# Radiometer Equation Applied to Telescopes

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## Radio Basics and Reflector Antennas

(presentation by Katey Alatalo; notes by Amber Bauermeister)

### Specific Intensity An antenna observing a blob.

We define the specific intensity as $I_{\nu }$ :

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

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

So,

${I_{\nu }={\frac {dE}{dtd\nu dAd\Omega }}=[erg\ s^{-1}\ cm^{-2}\ Hz^{-1}\ Sr^{-1}]}\,\!$ #### “Specific intensity is per EVERYTHING”

$\rightarrow$ to get something else useful, we must integrate over $I_{\nu }$ • Flux Density: $F_{\nu }=\int I_{\nu }d\Omega =[erg\ s^{-1}\ cm^{-2}\ Hz^{-1}]$ . The unit we use is the Jansky where 1 Jansky = $10^{-23}\ erg\ s^{-1}\ cm^{-2}\ Hz^{-1}$ . Note that the integral should technically have a $\cos \theta$ in it, but for small fields, $\cos \theta \sim 1$ .
• “Surface Brightness”: $\Sigma (\nu )=\int I_{\nu }d\nu$ (still a surface brightness without this integral)
• Flux: $F=\int \int I_{\nu }d\Omega d\nu =[erg\ s^{-1}\ cm^{-2}]$ • Luminosity: $L=\int FdA=[erg\ s^{-1}]$ #### Specific intensity is conserved along a ray

If you move a source twice as far away, the power received from the source is reduced by a factor of 4 (inverse square law), but the the beam of our telescope (the $\Delta \Omega$ ) now covers 4 times the area on the source. These two effects therefore cancel each other and the specific intensity is conserved. (Note that this is not true for cosmological distances in an expanding universe!)

#### Blackbody equation

$I_{\nu }={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{h\nu /kT}-1}}\,\!$ In radio, $h\nu < (usually), so that

$I_{\nu }\approx {\frac {2kT}{\lambda ^{2}}}\,\!$ Therefore, for any source (need not be thermal), radio astronomers like to define the brightness temperature, $T_{B}$ , as the temperature of the blackbody needed to produce the observed specific intensity at that frequency:

$T_{B}(\nu )={\frac {I_{\nu }\lambda ^{2}}{2k}}\,\!$ This notation is most frequently used for resolved sources. \It is also common to report $I_{\nu }$ in Jy per beam.

### Antenna Temperature, $T_{A}$ The antenna temperature is defined as the brightness temperature associated with the power received by the telescope:

{\begin{aligned}P_{\nu }&=I_{\nu }dAd\Omega \\&={\frac {2kT_{A}}{\lambda ^{2}}}dAd\Omega \\&=2kT_{A}\\\end{aligned}}\,\! The last step comes from the fact that $dAd\Omega =\lambda ^{2}$ by the antenna theorem (see section 2.1 below).

#### Resolved Sources

If the source is resolved (fills the beam), then the antenna temperature is simply the brightness temperature of the source:

$T_{A}=T_{B}\,\!$ The natural units to report observations of a resolved source are specific intensity or brightness temperature.

#### Unresolved Sources

If the source is unresolved (the solid angle of the source, $\Omega _{s}$ is less than the solid angle of the beam, $\Omega _{b}$ ) then the relation becomes (as a result of the convolution of $\Omega _{s}$ and $\Omega _{b}$ ):

$T_{A}=T_{B}{\frac {\Omega _{s}}{\sqrt {\Omega _{s}^{2}+\Omega _{b}^{2}}}}\,\!$ Relate this to the specific flux ($F_{\nu }$ or $S_{\nu }$ ):

{\begin{aligned}F_{\nu }=S_{\nu }&=\int I_{\nu }d\Omega =I_{\nu }\Omega _{s}={\frac {2kT_{B}\Omega _{s}}{\lambda ^{2}}}\\\Rightarrow &T_{B}\Omega _{s}={\frac {S_{\nu }\lambda ^{2}}{2k}}\end{aligned}}\,\! If $\Omega _{s}<<\Omega _{b}$ ,

$kT_{A}=\left({\frac {\lambda ^{2}}{\Omega _{b}}}\right){\frac {S_{\nu }}{2}}=\left({\frac {\lambda ^{2}}{(\lambda ^{2}/A_{dish})}}\right){\frac {S_{\nu }}{2}}\,\!$ ${kT_{A}={\frac {S_{\nu }A_{dish}}{2}}}\,\!$ The factor of $1/2$ comes from only detecting one polarization. We see that $T_{A}$ is just the total power (per Hz) collected by the telescope. Therefore, the most appropriate way to report observations of a source smaller than the beam is in flux density (Janskys).

## Practicalities and Performance Parameters

(presentation and notes by Amber Bauermeister)

### Some Important Equations

• The antenna theorem
$A_{e}\Omega _{a}=\lambda ^{2}\,\!$ where $\Omega _{a}$ is the antenna solid angle and $A_{e}$ is the effective area: $A_{e}=\eta _{a}A_{p}$ ($\eta _{a}$ is the aperture efficiency and $A_{p}$ is the projected area of the telescope).

• K / Jy\Recall that we relate the power received by the antenna at a frequency $\nu$ , $P_{\nu }$ , to the antenna temperature, $T_{A}$ , and the observed intensity, $I_{\nu }$ , as follows:
{\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 is just

$K/Jy={\frac {A_{e}}{2k}}\,\!$ ### The System Temperature

While the antenna temperature, $T_{A}$ describes the flux received from the source you are interested in, the system temperature, $T_{sys}$ describes the actual power received due to the sky (source plus atmosphere) and the receiver itself ($T_{R}$ ).

$T_{sys}=T_{A}+T_{atm}+T_{R}\,\!$ The main component is the receiver temperature, $T_{R}$ . Typically $T_{R}>>T_{A}$ , so how do we detect the source we are interested in? First point the antenna at the source (on-source), then point the antenna somewhere else, preferably a part of the sky with no sources (off-source). Then,

$T_{sys}(on-source)=T_{A}+T_{R}+T_{atm}\,\!$ $T_{sys}(off-source)=T_{blank\ sky}+T_{R}+T_{atm}\,\!$ So in order to detect the source, we need $T_{A}>\Delta T_{sys}$ (the noise in our measurement of $T_{sys}$ ). By ‘Root-N statistics’, the noise is just $T_{sys}/{\sqrt {N}}$ where N is the number of independent samples of the signal. 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,

$\Delta T_{sys}={\frac {T_{sys}}{\sqrt {\tau \Delta \nu }}}\,\!$ ### The Radiometer Equation

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

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

### SEFD

The SEFD is the ‘system equivalent flux density’. This is basically the flux 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!):

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