Transcript Lecture 1 - University of Cape Town

Lecture 11
• (Was going to be
– Time series
– Fourier
– Bayes
• but I haven’t finished these. So instead:)
• Radio astronomy fundamentals
NASSP Masters 5003F - Computational Astronomy - 2009
Radio Astronomy Fundamentals
R
Types of electron acceleration:
• Thermal (random jiggling)
• Synchrotron (spiral)
• Spectral line (resonant sloshing)
NASSP Masters 5003F - Computational Astronomy - 2009
Noise power spectrum
• Analyse the signal into Fourier
components. jth component is:
Vj(t) = Vj exp(-iωjt)
= Vj (cos[ωjt] + i sin[ωjt])
• The Fourier coefficient Vj is in general
complex-valued. Power in this component
is: P = V *(t)V (t)/R
j
j
j
= Vj *Vj/R (cos2[ωjt] + sin2[ωjt])
= Vj *Vj/R
• Closely related to the ‘power spectrum’
we’ve already encountered in Fourier
theory.
NASSP Masters 5003F - Computational Astronomy - 2009
Averaging the power spectrum
t=1
t = 16
t=4
t = 64
NASSP Masters 5003F - Computational Astronomy - 2009
Total noise power output by the antenna.
• “Noise is noise”: signal from an
astrophysical source is indistinguishable
from contamination from
– Background thermal radio noise.
– Ditto from intervening atmosphere.
– Noise generated in the receiver system.
• Each of these makes a contribution to the
total. Thus the total noise power is
Ptotal = Psource + Pbackground + Patmosphere + Psystem
NASSP Masters 5003F - Computational Astronomy - 2009
On-and-off source comparison
• The simplest way to determine the source
contribution is to make 1 measurement pointing
at the source, then a second pointing away from
(but close to) the source, then subtract the two.
• Scanning over the source is also popular.
• Uncertainty in total power measurements is:
P
P 
t 
• A low-pass filter with a time constant t is another
way of ‘averaging’.
NASSP Masters 5003F - Computational Astronomy - 2009
Antenna detection efficiency
• The source radiates at S W m-2 Hz-1.
• The antenna has an effective area Ae in
the direction of the source. (Eg for a dish
antenna pointed to the source, this is close
to the actual area of the dish.)
• Thus the power per unit frequency interval
picked up by the antenna is:
w = AeS watts per herz.
• However, antennas are only sensitive to
one polarisation...
NASSP Masters 5003F - Computational Astronomy - 2009
Decomposition into polarised components
The total power in the signal is the sum of the power in each
polarization. An antenna can only pick up 1 polarization though.
NASSP Masters 5003F - Computational Astronomy - 2009
Dependence on source polarisation
• If the source is unpolarized, the antenna
will only pick up ½ the power, regardless
of orientation.
• If the source is 100% polarized, the
antenna will pick up between 0 and 100%
of the power, depending on orientation
(and type of detector – eg is detector
sensitive to linear polarization, or circular).
• Obviously all values in between will be
encountered. Thus measurement of
source polarization is important.
NASSP Masters 5003F - Computational Astronomy - 2009
Directionality of antennas.
A radio telescope often (not always) incorporates a mirror.
These are supposed to be smooth
mirrors?
GMRT
Parkes
Ok as long as the roughness is << λ.
An optically ‘smooth’ surface
NASSP Masters 5003F - Computational Astronomy - 2009
Directionality of antennas.
Radio telescopes with a mirror can be analysed like
any other reflecting telescope...
Point Spread Function
Focal plane
Reflector
NASSP Masters 5003F - Computational Astronomy - 2009
A more usual treatment:
Side lobe
It is often conceptually easier to imagine
that the antenna is emitting radiation to
the sky rather than absorbing it.
Beam width
Side lobe
