THEORY OF MEASUREMENTS Mike Davis Fourth NAIC-NRAO School on Single-Dish Radio Astronomy Green Bank, WV July 2007

Thanks To Don Campbell, author of the original version of this talk, and to the editors of ASP Volume iarticle_query?2002ASPC C&data_type=PDF_HIGH &whole_paper=YES&type=PRINTER&filetype=.pdf

OUTLINE Antenna-Sky Coupling Flux Density Antenna Sensitivity The Radiometer Equation Performance Measures Beam Patterns as Spatial Filters

Summary Clean beams are better (Sky coupling) Bigger and more efficient is better (Effective Aperture) More observing time is better (Radiometer Equation - √n) Lower noise is better (Performance Measures) What you see is –not- what’s there  (Beam Patterns as Spatial Filters)

Antenna-Sky Coupling Power Received: Source ^ ^ Antenna Pattern Effective Area ^ Factor ½ comes from one of two polarizations

Antenna-Sky Coupling (cont’d) Power Received in bandwidth ∆ν :

Antenna-Sky Coupling (cont’d) For thermal sources (Planck’s Law):

Antenna-Sky Coupling (cont’d) For kT << hv (Rayleigh-Jeans):

Antenna-Sky Coupling (cont’d) Substituting, this gives the following Rayleigh-Jeans approximation:

Antenna Temperature Power available at terminal of a resistor: Replace the antenna with a matched resistor at a physical temperature that gives the same response: T A is defined as the ANTENNA TEMPERATURE.

Flux Density Spectral flux density for a discrete source (one with a clear boundary):

Flux Density (cont’d) Observed Flux Density: This is < S depending on the size of the source.

Flux Density (cont’d) Large Source:

Flux Density (cont’d) Small Source

Flux Density (cont’d) The standard unit of spectral flux density in radio astronomy is the Jansky: 0 dBJy = -260 dB Wm -2 Hz -1 In decibels:

Decibel Approximation Good to 1% or better dB~ValuePercent Error π/ π π2π

Antenna Sensitivity (K/Jy) From earlier equations: which gives: An effective aperture of 2760 m 2 is required to give a sensitivity of 1.0 K/Jy.

The Radiometer Equation Averaging n samples improves an estimate by √n : ∆T = T/√n. There are ∆ν independent samples per second for a measurement bandwidth ∆ν Hz. Averaging for ∆τ seconds gives n = ∆τ ∆ν : A 100 MHz bandwidth reduces noise by a factor 10,000 in 1 second.

Performance Measures Signal/Noise Hence Ae/Tsys (m 2 /Kelvin) is a useful measure of telescope performance.

System Equivalent Flux Density SEFD is defined as the point source flux density required to produce an antenna temperature equal to the system temperature: An antenna with 2 K/Jy sensitivity and system temperature 20 K has an SEFD = 10 Jy. Note that smaller SEFD is better.

Scanning the Antenna Moving the antenna pattern across the source results in a convolution. In one dimension: This convolution integral may be written as

Antenna Pattern as Spatial Filter The Convolution Theorem gives The spatial structure of the true sky signal is weighted by the transform of the antenna pattern. High spatial frequencies are lost.

Antenna Pattern and its Fourier Transform

Antenna Pattern as Spatial Filter (cont’d) It is generally true for any antenna that the spatial response of the far-field pattern is the autocorrelation of the aperture plane distribution. For an array, this has a hole at zero- spacing that eliminates low spatial frequencies. Accurate sky representation may require combining single dish and array data for this reason.

Thanks Again To Don Campbell, author of the original version of this talk, and to the editors of ASP Volume iarticle_query?2002ASPC C&data_type=PDF_HIGH &whole_paper=YES&type=PRINTER&filetype=.pdf