Selection of Homework Questions

   

Topic 7: Ellipticals

1. 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.

2. On closer scrutiny, ellipticals seem to divide into two distinct classes depending on the 2-D shape of their isophotes. How is this shape difference defined and in what other ways are these two types different from eachother?

3. It has been suggested that, while probably over-simplified, 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.

1. Give the equivalent expression using surface brightness measured in magnitudes, µ

2. 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).

3. 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_d² 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)].

4. Show that the definition of R_e, namely   ₀  2R I(R) dR  =  2 × ₀^R_e  2R I(R) dR, can be recast as ₀  x^2n-1 e^-bx dx   =   2 × ₀¹  x^2n-1 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].

5. 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)?

6. 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 D₂₅ 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 3-D shapes. Describe the observations and line of reasoning that has been taken to ascertain, statistically, the 3-D shapes of
  (a) ellipticals
  (b) spiral disks.