless values as follows: The following initial distribution of the angular velocity is assumed: After that all values on the external boundary are fixed because it is supersonic.
No boundary conditions are necessary on the inner circle, since the flow through it is supersonic.

Numerical scheme.
Point clustering towards the internal boundary circle is made to obtain a sufficiently fine flow resolution in the vicinity of the gravitating center. The following formula is used: with the clustering parameter β.
The assumption is made of a piecewise-parabolic distribution of the primitive gasdynamic parameters q inside the cells to specify values on their boundaries and slope delimiters are used to obtain the nonoscillatory property.
The averaged slope inside the cell in the radial direction is determined as ) The left and the right boundary value are then estimated as where the matrixR is a positive diagonal matrix with the entries equal to the spectral radius (the maximum of eigenvalue magnitudes) of ∂Ē ∂U . This scheme gives a drastic simplification of the numerical algorithm comparing to the methods based on precise characteristic splitting of the Jacobian matrices, while preserving nonoscillatory property.
Having the second order of accuracy, it is less dissipative than the original Lax-Friedrichs scheme. Similarly, the fluxes through another pair of cell surfaces are obtained.
As seen from Eq. (12) the source term is approximated implicitly for better stability of the numerical scheme. A proper linearization of this term is made to realize the numerical procedure of obtaining the time-converging steady-state solution: The promotion of the solution in time is performed with the first order of accuracy by the formula: for t = k ∆t, k = 0, 1, . . . , and ∆t defined by the CFL condition (I is the identity matrix).

Analysis of numerical results.
All The steady state results for this case are presented in Figs. 2-6 (the axis of rotation is aligned with the z-axis (see Fig. 1). In Fig. 2 decelerates to become zero at θ = π/2. Behind these shocks the region of a fast rotation is observed (Figs. 14, 17). This region of a highly rotating dense matter will form an accretion disk if S < 1.

Discussion.
Numerical modeling is performed for axisymmetric rotating gas accretion onto the star.