Radio Astronomy with a Single-Dish Radio Telescope

Michael Gaylard
Hartebeesthoek Radio Astronomy Observatory

August 28, 2012

1 Introduction

The aim of these notes is to provide some basic theory to help us understand how radio telescopes work and
how to do practical radio astronomy with them.

Radio waves are a type of electromagnetic wave. The spectrum of electromagnetic waves goes from Gamma
rays at short wavelengths = high frequencies to radio waves at long wavelengths = low frequencies, as shown
in Fig. 1.

Radio waves are those electromagnetic waves with a wavelength greater than a centimetre. For example,
commercial FM radio operates in the frequency range from 88 to 108 MHz, corresponding to a wavelength
band of about 3 metres. Cellphones operate at a frequency of 900 MHz, i.e. a wavelength of 33 cm. We can
refer to radio waves by either their wavelength λ or their frequency ν. All electromagnetic waves travel at
the speed of light c, and
c = νλ. (1)
Hence to convert between frequency and wavelength we have λ = c/ν, and ν = c/λ.

Radio telescopes are designed to detect natural radio emission from objects beyond the Earth. The 26-m
radio telescope at Hartebeesthoek is a microwave telescope, i.e. it operates in the part of the radio spectrum
where the wavelengths are from 30 to 1 cm, i.e. a frequency of 1 up to 30 GHz. Microwave ovens operate at
a wavelength of about 13 cm (2.4GHz), and can be detected by the Hartebeesthoek radio telescope.

Radio astronomy is generally carried out with telescopes that are large compared to the wavelength being
observed. This means that they pick up radio waves coming from a small area of the sky, in what is known as
the “main beam” of the telescope. For a single telescope, this main beam typically has an angular diameter
smaller than a degree, as shown for example in Fig. 2.

What do we mean by angular diameter or angular size? This is explained in Fig. 3. For example, the Sun
has a physical diameter of 1.4 million kilometres, while the Moon has a diameter of 3500 kilometres. Yet,

Figure 1: The spectrum of electromagnetic radiation, from high frequency (γ rays) to low frequency (radio
waves).

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Figure 2: Part of a map of the radio sky made with the Hartebeesthoek telescope at 2.3 GHz. The angular
scale is in degrees. The smallest patch of sky the telescope “sees” at this frequency is about one third of
a degree. Objects much bigger than this appear approximately their true size (e.g. the Barnard Loop).
However small objects all appear smeared out to about one third of a degree in size. Note the “blobs”
forming rings around the strong radio sources Orion A and Orion B. These effects are explained in the text.
Map by Jonas et al. 1998.

as seen from the Earth, the Sun and the Moon appear to be the same size, i.e. they have the same angular
diameter. How can this be? Although the Sun is 400 times bigger than the Moon, it is also 400 times further
away.

The beam size depends on the frequency in use and the diameter of the telescope. Where networks of radio
telescopes operate together, the diameter of the virtual telescope thus created can be up to 10 000 km as
opposed to tens of metres for a single telescope. The angular diameter of the synthesized beam of this
telescope array may be as small as one millionth of a degree.

Outside of the main beam, the telescope is still weakly sensitive to radiation coming from other directions,
in what are known as its “sidelobes”. Man-made radio signals become a problem if they can be detected
in these sidelobes, e.g. microwave ovens, wi-fi, bluetooth. These are specific examples of “radio frequency
interference” (RFI).

When we look at the sky, our eyes see light which has a continuous, varying distribution of brightness. In the

A B

2
1 3 4
T T

r1 r3 r4

r2

Figure 3: Comparison of physical radius r and angular radius θ. In diagram A the two objects have different
physical radii: r2 > r1, but the same angular radii as seen by telescope T : θ1 = θ2. This applies to the
Moon and Sun as seen from Earth. By contrast, in diagram B, r4 > r3, but θ4 < θ3.

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Figure 4: A focusing lens or reflector with a circular Figure 5: The diffraction pattern produced by a
aperture. circular focusing lens or reflector.

Figure 6: Cross-sections through the beam pattern Figure 7: Beam cross-sections of ideal and a typical
of an ideal antenna and real antenna, with a linear real antennas, with a logarithmic vertical scale to
vertical scale. Note the first nulls at 1.2λ/D. show the sidelobe structure.

same way, the radio telescope looks at a sky which has a continuous, varying, distribution of radio emission.
There are objects of large angular extent producing radio waves, such as hot gas clouds in the Milky Way,
and objects of small angular extent, such as distant galaxies, masers and pulsars. Some of the these radio
sources will be of larger angular size than the main beam of the telescope and some will be smaller. What
the telescope “sees” is the actual radio brightness distribution in the sky convolved with (“smeared out” by)
the beam of the telescope. The bigger the beam, the more it smears out.

2 Radio telescope antennas

A “classic” radio telescope comprises a circular parabolic reflector with a small receiving element such as a
microwave feed horn at the focus to collect the incoming radio waves and pass them to transistor amplifiers
in a receiver. A DSTV satellite dish works in exactly the same way and can be used as a mini-radio telescope.
The reflector telescope was first described by Scotsman James Gregory in 1663, but Englishman Isaac Newton
built the first in 1668. American Groote Reber built the first reflector radio telescope, in 1937.

To understand how this system responds to radiation coming from different angles, consider what happens
when a plane wave of wavelength λ is incident on a circular aperture of diameter D (Fig. 4). Constructive
and destructive interference produces a circularly symmetric diffraction pattern, with a central maximum
and concentric rings of decreasing strength (Fig. 5). This same pattern describes the response of a circular
antenna to plane waves coming from different angles. The first minimum or null occurs at a radius of about
1.2λ/D radians, so the beamwidth to first nulls is

BW F N ∼ 2.4λ/D [radians]. (2)

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Figure 8: The size of the main beam of the 26-m telescope depends on the operating frequency / wavelength.
Here the actual observed main beam at four wavelengths are shown with the angular size of the Moon for
comparison. Dual feeds on the 6 and 3.5 cm receivers produce two beams.

The beamwidth at the half-power points (HPBW), also called the Full Width at Half Maximum (FWHM)
is about half this, as shown in figs. 6 and 7.

HP BW = F W HM ∼ 1.2λ/D [radians]. (3)

The angular size of the main beam of the 26-m Hartebeesthoek telescope at four different wavelengths is
shown in Fig. 8. An “ideal” antenna would produce a beam that would capture 100% of the incoming
energy in the centre of the main beam and would have no sidelobes. This antenna would have an “aperture
efficiency” ǫap of 1. It is not possible to actually achieve this.

Practically, there two things we can do to make the antenna as efficient as possible. Firstly, we ensure that
nothing blocks the radio waves coming into the antenna. We can do this by placing the feed horn off to the
side of the reflector, as in a DSTV satellite dish. This is called an “offset paraboloid” design. Secondly, we
ensure that the surface is very smooth compared to the wavelength at which the antenna must work. When
both these conditions apply, the “best achievable” aperture efficiency is about 0.80. Figs. 6 and 7 show an
ideal and an actual beam pattern in cross section on linear and logarithmic scales. The “ideal” pattern has
been modelled here with a parabolic shape, while the mathematical form of the real pattern is a (sinX/X)2
function. This describes the diffraction pattern where nothing obstructs the path of the waves, providing
what is called an “unblocked aperture”. The large new Green Bank (Byrd) Telescope in the USA (Fig. 9)
is shaped like a giant DSTV satellite dish and has an unblocked aperture. Its aperture efficiency is 0.71 at
long wavelengths.

Often a small reflector of hyperbolic curvature is placed in front of the focus of the main reflector. This is
called a “subreflector”. The feed horn is then placed at the focus of this second reflector. This is called a
Cassegrain optical system and was developed in 1672 by the French sculptor Sieur Guillaume Cassegrain. It
is widely used for both optical and radio telescopes, including the Hartebeesthoek 26-m telescope (Fig. 10).
The blockage of the aperture by the hyperbolic subreflector, its supports, and the feed housing reduces the
maximum achievable aperture efficiency on this telescope to about 0.64.

3 Gain of a parabolic reflector antenna

If we regard a radio telescope as an electronic circuit, the antenna acts as an amplifier. What is the gain of
this amplifier? This depends on the area of sky that it sees. Angular area is called a “solid angle” and the
units are radians2 , or steradians (sr). An object with an angular radius θ radians has subtends a solid angle

Ω = 2π(1 − cos θ) [sr]. (4)

For small θ,
Ω = πθ2 [sr]. (5)

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Figure 9: The 110x100-m diameter Green Bank Figure 10: The 26-m Hartebeesthoek antenna,
Telescope is the largest steerable radio telescope. showing the aperture blockage from the subreflector
It has an offset feed and unblocked aperture, giving and its supports that reduces the aperture efficency
an aperture efficiency of 0.71 at longer wavelengths. compared to the GBT. This has a maximum possi-
ble aperture efficiency of 0.64.

First consider an antenna that is equally sensitive to radiation from all directions. It is defined to have again
of unity. It “sees” the whole sky. In this case θ = π hence cos θ = -1, and so the whole sky has a solid angle
of 4π sr. Large antennas are primarily sensitive to radiation coming from a small solid angle (fig. 11) and
so the gain in the main lobe is much greater than unity.

To explore this in more detail, first we define the total beam solid angle as
ZZ
ΩA = Pn (θ, φ)dΩ [sr] (6)
4π

where Pn (θ, φ) = beam (power) pattern of antenna, normalised so that Pn (0, 0) = 1. One can think of the
total beam solid angle as the solid angle that would be subtended by an “ideal” beam with the same gain
as at the centre of the actual beam, so that it receives the same amount of power as the actual antenna
does integrated over all angles (Fig. 12). If we only integrate to the first minimum of the beam pattern, we
obtain the main beam solid angle: ZZ
ΩM = Pn (θ, φ)dΩ [sr] (7)
mainlobe

It can be shown (e.g. Kraus 1986) that the beam solid angle ΩA depends on the ratio of the square of the
wavelength λ to the effective collecting area Ae , which is defined as the product of the physical area Ap and
the aperture efficiency ǫap :
λ2 λ2
ΩA = = [sr] (8)
Ae Ap ǫap
An isotropic receiving antenna is an antenna that receives equally from all directions, ie from a solid angle
of 4π steradians, and has a gain of unity. The antenna gain G is the ratio of the solid angles from which an
isotropic radiator and the actual antenna receive:
4π 4πAp ǫap
G= = (9)
ΩA λ2

For a typical radio telescope the gain can be 100 000 to 10 000 000, i.e. 105 to 107 , or 50 to 70 dB (Remember
that the gain G of an amplifier in the engineering units of decibels (dB) is 10 log10 [G]). If the aperture
efficiency of the Hartebeesthoek 26-m telescope is approximately 0.5, what is its gain at 1.6 GHz and at 12.5
GHz?

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Figure 11: The beam pattern in polar coordinates, Figure 12: The beam solid angle.
showing measures of the width of the main beam
(Kraus 1986).

4 Power received from radio emitters in the sky

We first need to comment on the polarization of the incoming signals. Detectors operating in the visible
part of the spectrum, such as photographic film or the CCD in a digital camera, capture incoming photons
regardless of their polarization state. By contrast, at radio wavelengths the receivers are sensitive to the
polarization state of the incoming radiation. It is common practice in radio astronomy to have two receivers
attached to each receiving feedhorn, with a splitter feeding left-circularly polarized radiation to one receiver
and right-circularly polarized radiation to the other. The total intensity is the sum of what is received in
each polarization. In the earlier days of radio astronomy often only one receiver was installed, and the
textbooks may assume this in their derivation of equations, although it is normally stated explicitly at some
point. This is what gives rise to the factor 12 in eqn. 10 below.

We can now define the power w received per unit bandwidth, in each polarization, from an element of solid
angle of the sky: ZZ
1
w = Ae B(θ, φ)Pn (θ, φ)dΩ [W Hz−1 per polarization] (10)
2
Ω

where Ae = effective aperture (= collecting area) of antenna [m−2 ]

B(θ, φ) = brightness distribution of radio emission across the sky [W m−2 Hz−1 sr−1 ]
Pn (θ, φ) = normalised power (beam) pattern of the antenna
dΩ = sinθ dθ dφ, element of solid angle [sr]

What we normally measure in radio astronomy is the integral of the brightness over a radio source. This
called the flux density S of the source: ZZ
S= B(θ, φ)dΩ (11)
source

When the source is observed with an antenna with the power pattern Pn (θ, φ), the observed flux density is
given by the integral of the brightness distribution multiplied by the antenna beam pattern:
ZZ
So = B(θ, φ)Pn (θ, φ)dΩ (12)
source

For sources small compared to the beam and in the centre of the beam, Pn (θ, φ) ≃ 1. For extended sources
with simple geometries, simple analytic functions enable So to be corrected, and these will be discussed later.

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Figure 13: (a) Simulated true radio source brightness distribution, convolved with beams of width (b) 5
units (c) 10 units (d) 15 units.

A simulation of how the finite width of the telescope beam “blurs” the true radio brightness distribution is
shown in Fig. 13.

The SI unit of flux density would be W m−2 Hz−1 . However, the radio emission is very weak as the radio
emitters are very far away, and the unit known as the Jansky [Jy], after the radio astronomy pioneer Karl
Jansky, was adopted in 1973. It is defined as 10−26 W m−2 Hz−1 .

5 Brightness Temperature and Antenna Temperature

Radio astronomy appears as something of a blend of astronomy and basic electric circuit theory. This derives
from the fact that the radio telescope can be considered as an electric circuit, and the object being observed
can be considered as a resistor at a particular temperature connected to the first amplifier in the receiver
(by radio waves rather than by wires). For some astronomical objects the temperature that we measure is
meaningful as a physical temperature. For others it is not, depending on the radiation mechanism involved,
and we therefore refer to the “brightness temperature” TB for these emitters. Astronomical masers provide
an example of this. The gas producing the emission may have a physical temperature of less than 200K, but
the very intense stimulated emission of the maser could have a brightness temperature of 1012 K.

For a black body radiator, the brightness B is given by

2hν 3 1
B= [W m−2 Hz−1 sr−1 ] (13)
c2 ehν/kT − 1

where h = Planck’s constant = 6.63 × 10−34 [J s]

ν = frequency [Hz]
c = velocity of light = 3 × 108 [m s−1 ]
k = Boltzmann’s constant = 1.38 × 10−23 [J K−1 ]
T = temperature [K].

Fig. 14 shows the brightness as a function of frequency for several black body radiators set to be of equal
size but at different temperatures. The frequency range over which the Hartebeesthoek telescope operates
is marked by vertical lines. The frequency range our eyes can see is marked by the colours of the rainbow
in a vertical column labelled “vis” near 1015 Hz. Clearly hotter objects produce more radiation than cooler
ones, and the brightness maximum occurs at a higher frequency. The wavelength or frequency at which the
intensity peaks is given by the well-known Wien displacement law.

From Fig. 14 we can see that for all objects with temperatures more than a couple of degrees above absolute
zero, the brightness peak occurs well above the operating range of radio telescopes. Hence we are working
in the range where hν << kT , so the Rayleigh-Jeans law applies and the brightness is proportional to the
temperature:
2kT
B= 2 [W m−2 Hz−1 sr−1 ] (14)
λ

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 Figure 14: Blackbody radiation from solid objects of the same angular size, at different temperatures

If we observe the radiation from a discrete source which has an effective temperature T and subtends a solid
angle Ωs , the source flux density (in SI units) is simply the product of the brightness and the source solid
angle:
2kT Ωs
S= [W m−2 Hz−1 ] (15)
λ2
As an example, if we take the brightness temperature of the Sun at a wavelength of 10 cm to be 10 000 K,
what is its flux density in SI units and in Jy? (Remember 1 Jy = 10−26 W.m−2 Hz−1 ). The nearest naked
eye star is α Centauri. This is a binary system of two Sun-like stars. What is its flux density at λ = 10 cm?

More generally, if the temperature distribution over the emitter is not uniform, the flux density becomes the
integral of the temperature distribution over the object:
ZZ
2k
S= 2 T (θ, φ)dΩ [W m−2 Hz−1 ] (16)
λ
Source

For a solid object such as a planet like the Earth, the optical depth through the object is very large, and
the observed temperature Tb ≃ Te , the actual temperature. For objects such as gas clouds, Tb will depend
on the optical depth τc , the physical temperature being denoted by the electron temperature Te :

Tb = Te (1 − e−τc ) [K] (17)

For small optical depths the brightness temperature approximates to:

Tb = Te τc [K] (18)

We can regard the observed radio source as being equivalent to a resistive load on the input to the first
amplifier in the receiver system on the telescope. The power w per unit bandwidth received at the terminals
of a resistor of temperature T would be

w = kT [W Hz−1 ] (19)

The radio source is not connected to the amplifier by wires, as a resistor would be. It is connected by
the radio waves emitted by the source and received by the antenna. This lets us consider it as a resistor,

8
and we can equate the power received from the source with power received from a resistor with the same
temperature. We call this temperature the measured “antenna temperature” TA of the radio source. It has
nothing to do with the physical temperature of the antenna.

The total energy received per unit bandwidth in the two polarizations is then the effective collecting area
times the observed flux density So , which we equate to the power received from the equivalent resistor:
ZZ
wlcp+rcp = Ae B(θ, φ)Pn (θ, φ)dΩ = Ae So = k(TAlcp + TArcp ) [W Hz−1 ] (20)
Ω

To obtain the true flux density S we introduce a size correction factor Ks . For sources that are very small
compared to the beam size, Ks = 1 and So = S, but the correction must be taken into account if the source
size is a significant fraction of the beam size. Details of the size correction are given later.

Simplifying eqn. 20, we obtain the true flux density of the source by summing the antenna temperatures
measured in left- and right-handed circular polarization and allowing if necessary for finite source size through
Ks :
k(TAlcp + TArcp )Ks
S= × 1026 [Jy] (21)
Ae
It is important to note that the flux density of a radio source is intrinsic to it, and the same flux density
should be measured by any properly calibrated telescope. However the antenna temperatures measured for
the same emitter by different telescopes will be proportional to their effective collecting areas.

We can only calculate the source flux density if we know the effective aperture (collecting area) at the
frequency being used, so we rewrite eqn. 21 and substitute the constants, to give:
1380(TAlcp + TArcp )Ks
Ae = [m2 ] (22)
So
This lets us calibrate the radio telescope at each frequency of interest. We carry out scans of standard
calibrator sources (Ott et al. 1994) and measure the peak antenna temperature in each polarization. The
calibrator flux densities are obtained from the formulae given by Ott et al. Substitution into the above
equation provides the effective aperture (collecting area). The physical collecting area Ap is obtainable from
the known diameter of the telescope (25.9 m for the Hartebeesthoek telescope). The aperture efficiency ǫap
can then be obtained at each frequency:
Ae
ǫap = (23)
Ap
For convenience, we often refer to the Point Source Sensitivity (P SS), which is (correctly) the number of
Kelvins of antenna temperature per polarization obtained per Jansky of source flux density. This is also
known as the ‘DPFU’ or ‘Degrees per Flux Unit’, the flux unit being the old term for the Jansky.

For small antennas (such as the Hart 26m) the P SS is often used in the inverse way, i.e. the number of
Janskys of flux density required to produce one Kelvin of antenna temperature in each polarization. For the
Hartebeesthoek telescope, the P SS, expressed in the latter form, is typically about 5 Jy/K per polariza-
tion. The P SS in each polarization is simple to determine experimentally, from the antenna temperatures
measured for calibrator sources of known flux density (remembering that for an unpolarized source, half the
total flux density is received in each polarization):
(S/2) (S/2)
P SSlcp = and P SSrcp = [Jy K−1 per polarization] (24)
Ks TAlcp Ks TArcp
Theoretically the values for the two polarizations should be the same; in practise there is always a small
difference between them, and data from each polarization should be corrected using the value appropriate
to that polarization.

If the angular size of emitting object is known and is small compared to the beam size, the emitter’s brightness
temperature Tb can be estimated from the ratio of the beam solid angle ΩA to the source solid angle Ωs :
ΩA
Tb = TA [K]. (25)
Ωs

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6 Microwave receiver systems

The Hartebeesthoek 26-m radio telescope is currently equipped with receivers to cover selected bands used
for radio astronomy: 18cm / 1.6GHz, 13cm / 2.3GHz, 6cm / 5GHz, 5cm / 6.0GHz, 3.5cm / 8.5GHz, 2.5cm
/ 12GHz, 1.3cm / 22GHz. In general, we want each receiver to cover as wide a band as possible. However,
some receivers are designed only to observe emission at one particular frequency from a specific atom or
molecule, and these have a relatively narrow bandwidth. For comparison, the bandwidth of the receiver in
an FM radio is 20MHz, as it covers 88 to 108MHz. When built by NASA in 1961 to track spacecraft, the
26-m antenna had one receiver, operating at 30cm / 960MHz.

The main components of a typical conventional microwave receiver system are shown in Fig. 15. The
incoming signals are very faint and noise-like. If the output of a receiver is connected to a loudspeaker, the
signal sounds like a hiss, the same hiss one hears if a radio is tuned off-station. The internal noise in the
amplifiers is generally much larger than the signal. To maximise sensitivity we need to minimise the noise of
the amplifiers. This is generally done with specially designed amplifiers that can be cooled in refrigerators,
in our case to 16 K, or −257oC.

Over a limited band, the noise-like signal from the sky and the noise from the amplifiers in the receiver can
be treated as though they were produced in resistors with specific absolute temperatures. The effective noise
temperature of a receiver TR depends on the noise temperature Tn and gain Gn of each amplifier:
T2 T3
TR = T1 + + [K] (26)
G1 G1 G2
In the example shown in Fig. 15, what are the contributions to the receiver temperature from the first three
stages? Which amplifier is most critical in keeping TR small?

The receiver noise temperature T is often quoted in the form of a “noise figure” F , relative to a nominal
ambient temperature of 290 K (17◦ C):
F = 1 + T /290 (27)
and can also be expressed in decibels:

FdB = 10 log10 (F ) [dB] (28)

What is the noise figure in dB of the receiver in Fig. 15?

Signal losses in waveguide and co-axial cable are large at microwave frequencies, so mixers are used to convert
signals down to a lower frequency that can be passed through many metres of cable to the control room
where the signal detecting instruments are located. The attenuation in passive components both reduces
(attenuates) the signal strength and introduces extra noise. Defining the signal loss L (∞ > L > 1) of the
component as the reciprocal of the gain G (0 < G < 1), the noise temperature TL of a lossy component at
a physical temperature TLP will be
TL = (L − 1)TLP [K] (29)
In a section of waveguide at ambient temperature, 85% of the signal is transmitted, so G = 0.85. What
noise temperature is introduced by the waveguide? If this waveguide is used to connect the feed horn to the
first amplifier, the noise temperature TRT of the waveguide plus receiver will become

TRT = TL + LTR = (L − 1)TLP + LTR [K] (30)

In the example given in Fig. 15, what would the receiver temperature become?

The power w at the output of the mixer in Fig. 15 is given by

w = G1 G2 G3 k(TA + T1 )∆ν + G2 G3 kT2 ∆ν + G3 kT3 ∆ν [W] (31)

where TA = apparent temperature of the sky as seen by the antenna [K]

k = Boltzmann’s constant = 1.38 × 10−23 [J K−1 ]
∆ν = bandwidth [Hz].

10
 Hyperbolic secondary reflector

Antenna: gain 50 − 70 dB,

bandpass 300 − 30 000 MHz

Parabolic main reflector

feedhorn: bandpass 1600 − 1750 MHz

waveguide polarisation splitter

left−circularly polarized signal right−circularly polarised signal (receiver duplicates everything shown for lcp)

noise diode produces 20 dB waveguide coupler injects 1/100 of calibration signal

white noise calibration signal
cryogenic amplifier: T = 30 K, gain = 23dB
under computer control
refrigerator at 16 K
uncooled amplifier: T = 150 K, gain = 26 dB

mixer: T = 1000 K, gain = 13 dB

intermediate frequency output = radio frequency − local oscillator frequency

local oscilllator signal,
computer−controlled I F amplifier: 60 dB gain, bandpass 10 − 400 MHz

to tune the receiver I F filter selector

I F filters: direct / 32 / 16 / 8 / 4 MHz bandwidth

I F signal, bandpass 144 − 176 MHz

attenuators to adjust signal level, computer−controlled

to spectrometer
to pulsar timer
to VLBI terminal
radiometer, comprising:
amplifier

square−law diode detector

output voltage proportional to input power

low−pass filter, time constant = 0.1 seconds

op−amp

voltage−to−frequency converter

counter computer

Figure 15: Main components of a typical microwave receiver and radiometer.

11
The radiometer is the basic instrument for measuring the power of the incoming signal. Radiometry is
analogous to photometry in optical astronomy. The simplest form of radiometer is the “total power” type
shown in Fig. 15.

The signal measured by the radiometer will vary if the gain changes in any of the amplifiers (or the loss
changes in any of the passive components such as waveguide or cables), and this could be mis-interpreted as
a change in the signal from the sky. However changes in gain can be measured if a noise signal of constant
strength is injected at regular intervals immediately after the feed horn, before the first amplifier. This
signal is detected by the software reading the output of the radiometer, which then adjusts the output to
keep the measured strength of the injected signal constant. This technique is is known as “noise-adding,
gain-stabilised” radiometry, and is available on the Hartebeesthoek 26-m telescope.

The varying water vapour content in the atmosphere acts as a variable attenuator for the incoming signal
from space, and adds its own noise to the signal. If two feeds are installed next to each other on the
telescope, the effects of the atmosphere will be largely common to both of them, but they will be looking
at different points in space. If we are measuring signals from radio sources whose angular sizes are smaller
than the separation of the beams from the two feeds, we can switch rapidly between the two feeds and just
detect the difference in signal. This technique is called “Dicke-switched” radiometry, after its inventor. On
the Hartebeesthoek 26-m telescope, the 6-cm and 3.5-cm receivers are equipped with dual feedhorns and
can operate in this mode. The old receiver mounted on the east wall in the Visitors Centre is the original
dual-feed system for 6 cm wavelength.

7 Detecting radio emission from space

When the telescope looks at a radio source in the sky, the receiver output is a combination of energy received
from several different sources:

• Behind the radio source whose flux density we want to measure is the cosmic microwave background
(CMB) coming from every direction in space. This is the relic radiation left as the first atoms formed
380 000 years after the Big Bang. The Black Body temperature of the CMB Tcmb has now decreased to
2.7 Kelvins, thanks to the expansion of the Universe. This produces a brightness temperature Tbcmb ,
which depends on frequency.
• The emission from the radio source to be measured, which produces a source antenna temperature TA .
• Radiation from the dry atmosphere Tat . Optical depth τ (eqn. 17) depends on elevation angle.
• Radiation from water vapour in atmosphere Twv (τ depends on weather); important above 10GHz.
Optical depth depends on elevation angle.
• Radiation from the ground in the beam sidelobes Tg , depends on elevation angle.
• The amplifiers in the receivers generate their own electronic noise and so produce a receiver noise
temperature TR .

The sum of these parts is called the “system temperature” Tsys . All the components are frequency-dependent.
Summing from the most distant noise contributor to the nearest we have:

Tsys = Tbcmb + TA + Tat + Twv + Tg + TR [K] (32)

The most basic measurement that can be made of a radio source is its signal strength over a defined band,
by radiometry. This is like what our eyes do when we look at light sources of different brightness, such as
the Sun, a light bulb, a candle, or a star. The output signal from the radiometer is proportional to Tsys ,
which is what we want to measure, and from which we then want to extract TA , the signal from the source
of interest. We will describe one way of extracting TA from Tsys in the next section.

12
However, because the input signal is noise-like, the output signal shows fluctuations. The output voltage in
each polarization will show fluctuations with a root mean squared size ∆Trms . The size of the fluctuations
is directly proportional to Tsys , but also depends on the square root of the receiver bandwidth ∆ν and the
length of time for which the signal is averaged, which we call the “integration time”, t:
Tsys
∆Trms = √ [K] (33)
∆ν t
This is called the “radiometer sensitivity equation”. The bigger the bandwidth and the integration time,
the smaller the noisey fluctuations will be in the output signal. We can generalise this to allow for losses
associated with specific types of receiver, which will increase the fluctuations, and for the averaging of n
repeated scans, which will reduce the fluctuations:
KR Tsys
∆Trms = √ [K] (34)
∆ν t n
where KR = sensitivity constant of the instrument. Its value is 1 for a simple radiometer.

The smallest change in antenna temperature ∆Tmin that can realistically be detected is normally taken as
three times the rms noise:
∆Tmin = 3∆Trms [K] (35)
In the same way, we can usefully define the minimum detectable flux density Smin , by making use of equations
21, 34 and 35:
3KR kTsys
∆Smin = √ × 1026 [Jy] (36)
Ae ∆ν t n
These equations let us check that the measured noise in the data matches that expected, ie that the receivers
are functioning correctly, and it lets us predict whether radio sources of a given flux density should be
observable within a given integration time.

8 Measuring the strength of radio sources in space

Now we need to put all this theory together to make actual measurements of radio sources in space. The
simplest way to measure the intensity of a compact radio source in the sky, i.e. one that has an angular size
much smaller than the telescope beam, is to park the telescope a little west of the current position of the
radio source in the sky, and use the rotation of the Earth to let the telescope beam drift steadily across the
source. Not surprisingly, this observing method is called a drift scan. The output of the radiometers will be
the convolution of the antenna beam pattern (e.g. see Fig. 16) with the brightness distribution of the source
(e.g. Fig. 13). Looking back at eqn. 32, we can see that this method has the advantage that Tcmb , Tat , Tg
and TR should all be constant, and only TA should change, this being what we want to measure.

If the radio source has an angular size much smaller than the angular size of the beam, the output from the
radiometer during the scan is effectively an east-west cross-section of the beam of the telescope. An example
is shown in Fig. 17. The passage of the main beam across the radio source is obvious in the centre, and the
first sidelobes are weakly seen on each side. The noise described by equation 33 is clearly visible. Looking
at the minima across the scan, we can see a slow drift in the signal level. This could be due to changing
atmospheric conditions, or to a slow change in gain of the receiver system. To measure the signal strength
we need to establish the slope between the first nulls by drawing a line between them. Then we measure the
height above that line at the centre of the beam. This gives us the antenna temperature TA of the source.

From a drift scan such Fig. 17, we can also measure the full width at half maximum (FWHM). This is
commonly used as a descriptor for the width of the telescope beam, together with the “beamwidth to first
nulls” (BWFN), and we can compare this to what was earlier calculated theoretically. Note that scans in
Right Ascension are broadened by the secant of the Declination of the source. The true FWHM or BWFN is
the value measured from a drift scan, multiplied by the cosine of the Declination of the source at the epoch
of observation.

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Figure 16: Actual beam pattern at 2300 MHz of Figure 17: Typical drift scan through an unresolved
the Hartebeesthoek telescope. Contour levels are radio source. The signal is equivalent to a horizon-
at 3dB intervals, so that each contour is half the tal cross-section through the centre of the antenna
level of the previous one. If we compare this with beam pattern. The first sidelobes can be seen on
fig. 5, why are the sidelobe rings broken up into a either side of the main beam - compare with Fig. 6.
symmetric pattern of “blobs”? The noise level rms value is given by eqn. 33.

If this is a calibrator source, then the point source sensitivity in this polarization is immediately obtained
from the flux density S at the observing frequency (Ott et al. 1994), using equation 24. Once this has been
determined, the flux density of unknown sources can be found from their observed antenna temperatures.
For an unresolved radio source, the full width at half maximum (FWHM) of the scan equals the half-power
beamwidth. If the source is somewhat extended, the width will be broadened, as discussed previously.

References

Several good books are available on radio astronomy. Miller is a free download from the internet, and is great
as a starter. Kraus is the classic reference on radio astronomy. Burke & Graham-Smith is a good modern
overview. Condon & Ransom provide the notes for a fourth year radio astronomy course. Rohlfs & Wilson
is the current standard reference for radio astronomers. Baars’ paper is a useful summary on practicalities.
Stanimirovic et al. is thorough on single-dish astronomy, while Taylor et al. cover synthesis imaging with
multiple radio telescopes.

Baars J W M, 1973, The Measurement of Large Antennas with Cosmic Radio Sources, IEEE Transactions
on Antennas and Propagation, AP-21, 461

Burke B F & Graham-Smith F, 2010, An introduction to Radio Astronomy, 3rd ed., Cambridge University
Press

Condon J J & Ransom S M, 2008, Essential Radio Astronomy, available at http://www.cv.nrao.edu/

course/astr534/ERA.shtml

Jonas J L, Baart E E & Nicolson G D, 1998, The Rhodes/HartRAO 2326-MHz radio continuum survey,
Monthly Notices of the Royal Astronomical Society, 297, 977

Kraus J D, 1986, Radio Astronomy, 2nd ed., Cygnus-Quasar Books

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Miller, D F, 1998, Basics of Radio Astronomy for the Goldstone-Apple Valley Radio Telescope, Jet Propulsion
laboratory JPL D-13835, free download from http://www2.jpl.nasa.gov/radioastronomy/

Ott M, Witzel A, Quirrenbach A, Krichbaum T P, Standke K J, Schalinski C J & Hummel C A, 1994, An

updated list of radio flux density calibrators, Astronomy & Astrophysics, 284, 331

Rohlfs K, Wilson T L, Huttemeister S, 2009, Tools of Radio Astronomy, 5th ed., Springer

Stanimirovic A, Altschuler, D R, Goldsmith, P F, Salter, C J, 2002, Single-Dish Radio Astronomy: Tech-

niques and Applications, Astron. Soc. Pacific Conf. Series 278, available at http://adsabs.harvard.edu/cgi-
bin/nph-toc query?journal=ASPC.&volume=278&fulltoc=YES

Taylor G B, Carilli C L, Perley R A, 1999, Synthesis Imaging in Radio Astronomy II, Astron Soc. Pacific
Conf. Series 180, available at http://cdsads.u-strasbg.fr/cgi-bin/nph-abs connect? bibcode=1999ASPC..180&amp;
db key=ALL&amp;sort=BIBCODE&amp;nr to return= 500&amp;data and=YES&amp;toc link=YES

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