Radiometer Equation Applied to Telescopes

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The Radiometer Equation

The radiometer equation is really all about signal to noise. Here’s the answer:

 \frac{S}{N} = \frac{T_{src}}{T_{rms}} = \frac{T_{src}}{T_{sys}} \sqrt {\tau \Delta \nu } \,\!

where:

  • Tsrc is the signal of the source you’re observing
  • Tsys is your system temperature
  • Trms is the noise in your system, or the RMS fluctuations in your system temperature
  • Δν is the bandwidth of your correlator (in Hz)
  • τ is integration time (seconds)

What’s the deal with all the temperatures?

All of the temperatures in the above equation have their origin in the Rayleigh-Jeans limit of the blackbody equation:

 I_\nu \approx \frac{2kT}{\lambda ^2} \,\!

Radio astronomers like to refer to the above temperature T as the "brightness temperature," TB, of a source. TB is the temperature of the blackbody needed to produce the observed specific intensity at that frequency:

 T_ B(\nu ) = \frac{I_\nu \lambda ^2}{2k} \,\!

Now, back to specific intensity. Iν is defined as:

 I_\nu = \frac{\Delta E}{\Delta \Omega \Delta A \Delta t \Delta \nu } \,\!

where:

  • Δt is the exposure time
  • Δν is a small interval in frequency
  • ΔE is the energy emitted in bandwidth Δν over time Δt
  • ΔA is the area of the telescope
  • ΔΩ is the observed solid angle on the source, or your beam size (steradians)

So,

  {I_\nu = \frac{dE}{dt d\nu dA d\Omega } = [erg\  s^{-1}\  cm^{-2}\  Hz^{-1}\  Sr^{-1}]} \,\!

It is common to report Iν in Jy/beam, where the beam has units of Sr − 1, and Janskys (Jy) are the units of flux density Sν, defined below.

To arrive at other quantities, we must integrate over Iν:

  • Flux Density: S_\nu = \int I_\nu d\Omega = [erg\  s^{-1}\  cm^{-2}\  Hz^{-1}]. The fundamental unit of flux density is the Jansky. 1 \textrm{Jy} = 10^{-23} erg\  s^{-1}\  cm^{-2}\  Hz^{-1}
  • Power received: P_\nu = \int S_\nu dA = [erg\  s^{-1}\  Hz^{-1}].

Source Temperature, Tsrc

The source temperature is defined as the brightness temperature associated with the power received by the telescope from the source you’re observing:

 \begin{align}  P_\nu &  = I_\nu dA d\Omega \\ &  = \frac{2kT_{src}}{\lambda ^2} dA d\Omega \\ &  = 2kT_{src}\\ \end{align} \,\!

The last step comes from the Antenna Theorem, which states that dAdΩ = λ2.

K / Jy, or "forward gain"

Now that we’ve related the power Pν received by the antenna at a given frequency, to the source temperature, Tsrc, and the observed intensity, Iν, we can arrive at a conversion between flux density and brightness temperature for an unresolved source:

 \begin{align}  P_\nu &  = I_\nu A_ e \Omega _ a \\ 2kT_ A &  = S_\nu A_ e\\ T_ A &  = \left(\frac{A_ e}{2k}\right) S_\nu \\ \end{align} \,\!

Therefore, the conversion between K and Jy, also known as the "forward gain" of an antenna, is just

 K / Jy = \frac{A_ e}{2k} \,\!

Thus, brightness temperature is just another measure of the brightness of a source. The forward gain is a physical property of a telescope, which dictates the telescope receivers’ response to a given increase in Janskys.

The System Temperature

While the source temperature Tsrc describes the energy received from the source you are interested in, the system temperature, Tsys describes the actual power received due to both the sky (Tsky) and the receivers (TRx):

 T_{sys} = T_{sky} + T_{Rx}  \,\!

The main component is the receiver temperature, TRx, which comes from the thermal (or Johnson) noise from the receiver electronics. Depending on the observation waveband, receivers are sometimes cooled to reduce TRx.

Tsky is the power generated by everything you don’t want to be looking at on the sky: background sources, water vapor in the atmosphere (especially in mm-wave astronomy), galactic backgrounds (especially in long-wavelength astronomy), etc.