Beam width ~λ/D, same as for any other reflector.
Eg Parkes 64m dish at 21 cm, beam width ~ 15’.
NASSP Masters 5003F - Computational Astronomy - 2009
Going into a little more detail...
• Essential quantities:
– The distribution of brightness B(θ,φ) over the
celestial sphere. (See next slide for definition of
θ,φ.) The units of this are W m-2 Hz-1 sr-1 (watts
per square metre per herz per steradian).
– The effective area Ae of the antenna, in m2.
(This is something which must be measured as
part of the antenna calibration.)
– The relative efficiency f(θ,φ) of the antenna,
which is normalized such that it has a maximum
of 1. (This shape must also be calibrated.)
– The received power spectrum w (units: W Hz-1).
NASSP Masters 5003F - Computational Astronomy - 2009
Going into a little more detail...
Pointing direction of the
antenna – NOT the zenith.
Kraus uses P
where I have f.
J D Kraus, “Radio
Astronomy” 2nd ed.,
Fig 3-2.
NASSP Masters 5003F - Computational Astronomy - 2009
Going into a little more detail...
• The general relation between these
quantities is:
1
w  Ae  d f  ,  B ,  
2
Remember that the ½ only applies where B
is unpolarized.
• Further useful relations:
S   d B  ,  
 A   d f  ,  
• It can be shown that ΩA = λ2/Ae.
NASSP Masters 5003F - Computational Astronomy - 2009
Going into a little more detail...
• Let’s consider two limiting cases:
– B(θ,φ) = B (ie, uniform over the sky);
– B(θ,φ) = S δ(θ-0,φ-0) (ie, a point source of
flux=S, located at beam centre).
B(θ,φ)=Sδ
B
f
w = ½ λ2 B
f
w = ½ Ae S
...the ½ still applies only for unpolarized B.
NASSP Masters 5003F - Computational Astronomy - 2009
Conversion of everything to temperatures.
• Suppose our antenna is inside a cavity
with the walls at temperature T (in kelvin).
• It can be shown that the power per unit
frequency picked up by the antenna is
w = kT watts per herz.
• Because of this linear relation between a
white noise power spectrum and
temperature, it is customary in radio
astronomy to convert all power spectral
densities to ‘temperatures’. Hence:
NASSP Masters 5003F - Computational Astronomy - 2009
System temperature
Ttotal = Tsource + Tbackground + Tatmosphere + Tsystem
• Tsource only says something about the real
temperature of the source if
– The source area is >>ΩA, and
– The physical process producing the radio waves
really is thermal.
• Tatmosphere is a few kelvin at about 1 GHz.
• Tbackground may be as much as 300 K if the
antenna is seeing anything of the surroundings!
Therefore avoid this.
• Tsystem again says nothing about the real
temperature of the receiver electronics. Rather it
is a figure of merit – the lower the better.
NASSP Masters 5003F - Computational Astronomy - 2009
The more usual way to write the measurement
uncertainty:
Ttotal
T 
t 
•
•
•
Thus the minimum detectable flux is
2k Ttotal
Smin 

Ae
t 
and the minimum detectable brightness:
2k Ttotal
Bmin  2 

Note:
t 
1. Bmin not dependent on Ae.
2. Factors of 2 only for unpolarized case.
NASSP Masters 5003F - Computational Astronomy - 2009
“Back end”
“Front end”
A more realistic system:
R M Price, “Radiometer Fundamentals”,
Meth. Exp. Phys. 12B (1976),
Fig 1, section 3.1.4.
NASSP Masters 5003F - Computational Astronomy - 2009
Jargon
• The ‘antenna’:
A 38 GHz feed horn. The
corrugations are good
for wide bandwidth.
– the reflecting surface
(ie the dish).
• The ‘feed’:
– usually a horn to
focus the RF onto the
detector.
• The ‘front end’:
– electronics near the
Rx (shorthand for
receiver).
• The ‘back end’:
– electronics near the
data recorder.
• The LO:
– local oscillator.
•RF:
–Radio Frequency.
•IF:
–Intermediate Frequency.
NASSP Masters 5003F - Computational Astronomy - 2009