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The Differential Equations of Stellar Evolution

  • Rudolf Kippenhahn
  • Alfred Weigert
Chapter
Part of the Astronomy and Astrophysics Library book series (AAL)

Abstract

Collecting the basic differential equations for a spherically symmetric star derived in Chap. I, we are then led by (1.6), (2.16), (4.27, 28), (7.32), and (8.4) to:
$$ \frac{{\partial r}}{{\partial m}} = \frac{1}{{4\pi {r^2}\varrho }} $$
(9.1)
$$ \frac{{\partial P}}{{\partial m}} = - \frac{{Gm}}{{4\pi {r^4}}} - \frac{1}{{4\pi {r^2}}}\frac{{{\partial^2}}}{{\partial {t^2}}} $$
(9.2)
$$ \frac{{\partial l}}{{\partial m}} = {\varepsilon_n} - {\varepsilon_v} - {c_P}\frac{{\partial T}}{{\partial t}} + \frac{\delta }{\varrho }\frac{{\partial P}}{{\partial t}} $$
(9.3)
$$ \frac{{\partial T}}{{\partial m}} = \frac{{GmT}}{{4\pi {r^4}P}}\nabla $$
(9.4)
$$ \frac{{\partial {X_i}}}{{\partial t}} = \frac{{{m_i}}}{\varrho }(\sum\limits_j {{r_{{ji}}}} - \sum\limits_k {{r_{{ik}}}} ) $$
(9.5)
In (9.5) we have a set of I equations (one of which may be replaced by the normal-ization ∑ i X i =1) for the change of the mass fractions X i of the relevant nuclei i=1,…, I having masses m i . An additional formula (8.26) regulates the mixing of the composition in convective regions. In (9.3), \( \delta \equiv - {(\partial \ln \varrho /\partial \ln T)_P} \), and in (9.4), \( \nabla \equiv d\ln T/d\ln P \). If the energy transport is due to radiation (and conduction), then \( \nabla \) has to be replaced by \( {\nabla_{{rad}}} \), which is given by (5.28):
$$ \nabla = {\nabla_{{rad}}} = \frac{3}{{16\pi acG}}\frac{{klP}}{{m{T^4}}} $$
(9.6)
If the enerty is carried by convection, then \( \nabla \) in (9.4) has to be replaced by a value obtained from a proper theory of convection; this may be \( {\nabla_{{rad}}} \) in the deep interir, or obtained from a solution of the cubic equation (4.26) for superadiabatic convection in the outer layers.

Keywords

Time Derivative Unknown Variable Stellar Evolution Hydrostatic Equilibrium Convective Region 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Rudolf Kippenhahn
    • 1
  • Alfred Weigert
  1. 1.GöttingenGermany

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