Question 35.5: Determine the scaling of the luminosity L with M and R for (......

By the principle of homology, a temperature increment dT scales in the same way as T, which eqn 35.58 gives as T(r) ∝ MR^{−1}. An increment in radius, however, scales with radius, i.e. dR ∝ R. Therefore the temperature gradient follows dT/dr ∝ MR^{−1}/R, giving

\frac{dT}{dr} ∝ MR^{−2} .           (35.61)

Equation 35.45 becomes

\frac{dT}{dr} = -\frac{3κ(r)ρ(r)L(r) }{64πr^{2} σ[T(r)]^{3} }          (35.45)

\frac{L(r) }{r^{2} }∝ − \frac{T(r)^{ 3} }{ρ(r)κ(r)} \frac{dT}{dr},          (35.62)

and hence in case (a), for which κ(r) ∝ ρ(r)T(r)^{ -3.5}, we find

L(r) ∝ \frac{M^{5.5} }{R^{0.5} }.          (35.63)

The assumption of homology means that if the luminosity at any radius r scales as M^{5.5}R^{-0.5}, then the surface luminosity scales in this way, so we may write

L ∝ \frac{M^{5.5} }{R^{0.5} }.           (35.64)

For case (b), since κ(r) is a constant, we find L(r) ∝ M³ and hence

L ∝ M³.            (35.65)