Selection of Homework Questions
Topic 7: Ellipticals
(1) General :
 To first order, elliptical galaxies appear to
be an extremely homogeneous group.
Describe some of the 2 and
3 parameter correlations which support this claim, explaining what
physical principals lie behind the correlations.
 On closer scrutiny, ellipticals seem to divide into two distinct
classes depending on the 2D shape of their isophotes.
How is this shape difference defined and in what other ways are these
two types different from eachother?
 It has been suggested that, while probably oversimplified, each type
has experienced a different formation scenario. Outline these
two possible scenarios and state which one goes with which type of
elliptical galaxy.
(2) The Sersic Brightness Profile
The radial surface brightness distribution (in flux units) for a wide range of
systems can be described by the Sersic Law:

(Q7.1) 
where R_{e} encloses half the total light.
 Give the equivalent expression using surface brightness measured in magnitudes, µ
 Express the central surface brightness, I(0) and µ(0), in terms of
the surface brightness at the effective radius, I(R_{e}) and µ(R_{e}). Hence find equivalent expressions for I(R) and µ(R) using I(0) and µ(0).
 The special case n=1 yields an exponential profile which is usually expressed in terms of the scale length R_{d} : I(R) = I(0) exp(R/R_{d}), so that I(R_{d}) = 1/e ×: I(0).
Show that L_{tot} = 2 R_{d}^{2} I(0), and that R_{e} = 1.67 R_{d}. Hence show that b = 1.68 for this exponential disk. [Please don't be confused by
the subscripts here: R_{e} encloses half the light (e for effective), while R_{d} marks
the radius at which the surface brightness has fallen to 1/e of the central surface brightness (d for
disk, since disks are close to exponentials)].
 Show that the definition of R_{e}, namely _{0}^{} 2R I(R) dR = 2 × _{0}^{Re} 2R I(R) dR, can be recast as _{0}^{} x^{2n1} e^{bx} dx = 2 × _{0}^{1} x^{2n1} e^{bx} dx. Hence, using numerical methods, show that b = 1.999n  0.327 (for n = 1  8), giving b in terms of n. [Hint: you will need to evaluate the integrals numerically, as well as hunt
for the appropriate b that satisfies the equation for each n. Having done that, plot b vs n and confirm that the line b = 1.999n  0.327 goes through the points].
 Using your previous results, how does I(0) / I(R_{e}) [or equivalently, µ(0)  µ(R_{e}) ] depend on n? Are ellipticals (n ~ 4) more or less concentrated than disks (n ~ 1)?
 For M87 and M32, look up B_{T} and A_{e} in RC3 and,
assuming they both fit the R^{1/4} law at all radii, calculate
the factor by which their centers outshine the moonless sky, which has
µ_{B} = 22.7. Also, estimate D_{25} and compare it with the
value given in RC3.
(3) Galaxy Shapes
Of course, we only ever see galaxies in
projection on the sky. And yet, we feel we have significant knowledge
of their 3D shapes. Describe the observations and line of reasoning
that has been taken to ascertain, statistically, the 3D shapes of
(a) ellipticals
(b) spiral disks.