Our old friends, Lyman-alphas: Blog 24 & 25, Worksheet 7.2, Problem 4 & 5

This week, we learned about active galactic nuclei and all of their fun applications. One of those applications in particular is especially dear to me, because the summer before tenth grade, I did a research project (and wrote a blog post about it) in a lab where I calculated the distance to ancient galaxies using redshift from Lyman-alpha emitting active galactic nuclei. And wouldn't you know, that's what we're doing in class this week:

Problem 4

One feature you surely noticed in a spectrum was the strong, broad emission lines. Here is a closer look at the strongest emission line in the spectrum:

This feature arises from hydrogen gas in the accretion disk. The photons radiated during the accretion process are constantly ionizing nearby hydrogen atoms. So there are many free protons and electrons in the disk. When one of these protons comes close enough to an electron, they recombine into a new hydrogen atom, and the electron will lose energy until it reaches the lowest allowed energy state, labeled n = 1 in the model of the hydrogen atom shown below (and called the ground state):

On its way to the ground state, the electron passes through other allowed energy states (called excited states). Technically speaking, atoms have an infinite number of allowed energy states, but electrons spend most of their time occupying those of lowest energies, and so only the n = 2 and n = 3 excited states are shown above for simplicity. Because the difference in energy between, e.g., the n = 2 and n = 1 states are always the same, the electron always loses the same amount of energy when it passes between them. Thus, the photon it emits during this process will always have the same wavelength. For the hydrogen atom, the energy difference between the n = 2 and n = 1 energy levels is 10.19 eV, corresponding to a photon wavelength of λ = 1215.67 Angstroms. This is the most commonly-observed atomic transition in all of astronomy, as hydrogen is by far the most abundant element in the Universe. It is referred to as the Lyman α transition (or Lyα for short). It turns out that that strongest emission feature you observed in the quasar spectrum above arises from Lyα emission from material orbiting around the central black hole.

Part A: Recall the Doppler equation: \[ \frac{ \lambda_{observed} - \lambda_{emitted}}{\lambda_{emitted}} = z \approx \frac{v}{c} \]Using the data provided, calculate the redshift of this quasar. \[ z =  \frac{ 1410 - 1215.67}{1215.67} \approx \frac{1}{6}  \] So that \[ v \approx \frac{c}{6}\]

Part B: Again using the data provided, along with the Virial Theorem, estimate the mass of the black hole in this quasar. It will help to know that the typical accretion disk around a \(10^8 M_{\odot} \) black hole extends to a radius of r = \(10^{15} \) m. 

Ideally, our emission lines would be infinitely thin and tall, but they have width. This happens because there is some dispersion due to the rotation of the galaxy. This means some of the galaxy will be redshifted and some will be blueshifted. 

So we can use this to find rotational velocity.

\[ z = \frac{ \lambda_{observed} - \lambda_{emitted}}{\lambda_{emitted}} = \frac{1415 - 1406}{1406} \approx 0.00640 \]

\[ z = \frac{1415 - 1406}{1406} = 0.0064 \] \[ v \approx 0.0064 c \]

Now we can use the virial theorem: \[ K = - \frac{1}{2} U \] \[ Mv^2 = \frac{G M^2}{R}\] \[M_{BH} = \frac{ v^2 R}{G} = \frac{( 0.0064 \cdot 3 \times 10^{10})^2 (10^{17})}{6.67 \times 10^{-8}} = 5.5 \times 10^{40} \: g \approx 3 \times 10^7 M_{\odot} \] 

Problem 5

You may also have noticed some weak “dips” (or absorption features) in the spectrum:

Part A: Suggest some plausible origins for these features. By way of inspiration, you may want to consider what might occur if the bright light from this quasar’s accretion disk encounters some gaseous material on its way to Earth. That gaseous material will definitely contain hydrogen, and those hydrogen atoms will probably have electrons occupying the lowest allowed energy state.

As the question implies, we are probably looking at instances where light is being absorbed or emitted by hydrogen gas on its way to us. These dips themselves, occurring before the peak, imply that this gas is between the galaxy and us, therefore they are not as redshifted as the distant galaxy. 

Part B: A spectrum of a different quasar is shown below. Assuming the strongest emission line you see here is due to Lyα, what is the approximate redshift of this object?

\[ \frac{ \lambda_{observed} - \lambda_{emitted}}{\lambda_{emitted}} = z \approx \frac{v}{c} \]

\[ z = \frac{5650 - 1215.67}{1215.67} = 3.347 \approx 3\]

Part C: What is the most noticeable difference between this spectrum and the spectrum of 3C 273? What conclusion might we draw regarding the incidence of gas in the early Universe as compared to the nearby Universe?

In this galaxy, there are many more variation, especially a larger number of deeper "dips." Based on the large redshift of the galaxy, we know it is quite old and far away. This means the light had to have traveled through a lot more gas to get here, and some of that gas had hydrogen atoms occupying the lowest energy state. Additionally, this could tell us that there was a lot more gas in the early universe. This fits in well with our current ideas of the early universe.