Question 35.4: Using the principle of homology for a star of total mass M a......
Using the principle of homology for a star of total mass M and radius R, show that (a) p(r) ∝ R^{−4 } and (b) T(r) ∝ R^{−1 }.
(a) The equation for hydrostatic equilibrium, eqn 35.16, states that
\frac{dp}{dr} = -\frac{Gm(r)ρ(r) }{r^{2} }. (35.16)
\frac{dp}{dr} = \frac{Gm(r)ρ(r) }{r^{2} }, (35.54)
so using ρ ∝ MR^{−3 } and writing dp/dr = p_{c}/R, we deduce that
\frac{p_{c} }{R } ∝ M^{2} R^{−5} . (35.55)
Equation 35.55 means that p_{c} ∝ M^{2} R^{−4}, and using the principle of homology
p(r) ∝ M^{2} R^{−4} (35.56)
(b) We next consider a relationship for scaling the temperature throughout a star. Our starting point this time is the ideal gas law, which we met in eqn 6.18, from which we may write the following:
\boxed{pV = Nk_{B} T, } (6.18)
T(r) ∝ \frac{p(r) }{ρ(r)}. (35.57)
Using ρ ∝ MR^{−3 } and eqn 35.56, we have
T(r) ∝ MR^{−1 }. (35.58)
Hence as the star shrinks, its central temperature increases. Note that this does not give information on the surface temperature T(R), since this depends on the precise form of T(r).