From the CCD images of a nearly edge-on, disk galaxy NGC 666, it is known that it has a negligible bulge (type Sc), no color gradients, and a disk with an exponential brightness profile corresponding to surface density
Sigma(R) = Sigma_0(R) exp[ -R/R_d ]
where R_d = 2.8 +-0.15 kpc. (Notice this value does have
uncertainty!).
The total bolometric luminosity of the disk is L_{tot}=(3+-0.1) 10^{10}
L_sun (solar luminosities).
The H I radial velocity profile yields the following average rotation curve, which has an intrinsic +-3.5 km/s error on V_c (please note that the figure below shows a larger, 7 km/s error bar - it's the "2-sigma" value you shouldn't use in the calculations).
R(kpc) V_c(km/s) 1.3 123. +-3.5 2.6 183. 3.6 205. 5.0 226. 6.1 238. 7.4 238. 8.8 230. 10.0 230. 11.4 223. 12.5 222. 13.8 220. 15.1 220. 16.3 218. 17.6 221. 18.8 219. 20.1 224. 21.2 221. 22.5 223. 23.8 217. 24.9 219. 26.1 219. 27.5 219. 28.6 221. 30.0 221. 31.3 221.
This rotation curve suggests the presence of a (dark) halo in NGC666. Perform the decomposition of the rotation curve into the parts due to the disk and the halo, along the following lines.
0. Read van Albada et al (1985, ApJ 295, 305), and Kent (1986, AJ 91,
1301), and refer to them if in doubt about the idea of the "maximum disk"
method. See also chapter 10.4 of the Gilmore, King, and van der Kruit book
"The Milky Way as a Galaxy".
Why do we insist on maximizing the disk contribution?
1. Using an appropriate formula for v_c(R) due to the infinitely thin
and radially untruncated exponential disk write a program (subroutine)
that computes the rotation curve for any pair of disk parameters: R_d and
Sigma_0, or equivalently R_d and M_d (total disk mass).
Hint: You may want to use the Bessel
functions subroutines from Numerical Recipes.
2. Fit the inner part of the rotation curve (roughly until the kink or maximum, not further than r ~ 2 R_d using as massive disk as possible, by starting from too high disk mass, and lowering it until a good fit is found. To determine how good the fit is use the standard chi^2 error function.
3. The remaining discrepancy between your theoretical and the observed
v_c can be modeled using the dark halo density-potential pair given by
Kent(1986):
rho(r) = rho_0 (1+r^2/a^2)^{-1}
and
v_h^2(r) = 2 sigma^2 [1 - (r_h /r) arctan (r/r_h)]
Verify this pair! Use either rho_0 or the dispersion sigma (and the core radius r_h) as free parameters while fitting the data. Minimize chi^2 to find the values of two halo parametrs.
4. Plot v_c^2 (not v_c): the observations, the total model, and its components, in the same figure. Compute Sigma_0 in M_sun / pc^2 from M_d or vice versa, and the same with sigma and rho_0 (again, don't use c.g.s. when you quote results!). What is the surface density of the disk at R=8.5 kpc from the center of NGC 666? How does it compare with the density of the Galaxy at the solar radius?
5. Plot the mass-to-light ratio Upsilon (r<R) of the matter inside radius R. Assume completely non-luminous halo and Upsilon_disk following from your maximum-disk model. Discuss results, compare with known galaxies.
6. Estimate the standard error in each quantities you have determined for the disk and halo, explain how you did this. Discuss also the value of chi^2 you achieved.
7. How unique is your decomposition? To find out reduce the disk density to 1/3 of the maximum value you have found, and then find the halo necessary in that case. Is the fit (chi^2) much worse?
8. How small a disk can you assume and still get reasonable fit? Can you get a good fit with "zero disk"?
The last two points are somewhat optional, but you will earn much extra
credit (equivalent to a
separate problem) when you do them:
9. Do not use maximum disk method. Instead, invent and perform a procedure
for fitting many parameters at the same time (for both disk and halo).
Describe how much the values and the uncertainties of those
values changed with respect to the maximum disk procedure. Numerical
Recipes will, as always, provide
a useful guide to error estimation.
10. For extra credit, generalize your exponential disk subroutine to allow for a finite thickness of the disk, and/or a truncated disk. Describe the relative changes to the v_c in those generalized cases. Establish and motivate some reasonable values of the thickness, and the sharp optical edge. What are the inferred halo parameters, mass-to-light ratio etc. that they imply? Did you get a better fit than with a thin disk?
NGC 666 was invented for the purpose of this problem - then actually discovered by Anja Grigorieva! And it kind-of fits the invented description. I don't see any bulge or color gradients, do you? Congratulations, Anja! (and yes, that's an extra credit :-)
Next challenge: who'll confirm that it has an exponential disk
with scale length 2.8+-0.15 kpc?
1. Derive the potential and the rotation curve of a 2-D Kuzmin disk with
Sigma(R) = Sigma_0 (1+R^2/R_0^2)^{-3/2}
using the transform methods in the cylindrical coordinates (Hankel transform):
Sigma(R)---Hankel-transform------> Sigma(k) -------->
-------->Phi(k) = -2 pi G Sigma(k) /|k| ----Hankel-transform--------->
-----> Phi(R) ----------> v_c(R).
Hint: Some integrations involved can be done analytically using the Gradsteyn-Ryzhik tables of integrals.
2. Try to generalize to finite-thickness Kuzmin disk with effective thickness H. The specific choice of the vertical disk density profile rho(R,z) as a function of z is up to you; try simple functions (top-hat, exponential) that make the analytical integration possible. Discuss the source of any difficulties you encounter. Can you find potential analytically? What about the forces (accelerations)? Compare the 2-D and the "somewhat thick" Kuzmin disk rotation curves (chose reasonable H).
(As always, do not assume that a problem either (i) has a simple
solution, (ii) has a complicated solution,
or (iii) has a solution. Like in a real research, do your best, go
as far as you can... then describe what you found.)
Using the results of BT problem 2-9, find the potential of the 2-D and 3-D "Kuzmin bar" rotating with angular velocity Omega_p. The 3-D object has vertical exponential profile with scale-heigth H:
rho (R,theta, z, t) = rho_2 (1+R^2/R_0^2)^{-3/2}
exp (-|z/H|) exp [i(m theta-Omega_p t) ]
where m=2 (a bi-symmetric bar).
How does one compute radial and azimuthal forces from the potential Phi(R,theta, z, t) ? Rotation curve?