Week 6 (Nov 7 and 9) In-Class Problems

-------------------------Problem 1: The Spectra!! Emission-Line Diagnostics in Practice

For this problem, we are going to be working with ACTUAL spectra and measuring some emission lines to determine classic emission-line diagnostics for real galaxies. You will want to make sure you have the python package linetools installed. For instructions, see: https://linetools.readthedocs.io/en/latest/install.html Links to an external site.

For those of you not terribly familiar with linetools, I will show an example of how to use the GUI to measure lines. 

You are going to want to download the following three fits files: 

This is an SDSS optical spectrum: 

J0929+4644_172_157_spec.fits Download J0929+4644_172_157_spec.fits

This is from Keck/LRIS so it is split between blue and red sides. 

J0943+0531_106_34_bluespec.fits Download J0943+0531_106_34_bluespec.fits  and J0943+0531_106_34_redspec.fits Download J0943+0531_106_34_redspec.fits 

First, read in the SDSS spectrum of the galaxy J0929+4644 172_157 using the xspecgui that is part of the python package linetools.

For those of you not familiar with linetools or GUIs in python, here's an example of what I am able to start by running a simple ipython session: 

from linetools.spectra.xspectrum1d import XSpectrum1D

from linetools.guis import xspecgui as ltxsg

fname = 'J0929+4644_172_157_spec.fits'

sp = XSpectrum1D.from_file(fname)

ltxsg.main(sp)

That should give you a GUI. Typing "?" will list the possible commands. This link Links to an external site. is helpful. Play around with the GUI a bit! It is fun! 

Also, for book-keeping purposes here, it will be helpful if you make a table of your line measurements. Not a requirement, but it helps to start out being organized. You'll be measuring 6 emission lines per spectrum. And taking various ratios. 

a) Using the emission lines present in the spectrum, find the redshift of the galaxy J0929+4644 172_157. Hint: you can toggle to the "galaxy" line list, and adjust the redshift until you find a good fit. I always find that identifying [OIII] 5007 and Hbeta 4861 is the easiest way to get a redshift with these optical emission line spectra. If you click on a line that you think is the right line, you can select it from a pop up list by right-clicking...

b) For the galaxy  J0929+4644 172_157 measure the emission line ratio [SII] 6716/[SII] 6731. Hint: Use the 'G" key to fit a gaussian to the emission line and measure a line flux! Note this is the cooling luminosity!! Are we looking primarily at the high density or low density limit? 

c) For the galaxy  J0929+4644 172_157 measure the emission line ratio [OIII] 5007 / H$\beta$ . For H$\beta$ , it may be in an absorption trough, and you'll want to measure the line flux from the very bottom to make sure to get all the line flux. We'll call this ratio O3HB. 

d). For the galaxy  J0929+4644 172_157 measure the emission line ratio [NII] 6583/ H$\alpha$. We'll call this N2. 

e). Pettini and Pagel (2004) developed a nice abundance formula, using Cloudy, that uses O3HB and N2 to determine the average oxygen abundance of the HII regions that dominate the emission line fluxes in a galaxy. So, let's use their formulae to determine the oxygen abundance of J0929+4644 172_157. 

First, note that the Log (O3HB/N2) will be called O3N2 in the below equation. 

Pettini and Pagel's equations are as follows: 

For problems on heating and cooling this week, we'll need to be running some kind of computer program -- python notebook, IDL, Mathematica, whatever your preference, but we'll have to do a numerical calculation for these. It'll be an adventure! There are very few heating/cooling problems one can solve analytically, unfortunately. 

