THEORY OF MEASUREMENTSMike Davis

Fourth NAIC-NRAO School on Single-Dish Radio Astronomy

Green Bank, WVJuly 2007

Thanks

To Don Campbell, author of the original version of this talk, and to the editors ofASP Volume 278.

http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?2002ASPC..278...81C&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 PatternEffective 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 TemperaturePower available at terminal of a resistor:

Replace the antenna with a matched resistor at a physical temperature that gives the same response:

TA 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-2Hz-1

In decibels:

Decibel ApproximationGood to 1% or better

dB ~Value Percent Error

0 1 0.001 1.25 -0.712 π/2 -0.903 2 0.244 2.5 -0.485 π -0.666 4 0.477 5 -0.248 2π -0.429 8 0.7110 10 0.00

Antenna Sensitivity (K/Jy)From earlier equations:

which gives:

An effective aperture of 2760 m2 is required to give a sensitivity of 1.0 K/Jy.

The Radiometer EquationAveraging 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 (m2/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 ofASP Volume 278.

http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?2002ASPC..278...81C&amp;data_type=PDF_HIGH&amp;whole_paper=YES&amp;type=PRINTER&amp;filetype=.pdf