Spherical harmonics and Earth-rotation synthesis in radio astronomy

That spherical harmonic functionsY_{nm}(\theta,\phi)can be used to advantage in the Earth-rotation synthesis of radio astronomy maps is shown in this paper. As Earth rotates the baseline of a radio interferometer generates a cone whose angle\theta, measured from theN-Spolar axis, can be varied by changing the baseline's azimuthal direction on the surface of Earth. A series of Earth-rotation measurements, at different cone angles but with baselines of equal lengthb, can be regarded as being made on a baseline sphere of radiusb, the analog of theuvplane in Fourier-type synthesis. The measured output distribution can be expanded as a spherical harmonic series on the baseline sphere. The coefficients of the series are related to the coefficients of the spherical harmonic series expansion of the source distribution on the celestial sphere by a matrix transformation. The matrix [B] is a function only of the baseline configuration (it does not vary with source declination). Inversion of the matrix leads to the solution for the source coefficients, from which a spherical harmonic map is formed of the source distribution.