Effective Area Ae

Above we have written the area of an antenna as Ae, or the "effective area."

 A_ e = \eta _ a A_ p \,\!

where (ηa is the aperture efficiency and Ap is the projected area of the telescope). ηa is a number less than one, and signifies that not all radiation incident upon the telescope actually makes it to the receiver because of the dish’s finite reflectivity. This loss of signal requires that the signal later be amplified. There are two scenarios that can be encountered:

  • Tsys is dominated by TRx: in this case aperture efficiency matters, because in amplifying the signal, noise from the receiver is also amplified.
  • Tsys is dominated by Tsky: in this case, which might happen on a cloudy day at a millimeter telescope where the noise from the atmosphere is dominant, aperture efficiency doesn’t matter. This is because the dominant sky noise is first cut down by a low aperture efficiency, and then multiplied back up to its original level by the amplifiers.

Detecting the signal...

Typically Tsys > > Tsrc, so how do we detect the source we are interested in? Beat down the noise!

In order to detect the source, we need Tsrc > Trms, where the observation-dependent quantity Trms is the noise in our measurement of the observation-independent quantity Tsys:

 T_{rms} = \frac{T_{sys}}{\sqrt {N}} \,\!

where N is the number of independent data points.

For a telescope, the number of independent samples is \Delta \nu \cdot \tau , where Δν is the bandwidth (Hz) and τ is the integration time (seconds). With a bandwidth Δν, the signal is statistically independent over a time interval 1 / (2Δν) (the factor of 2 comes from the Nyquist sampling rate), so that the number of independent samples is just τ divided by 1 / (2Δν), so N = 2τΔν. Then, as explained in the NRAO course, the fluctuations in a given measurement of Tsys are \sqrt {2}~ T_{sys}, and the factor of two disappears, giving us the final result:

 T_{rms} = \frac{T_{sys}}{\sqrt {\tau \Delta \nu }} \,\!

It is important to note that the "noise" (or RMS variations) in any given measurement is proportional to the uncertainty in each measurement, divided by \sqrt {N_{samples}}. The fact that T_{rms} \propto T_{sys} is not because Tsys represents an uncertainty in each measurement! It is because, in the Rayleigh-Jeans limit of the blackbody equation, the uncertainty in a measurement in a given mode is proportional to the number of photons Nγ in that mode. And since T_{sys} \propto N_\gamma , it follows that T_{rms} \propto T_{sys}. Even though Tsys is sometimes called "noise temperature," we emphasize here that Tsys is a real signal; it does not represent real "noise," or fluctuations in measurements.

The Radiometer Equation

Now we can write down an expression for the signal to noise ratio (the radiometer equation):

 \frac{S}{N} = \frac{T_{src}}{T_{rms}} = \frac{T_{src}}{T_{sys}} \sqrt {\tau \Delta \nu } \,\!

Typical values might be Δν = 10 MHz, τ = 1 sec so that  \sqrt {\tau \Delta \nu } \sim 3\times 10^3. Typical values for Tsys are 40 - 200 K.

SEFD

The SEFD is the ‘system equivalent flux density’, which is the flux density equivalent of Tsys:

 \textrm{SEFD} = \frac{T_{sys}}{(K/Jy)} = \frac{T_{sys}}{A_ e / 2k} = \frac{2kT_{sys}}{A_ e} \,\!

The SEFD is a useful way to compare the sensitivity of two different systems since it folds in both Tsys and Ae. This also greatly simplifies the sensitivity calculation: if you know the flux in Jy of the source you want to detect, and you know the SEFD, then you can easily calculate the integration time you need to make a given S/N detection (for an unresolved source):

 \frac{S}{N} = \frac{S_\nu (Jy)}{\textrm{SEFD}} \sqrt {\tau \Delta \nu } \,\!

After substituting in the temperature-based Radiometer Equation for S, we arrive at an intuitive expression for the RMS variations in flux density Sν,rms:

 S_{\nu , rms} = \frac{\textrm{SEFD}}{\sqrt {\tau \Delta \nu }} \,\!

In the temperature-based Radiometer Equation, signal-to-noise increases with increased bandwidth and integration time. In the flux-density-based Radiometer equation, RMS flux density varations decrease with increased bandwidth and integration time.

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