1

Problems & Solutions in Radio Astrophysics (compiled by Dr Dilip G Banhatti)

for use with B F Burke & F Graham-Smith’s

An Introduction to Radio Astronomy (1997, 2002 & 2010 (1st, 2

nd & 3

rd) edns –

Cambridge University Press) (References to page numbers & equations are from 1st edn.)

[(1) For some entries, use J D Kraus’s Radio Astronomy (2nd

1986 Cygnus-Quasar edn or

1st McGraw Hill 1966 edn) for notation & details. Such entries are marked with (JDK).

(2) For some entries, Burke & Graham-Smith’s book may not have all background, which

should, however, be commonly known by students. In case of any need, Kraus’s book

generally gives needed background. These entries are not marked specially.]

--------------------------------------------

# (Size of Earth) Erastothenes observed that sunrays which fall straight down a pit at

Syene in Egypt make an angle of 7º to vertical at Alexandria in Greece. Given that

distance between these two locations is 800 km, calculate Earth’s radius. Draw diagram.

# (Relative distances of Moon & Sun) Aristarchos attempted to determine relative

distances from Earth to Moon & Sun by observing Moon from Earth at its first & third

quarter phases. Draw diagram to explain this method. However, this method isn’t

practical. Why?

# (Distance to Sun) Using simple geometry, discuss how radii of Venus’ & Earth’s orbit

around Sun can be found. Draw diagram. Hence find distance from Earth to Sun.

# (Stellar parallax) Define trigonometric parallax of a star. (It is also called heliocentric

parallax.) How is the unit of distance parsec defined in terms of this parallax? Find the

relation between parsec & light year.

# (Sky observation) Find region of sky accessible to a telescope observing from the tallest

place (to avoid local obstructions) at latitude ℓ on Earth’s surface.

# (Astronomical coordinates) At a place on Earth, North celestial pole is 48º from zenith.

At what altitude is the pole? Find latitude of the place. Draw diagram.

# (Astronomical coordinates) For an observer on Earth’s equator, a star rises exactly

south-east. What is the maximum elevation it attains? In which direction does it set?

Draw diagram.

# (Astronomical coordinates) Define & mark on a diagram: zenith, nadir, astronomical

horizon, great circle, celestial poles, celestial meridian, a star’s hour circle, its right

ascension & declination.

# (Celestial coordinates) Define & mark on a diagram coordinates of a star along with the

relevant reference circles & directions.

# (Astronomical coordinates) Explain with diagram the relation between mean sidereal

day & mean solar day.

2

# (Astronomical coordinates) Show in a diagram RA (right ascension) & declination

(dec) of an object. What are the RA & dec of vernal equinox? Of autumnal equinox? Of

summer & winter solstices? [Assume observer to be in northern hemisphere of Earth.]

# (Circumpolar stars) What are circumpolar stars? Which of the following stars are

circumpolar at latitude 20º N?

Star | Vega | Dubhe | Sirius | Canopus | Procyon |

Declination / deg | +38 | +62 | -17 | -53 | +5 |

# (Astronomical coordinates) Draw the celestial sphere & indicate on it objects with the

coordinates (1) 05h30

m RA, 0º dec; (2) 00

h RA, +30º dec; (3) 22

h RA, -10º dec. Draw the

spherical triangle formed by the three objects. Find the HAs (hour angles) of the 3 objects

at 06h LST (local sidereal time). Are all the objects above local horizon at this time?

# (JDK) (Astronomical coordinates) What is precession of equinoxes? What is its cause?

A radio source has coordinates 4h30

m15

s RA & 30º 15’ dec for epoch 1950.0. Find its

coordinates for 1992.0. (Precession constants: m = 3s.07327, n = 1

s.33167 = 20”.0426.)

# (Astronomical coordinates) Establish relation between local sidereal time (LST) & hour

angle (HA). Vega’s RA is 18h35

m. Find the LST when its HA is 3

h48

m.

# (Synodic & sidereal periods) As observed from Earth’s surface, a sunspot group moves

across Sun’s disk at a rate equivalent to rotation period of 27 days (called synodic

period). Calculate corresponding period relative to distant stars (called sidereal period).

Draw diagram to help with answer.

# (Stellar magnitude scale) Distinguish between apparent (m) & absolute (M) magnitudes

of a star. Find the relation between these for a star at d parsec. What is its distance

modulus? A star has m = +1.5 & M = -1.5. What is its distance? Calculate its parallax.

# (Stellar magnitudes) Define apparent & absolute magnitudes m & M of a star. How is

distance to star calculated from these? A star has m = +1.0 & M = -3.0. Calculate its

distance. What is its parallax?

# (Stellar magnitude scale) Two stars have the same apparent magnitude +2.0. What is

the apparent magnitude of their combined light?

# (Stellar magnitude scale) Calculate apparent magnitude of the combined light of two

stars of apparent magnitudes +1.0 & +3.0.

# (Universal Planck curve) Starting from Planck blackbody radiation law, show that the

relative brightness (normalized to 1 at the peak) is a function of wavelength λ and

temperature T only as the combination λT. [Hint: Use Wien’s displacement law λmaxT =

constant.]

# (Wien’s displacement law) Calculate the ratio of the temperatures of two blackbodies

with spectral peaks at 500 Ǻ & 5000 Ǻ. Calculate the ratio of their total brightnesses.

3

[Hint: Use Stefan-Boltzmann law that the radiation integrated over the whole

electromagnetic spectrum is proportional to T4.]

# (Radiative transfer & Kirchhoff’s law) From equation of radiative transfer

dB/dr + KρB = jρ/4π, derive Kirchhoff’s law & state it in words. Solar brightness at 0.7 µ

is 3.5 x 10-8

W/(m2Hz.Sr) & absorption coefficient in photosphere is 0.1 m

2/kg.

Assuming local thermodynamic equilibrium (LTE), calculate the emissivity. Hence find

solar photospheric temperature.

# (Radiative transfer & optical depth) Define optical depth relative to radiation passing

through an emitting & absorbing medium. Write equations. Solve them formally with

optical depth as independent variable.

# (Far field domain) When a radiation source is so far that the wavefront from it has

negligible curvature relative to the wavelength across the receiving antenna size, the

instrument or telescope is said to be measuring in far field domain. Quantify this. (Cf. pp.

13-4 & Fig. 2.4 of 1997 (1st) edn of book.) Compare near field & far field observing /

measuring domains to Fresnel & Fraunhofer diffraction domains.

# (Near & far field domains) For a radiation source at distance R observed with an

antenna of size d at wavelength λ, far field observation domain is when R >> Rff = d2/ λ.

Examine if observing Moon with space VLBI is in near field or far field domain.

# (JDK) (Circular polarization conventions) Learn about conventions for left & right

circular polarizations in classical physics (tip of E traces left handed helix for left

circular) & as per Institution of Radio Engineers (1942) (type of helical beam antenna for

generating or receiving). IRE (1942) convention is adopted in radio astronomy.

# (Polarization ellipse) Show that the equations Ex = E1sin(ωt) & Ey = E2sin(ωt + δ) represent an ellipse in the variables Ex & Ey. Find the eccentricity of the ellipse.

# (Polarization ellipse) Show that the equations Er = ERexp(jωt) & Eℓ = ELexp[-j(ωt + δ)] represent an ellipse in the variables Ex = Re(Er+ Eℓ) & Ey = Im(Er+ Eℓ). Find the

eccentricity of the ellipse.

# (JDK) (Stokes parameters of polarized light) A wave has normalized Stokes parameters

1, 0, 1/3, 0. What is the fraction of unpolarized power? What type of polarization does the

polarized power show? [Stokes parameters I, Q, U, V give the total & polarized

intensities of an electromagnetic wave. I, the total intensity is also called S, the flux

density. (I = S.) Q, U specify the linearly (or plane) polarized intensity & polarization

position angle. V specifies the circularly polarized intensity. Normalized Stokes

parameters are the ratios s0 = I/S = 1, s1 = Q/S = Q/I, s2 = U/S = U/I & s3 = V/I = V/S.

Find out more to answer the question.]

# (JDK) (Stokes parameters) Find normalized Stokes parameters for d = ½, AR = 4, τ =

135º.

4

# (JDK) (Measuring polarization) Explain how to use dipole antenna to measure degree

of polarization. If responses due to incident wave in two oppositely handed helical

antennas is in ratio 1:3, calculate degree of circular polarization of wave. Calculate

fraction of unpolarized power for wave of normalized Stokes parameters 1, 0, 1/3, 0. Find

type of polarization of polarized power.

# (JDK) (Poincaré sphere) Find out how polarized light is described using angles relative

to Poincaré sphere. Relate this to description using Stokes parameters.

# (JDK) (Poincaré sphere) Draw & label Poincaré sphere showing different polarization

states.

# (JDK) (Polarization measurement) Four linearly polarized antennas at angles 0º, 45º,

90º & 135º and two circularly polarized antennas left & right handed, with responses Wx,

Wx’, Wy, Wy’, WL & WR, have equal effective apertures. Is there any type of wave for

which all 6 responses are equal? If so, what are the wave parameters? [Hint: Normalized

Stokes parameters are s0 = 1, s1 = (Wx – Wy)/( Wx + Wy), s2 = (Wx’ – Wy’)/( Wx + Wy),

s3 = (WL – WR)/( Wx + Wy).]

# (JDK) (Degrees of circular & linear polarization) Show that total degree of polarization

d & degrees of circular (dc) & linear (dℓ) polarization are related by d2 = dc

2 + dℓ

2, and a

general state of elliptical polarization is a superposition of circular & linear polarizations.

Show that (Q2 + U

2)1/2

= I.dℓ, & V = I.dc. Also show that dℓ = cos(2ε) & dc = |sin(2ε)|, where 2ε is the “latitude” on Poincaré sphere.

# (JDK) (Polarization matrix) Find out how polarized light is described using polarization

matrix. Relate this to descriptions using Stokes parameters & Poincaré sphere.

# (JDK) (Stokes parameters & coherency matrices) Find normalized Stokes parameters &

coherency matrices for waves with (i) d = ½, AR = 4, τ = 135º; (ii) d = 1, AR = 4, τ = 45º.

# (JDK) (Stokes parameters) A wave of 10-10

Jy & normalized Stokes parameters 1, 0,

1/3, 0 is incident on an antenna of 1000 m2 effective aperture & noramalized Stokes

parameters 1, 0, 1, 0. (a) Find received spectral power. (b) Give the wave & antenna

characteristics in terms of polarization types. (1 Jy = 10-26

W/(m2.Hz).)

# (Partial polarization) Explain why it is necessary to consider partial polarization. [Most

astronomical objects / celestial sources emit partially polarized electromagnetic waves.

For example, synchrotron radiation is often polarized as much as 60%.]

# (JDK) (Partial polarization) A partially polarized wave is composed of a completely

polarized wave & a completely unpolarized wave. For a wave of flux density S & Stokes

parameters I (= S), Q, U, V, degree of polarization d, and angles 2ε & 2τ specifying

polarization state (2ε is “latitude” & 2τ “longitude” on Poincaré sphere), show that

(I, Q, U, V) = S.(s0, s1, s2, s3)

= S.(1 – d, 0, 0, 0) + S.d.(1, cos(2ε).cos(2τ), cos(2ε).sin(2τ), sin(2ε)), &

5

d = (s12 + s2

2 + s3

2)1/2

. Show further that dℓ = (s12 + s2

2)1/2

& dc = s3 are degrees of linear &

circular polarizations.

# (Electromagnetic spectrum) Calculate frequencies, wavelengths, energies &

wavenumbers for different regions of electromagnetic spectrum from γ-rays to longwave

radio, especially HF (high frequency) (just below 30 MHz), VHF (very high frequency)

(30 to 300 MHz), UHF (ultrahigh frequency) (300 to 1000 MHz), microwaves (1 to 30

GHz), millimeter-wave, & sub-millimetre-wave. The names HF, VHF & UHF come from

early radio science, which also uses terms like metre-wave & shortwave (wavelengths of

10s of cm). Certain microwave bands have acquired names like L-band (wavelength ≈ 20

cm), S-band (≈ 10 cm), X-band (≈ 3 cm), Ku-band (sometimes U-band, ≈ 2 cm), & K-

band (≈ 1 cm).

# (Feebleness of radio photons) There have been (on average) 50 radio telescopes of

(average) diameter 25 m operating since 1960. If the average power received by each is

10-16

W over this period, what is the total energy received? What is the number of radio

photons of wavelength 20cm that corresponds to this energy? Compare the energy to the

energy needed to lift an A4 sheet of paper (say, 0.1 g mass) over 50 cm height against

Earth’s gravity? (g = 980 cm/s2.)

# (Plasma frequency) What is the lowest frequency for radio astronomical observations

from Earth? How is it determined?

# (Molecular absorption) How is the highest radio astronomically useful frequency

determined? (Cf. Fig. 1.1, p. 3 of 1997 (1st) edn book.)

# (Plasma frequency) Using simple dynamics & standard constants, show that a plasma

of electron density N has characteristic frequency of oscillation (ν / kHz) = 9 (N / cm-3

)1/2

.

Derive a similar numerical relation for proton oscillation.

# (Ionospheric & interstellar cut-off) The lowest frequency ν for radio astronomical

observations from Earth is about 30 MHz. Calculate the electron density N in ionosphere.

The interstellar medium (ISM) has N = 0.03 cm-3

. Calculate the ISM cut-off frequency.

# (Plasma frequency) Using relation between refractive index & plasma frequency,

explain how low frequency cut-off arises for Earth’s ionosphere. For typical electron

density 106 /cc, calculate this cut-off.

# (Solar coronal radiation) What is Alfvén speed? Calculate its value in solar

photosphere, where ρ = 2 x 10-4

kg/m3, B = 0.1 Weber/m

2, & µ = 1.2 x 10

-6 henry/m.

What is plasma frequency? Using values of standard constants (from Clark’s Tables, for

example), show that plasma frequency (νe/Hz) = (electron density/m-3

)1/2

. Radio waves of

a given frequency from quiet Sun arise from solar corona at heights above the radius at

which this frequency = plasma frequency. Explain why. Calculate ratio of photospheric

plasma frequency to that at RSun/2 above photosphere, if electron density of solar plasma

is (N/1014

m-3

) = 1.55(R/RSun)-6

+ 2.99(R/RSun)-16

, RSun being solar photospheric radius.

6

# (Propagation through ionized gas) Describe briefly propagation of radio waves through

ionized gas in different astrophysical settings: ionosphere, solar wind = interplanetary

medium, circumstellar Emden sphere, interstellar medium, intergalactic medium,

intracluster medium & so on. Name the key properties & parameters which determine the

(astro)physics in each case.

# (Radio waves in Earth’s atmosphere) Write a note on propagation of radio waves in

Earth’s atmosphere.

# (Telescope resolution) Calculate the diffraction limit (i.e., angular resolution) in

arcseconds of a 20 cm diameter optical telescope at λ500 nm. Compare this with

atmospheric “seeing”, the limit set by twinkling of stars. Calculate the diameter of a radio

telescope operating at λ1 m to have the same resolution.

# (Width of Gaussian) From the Gaussian exp(-x2/σ2

) variation, find x1/2, the half-

maximum point. Define & understand terms like FWHM (i.e., full width at half

maximum).

# (Fourier transform) Define & graph a simple function. Calculate its Fourier transform

& graph it. Write down & prove the relation between the convolution of two functions &

their Fourier transforms.

# (Fourier transform) Write down expressions for the three F(t) functions shown on left

side of Fig. A.1.1 (p.262, 1997 (1st) edn). Transform these to f(ν) using Fourier transform

relation, & convince yourself that the transforms are represented by the graphs on right

side in the figure.

# (Fourier transform) Write down expressions for the three F(t) functions shown on left

side of Fig. A.1.2 (p.263, 1997 (1st) edn). Transform these to f(ν) using Fourier transform

relation, & convince yourself that the transforms are represented by the graphs on right

side in the figure.

# (Fourier transform) Explain with examples relation between convolution & Fourier

transform.

# (Fourier transform) Define & distinguish between & find a relation between the cross-

correlation of two functions and the auto-correlation of a function. Find & comment on

corresponding relations in the (Fourier) transform domain.

# (Fourier transform) Understand & explain Wiener-Khinchin theorem in as simple terms

as possible.

# (Antenna receiver system) Describe a radio astronomical antenna receiver system.

Draw a diagram of the signal flow in time & frequency domains.

7

# (Receiver calibration) How is the receiver in a radio telescope receiver system

calibrated?

# (Nyquist relation) State Nyquist relation & explain its meaning.

# (Receiver) Find the noise temperature of an amplifier stage of noise figure 2 dB.

# (Noise temperature) Define noise temperature of receiver-transmission line

combination.

# (Noise figure & noise temperature) Define noise figure of a radio telescope receiver.

Measured noise figure of a radiometer is 11.5 dB. Find system noise temperature. If its

bandwidth is 1 GHz & integration time is 100 sec, what is its rms noise temperature?

# (Noise figure & noise temperature) What is noise figure of a receiver? Measured noise

figure of a receiver is 10 dB. Calculate receiver noise temperature for ambient

temperature 300 K. For a 2-stage receiver with equal gain per stage of 13 dB, calculate

first stage noise temperature if 2nd

stage has noise temperature 2000 K.

# (Receiver) Find overall noise temperature of a receiver whose first & second stages

have noise temperatures 45 K & 140 K and equal gains of 13 dB each. Rest of receiver

has noise temperature 800 K.

# (Receiver) Write the relation between minimum detectable signal in terms of system

noise temperature / flux density, bandwidth & integration time of a radio telescope

receiver. Give an example.

# (Receiver) A total power receiver with 100 K system noise temperature has equivalent

predetection bandwidth 5 MHz & postdetection equivalent integration time 10 s. For

average predetection gain 20 dB & effective receiver gain fluctuation 0.5 dB, calculate

minimum detectable temperature for the receiver.

# (Receiver) Describe total power & interferometer receivers & compare their properties.

# (Receiver) Explain how an interferometer signal can also be obtained as a phase-

switched signal.

# (Receiver) Describe a two-port receiver using a block diagram.

# (Noise temperature & noise figure) Define noise temperature & noise figure of a linear

two-port.

# (Receiver) Explain principle of heterodyne receiver using phenomenon of beats.

# (Antenna theory) Find far field in direction φ for two equal in phase point sources

separated by λ/2. Draw the field pattern.

8

# (Antenna theory) A uniform linear array has N isotropic in-phase point antennas with

spacing λ/2. Derive expressions for (i) half power beam width, (ii) level of fist side lobe,

(iii) beam solid angle, (iv) beam efficiency, (v) directivity, & (vi) effective aperture.

# (Antenna theory) For Hertz dipole antenna, Pn(θ) = P0.sin2θ gives power pattern.

Calculate (i) antenna beam solid angle, (ii) main beam solid angle, (iii) beam efficiency,

(iv) effective area.

# (Antenna theory) Using various relations for an antenna in terms of its power pattern,

along with W = Ae.(kT/λ2).∆ν.ΩA & W = kTA.∆ν for power received, show that for a

source of angular size << telescope beam, TA = Sν.Ae/2k. Also find antenna temperature

TA in terms of TB, observed brightness temperature.

# (Antenna theory) For a Gaussian source of angular size θS & peak brightness

temperature T0, measured with a Gaussian telescope beam of angular size θB, find

measured brightness temperature TB using the fact that integrating over the entire source

gives its flux density Sν, which is an intrinsic source property, & hence independent of

telescope beam. Relate TB & T0 & hence TA & T0 for a source small compared to

telescope beam.

# (Antenna theory) Draw diagrams to represent triangular antenna pattern P(φ) of width

2º & rectangular source brightness distribution B(φ) of the same width & maximum value

B0. Hence draw accurate graph of measured flux density as antenna sweeps across

source.

# (Antenna theory) Relate 1D aperture distribution E(xλ) & its far field distribution

E(sinφ). Draw both for (a) uniform, (b) cosine & (c) Gaussian variation for E(xλ). [xλ is

distance along aperture in units of wavelength, & φ is angle of pointing.]

# (Antenna theory) Explain with reference to context the following relations: (1) Pνdν =

kTdν, (2) Aeff = λ2G/4π, (3) ∆T = Ts/√(Bτ), (4) ∆S = 2kTn/A√(Bτ), (5) ∆Sxy =

2k√(Tn1Tn2)/(2A1A2 Bτ).

# (Antenna theory) Calculate antenna surface efficiency for rms surface error λ/20.

# (Antenna theory) Calculate field pattern for a rectangular 1D aperture of uniform

illumination abruptly cut to zero outside.

# (Antenna theory, Fourier transform) The aperture field distribution V(x) for a 1D

aperture is uniform (taken = 1) for |x| ≤ ½ & zero for |x| > ½. (i) Calculate far field

distribution E(s) = FT of V(x). (ii) In FT relation V(x) E(s), what do x & s

represent? (iii) What is the significance of |s| > 1? If the baseline of an interferometer is

doubled, its fringe spacing is halved. Similarly, if the width of a 1D aperture is changed

by factor K, width of its far field pattern changes by factor 1/K. (iv) Which theorem in FT

theory is thereby illustrated? (v) State & prove this theorem.

9

# (Scintillation / twinkling) Explain how scintillation or twinkling arises due to

irregularities in medium between radio source & telescope. For scintillation time scale of

1 min at observing wavelength 1 m due to solar wind flowing across at 400 km/s,

calculate spatial scale of solar wind irregularities.

# (Dicke switching) Explain briefly how Dicke switching eliminates instability & drift of

signal from a radio telescope receiver system.

# (Dicke & Ryle-Vonberg receivers) Explain with diagram principle of Dicke switched

receiver. By what factor is effect of gain variation reduced? How is it converted to a null

system, viz, Ryle-Vonberg receiver? Explain how effect of gain variation is thereby

eliminated.

# (Radio spectrometry) Describe briefly with diagrams principles of radio spectrometry.

# (Interferometry) Explain salient points of simple adding interferometer and phase

switched interferometer with block diagrams & input / output waveforms.

# (Interferometer receiver) Write sensitivity relation for a cross-correlation radiometer

(i.e., an interferometer) & explain its meaning with an example.

# (Autocorrelator) Find the clock frequency of of a digital autocorrelator to analyze

maximum bandwidth of 500 MHz.

# (Acousto-optical spectrometer) (i) Discuss acousto-optical spectrometer with a

diagram. (ii) The acousto-optical medium in the Bragg cell is water. What is the

frequency resolution if the aperture is D = 96 mm? Sound speed in water = 1480 m/s.

# (Cylindrical aperture) A λ1 m cylindrical paraboloidal telescope is 30 m E-W by 530 m

N-S in extent. Calculate nominal beam size & shape.

# (Finite radio source) Write an expression for total observed power received over a

bandwidth from a source of finite extent.

# (Finite radio source) A 1 deg2 radio source of 200 K is blocked by a 5 deg

2 cloud of

optical depth 1 & brightness temperature 100 K. Calculate antenna temperature for a 50

m2 telescope.

# (Large radio source) A radio telescope of beam efficiency 0.7 points at a large radio

source of temperature 10 K which completely fills the beam. Half the minor lobe beam

area is in backward direction towards ground of temperature 300 K. Calculate antenna

temperature.

# (Finite radio source) Find flux density of Sun at λ500 nm assuming T = 6000 K, and

calculating ω, the solid angle Sun subtends, from Sun’s photospheric diameter of ½ deg

in optical band.

10

# (Interferometer fringes) A two-element λ1 m interferometer has E-W spacing 30 m.

The observed fringe period is 16 min. Calculate the zenith angle for a 0º declination

source. What is the fringe period for such a source at the meridian?

# (Interferometer fringes) Find fringe period for E-W interferometer of 30 m spacing

operating at λ1 m when observing a source at 60º from zenith. What would be fringe

period for source at zenith?

# (Interferometer fringes) A two-element radio interferometer operating at 200 MHz has

E-W spacing 500 m. Calculate time between fringes for point source at declination 50º at

meridian. Also for dec 10º.

# (Two-element interferometer) Draw Fig. 5.4 & write eqn. (5.14) on p.57 of book (1997

(1st) edn). Explain how an interferometer with 2 elements observes a source of finite size.

# (Interferometry) Draw Fig. 5.5 & write eqns (5.23) & (5.24) on p.59 of book (1997 (1st)

edn). Explain relation between a two-element interferometer, a celestial source brightness

distribution & visibilities in uv-plane.

# (Interferometry) Draw diagrams to relate 1D visibility 1D sky brightness for the

following qualitative descriptions: (i) Discontinuous outer edge Near-in sidelobes,

(ii) Missing short spacings Negative bowl, (iii) Regular gaps Grating response.

# (Aperture synthesis) Explain briefly principle of aperture synthesis telescopes. Show

how closure phase is useful for synthesis imaging using self-calibration. Discuss briefly

closure amplitudes.

# (Thermal & nonthermal processes) Thermodynamic temperature is defined for some

sort of local equilibrium or quasi-equilibrium, generally in case of thermal processes. It is

then more or less independent of wavelength or frequency (blackbody approximation).

However, the concept of temperature has been stretched in meaning to apply in context of

nonthermal processes in (radio) astrophysics. Define & explain briefly all uses of the

term temperature that you have learnt.

# (Temperatures in astrophysics) Explain the terms antenna temperature & brightness

temperature. Describe physical principles used for their definition.

# (Ionized hydrogen) Explain in detail emission mechanism for ionized hydrogen. An

ionized hydrogen cloud has brightness temperature 150 K at 600 MHz. Calculate

emission measure EM & average free electron density if the Strömgren sphere is of 25 pc

diameter. Assume electron temperature 104 K, & use EM (in cm

-6.pc) = 2.5τ.(ν/MHz)

2.

# (Synchrotron radiation) Give a brief account of synchrotron radiation mechanism as

relevant in astrophysics.

11

# (Continuum radio emission) Name three main mechanisms of continuum radio

emission. How do their spectra differ from each other? Three radio sources have flux

densities at 100 MHz & 1 GHz as follows:

------------|---------------------------

Rad src | Flux density (Jy) at

|----------------------------

| 100 MHz | 1 GHz

-----------|----------------|------------

A | 10 | 0.1

-----------|----------------|------------

B | 3 | 3

-----------|----------------|------------

C | 5 | 500

-----------|----------------|---------

# (Masers) Explain principle of maser action. Find out how astrophysical masers are

possible. Connect celestial maser observation to laboratory maser theory.

