D3 - power laws, behaviour at small radii

surface brightness profiles often deviate from the Sersic law at very small radii, eg: close to nuclei
they behave as a power law:

I (r) = I_{o} r^{- α}

if steeper than Sersic: cusp; if shallower: core

image: Graham, et al (2003)

considering a hypothesis that the power law in the surface brightness comes from a spherical distribution of star with a density profile that is also a power law

the number density of star: $N (R) \propto R^{- γ}$

I (R) = N (R) ⟨ \frac{L}{s t a r} ⟩ \propto N (R)

this is valid for a constant average luminosity per star
$h$ is along the line of sight, perpendicular to $R$
the circle defining the spherical distribution has distance $r = \sqrt{R^{2} + h^{2}}$ from the galactic centre

N (R) = \int_{0}^{\infty} n (r) d h \propto \int_{0}^{\infty} [r (h)]^{- γ} d h = \int_{0}^{\infty} [\sqrt{R^{2} + h^{2}}]^{- γ} d h

making a change of variable $u = h / R :$

N (R) = R^{1 - γ} \int_{0}^{\infty} (u^{2} + 1)^{γ / 2} d u

the integral is now a number (dependent n $γ$ )

I (R) \propto R^{1 - γ}

therefore, the a spherical distribution with a power law $e^{- γ}$ results in a power law in $I$ with $e^{1 - γ}$
eg: NGC 596 with $I (R) \propto R^{- 0.55}$ would have $n (r) = r^{- γ}$ with $γ = - 1.55$
this is a very steep distribution
they are thought to experience circumnuclear outbursts, possibly due to gas falling onto a massive central object (SMBH), getting compressed, and triggering an episode of enhanced star formation