Analytic and numerical calculations of the radial stability of the isothermal sphere
A. C. Raga,1 J. C. Rodríguez–Ramírez,1 A. Rodríguez–González,1 V. Lora,2 and A. Esquivel1
1 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apdo. Postal 70–543, 04510 D.F., México. (raga@nucleares.unam.mx, juan.rodríguez@nucleares.unam.mx, ary@nucleares.unam.mx).
2 Astronomisches Rechen–Institut Zentrum für Astronomie der Universität Heidelberg, Mönchhofstr. 12–14, 69120 Heidelberg, Germany. (verolora@gmail.com).
Received 2012 December 3. ]]> Accepted 2013 February 14.
RESUMEN
Usamos una solución analítica aproximada de toda la extensión radial de una esfera no singular, isotérmica, autogravitante para derivar analíticamente las propiedades generales de las esferas resultantes, y su estabilidad a perturbaciones radiales. Derivamos el criterio de estabilidad de Bonnor y Ebert, y confirmamos analíticamente sus resultados (numéricos). Finalmente, calculamos simulaciones esféricamente simétricas de las ecuaciones de dinámica de gases Lagrangeanas, con dependencia temporal, mostrando que la transición entre soluciones estables e inestables sí ocurre en un valor del radio exterior de la esfera cercano al obtenido del criterio de estabilidad de Bonnor.
ABSTRACT
We use an approximate, analytic solution to the full radial extent of the non–singular, isothermal, self–gravitating sphere to derive analytically the general properties of the resulting spheres, and their stability to radial perturbations. We rederive the stability criterion of Bonnor and Ebert, and confirm analytically their (numerical) results. Finally, we compute spherically symmetric simulations of the time–dependent, Lagrangean, gas–dynamic equations, showing that the transition between stable and unstable solutions does occur for a value of the outer radius of the sphere close to the one obtained from Bonnor's stability criterion.
Key Words: galaxies: halos — ISM: clouds — stars: formation.
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]]>ACKNOWLEDGMENTS
We acknowledge support from the Conacyt grants 61547, 101356, 101975, 165584 and 167611, and the DGAPA–Universidad Nacional Autónoma de México grants IN105312 and IN106212. We thank an anonymous referee for several sugestions (out of which arose the discussion at the end of § 4, with equations 18 and 19, and to equations 24–26 in § 6).
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