-------------------------Problem 2: 

For this problem, we will consider an HII region consisting of only Hydrogen. The source of ionizing photons is a blackbody of temperature $T\:=\:32,000\:K$. We will assume the HII region is in thermal and ionization equilibrium. 

a) Write down the equation for the primary heating source of the HII Region. (hint: Check out Lecture 8 PDF, slide 12). Explain what each term means in this equation. You may assume, as I did in my notes, that: 

$\alpha_B\left(T\right)\:\approx\left(4\:\times10^{-13}\:cm^3\:s^{-1}\right)\left(\frac{10^4K}{T}\right)^{^{0.73}}$

b) Write down the expression for the cooling due to recombination (hint: slide 17 of Lecture 8 PDF), and, in order to simplify things, we will assume that the recombination rate coefficient, $\beta_H\left(T\right)\:=\:0.86\:\alpha_B\left(T\right)$. Again, make sure you explain each term in the expression. 

c) Finally, we will assume the only other significant source of cooling in this HII region (recall: there are no metal lines!) is cooling due to free-free emission, or free electrons that scatter off of free protons and produce emission with a known power-law scaling. We will assume it scales as Draine gives in equation 27.24: 

$\Lambda_{free-free}=0.54\:n_en_p\alpha_B\left(T\right)\left(\frac{T}{10^4K}\right)^{0.37}kT$

Set up the heating and cooling balance equations, and simplify as much as possible. 

d) Plug this simplified equation into some program of your choosing, and solve for the equilibrium temperature of this H-only nebula. You may wish to solve this graphically, by plotting the heating and cooling and finding the temperature at which the curves intersect. 

e) Now imagine the nebula has some metallicity. Describe in a few sentences the additional cooling that results when we consider metal-line transitions, and how we would have to go about including it in our calculation for the HII region equilibrium temperature. 

------------------------(Probably will not get to this part...) --- STOP HERE. 

Now let's consider the cold-phase of the neutral ISM (Cold Neutral Medium, aka CNM). In the CNM, we learned that the cooling is strongly dominated by one particular metal species: the [CII] fine structure line at $158\:\mu m$ (see Draine Figure 30.1). In this problem, we will compute the equilibrium temperature for a gas in which the main coolant is [CII], for which $\frac{\Delta E}{k}\:=\:92\:K.$

[CII] has a critical density that is rather high, and thus we can assume it's cooling function behaves as we derived for metal-line cooling in the low-density regime (e.g. has a density-squared dependence). Specifically, any collisions that excite CII yield immediate radiative de-excitations that remove energy. We will solve for our equilibrium temperature assuming that the thermal pressure of the CNM is equal to the mean value in the solar neighborhood, $\frac{P}{k}\approx3000\:K\:cm^{-3}$ and that the fractional abundance of carbon relative to hydrogen is 0.00014 (Lecture 8, Slide 25 may be helpful to you) and that all free electrons come from carbon such that $n_{CII}\:=\:n_e\:=\:n_C$. For reference, the degeneracies of this transition are: $g_u=\:4;\:g_l\:=2$.  It may also be a useful reminder to note that the Boltzmann constant,  $k\:=\:1.38\times10^{-16}\:erg\:K^{-1}$. The collision rate coefficient $\Omega_{ul}\:=\frac{\:\left(1.55\:+\:1.25T_4\right)}{\left(1+0.35T_4^{1.25}\right)}$where $T_4\:\equiv\frac{T}{10^4\:K}$. 

a)  Write down the equation for the total cooling due to the [CII] fine structure line at $158\:\mu m$, $\Lambda_{\left[CII\right]}$. Reduce it as much as you can (e.g. to something easy to input into a program/notebook). 

b) Assume the only heating source is the dust grain photoelectric heating rate, $\Gamma_{dust}\:\approx n_H\:\left(6\:\times10^{-26}\:ergs\:s^{-1}\right)$. Solve for the equilibrium CNM gas temperature as a function of density. Recall that $nT\:=\frac{\:P}{k}$. You can plot T versus n and P/k versus n for $n\:=\:10\:-\:300\:cm^{-3}$. Numerically or graphically solve for the density that corresponds to the gas thermal pressure, given above, $\frac{P}{k}\approx3000\:K\:cm^{-3}$.