THEORY OF MEASUREMENTS Mike Davis
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Transcript THEORY OF MEASUREMENTS Mike Davis
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 278.
http://articles.adsabs.harvard.edu/cgi-bin/nphiarticle_query?2002ASPC..278...81C&data_type=PDF_HIGH
&whole_paper=YES&type=PRINTER&filetype=.pdf
OUTLINE
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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:
Effective Area ^
Source ^
^ Antenna Pattern
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:
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:
In decibels:
0 dBJy = -260 dB Wm-2Hz-1
Decibel Approximation
Good to 1% or better
dB
0
~Value
1
1
0.00
1.25
2
3
2
2.5
4
-0.66
0.47
5
8
10
-0.48
π
7
-0.90
0.24
5
9
-0.71
π/2
4
6
Percent Error
-0.24
2π
8
-0.42
0.71
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 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 (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 zerospacing 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 278.
http://articles.adsabs.harvard.edu/cgi-bin/nphiarticle_query?2002ASPC..278...81C&data_type=PDF_HIGH
&whole_paper=YES&type=PRINTER&filetype=.pdf