A measure of the sensitivity of an observation in radio astronomy is provided by the increase in power level at the receiver input which causes a change in the output equal to the r.m.s. noise at the output. The output of the receiver detector is a function of the total power at the input of the receiver. The total input power consists of the wanted signal power and the unwanted noise power (*e.g.*, thermal and receiver noise). Both contributions are caused by random processes, and it is not possible to distinguish between them quantitatively. However, both have an average power level, and if these levels can be established with sufficient precision, the presence of the wanted signal can be detected. (It is assumed that the gain and other parameters of the receiving system remain constant during the observation.) The statistical average of a stationary random variable such as noise power (*P*) can be found with a precision which is inversely proportional to the square root of the number of independent samples (*N*), and the standard deviation of this average is:

d(1)P~PN^{-0.5}

The standard deviation (d*P*) is also a root mean square (r.m.s.) quantity. By observting a sufficient number of samples (*N*), the measurement of the radio noise power can be made with high precision. By reducing the fluctuations d*P* to a value less than the wanted signal power, detection of very weak signals is possible. Within a band of width d*f*, approximately 2d*f* samples per second can be measured by the receiver, and by extending the observing time (*t*), (also called: “integration time”), *N* can be made very large. Now,

(2)N~ 2 df t

and if this relation is combined with (1),

d--PK=--------- (3)P(df t)^{0.5}

where *K* is a proportionality factor which is dependent on details of the equipment and the observing technique, and is discussed by Kraus (1966, *Radio Astronomy*, McGraw Hill, New York, NY – 2nd edition 1986, Cygnus-Quasar Books, Powell, OH). For a basis total power system (*i.e.* one that measures the total noise power delivered by an antenna) *K* = 1, and this value will be adopted is adopted for generality. (Note, however, that for the case where the observing time is divided equally between a source and a reference position, the required value of *t* is equal to the ratio of the time the source is observed and the time the reference position is observed, d*t*. Usually d*t* is of the order of half the total observing time. The r.m.s. error in the difference between the measurements on the source and on the reference position is equal to the error in the measurement on the source multiplied by d*t*^{-0.5}).

As used above, *P* and d*P* refer to noise powers, but equation (3) also holds if these quantities are power spectral densities, which will be denoted by subscript *s*. Thus, d*P*_{s}, the noise fluctuation in power spectral density in the sensitivity equation (3), is related to the total system sensitivity (noise fluctuations) expressed in temperature fluctuations (d*T*) through the Boltzmann constant, k, as shown in equation (4):

d(4)P_{s}= k dT

and the sensitivity equation is expressed by

TdT=--------- (5)(df t)^{0.5}

where:

(6)T=T_{A}+T_{R}

*T* is the system temperature and is the sum of *T*_{A}, the antenna noise temperature which results from the cosmic emissions, the Earth’s atmosphere and radiation from the Earth, and *T*_{R}, the receiver noise temperature (* from:* “ITU-R

*Handbook on Radio Astronomy*“, 1995, section 4.3.1).