Advertisement

Celestial Mechanics and Dynamical Astronomy

, Volume 118, Issue 3, pp 291–298 | Cite as

Approximate analytic solutions to the isothermal Lane–Emden equation

  • R. IaconoEmail author
  • M. De Felice
Original Article

Abstract

We derive accurate analytic approximations to the solution of the isothermal Lane–Emden equation, a basic equation in Astrophysics that describes the Newtonian equilibrium structure of self-gravitating, isothermal fluid spheres. The solutions we obtain, using analytic arguments and rational approximations, have simple forms, and are accurate over a radial extent that is much larger than that covered by conventional series expansions around the origin. Our best approximation has a maximum error on density of 0.04 % at 10 core radii, and is still within 1 % from an accurate numerical solution at a radius three times larger.

Keywords

Isothermal Lane–Emden equation Bonnor–Ebert gas spheres  Padé approximants 

References

  1. Baker Jr, G.A.: Theory and application of the Padé approximant method. In: Brueckner, K.A. (ed.) Advances in Theoretical Physics, vol. 1, pp. 1–58. Academic Press, New York (1965)Google Scholar
  2. Bender, C.M., Milton, K.A., Pinsky, S.S., Simmons Jr, L.M.: A new perturbative approach to nonlinear problems. J. Math. Phys. 30, 1447 (1989)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. Binney, J., Tremaine, S.: Galactic Dynamics. Princeton University Press, Princeton (1987)zbMATHGoogle Scholar
  4. Bonnor, W.B.: Boyle’s law and gravitational instability. Mon. Not. R. Astron. Soc. 116, 351 (1956)ADSMathSciNetGoogle Scholar
  5. Ebert, R.: Uber die Verdichtung von H I-Gebieten. Zeitschrift fur Astrophysik 37, 217 (1955)ADSzbMATHGoogle Scholar
  6. Hunter, C.: Series solutions for polytropes and the isothermal sphere. Mon. Not. R. Astron. Soc. 328, 839847 (2001)CrossRefGoogle Scholar
  7. King, I.R.: The structure of star clusters. I. An empirical density law. Astron. J. 67, 471 (1962)ADSCrossRefGoogle Scholar
  8. Lampert, M.A., Martinelli, R.A.: Solution of the nonlinear Poisson–Boltzmann equation in the interior of charged, spherical and cylindrical vesicles. I. The high-charge limit. Chem. Phys. 88, 399–413 (1984)ADSCrossRefGoogle Scholar
  9. Liu, F.K.: Polytropic gas spheres: an approximate analytic solution of the Lane–Emden equation. Mon. Not. R. Astron. Soc. 281, 1197–1205 (1996)ADSCrossRefGoogle Scholar
  10. Mirza, B.M.: Approximate analytical solutions of the Lane–Emden equation for a self-gravitating isothermal gas sphere. Mon. Not. R. Astron. Soc. 395, 2288–2291 (2009)ADSCrossRefGoogle Scholar
  11. Natarajan, P., Lynden-Bell, D.: An analytic approximation to the isothermal sphere. Mon. Not. R. Astron. Soc. 286, 268–270 (1997)ADSCrossRefGoogle Scholar
  12. Pandey, R.K., Kumar, N.: Solution of Lane–Emden type equations using Bernstein operational matrix of differentiation. New Astron. 17, 303–308 (2012)ADSCrossRefGoogle Scholar
  13. Van Gorder, R.A.: Analytical solutions to a quasilinear differential equation related to the Lane–Emden equation of the second kind. Celest. Mech. Dyn. Astron. 109, 137–145 (2011)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.ENEA, CR CasacciaRomeItaly

Personalised recommendations