Week 8 (Nov 28th, Nov 30th) In-Class Problems

Problem 1: Fun with the SiIV Doublet

SiIV is an 11-electron ion often observed along QSO sightlines in the rest-frame UV as a doublet at $\lambda_1\:=\:1393.755\:\:\left(^2P_{\frac{3}{2}}\longrightarrow\:^2S_{\frac{1}{2}}\right);\:\lambda_2\:=\:1402.770\:\left(^2P_{\frac{1}{2}}\longrightarrow\:^2S_{\frac{1}{2}}\right)$

with wavelengths, above, given in Angstroms. The oscillator strengths of the two transitions are: $f_{1393 } = 0.5280$ and $f_{1402 } = 0.2620$. And the collisional coefficients are $\gamma_{ul} = 8.8 \times 10^{8} s^{-1}$ for the SiIV 1393 line and $\gamma_{ul}=8.7\times10^8s^{-1}$ for the 1402 line; for simplicity, you can assume they are roughly equal. The below equation describes the ratio of the equivalent widths of the two lines of the SiIV doublet in each of the three regimes of the curve of growth: 

In fact, the approximation above can be applied to any doublet (this will be helpful in Problem 2)! Just replace the wavelength subscripts with the appropriate strong and weak line wavelengths of the doublet.

Part A: 

Determine the value of $\frac{W_{1393}}{W_{1402}}$ for the SiIV doublet shown in the spectrum plotted in the PDF below. 

SiIVAbsorptionLinesEWMeasurement.pdf Download SiIVAbsorptionLinesEWMeasurement.pdf

You will want to make as precise a measurement as you can for the dimensionless ratio of the equivalent widths of the SiIV doublet, and thus grid lines are provided atop the actual HIRES spectra, which have already been continuum-normalized for you. Note that there are a number of automated tools (e.g. python linetools) that will measure an EW for you given an input spectrum fits file with wavelength and continuum-normalized flux, but we are doing it this way for fun and to make sure we all know what an equivalent width actually is. 

Part B: 

As you can see in the given equation for $\frac{W_{1393}}{W_{1402}}$, the ratio of the two equivalent widths of the lines in the SiIV doublet is a sensitive indicator of whether we are on the "optically thin" (aka "linear") part of the curve of growth are not.

What regime of the curve of growth are we in given the ratio you measured? Discuss in a sentence or two what this means in terms for our measurements,  especially for any SiIV column density we may eventually measure. 

Part C: 

What constraints, if any,  can we place on $\tau_0$ in this case? 

Part D: 

Assume the EW you measured (as an area) for the 1393 line in Part A corresponds to a value of 50 mA (milli Angstroms). The Apparent Optical Depth Method (AODM) can be used to measure the column density of a transition in the optically thin limit (where $\tau_{0} \ll 1$), and Savage and Sembach (1996) equation 3 provides a convenient form of this equation commonly used by absorption-line spectroscopists (it can be derived from equation 9.12 in Draine's textbook for fun, but is not required here): 

$N (cm^{-2} ) = 1.13 \times 10^{17} \frac{W_{\lambda } (mA)}{f_{\lambda } \lambda^{2}(A) }$

Note that when a line is saturated (e.g. we are not in the optically thin limit) this method provides only a lower limit to the column density of an ion (e.g. SiIV). 

Use the AODM to determine the column density of SiIV in our given spectrum and discuss whether it is a true measurement or a lower limit.

Part E: 

Now, let’s assume you’ve run a Cloudy model on a full suite of ionic transitions, and you have determined that the amount of N_SiIV  in Part D represents about 10% of the total Silicon column density, N_Si. Assume you know the total Hydrogen column density is N_H = 10¹⁹ cm^-2 along this line of sight. Assume the solar Si abundance by number is 12 + log₁₀(Si/H) = 7.51 (Asplund 2009).  What can you say about  the abundance of silicon relative to solar of this SiIV absorber?    

Part F: 

Recall that the FWHM of an absorption line due to a purely thermal velocity distribution (e.g. Doppler broadening) is given by the following relation: 

$\left(\Delta v\right)_{FWHM}^{therm}=\sqrt{8\ln2}\sigma_{v,\:therm}\:=\:2\sqrt{2\ln2}\left(\frac{kT}{M}\right)^{\frac{1}{2}}\:=\:2.15\:\left(\frac{\frac{T}{100K}}{\frac{M}{m_H}}\right)^{\frac{1}{2}}\:km\:s^{-1}$

Where M is the atomic mass and T is the temperature of the gas assumed to be in thermal equilibrium (or at least kinetic equilibrium where the velocities obey a Maxwellian distribution). 

Also, recall that true observed line profiles are the result of a combination of thermal broadening, $\sigma_{v,\:therm} =\sqrt{\frac{KT }{M } }$,  plus some non-thermal broadening (i.e. turbulence) that has Gaussian velocity distribution with one-dimensional velocity dispersion: $\sigma_{v,\:turb}$. 

If the turbulence has a Gaussian velocity distribution, the observed overall velocity distribution function of atoms of mass M will be Gaussian, with the total one-dimensional velocity dispersion: 

$\sigma_v^2\:=\:\sigma_{v,turb}^2\:+\frac{\:kT}{M}$

Now let's assume you are measuring this SiIV in a gas cloud that you suspect is approximately 10,000 K. Let's also assume for this part only that the SiIV is optically thin (e.g. not saturated) and that you have measured the FHWM of HI along the same line of sight (neutral hydrogen) to be 40 km s^-1  .

