# Radiometer Equation Applied to Telescopes

### Reference Material

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

where:

• ${\displaystyle T_{src}}$ is the signal of the source you’re observing
• ${\displaystyle T_{sys}}$ is your system temperature
• ${\displaystyle T_{rms}}$ is the noise in your system, or the RMS fluctuations in your system temperature
• ${\displaystyle \Delta \nu }$ is the bandwidth of your correlator (in Hz)
• ${\displaystyle \tau }$ 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:

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

Radio astronomers like to refer to the above temperature ${\displaystyle T}$ as the "brightness temperature," ${\displaystyle T_{B}}$, of a source. ${\displaystyle T_{B}}$ is the temperature of the blackbody needed to produce the observed specific intensity at that frequency:

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

Now, back to specific intensity. ${\displaystyle I_{\nu }}$ is defined as:

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

where:

• ${\displaystyle \Delta t}$ is the exposure time
• ${\displaystyle \Delta \nu }$ is a small interval in frequency
• ${\displaystyle \Delta E}$ is the energy emitted in bandwidth ${\displaystyle \Delta \nu }$ over time ${\displaystyle \Delta t}$
• ${\displaystyle \Delta A}$ is the area of the telescope
• ${\displaystyle \Delta \Omega }$ is the observed solid angle on the source, or your beam size (steradians)

So,

${\displaystyle {I_{\nu }={\frac {dE}{dtd\nu dAd\Omega }}=[erg\ s^{-1}\ cm^{-2}\ Hz^{-1}\ Sr^{-1}]}\,\!}$

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

To arrive at other quantities, we must integrate over ${\displaystyle I_{\nu }}$:

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

### Source Temperature, ${\displaystyle T_{src}}$

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

{\displaystyle {\begin{aligned}P_{\nu }&=I_{\nu }dAd\Omega \\&={\frac {2kT_{src}}{\lambda ^{2}}}dAd\Omega \\&=2kT_{src}\\\end{aligned}}\,\!}

The last step comes from the Antenna Theorem, which states that ${\displaystyle dAd\Omega =\lambda ^{2}}$.

### K / Jy, or "forward gain"

Now that we’ve related the power ${\displaystyle P_{\nu }}$ received by the antenna at a given frequency, to the source temperature, ${\displaystyle T_{src}}$, and the observed intensity, ${\displaystyle I_{\nu }}$, we can arrive at a conversion between flux density and brightness temperature for an unresolved source:

{\displaystyle {\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

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

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 ${\displaystyle T_{src}}$ describes the energy received from the source you are interested in, the system temperature, ${\displaystyle T_{sys}}$ describes the actual power received due to both the sky (${\displaystyle T_{sky}}$) and the receivers (${\displaystyle T_{Rx}}$):

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

The main component is the receiver temperature, ${\displaystyle T_{Rx}}$, which comes from the thermal (or Johnson) noise from the receiver electronics. Depending on the observation waveband, receivers are sometimes cooled to reduce ${\displaystyle T_{Rx}}$.

${\displaystyle T_{sky}}$ 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 ${\displaystyle A_{e}}$

Above we have written the area of an antenna as ${\displaystyle A_{e}}$, or the "effective area."

${\displaystyle A_{e}=\eta _{a}A_{p}\,\!}$

where (${\displaystyle \eta _{a}}$ is the aperture efficiency and ${\displaystyle A_{p}}$ is the projected area of the telescope). ${\displaystyle \eta _{a}}$ 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:

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

### Detecting the signal...

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

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

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

where ${\displaystyle N}$ is the number of independent data points.

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

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

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 ${\displaystyle {\sqrt {N_{samples}}}}$. The fact that ${\displaystyle T_{rms}\propto T_{sys}}$ is not because ${\displaystyle T_{sys}}$ 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 ${\displaystyle N_{\gamma }}$ in that mode. And since ${\displaystyle T_{sys}\propto N_{\gamma }}$, it follows that ${\displaystyle T_{rms}\propto T_{sys}}$. Even though ${\displaystyle T_{sys}}$ is sometimes called "noise temperature," we emphasize here that ${\displaystyle T_{sys}}$ is a real signal; it does not represent real "noise," or fluctuations in measurements.

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

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

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

#### SEFD

The SEFD is the ‘system equivalent flux density’, which is the flux density equivalent of ${\displaystyle T_{sys}}$:

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

The SEFD is a useful way to compare the sensitivity of two different systems since it folds in both ${\displaystyle T_{sys}}$ and ${\displaystyle A_{e}}$. 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):

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

After substituting in the temperature-based Radiometer Equation for S, we arrive at an intuitive expression for the RMS variations in flux density ${\displaystyle S_{\nu ,rms}}$:

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

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.

#### Extending to interferometric arrays

You should refer to the lecture on 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 ${\displaystyle N}$ 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:

{\displaystyle {\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}}\,\!}

${\displaystyle S_{\nu ,rms}}$ are the RMS variations in the flux density that appear in an image, or in a synthesized beam. These fluctuations dictate the faintest sources you can reliably detect in an image. The factor of 2 in the denominator comes from the amplitude and phase measurements, and ${\displaystyle N(N-1)/2}$ is the number of baselines between the ${\displaystyle N}$ antennas.