View source for Radiometer Equation Applied to Telescopes

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Much of this is based on the Radio 101 single-dish lecture, [[Single_Dish_Basics | Single Dish Basics]].

[http://www.youtube.com/watch?v=dwT-tMoscsY Click here] to watch Chat Hull's video on the Radiometer Equation.

[[Media:Radiometer_equation_notes.pdf | Click here]] to see the handwritten notes that I used to make the video and edit the reference material below.

\section*{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:
\begin{itemize}
\item $T_{src}$ is the signal of the source you're observing
\item $T_{sys}$ is your system temperature
\item $T_{rms}$ is the noise in your system, or the RMS fluctuations in your system temperature
\item $\Delta\nu$ is the bandwidth of your correlator (in kHz, MHz, GHz...)
\item $\tau$ is integration time (seconds)
\end{itemize}


\subsection*{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," $T_B$, of a source.  $T_B$ 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.  $I_\nu$ is defined as:

$$I_\nu = \frac{\Delta E}{\Delta \Omega \Delta A \Delta t \Delta \nu}$$
where:
\begin{itemize}
\item $\Delta t$ is the exposure time
\item $\Delta \nu$ is a small interval in frequency
\item $\Delta E$ is the energy emitted in bandwidth $\Delta \nu$ over time $\Delta t$
\item $\Delta A$ is the area of the telescope
\item $\Delta \Omega$ is the observed solid angle on the source, or your beam size (steradians)
\end{itemize}

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

It is common to report $I_\nu$ in Jy/beam, where the beam has units of $Sr^{-1}$, and Janskys (Jy) are the units of flux density $S_\nu$, defined below.

To arrive at other quantities, we must integrate over $I_\nu$:
\begin{itemize}
\item {\bf 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.  $1 \textrm{Jy} = 10^{-23} erg\ s^{-1}\ cm^{-2}\ Hz^{-1}$
\item {\bf Power received:} $P_\nu = \int S_\nu dA = [erg\ s^{-1}\ Hz^{-1}]$. 
\end{itemize}

\subsection*{Source Temperature, $T_{src}$}
The source temperature is defined as the brightness temperature associated with the power
received by the telescope from the source you're observing:
$$\begin{aligned}
P_\nu & =  I_\nu dA d\Omega \\
& =  \frac{2kT_{src}}{\lambda^2} dA d\Omega\\
& =  2kT_{src}\\
\end{aligned}$$
The last step comes from the Antenna Theorem, which states that $dA d\Omega = \lambda^2$.

\subsection*{K / Jy, or "forward gain"}
Now that we've related the power $P_\nu$ received by the antenna at a given frequency, 
to the source temperature, $T_{src}$, and the observed intensity, $I_\nu$, we can arrive at a conversion between 
flux density and brightness temperature:
$$\begin{aligned}
P_\nu & =  I_\nu A_e \Omega_a \\
2kT_A & =  S_\nu A_e\\
T_A & =  \left(\frac{A_e}{2k}\right) S_\nu\\
\end{aligned}$$
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.

\subsection*{The System Temperature}
While the source temperature $T_{src}$ describes the energy received from the source you are interested in, 
the system temperature, $T_{sys}$ describes the actual power received due to both the sky ($T_{sky}$) and the receivers ($T_{Rx}$):
$$T_{sys} = T_{sky} + T_{Rx} $$
The main component is the receiver temperature, $T_{Rx}$, which comes from the [http://en.wikipedia.org/wiki/Johnson%E2%80%93Nyquist_noise
| thermal (or Johnson) noise] from the receiver electronics.  Depending on the observation waveband, receivers are sometimes cooled to reduce $T_{Rx}$.

$T_{sky}$ is the power generated by everything you \textit{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.

\subsubsection*{Effective Area $A_e$}

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

$$A_e =  \eta_a A_p$$ 

where ($\eta_a$ is the aperture efficiency and $A_p$ is the projected area of the telescope).  $\eta_a$ 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:

\begin{itemize}
\item $T_{sys}$ is dominated by $T_{Rx}$: in this case aperture efficiency matters, because in amplifying the signal, noise from the receiver is also amplified.
\item $T_{sys}$ is dominated by $T_{sky}$: 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.
\end{itemize}

\subsection*{Detecting the signal...}

Typically $T_{sys} >> T_{src}$, so how do we detect 
the source we are interested in? Beat down the noise!

In order to detect the source, we need $T_{src} > T_{rms}$, where the observation-dependent quantity $T_{rms}$ is the noise in our measurement of the observation-independent quantity $T_{sys}$:

$$T_{rms} = \frac{T_{sys}}{\sqrt{N}}$$

where $N$ is the number of independent data points.

For a telescope, the number of independent samples is $\Delta \nu \cdot \tau$, where $\Delta \nu$ is the 
 bandwidth (Hz) and $\tau$ is the integration time (seconds). With a bandwidth $\Delta \nu$, the signal is 
 statistically independent over a time interval $1/\Delta\nu$, so that the number of independent samples is 
 just $\tau$ divided by $1/\Delta\nu$, so $N = \tau \Delta\nu$. Therefore,
$$T_{rms} = \frac{T_{sys}}{\sqrt{\tau \Delta\nu}}$$

\subsection*{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 $\Delta \nu = 10$ MHz, $\tau = 1$ sec so that $ \sqrt{\tau \Delta\nu} \sim 3\times10^3$.
Typical values for $T_{sys}$ are 40 - 200 K.

\subsubsection*{SEFD}
The SEFD is the `system equivalent flux density', which is the flux density equivalent of $T_{sys}$:

$$\textrm{SEFD} = \frac{T_{sys}}{(K/Jy)} = \frac{T_{sys}}{A_e / 2k} = \frac{2kT_{sys}}{A_e}$$

The SEFD is a useful way to compare the sensitivity of two different systems since it folds in both
$T_{sys}$ and $A_e$. 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 $S \ N$, we arrive at an intuitive expression for the RMS variations in flux density $S_{\nu, rms}$:

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

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

\subsubsection*{Extending to interferometric arrays}

You should refer to the lecture on [[Basic_Interferometry | Basic Interferometry]] for explanations of the multi-antenna concepts presented here.

Making more independent measurements is one way to increase signal-to-noise.  With an array of $N$ antennas, every baseline (involving two antennas) adds two more independent measurements: a measurement of the signal's amplitude, and its phase.  Thus, for an array of antennas, the Radiometer Equation becomes:

$$\begin{aligned}
S_{\nu, rms} & = \frac{\textrm{SEFD}}{\sqrt{ \frac{N(N-1)}{2} \tau (2\Delta\nu)}} \\
& = \frac{\textrm{SEFD}}{\sqrt{ N(N-1) \tau \Delta\nu}}
\end{aligned}$$

The factor of 2 in the denominator comes from the amplitude and phase measurements, and $N(N-1) / 2$ is the number of baselines between the $N$ antennas.

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