 - What can you say about $\sigma_{v,\:turb}$ in this case? 

 - What do you expect to measure for the line width of SiIV in this case assuming that turbulence affects SiIV and HI-bearing gas equally?

 - If you instead measure the SiIV line width to be 50 km s^-1 what are two possible physical interpretations for this SiIV-absorbing cloud? 

Problem 2: Continuing the Fun with Doublets and Absorption Lines

In this problem, we will analyze a real Quasar spectrum from the HST Cosmic Origins spectrograph. You can find it here: J1241+5721_nbin3_jwnorm.fits Download J1241+5721_nbin3_jwnorm.fits 

It is already uploaded to dirac.lsst.dev if you need to use the virtual machine to get xspecgui to work! You'll find it in lincc/groups/uw-werksquad/data.

**Remember, on the virtual machine once you are signed in with your github credentials, you start a new desktop, go to the proper directory (see above), and then type "conda activate werksquad" in a terminal. 

Otherwise go ahead and download the spectrum to wherever you work for this class. 

Everybody should be able to run the script lt_xspec from the command line! Just try typing, "lt_xspec J1241+5721_nbin3_jwnorm.fits --norm" ! 

There are a few things to know about this spectrum. 

1. The QSO itself is at a redshift of 0.583. 

2. There are 4 strong intervening absorption systems detected along this line of sight at z = 0.147, 0.2055, 0.218, and 0.35566

3. I've already calibrated it, binned it, and normalized it for you so it is ready to analyze! 

Part A: 

Load the spectrum into the Xspec GUI (note: the lt_xspec script works well for this!), and if you like, use the continuum-normalized version by executing the:    " --norm option." Look at the absorption lines that are detected at each redshift. Which absorption system is your favorite, and why? Do any of these absorption systems show SiIV? Which ones? Which ones show OVI? 

Part B: 

Consider the OVI doublet in each absorption system (where available).  Measure the equivalent width ratio of the OVI 1031 and OVI 1037 lines for the four systems using the "E" key, and determine on what part of the curve of growth these OVI transitions lie for the relevant systems, recognizing that the OVI 1031 line should have about two times the rest-frame equivalent width of the 1037 line if they are optically thin (linear part of the COG).  

Part C: 

Now, focus on the absorption system at z = 0.2055. Construct a curve-of-growth for the HI Lyman series absorption lines and the four detected SiII absorption lines (1190, 1193, 1260, 1304).   Recall that for the linear part of the curve of growth:

$N (cm^{-2} ) = 1.13 \times 10^{17} \frac{W_{\lambda } (mA)}{f_{\lambda } \lambda^{2}(A) }$

Note the units above are specified so you don't have to struggle with them. 

Specifically,  you will want to plot Log $(W_{\lambda}/\lambda)$ in (mA/A) vs Log $(Nf\lambda)$ in (cm^-2 A).   Overplot the theoretical form for the linear  part of the curve of growth (given above). For the equivalent widths, note that when you measure them the Xspec GUI with the "E" key, it gives you a rest-frame equivalent width in milli-Angstroms. In addition, the screen print out gives you the transition wavelength in Angstroms and the dimensionless oscillator strength. You'll need to record these values to make your plot. For the column densities note you can use the "N" key in the same way as the "E" key. For the column densities you use, I want you to use the same column density value for each transition, and base it off of the weakest line in the series available to measure (e.g. SiII 1304) so that it is not saturated. HI is likely saturated for all of the Lyman series lines in the system at z = 0.2055. I have some other COS data that tells me log N_HI for this system is actually well-constrained to be 10^18.4 cm^-2 - go ahead and use this value. Once you have your plot, comment on which SiII and HI transitions are on the linear part of the COG, and which ones veer into the flat part of the curve of growth. (also note for the SiII 1190 line you will have to carefully measure around the SII 1190 line - yes, it's annoying!). Explain in words what absorption-line saturation means, now in the context of the curve of growth. 

STOP HERE, next parts are totally not required. 

Part  D: 

Take the best column density measurements of the "ion species" of SiII, SiIII, and SiIV for the absorption system at z = 0.2055 (e.g. the highest log N you measure from the weakest transition of each species). Sum the three column density values together to get an estimate of the total Silicon column density. What is this value and do you think it is likely to be a lower limit or an upper limit? What is the fraction of total Silicon in SiII? What fraction of the total Silicon is in SiIV? 

Part E: 

Use log N_HI for this system = 10^18.4 cm^-2. And assume this represents only 1% of the total Hydrogen column density of this absorber - the rest is in ionized hydrogen HII. What is the total Hydrogen column? 

Part F: 

As in Problem 1, Assume the solar Si abundance by number is 12 + log₁₀(Si/H) = 7.51 (Asplund 2009).  What can you say about  the abundance of silicon relative to solar of this z = 0.2055 absorber? Does your value agree with what Prochaska et al. 2017 Links to an external site. report in Table 3 for the [Z/H] of the system called "J1241+5721_199_6"? Why or why not (feel free to speculate)?