# Harvard Astronomy 201b

In Uncategorized on April 26, 2013 at 9:18 pm

How were the observations done?

If you are a theorist, you probably don’t like words like “bandpass”, and hate words like “calibration.”  If you are an observer, you probably know and love these words.  But even theorists have to understand how observations are done: they’re what keep science science.  Here I want to briefly explain some aspects of the paper’s discussion of how the data were taken that were on first read opaque.

First of all, given that the $NH_3$ rest frequency is roughly 24 GHz, we can find the energy of the transition: $E=h\nu = 10^{-4}\;\rm{eV}$.  This corresponds to an excitation temperature of 1.2 K, which we can also translate into a velocity via $v\sim (E/m)^{1/2} \sim 24\; {\rm m/s}$.  A typical molecular cloud temperature is on the order of 20 K, high enough that there should be many molecules in the upper energy state (think of it as a toy-model 2 level system) and hence many upper to lower transitions occurring (leading to emission at 24 GHz).

A 1 MHz window means the observations were done between 24 GHz – 500 MHz and 24 GHz + 500 MHz, and dividing this interval by 256 gives the 3.9 kHz per channel quoted.   This can be translated into a velocity by thinking of it using the small-z redshift formula $z\approx v/c$. Here, we have a fractional frequency shift $\Delta \nu / \nu \simeq v/c$, and setting $\Delta \nu / \nu$ =  3.9  kHZ/24 GHz leads to the velocity quoted (.049 km/s).

The baselines given (35 meters up to 1 km) determine the spatial resolution: just good old Carroll and Ostlie diffraction limit, $\theta = 1.22 \lambda/D$, where $\theta$ is the angular resolution, $\lambda$ the wavelength of light, and $D$ the diameter of the aperture (here, the baseline).  24 GHz has $\lambda = 1.2$ cm, so with a 1 km baseline the angular resolution is about .05’’ (for comparison, this is 1.3 times the angular resolution of the Hubble Space Telescope at 500 nm).

Nepers is a unit of optical depth; .0689 is optically thin (and measures the optical depth due to the Earth’s atmosphere, since the authors note this corresponds to fair weather; there is negligible optical depth from sources in space at this frequency).  It is interesting to note that the optical depth is given at 22 GHz, but the observations are made at 24 GHz; this is because 22 GHz is a standard fiducial frequency to allow comparison with the weather conditions under which other radio observations might have been made.

Now, calibration—theorists’ bane.  The paper’s discussion is compact and therefore somewhat confusing to those uninitiated into the secrets of radio astronomy, but understanding it is worthwhile for the insight it can yield into what actually goes on at these remote “large arrays” in the desert (alien storage?)

In a radio interferometric observation, what you actually measure is the amplitude and phase of the signal in each channel (here, there are 256): i.e., at each different frequency corresponding to the channels in the window about 24 GHz.  (Note: interferometric means that phases are being compared between signal received at spatially separated points on Earth: this allows the difference in path length from the source to two antennae to be computed, and hence the direction to the source). The amplitude and phase at these different frequencies is desirable because it gives (eventually) intensity as a function of frequency, or equivalently a spectral energy distribution (SED).  This in turn allows inferences about the properties of the source (e.g. dust grain size from SEDs of debris disks, cf. problem set 2, problem 4 ( $\beta$ describes the shape of the SED)).There will be some intrinsic noise in both amplitude and phase, which you want to eliminate.  For a source that has a flat spectrum, you know any bumps you see at a particular frequency are due to noise in the channel at that frequency.  This is the purpose of having an amplitude calibrator: typically quasars are used because they have flat spectra.

Now, the amplitude is measured by the amount of electrical current coming from a given channel: the more photons at that frequency hit the antenna, the more current.  But astronomers care about flux, not current.  The absolute flux calibrator is a source with a known flux. Measuring the voltage this source produces allows the conversion factor between voltage and flux to be deduced.  Using a source with known flux also allows one to calibrate the bandpass: the flux should be zero outside the window, but to know this is because the window is working, you have to be sure the flux is not just coincidentally zero at frequencies outside the window anyway.

Finally, the astute reader may notice that noise is reported as mJy/beam.  The noise should be linear in the area of the beam, so a larger beam would have more noise.  Thus two sets of observations done with different beam sizes would have different noise levels due to this; reporting noise/beam allows direct comparison of how good observations done with different beam sizes are.