Numerical Runge- Kutta (4, 4) Method to determine the Initial Configurations of the Jupitar Proto Planet Formed Via Disk Instability

Authors

  • M. S.Ali Assistant Professor, Department of Mathematics, Bangabandhu Sheikh Mujibur Rahman Science and Technology University, Gopalganj-8100, Dhaka, Bangladesh
  • M.A A.Mamun Lecturer of Mathematics, Bangladesh Army University of Engineering & Technology (BAUET) , Natore-6431, Bangladesh
  • M. M. Rahman Lecturer of Mathematics, Bangladesh Army University of Engineering & Technology (BAUET) , Natore-6431, Bangladesh
  • R. R. Mondal Research student, Department of Mathematics, Bangabandhu Sheikh Mujibur Rahman Science and Technology University, Gopalganj-8100, Dhaka, Bangladesh

Keywords:

Runge- Kutta (4, 4) Algorithm, Jupitar Protoplanet, Schwarzschild transformations, Clapeyron Equation, Disk Instability Models.

Abstract

Numerical computational technique(s) with simulation tool is one of the most important difficult tasks in order to carry out real time scientific astronomical and other sophisticated problems. The main focus and highlight of this paper is concerned with the introduction of a method Runge-Kutta (4, 4) technique to determine the distribution of thermodynamic variables inside protoplanets during pre-collapse stage, formed by gravitational instability, for protoplanetary masses between 0.3 to 10 Jupiter. The case of convection is a significant concern for transference of heat inside the protoplanets and the graphical solution demonstrates positively better performance by the RK (4, 4) algorithm for any length of time. A viable quantitative analysis has been carried out to clearly visualize the goodness and robustness of the Runge-Kutta (4, 4) algorithm.

References

[1] Boss, A. P. "Formation of extrasolar giant planets: Core accretion or disk instability?." Earth, Moon, and Planets 81.1 (1998): 19-26.
[2] Helled, Ravit, and Gerald Schubert. "Core formation in giant gaseous protoplanets." Icarus 198.1 (2008): 156-162.
[3] Boley, Aaron C., et al. "Clumps in the outer disk by disk instability: Why they are initially gas giants and the legacy of disruption." Icarus 207.2 (2010): 509-516.
[4] Pollack, James B., et al. "Formation of the giant planets by concurrent accretion of solids and gas." icarus 124.1 (1996): 62-85.
[5] Hubickyj, Olenka, Peter Bodenheimer, and Jack J. Lissauer. "Accretion of the gaseous envelope of Jupiter around a 5–10 Earth-mass core." Icarus 179.2 (2005): 415-431.
[6] Matsuo, Taro, et al. "Planetary formation scenarios revisited: Core-accretion versus disk instability." The Astrophysical Journal 662.2 (2007): 1282.
[7] Boss, Alan P. "Testing disk instability models for giant planet formation." The Astrophysical Journal Letters 661.1 (2007): L73.
[8] Durisen, R. H., et al. "Protostars and Planets V, ed." B. Reipurth, D. Jewitt, & K. Keil (Tucson: University of Arizona Press) p 607 (2007).
[9] Cha, Seung-Hoon, and Sergei Nayakshin. "A numerical simulation of a ‘Super-Earth’core delivery from? 100 to? 8 au." Monthly Notices of the Royal Astronomical Society 415.4 (2011): 3319-3334.
[10] Nayakshin, Sergei. "Grain sedimentation inside giant planet embryos." Monthly Notices of the Royal Astronomical Society 408.4 (2010): 2381-2396.
[11] Paul, G. C., J. N. Pramanik, and S. K. Bhattacharjee. "Gravitational settling time of solid grains in gaseous protoplanets." Acta Astronautica 76 (2012): 95-98.
[12] Senthilkumar Sukumar, and Gour Chandra Paul. "Application of new RKAHeM (4, 4) technique to analyze the structure of initial extrasolar giant protoplanets." Earth Science Informatics 5.1 (2012): 23-31.
[13] Evans, D. J., and A. R. Yaakub. "A new Runge Kutta RK (4, 4) method." International journal of computer mathematics 58.3-4 (1995): 169-187.
[14] Yaakub, A. R., and David J. Evans. "A fourth order Runge–Kutta RK (4, 4) method with error control." International journal of computer mathematics 71.3 (1999): 383-411.
[15] De Campli, William M., and A. G. W. Cameron. "Structure and evolution of isolated giant gaseous protoplanets." Icarus 38.3 (1979): 367-391.
[16] Bodenheimer, Peter, Gregory Laughlin, and Douglas NC Lin. "On the radii of extrasolar giant planets." The Astrophysical Journal 592.1 (2003): 555.

Downloads

Published

2018-11-03

How to Cite

S.Ali, M., A.Mamun, M., M. Rahman, M., & R. Mondal, R. (2018). Numerical Runge- Kutta (4, 4) Method to determine the Initial Configurations of the Jupitar Proto Planet Formed Via Disk Instability. American Scientific Research Journal for Engineering, Technology, and Sciences, 48(1), 115–127. Retrieved from https://asrjetsjournal.org/index.php/American_Scientific_Journal/article/view/3899

Issue

Vol. 48 No. 1 (2018)

Section

Articles

License

Authors who submit papers with this journal agree to the following terms.