D2 - surface brightness profiles

quantify the behaviour of $I$ or $μ$ as a function of radial distance from the galactic centre
the Sersic law is a general law that describes the radial surface brightness profiles of both elliptical and spiral galaxies:

I (r) = I_{0} \exp (- {(\frac{r}{a})}^{\frac{1}{n}})

where, $I_{0}$ is the surface brightness at the centre, and $a$ is the scale length, and $n$ is the Sersic index

$a$ is the typical radial distance over which $I$ decreases by a factor of $e$
the Sersic index describes the steepness of the surface brightness falloff

image: Gao, et al (2009)

galaxies fill a whole distribution in $n$
typically, spirals $n = 1 - 2$ and ellipticals $n = 4 - 6$ at the peak of the distribution

exponential profile ( $n = 1$ )

I (r) = I_{0} \exp (- \frac{r}{a})

this is known to fit well with disks of spirals, dSph, and dE

L_{t o t} = \int_{0}^{\infty} I_{0} \exp (- \frac{r}{a}) 2 π r d r = 2 π a^{2} I_{0}

in practice, define an effective radius $R_{e}$ that contains $50 %$ of $L_{t o t}$
can simply replace the upper bound of $\infty$ with $X = R_{e} / a$

\begin{aligned} L_{e f f} & = 0.5 L_{t o t} = π a^{2} I_{0} \\ = - 2 π a^{2} I_{0} [e^{- x} (x + 1)]_{0}^{X} \\ 1 & = - 2 e^{- X} (X + 1) + 2 \\ \frac{1}{2} & = (1 + X) e^{- X} \end{aligned}

this can be solved numerically, resulting in $X = 1.69$ , this $R_{e} = 1.69 a$

I (R_{e}) = I_{0} e^{- 1.69} \approx 0.184 I_{0}

de Vaucouleur's profile ( $n = 4$ )

I (r) = I_{0} \exp [- {(\frac{r}{a})}^{\frac{1}{4}}]

fits well for gE, E and bulges of spirals

L_{t o t} = 8! π a^{2} I_{0} R_{e} = 3461 a

image: Sanchez-Portal, et al (2004)

$n = 4$ has a much steeper falloff

\frac{L_{b u l g e}}{L_{d i s k}} = \frac{8! I_{0, b u l g e}}{2 I_{0, d i s k}}

if the ratio is close to $1 :$ very early spirals/lenticulars
if the ratio is close to $1 : 0.$ late spirals (Sd/SBd)