Week 7 Nov. 14th and 16th In-Class Problems

Due Nov 21, 2022 by 12pm
Points 10
Submitting a text entry box, a media recording, or a file upload

Problem 1, Monday November 14th:

Radio Observations of the HI 21-cm absorption line

Suppose we are observing a QSO with a strong radio continuum through a uniform, neutral gas cloud of HI that has a column density $LaTeX: N_{HI}$ $N_{HI}$ in the radial velocity interval LaTeX: dv $dv$ of $LaTeX: \frac{dN_{HI}}{dv}\:=\frac{\left(3\times10^{20}\:cm^{-2}\right)}{\left(20\:km\:s^{-1}\right)}$ $\frac{dN_{HI}}{dv}\:=\frac{\left(3\times10^{20}\:cm^{-2}\right)}{\left(20\:km\:s^{-1}\right)}$ .

The QSO has an observed flux density at 21-cm through the cloud of $LaTeX: F_{\nu}\:=\:F_{\nu}\left(0\right)e^{-\tau_{\nu}}$ $F_{\nu}\:=\:F_{\nu}\left(0\right)e^{-\tau_{\nu}}$ , where $LaTeX: \tau_{\nu}$ $\tau_{\nu}$ is the optical depth of our cloud at 21-cm.

From some of the equations discussed in the video lectures, it is fairly straightforward to show that:

$LaTeX: \tau_{\nu}\:=\:0.552\:\left(\frac{100\:K}{T_{spin}}\right)\left(\frac{\frac{dN_{HI}}{d\nu}}{\frac{10^{20}\:cm^{-2}}{km\:s^{-1}}}\right)$ $\tau_{\nu}\:=\:0.552\:\left(\frac{100\:K}{T_{spin}}\right)\left(\frac{\frac{dN_{HI}}{d\nu}}{\frac{10^{20}\:cm^{-2}}{km\:s^{-1}}}\right)$

We have taken both an "on-line" and an "off-line" measurement of the flux density of our background source.

a) discuss what is meant by "on-line" and "off-line" (you may wish to draw a picture) and write down the flux density of the QSO "on-line" and the flux density of the QSO "off-line."

b) Suppose we measure that the flux density of our QSO changes by less than 1% on-line to off-line. What is the upper limit we can place on $LaTeX: \tau_{\nu}$ $\tau_{\nu}$ , the optical depth of our cloud at 21-cm?

c) What can we then say about the lower limit to the spin temperature, $LaTeX: T_{spin}$ $T_{spin}$ , of the HI?

d) Suppose we find another cloud of exactly the same column density in the same radial velocity interval; that is, $LaTeX: \frac{dN_{HI}}{dv}\:=\frac{\left(3\times10^{20}\:cm^{-2}\right)}{\left(20\:km\:s^{-1}\right)}$ $\frac{dN_{HI}}{dv}\:=\frac{\left(3\times10^{20}\:cm^{-2}\right)}{\left(20\:km\:s^{-1}\right)}$ . However, we are able to detect this cloud in absorption. It's flux density changes by 50% on-line to off-line. What is $LaTeX: T_{spin}$ $T_{spin}$ of this second cloud?

e) Can you make a generic statement about how the spin temperature relates to the detectability of HI-21cm in absorption?

Problem 2, Wednesday November 16th: Thermal vs. Turbulent Line Broadening

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

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

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).

Suppose that we observe a radio-bright QSO and detect absorption lines from the Milky Way ISM in its spectrum. The 21-cm absorption line is observed to be optically thin with a profile FWHM(HI) = 10 km s^-1. The NaI D₂ line at 5892 Angstroms is observed to have a FWHM(NaI D₂) = 5 km s^-1. Both observed line profiles are the result of a combination of thermal broadening, $LaTeX: \sigma_{v,\:therm}$ $\sigma_{v,\:therm}$ plus turbulence with a Gaussian velocity distribution with one-dimensional velocity dispersion: $LaTeX: \sigma_{v,\:turb}$ $\sigma_{v,\:turb}$ .

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:

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

Part I:

If we assume both the NaI D₂ and HI 21-cm lines are optically thin, estimate the kinetic temperature T and $LaTeX: \sigma_{v,\:turb}$ $\sigma_{v,\:turb}$ .

Part 2:

Now let's assume the NaI D₂ line is not optically thin, and is instead saturated. What are the limits on T and $LaTeX: \sigma_{v,\:turb}$ $\sigma_{v,\:turb}$ ? Hint: the key here is to figure out which way the inequalities go! In this case, is our measurement of FWHM(NaI D₂) = 5 km s^-1an upper or lower limit on the velocity dispersion?

Forget about Part 3!!!!

Part 3:

Now let's assume you are measuring SiIV and HI absorption line widths in a gas cloud that you suspect is approximately 10,000 K. Let's also assume both lines are optically thin (e.g. not saturated) and you have measured the FHWM of HI to be 40 km s^-1 . What can you say about $LaTeX: \sigma_{v,\:turb}$ $\sigma_{v,\:turb}$ in this case? What do you expect to measure for the line-width of SiIV in this case? What if you instead measure the line-width to be 50 km s^-1 ? What would be the most likely physical interpretation in this case?