The Differential Equations of Stellar Evolution

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


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:
$$ \frac{{\partial r}}{{\partial m}} = \frac{1}{{4\pi {r^2}\varrho }} $$
$$ \frac{{\partial P}}{{\partial m}} = - \frac{{Gm}}{{4\pi {r^4}}} - \frac{1}{{4\pi {r^2}}}\frac{{{\partial^2}}}{{\partial {t^2}}} $$
$$ \frac{{\partial l}}{{\partial m}} = {\varepsilon_n} - {\varepsilon_v} - {c_P}\frac{{\partial T}}{{\partial t}} + \frac{\delta }{\varrho }\frac{{\partial P}}{{\partial t}} $$
$$ \frac{{\partial T}}{{\partial m}} = \frac{{GmT}}{{4\pi {r^4}P}}\nabla $$
$$ \frac{{\partial {X_i}}}{{\partial t}} = \frac{{{m_i}}}{\varrho }(\sum\limits_j {{r_{{ji}}}} - \sum\limits_k {{r_{{ik}}}} ) $$
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):
$$ \nabla = {\nabla_{{rad}}} = \frac{3}{{16\pi acG}}\frac{{klP}}{{m{T^4}}} $$
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.


Time Derivative Unknown Variable Stellar Evolution Hydrostatic Equilibrium Convective Region 
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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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