Sensitivity and System Temperature

For radiation emitted by a randomly polarised source, the power received by a radio telescope is given by:

$$P_{received} = \frac{1}{2} A S \Delta \nu ,$$ (2)

where $A$ is the effective area of the antenna, $S$ is the spectral power flux density and $\Delta \nu$ is the range of frequencies observed (the effective bandwidth). The factor of $\frac{1}{2}$ occurs because a detector can only respond to one polarisation component of the randomly polarised wave. The ATCA overcomes this limitation by providing separate detectors and electronics for two orthogonal polarisations, thus allowing all the power in the wave to be detected.

The sensitivity of a radio telescope is the minimum signal power that can be distinguished from the random fluctuations at the output of the receiving system which are caused by noise inherent in the system. The sensitivity is usually defined as the spectral power flux density (measured in Janskys, with 1Jy = $10^{-26}$Wm$^{-2}$Hz$^{-1}$) of a source that would produce the same signal power as the noise power.

The noise power consists of two main components: the noise power due to the receiving amplifier and other electrical system components, and the noise due to ground radiation, thermal emission from the atmosphere (which varies with elevation, cloud cover, etc), background radio emission from our Galaxy and other sources detected by the antenna. These noise powers are usually referred to as equivalent temperatures, although at no point is a physical temperature measured. The `temperature' due to the noise power is called the system temperature, $T_{sys}$, and is related to the noise power by:

$$T_{sys} = \frac{P_{noise}}{k \Delta \nu} ,$$ (3)

where $P_{noise}$ is the noise power and $k$ is Boltzmann's constant. As both the source and noise signals are random in nature, measurements of the power levels made at time intervals separated by $(2 \Delta \nu)^{-1}$ can be considered independent (e.g., Thompson, Moran & Swenson 1986, 2001). If the signal level is averaged for $t$ seconds then $2 \Delta \nu t$ samples have been measured.

The signal to noise ratio is the ratio of the power in the output that is due to the radio source being observed to that caused by the noise, and is given by:

$$\frac{S}{N} = \frac{T_{ant}}{T_{sys}} \times \sqrt{{\mathrm number\; of\; independent\; samples}} .$$ (4)

In this expression, $T_{ant}$ is the antenna temperature, the equivalent temperature of the radio source being observed. Note that ``antenna temperature'' is sometimes also taken to mean the contribution to the noise power from radio noise detected by the antenna, as described above. For typical values for bandwidth (128MHz) and integration time (12 hours), it is possible to detect a signal for which the power level is little more than $10^{-6}$ times the noise level. A table (on page 24) lists sensitivity values for the ATCA at various wavelengths. Christiansen and Högbom (1985) has a chapter on sensitivity.