E = - E = Ψ - v^{2} /2

. At fixed

r

, write velocity-space spherical coordinates with polar angle

α

from the radial direction, so

L = r v sin α

. Then

ρ = 2 π f_{0} r^{- 2 β} \int_{0}^{2Ψ} v^{2 - 2 β} (Ψ - v^{2} /2)^{n}, d v \int_{0}^{π} sin^{1 - 2 β} α d α .

ρ (r) = A r^{- 2 β} Ψ^{n + 3/2 - β},

where

A = f_{0}, 2^{3/2 - β} π^{3/2} \frac{Γ ( 1 - β ) Γ ( n + 1 )}{Γ ( n + 5/2 - β )} .

Thus

γ = 2 β

and

p = n + 3/2 - β

, with convergence for

β < 1

and

n > - 1

Comparison with part a gives

(β_{1}, n_{1}) = (\frac{3}{4}, \frac{5}{4}), (β_{2}, n_{2}) = (\frac{1}{2}, 2) .

f (E, L) = f_{01} L^{- 3/2} E^{5/4} + f_{02} L^{- 1} E^{2},

where matching the two density coefficients gives

f_{01} = \frac{3 2 ^{5/4} b}{5 π ^{5/2} G ^{2} M Γ ( 1/4 ) ^{2}}, f_{02} = \frac{3 b}{4 π ^{3} G ^{3} M ^{2}} .

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