 Exponential spherical mass factor

ID: exponential-spherical-mass-factor

ρ (r) = A r^{- 2} (1 - e^{- r / r_{0}})

within radius

R > 0

, integration against the spherical volume element gives

M = 4 π A R f (r_{0} / R)

q = r_{0} / R > 0

has

f (q) = \int_{0}^{1} (1 - e^{- s / q}) d s, f^{'} (q) = - 1 + (1 + 1/ q) e^{- 1/ q} < 0, f^{''} (q) = q^{- 3} e^{- 1/ q} > 0.

Thus

f

is positive, strictly decreasing and convex, with

f (0^{+}) = 1

and

f (\infty) = 0

. Its asymptotic forms are

f (q) = 1 - q + q e^{- 1/ q}

near zero and

f (q) = 1/ (2 q) - 1/ (6 q^{2}) + O (q^{- 3})

at infinity. The density's central

1/ r

divergence is integrable against the factor

r^{2} d r

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