### Physics

I promised earlier this week to talk a little bit more about how radio telescope arrays work, and I try to keep my promises. If anything in this explanation isn’t clear, feel free to ask questions.

One thing astronomers like to have is images that can resolve fine detail in structure. In optical wavelengths, the resolution of images is limited by the atmosphere, which blurs optical light. This is why Hubble is such an amazing telescope and why there is a lot of work on adaptive optics to correct for atmospheric blurring for ground telescopes. At radio wavelengths, the atmosphere doesn’t blur the signals we are trying to detect. This means we are operating in “diffraction-limited” mode. The physics of optics sets an absolute limit to the achievable resolution of a system, which is called the diffraction limit. The diffraction limit depends on the size of your system and the wavelength you want to give at. (The diffraction-limited resolution, in radians, is given by $\theta = \frac{1.22 \, \lambda}{D}$ where D is the size of your telescope and $\lambda$ is the wavelength of observations.)

Now, you might think that since radio telescopes don’t have to worry about atmospheric blurring, they would offer good resolution. However, the wavelength difference between radio and optical wavelengths is substantial. Typical resolution limits achieved for optical telescopes are on the order of one arcsecond. (There are sixty arcseconds in one arcminute and sixty arcminutes in one degree. The moon and sun are both about half a degree across.) Arecibo, the largest single dish radio telescope, has a resolution of about 3.5 arcminutes at a wavelength of 21 cm (which is where we observe). This is almost 200 times worse than the resolution one would have with a small optical telescope. In order to achieve comparable resolution at 21 cm, you would need a radio telescope with a diameter of over 40 km. Clearly, building such a telescope is not feasible.

Radio astronomers wanted a way to achieve good resolutions, though. Since it’s not possible to build a single telescope large enough to achieve high resolution, they came up with the idea of using multiple smaller antennas to synthesize a larger dish. This is the motivation behind the VLA and other radio telescope arrays. Some number of telescopes work together, observing the same source. The signals from the separate telescopes are then combined and processed so that you can produce an image with a resolution set by the largest baseline (separation between any two dishes) rather than the size of the radio dishes.  This means that you don’t need one giant dish; rather, you can have lots of smaller (and cheaper) telescopes spread over a large area.

Now, the details of how you combine the data and process it are quite complicated – that’s part of the reason why I’ve been in Socorro for the past few days. In fact, a Nobel Prize was awarded to Martin Ryle and Tony Hewish for figuring out the details of this process and how to reconstruct an image.  Therefore, I’m not going to go into the details of how this process works.  Rather, I’m going to be happy that it does work.

The ALFALFA Survey is trying to detect galaxies through emission at the ’21cm line’. In fact, you’ll see mention of the 21cm line throughout extragalactic radio astronomy as it is one the most common observations. This might lead you to wonder: What is the 21cm line? And why do we want to observe it? For now, I’ll address the first of those questions; you’ll learn all about why the 21cm line is interesting in a later post.

Alternately, this post could be titled: “A Quick and Dirty Introduction to the Hydrogen Atom”.  The ’21cm line’ is a specific transition in the hydrogen atom, so called because the energy of the transition corresponds to a photon with a wavelength of 21 cm.  Physicists love hydrogen because it’s the simplest atom – one proton and one electron – meaning it can be solved exactly.  Astronomers love hydrogen because it fills the universe and observing it in different states offers lots of information about conditions found in astronomical situations.

Most people are familiar with the simple Bohr model of the atom where electrons orbit the nucleus in quantized orbits.  Of course, like many physics explanations, the Bohr model isn’t accurate but it does provide a convenient picture for much of the physics so we continue to use it.  The transitions in atoms that most people are familiar with are electronic transitions.  The orbits of electrons in an atom have quantized energy states and there is a specific energy associated with a move from one orbit to another.  For hydrogen, these electronic transitions have energies that release photons in the ultraviolet, optical and infrared parts of spectra.  The energies associated with the 21 cm line are much, much lower.  In order to understand where the 21cm line comes from we have to look further at the structure of hydrogen. The first set of corrections made are termed “fine structure” and account for the fact that the base calculation ignores relativistic effects.  These aren’t the corrections that cause the 21 cm line, though.  For that we need to go to a second set of corrections (which have an even smaller change in energy levels) – the “hyperfine structure” of hydrogen.

An illustration of the Bohr model of the atom

The hyperfine structure refers to the weak interaction between the spins of the proton and electron.  Spin is an extremely quantum idea and hard to describe with a classical analogy.  The best way to think of it is as particles spinning on their own axis – it behaves as an angular momentum term in many ways.  However, it does have some odd properties, which we will ignore for now.  If you keep the simple picture of a charged particle spinning in mind, you can imagine a slight magnetic field resulting as moving charges cause magnetic fields.  Similarly, the proton creates a slight magnetic field.  The interaction of these two magnetic fields results in the 21 cm transition.

The electron and proton can both be thought of as magnetic dipoles generating magnetic fields.  There is then an energy difference between when the dipole moments of the electron and proton are aligned or anti-aligned. The state where the two spins (or magnetic moments) are anti-aligned is a lower energy state; when the atom transitions from a “spin-up” electron to a “spin-down” electron, a photon with a wavelength of 21 cm (corresponding to the energy difference between the levels) is released.

Illustration of the 21 cm transition

I’m going to try ending posts with the math so that it’s available if you’re interested and easy to skip if you’re not – hopefully everything I write will be clear without any equations.

The energy of a photon is related to its frequency by $E=h\nu$ where $\nu$ is the frequency and $h$ is Planck’s constant.

If you would like to know the wavelength of this photon, you can find it from the speed of light: $c=\nu \lambda$ where $\lambda$ is the wavelength.

If you would like to calculate the energy of the hyperfine transition a back-of-the-envelope approximation is:

$\Delta E \sim \frac{2 \mu_e \mu_p}{a_0^3}$

where $a_0$ is the Bohr radius of an atom and $\mu_{e,p}$ are the magnetic moments of the electron and proton, respectively.

Substituting in for the magnetic moments this becomes,

$\Delta E \sim \frac{1}{2} g_p \frac{m_e}{m_p} \alpha^2 \frac{e^2}{a_0}$

where $g_p$ is the degeneracy of the proton ( ~2.79), $m_{e,p}$ are the masses of the electron and proton, respectively, $\alpha$ is the fine structure constant (~1/137), and $e$ is the charge of the electron.

This approximation is very close to the true calculation using quantum mechanics:

$\Delta E = \frac{8}{3} g_p \frac{m_e}{m_p} \alpha^2 \frac{e^2}{a_0}$