Numerical simulation of gravitational instability development and clump formation in massive circumstellar disks using integral characteristics for the interpretation of results




circumstellar disk, structure formation, Smoothed-Particle Hydrodynamics (SPH), gravitational gas dynamics


Results of numerical simulation of instability development and formation of self-gravitating clumps (embryos of protoplanets) in a thin circumstellar gaseous disk are analyzed and systematized. Numerical experiments are performed using a disk model based on a combination of Smoothed Particle Hydrodynamic (SPH) and Hockney methods to solve Poisson’s equation on a uniform Cartesian grid. It is shown that the process of clump formation can be characterized by an average growth rate of the total mass of fragments in the disk; this rate is strongly dependent on the physical parameters of the disk and is slightly dependent on the parameters of the numerical model. It is confirmed that there exists a range of the disk parameters such that the appearance or absence of clumps in the disk depends on the resolution in use and on the details of the numerical algorithm, whereas beyond this range such a dependence is not observed. It is shown that, for a combination of the SPH method with grid-based method to calculate the gravitational force, it is necessary that the hydrodynamic smoothing length does not exceed the grid cell length, otherwise we obtain the following numerical effects in the solutions: the disk shape becomes a square and an artificial grouping of model particles takes place due to the evolution of pairing (clumping) instability in SPH.

Author Biography

O.P. Stoyanovskaya


  1. A. C. Boley, T. Hayfield, L. Mayer, and R. H. Durisend, “Clumps in the Outer Disk by Disk Instability: Why They are Initially Gas Giants and the Legacy of Disruption,” Icarus 207 (2), 509-516 (2010).
  2. A. P. Boss, “Possible Rapid Gas Giant Planet Formation in the Solar Nebula and Other Protoplanetary Disks,” Astrophys. J. 536 (2), L101-L104 (2000).
  3. S. Nayakshin, S.-H. Cha, and J. C. Bridges, “The Tidal Downsizing Hypothesis for Planet Formation and the Composition of Solar System Comets,” Mon. Not. R. Astron. Soc. 416 (1), L50-L54 (2011).
  4. M. N. Machida, S.-I. Inutsuka, and T. Matsumoto, “Recurrent Planet Formation and Intermittent Protostellar Outflows Induced by Episodic Mass Accretion,” Astrophys. J. 729 (1), 1-17 (2011).
    doi 10.1088/0004-637X/729/1/42
  5. F. Meru, “Triggered Fragmentation in Self-Gravitating Discs: Forming Fragments at Small Radii,” Mon. Not. R. Astron. Soc. 454 (3), 2529-2538 (2015).
  6. E. I. Vorobyov and S. Basu, “The Burst Mode of Protostellar Accretion,” Astrophys. J. 650 (2), 956-969 (2006).
  7. Z. Zhu, L. Hartmann, R. P. Nelson, and C. F. Gammie, “Challenges in Forming Planets by Gravitational Instability: Disk Irradiation and Clump Migration, Accretion, and Tidal Destruction,” Astrophys. J. 746 (1), 1-26 (2012).
    doi 10.1088/0004-637X/746/1/110
  8. M. D. Young and C. J. Clarke, “Dependence of Fragmentation in Self-Gravitating Accretion Discs on Small-Scale Structure,” Mon. Not. R. Astron. Soc. 451 (4), 3987-3994 (2015).
  9. S. Z. Takahashi, Y. Tsukamoto, and S. Inutsuka, “A Revised Condition for Self-Gravitational Fragmentation of Protoplanetary Discs,” Mon. Not. R. Astron. Soc. 458 (4), 3597-3612 (2016).
  10. F. Meru and M. R. Bate, “Non-Convergence of the Critical Cooling Time-Scale for Fragmentation of Self-Gravitating Discs,” Mon. Not. R. Astron. Soc. 411 (1), L1-L5 (2011).
  11. O. P. Stoyanovskaya, N. V. Snytnikov, and V. N. Snytnikov, “An Algorithm for Solving Transient Problems of Gravitational Gas Dynamics: A Combination of the SPH Method with a Grid Method of Gravitational Potential Computation,” Vychisl. Metody Programm. 16, 52-60 (2015).
  12. E. I. Vorobyov, “Embedded Protostellar Disks around (Sub-)Solar Protostars. I. Disk Structure and Evolution,” Astrophys. J. 2010. 723 (2), 1294-1307 (2010).
  13. V. L. Polyachenko and A. M. Fridman, Equilibrium and Stability of Gravitating Systems (Nauka, Moscow, 1976) [in Russian].
  14. R. H. Durisen, A. P. Boss, L. Mayer, et al., “Gravitational Instabilities in Gaseous Protoplanetary Disks and Implications for Giant Planet Formation,” in Protostars and Planets (Univ. Arizona Press, Tucson, 2007), pp.607-622.
  15. J. K. Truelove, R. I. Klein, C. F. McKee, et al., “The Jeans Condition: A New Constraint on Spatial Resolution in Simulations of Isothermal Self-Gravitational Hydrodynamics,” Astrophys. J. 489 (2), L179-L183 (1997).
  16. I. Thies, P. Kroupa, S. P. Goodwin, et al., “Tidally Induced Brown Dwarf and Planet Formation in Circumstellar Disks,” Astrophys. J. 2010. 717 (1), 577-585 (2010).
  17. D. J. Eisenstein and P. Hut, “HOP: A New Group-Finding Algorithm for N-Body Simulations,” Astrophys. J. 498 (1), 137-142 (1998).
  18. M. R. Bate and A. Burkert, “Resolution Requirements for Smoothed Particle Hydrodynamics Calculations with Self-Gravity,” Mon. Not. R. Astron. Soc. 288 (4), 1060-1072 (1997).
  19. A. F. Nelson, “Numerical Requirements for Simulations of Self-Gravitating and Non-Self-Gravitating Discs,” Mon. Not. R. Astron. Soc. 373 (3), 1039-1073 (2006).
  20. B. E. Robertson, A. V. Kravtsov, N. Y. Gnedin, et al., “Computational Eulerian Hydrodynamics and Galilean Invariance,” Mon. Not. R. Astron. Soc. 401 (4), 2463-2476 (2010).
  21. I. Schuessler and D. Schmitt, “Comments on Smoothed Particle Hydrodynamics,” Astron. Astrophys. 97 (2), 373-379 (1981).



How to Cite

Стояновская О. Numerical Simulation of Gravitational Instability Development and Clump Formation in Massive Circumstellar Disks Using Integral Characteristics for the Interpretation of Results // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2016. 17. 339-352. doi 10.26089/NumMet.v17r332



Section 1. Numerical methods and applications