**Introduction:
**

_{}
wHz^{-1}

where A = geometrical
area of the dish 7.3 m^{2} for 10 foot dish

h = efficiency of the dish » 50%

F = Flux
density of radio source in Janskys [1 Jy = 10^{-26} w m^{-2} Hz^{-1}]

The factor of ½ arises because the antenna receives only one
polarization and by convention the flux density that in both
polarizations. Using the knowledge that
a resistor at temperature T produces a spectral power density of kT, we can
express the power density into units of temperature T_{A} so that

_{}

where k = Boltzmann’s constant
[1.38 x 10^{-23} w Hz^{-1}].
For example, a 1000 Jy source will produce a 1.3 K signal in a 10 foot
antenna with 50% aperture efficiency.
If a radio source radiates thermal radiation as a “black body” of
temperature T_{B}, the flux density in the radio wavelength require,
given by the Rayleigh-Jeans law.

_{}

T_{B}
= black body temperature K

_{} = solid angle
subtended by the source rad^{2}

l = wavelength m

For example, the moon radiates as an approximately 190 K black body at 21 cm wavelength and since it subtends about 0.5 degrees at the Earth, the flux is 710 Jy. This flux will produce an output on the SRT of about 1K. From another viewpoint, the moon only covers about 1% of the beam area and so the 190 K is diluted down to 1 K (accounting for the efficiency). The 1 K signal from the moon is not a strong signal for the SRT since the system temperature is over 100 K, it is less than 1% of the receiver noise power.

_{s} the 1 sigma noise DT_{A} in the
average is given by

_{}

For example the SRT hardware averages the power for each frequency in a scan for 0.1 sec, so that with a 200 K system temperature the noise is about 3 K. Further reduction in noise can be obtained by averaging the averages so that

_{}

where N is the number of data averages. Thus if the receiver is limited only by its own noise we will have to average 800 0.1 second data points to lower the noise to 0.1 K. From this analysis it looks like the sensitivity can be increased by observing longer and longer. This is generally true but there is often a limit reached when the fluctuations are no longer random but become systematic. For example in order to observe the signal from the moon we need to move the dish so that it points at the moon and then points off the moon at some comparison region. This is known as “beam switching”. When we move the dish on and off the moon, other things change, like the surroundings in the spillover from the feed. We need to switch back and forth from “signal” to “comparison”. In this case the effective noise will be doubled for a given total averaging time because we have to difference the “on” and “off” averages and we can only spend half the time observing the signal. This factor of 2 arises because

- The
standard deviation of the difference of 2 independent random variable each
with standard deviation s is
_{ }s. - The
total time is cut in half so that the standard deviation increases by
another factor of
_{ }.

**Procedure:
**

**Analysis:
**

_{
}

where A_{k} =
the k^{th} average

_{
} = overall average of
all points

M = number of averages in sum

Make a plot of s (N)
vs N for N=1, 3, 10, 30,100, 300 and see if it follows the N^{-1/2}
dependence. One way to do this is to
plot 10 log_{10} s(N) vs 10 log_{10 }N^{1/2} and check
that the slope is –1/2.

**Comments:
**

_{
}

**List of Radio Sources:**

Source |
Expected T |
Comments |

Sun |
250-3000 |
Strong source can use 25 point map |

Moon |
1 |
Requires beamswitching |

Cass A |
3 |
Requires beamswitching |

Cygnus X |
7 |
Requires beamswitching |

Galaxy |
1-50 |
Strong signals only a few minutes needed to obtain good spectra |

Andromeda |
~0.5 |
Very weak - difficult experiment requires days of observing |

Discussion questions / exercises

- Explain why the standard deviation can be less than one even though the data in the file is written out as whole numbers
- Perform the same analysis on real data from the SRT radiometer. [If you have a large data set you will find that unless you remove long term drifts the standard deviation will not decline beyond N ³ 30. For spectral line it is appropriate to first average the spectra which removes the drifts in the radiometer gain. For continuum observations the position switching removes the radiometer drifts.]

Updated: 10 August 1999 |