c) Derive the Schwarzschild criterion for the onset of convection in an ideal gas, namely dln T d ln P 7-1 Y Explain all steps in your derivation, and justify any assumptions that you make.
c) Derive the Schwarzschild criterion for the onset of convection in an ideal gas, namely dln T d ln P 7-1 Y Explain all steps in your derivation, and justify any assumptions that you make.
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Transcribed Image Text:c) Derive the Schwarzschild criterion for the onset of convection in an ideal gas, namely
d ln T
d ln P
7-1
Y
Explain all steps in your derivation, and justify any assumptions that you make.
d) In a region of convective instability near the surface of a solar-type star of total mass M, the temperature and pressure are
related approximately by the expression P KT5/2. Show that the temperature gradient for an ideal gas in hydrostatic
=
equilibrium in this convection zone is given by
dT
dr
2Gm(r)μ
5Rr²
Further, assuming that the mass in the convection zone is small compared to M, show that at a depth h measured from the top of
the convection zone, the temperature is approximately given by
T = Ts +
2GMμ
-h₂
5RR²
when his small compared to R, and Ts is the temperature at the top of the convection zone.
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