Closure phase

The closure phase is an observable quantity in imaging astronomical interferometry, which allowed the use of interferometry with very long baselines. It forms the basis of the self-calibration approach to interferometric imaging. The observable which is usually used in most 'closure phase' observations is actually the complex quantity called the triple product (or bispectrum). The closure phase is the phase of this complex quantity, but the phrase 'closure phase' is still more commonly used than the more accurate phrase 'triple product'. The closure phase is an observable quantity in imaging astronomical interferometry, which allowed the use of interferometry with very long baselines. It forms the basis of the self-calibration approach to interferometric imaging. The observable which is usually used in most 'closure phase' observations is actually the complex quantity called the triple product (or bispectrum). The closure phase is the phase of this complex quantity, but the phrase 'closure phase' is still more commonly used than the more accurate phrase 'triple product'. Roger Jennison developed this novel technique for obtaining information about visibility phases in an interferometer when delay errors are present. Although his initial laboratory measurements of closure phase had been done at optical wavelengths, he foresaw greater potential for his technique in radio interferometry. In 1958 he demonstrated its effectiveness with a radio interferometer, but it only became widely used for long baseline radio interferometry in 1974. A minimum of three antennas are required. This method was used for the first VLBI measurements, and a modified form of this approach ('Self-Calibration') is still used today. The 'closure-phase' or 'self-calibration' methods are also used to eliminate the effects of astronomical seeing in optical and infrared observations using astronomical interferometers. A minimum of three antennas are required for closure phase measurements. In the simplest case, with three antennas in a line separated by the distances a1 and a2 shown in diagram at the right. The radio signals received are recorded onto magnetic tapes and sent to a laboratory such as the Very Long Baseline Array. The effective baselines for a source at an angle θ {displaystyle heta } will be x 1 = a 1 cos ⁡ θ {displaystyle x_{1}=a_{1}cos heta } , x 2 = a 2 cos ⁡ θ {displaystyle x_{2}=a_{2}cos heta } , and x 3 = ( a 1 + a 2 ) cos ⁡ θ {displaystyle x_{3}=(a_{1}+a_{2})cos heta } . When one mixes signals from two of antennas (compensating for a delay for the angle θ 0 {displaystyle heta _{0}} ) one observes interference signal with phase x ( θ ) − x ( θ 0 ) . {displaystyle x( heta )-x( heta _{0}).} Taking into account that signals may come from several sources, the complex interference signal is the Fourier transform P {displaystyle P} of the power density of the sources. The phases of the complex visibility of the radio source corresponding to baselines a1, a2 and a3 are denoted by ϕ 1 {displaystyle phi _{1}} , ϕ 2 {displaystyle phi _{2}} and ϕ 3 {displaystyle phi _{3}} respectively. These phases will contain errors resulting from εB and εC in the signal phases. The measured phases for baselines x1, x2 and x3, denoted ψ 1 {displaystyle psi _{1}} , ψ 2 {displaystyle psi _{2}} and ψ 3 {displaystyle psi _{3}} , will be: Jennison defined his observable O (now called the closure phase) for the three antennas as: