Numerical Simulation of the Interaction between an L1 Stream and an Accretion Disk in a Close Binary System

729 Progress of Theoretical Physics, Vol. 106, No. 4, October 2001 Numerical Simulation of the Interaction between an L1 Stream and an Accretion Disk in a Close Binary System Hidekazu Fujiwara, Makoto Makita, Takizo Nagae and Takuya Matsuda (Received January 23, 2001) Numerical simulation of the hydrodynamic behavior of an accretion disk in a close binary system is reported. Calculations were carried out for a region including a compact star and its gas-supplying companion. The equation of state is that of an ideal gas characterized by a specific heat ratio γ. Two cases, with γ = 1.01 and γ = 1.2, are studied. Our calculations show that the gas, flowing from the companion via a Lagrangian L1 point towards the accretion disk, forms a fine gas beam (L1 stream), which penetrates into the disk. Thus, no hot spot forms in these calculations. Another result is that the gas rotating with the disk forms —upon collision with the L1 stream— a bow shock wave, which we call an ‘L1 shock’. The disk becomes hot because the L1 shock heats the disk gas in the outer parts of the disk, so that the spiral shocks wind loosely, even with γ = 1.01. The L1 shock enhances axial asymmetry of the density distribution in the disk, and therefore angular momentum is transferred through the tidal torque more effectively. The maximum value of the effective α becomes ∼ 0.3. A ‘hot spot’ is not formed in our simulations, but our results suggest the formation of a ‘hot line’, which is the L1 shock elongated along the penetrating L1 stream. §1. Introduction and summary 1.1. Discovery of spiral structure Recently, Steeghs, Harlaftis & Horne 1) discovered, using the Doppler tomography technique, a spiral structure in the accretion disk of the dwarf nova IP Pegasi, appearing in its outburst phase. Since this discovery, similar spiral structures have been found on many other accretion disks. 2) - 4) Such spiral structures are generally thought to be unobservable during quiescent phases of dwarf novae, but Neustroev & Borisov 5) observed them in a quiescent phase in U Gem. Formation of spiral-shaped shocks in an accretion disk of a close binary was predicted on the basis of two-dimensional numerical simulations by Sawada, Matsuda & Hachisu, 6), 7) and Sawada et al. 8) Similar two-dimensional numerical simulations performed by many people thereafter all confirm the presence of the spiral shock waves. 9) - 14) Self-similar solutions of the spiral shocks were also obtained. 15) Angular momentum loss at the spiral shocks provides a mechanism for angular momentum transfer that differing from the conventional α-viscosity mechanism. Three-dimensional simulations have also been carried out for this object, but their results are not completely in agreement with one another. Most of the threedimensional numerical simulations of accretion disks performed to this time have used particle methods such as the SPH method. 16) - 23) Most have failed to find any spiral shock. However, using the SPH method, Yukawa, Matsuda & Boffin 24) observed spiral Department of Earth and Planetary Sciences, Kobe University Kobe 657-8501, Japan 730 H. Fujiwara, M. Makita, T. Nagae and T. Matsuda 1.2. Difference between 2D and 3D results Although radiative cooling has an important effect on the behavior of an accretion disk, it is very difficult to incorporate this effect accurately in three-dimensional calculations. To avoid this difficulty, we use as the equation of state of the gas that for an ideal gas and, as the specific heat ratio of the gas γ, a value smaller than 5/3, thereby simulating the effect of radiative cooling to some extent. What has puzzled us most in interpreting the results obtained from our threedimensional calculations is that they differ greatly from those obtained from twodimensional calculations (see Fig. 1). With the two-dimensional calculations, we can obtain a clear correlation between γ and the degree of winding-in of spiral shocks: the pitch angle decreases and winding angle becomes tighter as γ decreases (Makita et al., 31) Matsuda et al. 37) ). A smaller value of γ corresponds, with the gas undergoing an adiabatic change, to a smaller temperature elevation. The limit γ → 1 corresponds to an isothermal change. In the presence of shocks, this is not necessarily true, but with two-dimensional calculations, a small value of γ generally implies a low temperature of the accretion disk, which corresponds to a small sound speed shocks employing a specific heat ratio γ of 1.2. Lanzafame & Belvedere 25), 26) and Boffin, Haraguchi & Matsuda 27) also observed spiral shocks in their recent SPH calculations. This disagreement in the above-mentioned studies is thought to be due to the difference in resolution, or, more concretely, in the number of particles: It is found that the use of a sufficient number of particles, which enhances the resolution, leads to the appearance of spiral shocks. Because the finite difference/volume method has a high accuracy compared to the SPH method, it is expected that this method will provide a picture of the detailed structure of an accretion disk. Sawada & Matsuda 28) performed simulation employing a three-dimensional finite volume method for the first time. Although the result of this simulation shows the presence of spiral shocks, no definite statement could be made with respect to this phenomenon, because the calculation time used corresponds to only about half an orbital period. In recent years, our group has conducted a series of three-dimensional finite volume calculations and confirmed the presence of spiral shocks in all cases. 29) - 31) These calculations, however, restricted the computational region to the vicinity of the compact star and did not include the companion. The gas was assumed to flow into this region through a rectangular hole located at the Lagrangian L1 point. Since most of the computational region was filled with a low-pressure gas, the gas flowing into the region from the hole underwent a rapid expansion, thereby forming what is known as an under-expanded jet. The flow (L1stream) thus formed appeared to interact with the accretion disk in a complex manner. The hole, being artificial, caused the L1 stream to be an under-expanded jet with a square cross section, which is unnatural. To overcome these problems, it was necessary to carry out a simulation with the computational region extending up to the companion. In such a simulation, the true shape of the L1 stream could be observed. Bisikalo et al. 32) - 36) carried out studies very similar to the present one, and the relation between our results and theirs will be discussed in the last section. Numerical Simulation of an Accretion Disk 731 and thus to a large Mach number. It is natural that in this case the spiral shocks wind in tightly. However, as is clear from Fig. 1, with three-dimensional calculations, the obtained spiral shocks seem to wind in very loosely, in particular when γ = 1.01. This finding has long been a mystery to u (...truncated)