A statistical method for analysing observations of faint radio stars

<jats:title>ABSTRACT</jats:title><jats:p>The observation of faint radio sources is often limited by the angular resolving power of the radio telescope, and not by its sensitivity. Continuous recording of the output from the receiving apparatus then produces a trace which always shows the sum of the effects of a number of sources in the reception pattern, instead of showing sources passing through the reception pattern one at a time. The apparent intensities of sources found on the records are affected by the presence of several other, fainter, sources in the reception pattern, and even the existence of many of the faint sources is doubtful. Counts of numbers of sources at given intensity levels are therefore unreliable.</jats:p><jats:p>Any assumed number-intensity law for radio sources can, however, be tested by a method free from the uncertainties due to confusion between adjacent sources, using a direct relation between the number-intensity law and the frequency distribution of deflexions of various sizes on the observed records. The present paper shows the derivation of the relation between <jats:italic>M (I) dI</jats:italic>, the mean number of radio sources per steradian in the intensity interval <jats:italic>dI</jats:italic>, and <jats:italic>P(D)dD</jats:italic>, the probability that the deflexion <jats:italic>D</jats:italic> recorded at an arbitrary instant lies in the interval <jats:italic>dD</jats:italic>. Expressions for <jats:italic>P(D)</jats:italic> are worked out for some specific models of <jats:italic>M (I)</jats:italic>; a comparison with observation has been given in an earlier paper (<jats:bold>6</jats:bold>).</jats:p>