# (Synchrotron radiation) Calculate frequency of maximum synchrotron emission of a 1

GeV electron moving in a magnetic field 1 microGauss. Calculate Lorentz factor for the

electron. Also find the cone angle within which most radiation is concentrated.

# (Synchrotron radiation) How far can an electron moving at 99.99% speed of light travel

in magnetic field 1 microGauss before losing half its energy?

# (Synchrotron radiation) Find cone angle of synchrotron radiation of 500 MeV electron

in 1 nT magnetic field. Calculate its speed in terms of speed of light.

# (Rotation & dispersion measures) Explain how magnetic field along line of sight is

found using rotation measure (RM) & dispersion measure (DM) of a nonthermally

emitting plasma. (See eqn (8.25) on p. 111 – 1997 (1st) edn of book.)

# (Faraday rotation) Write a brief note on polarization of synchrotron radiation &

Faraday rotation of plane of polarization due to passage through magnetoionic plasma.

How are these used for inferring physical conditions in astrophysical objects?

# (Galactic continuum radiation) Use Figs. 8.1 to 8.4 (pp.97-8 of 1997 (1st) edn of book)

to identify discrete sources which appear in the various bands from metrewave radio (Fig.

8.3), centimetrewave radio (Fig. 8.4) through infrared (Fig. 8.2) to optical (Fig. 8.1).

Make a table with rough Galactic coordinates of the objects, putting a dash (--) when an

object is not seen in a map. Now read Sec. 8.1 carefully in the light of this table.

# (Rayleigh scattering off dust) Assuming there is 99.9% absorption in a certain direction

at λ500 nm, calculate the wavelength λ at which it becomes 0.01%, assuming Rayleigh

scattering off interstellar dust (i.e., absorption varying as λ–4).

12

# (Galactic continuum radiation) Use Fig. 8.5, which is a coarse resolution map at

decameter waves (equivalent to 30 MHz frequency), in conjunction with higher

frequency maps upto optical from Figs 8.1 to 8.4, to see common sources. Use more

recent decameter maps (e.g., those made with Gauribidanur & Mauritius arrays) from

literature for a better comparison. Find out about more recent all-sky surveys at higher

frequencies right to satellite-borne telescope surveys in X-rays & gamma-rays.

# (Hubble sequence for galaxies) Draw & label Hubble’s “tuning fork” diagram for

galaxy shapes.

# (Cyclotron frequency) Derive the cyclotron frequency νe of a nonrelativistic electron by

applying Newton’s law to its motion in a uniform magnetic field B. Calculate νe for

B = 1 µG. Find the cyclotron frequency νp for a proton & compare it with νe. How far can

an electron travel at 99.99% the speed of light in a magnetic field of 1 µG before losing

half its energy? [Use the formula (t1/2 / yr) = 25/[(B/Gauss)2.γ], where Lorentz factor

γ = (1– v2/c

2)

–1/2.]

# (Cyclotron trajectory) Estimate radius of curvature of solar wind proton trajectory.

(Perpendicular speed = 400 km/s, parallel speed = 100 km/s, magnetic field = 10–7

Tesla.)

Sketch trajectory.

# (Spectral line radiation) Given that the frequency of H109α is 5009 MHz, calculate the

ionization energy of the hydrogen atom.

# (Spectral line radiation) Given that ionization energy of H is 13.6 eV, calculate

frequency of line n = 110 109 called H109α.

# (Line formation in interstellar medium) Wrie a brief account of line formation in

interstellar medium.

# (Neutral hydrogen line in Milky Way) Write a brief note on neutral hydrogen

(H0 ≡ H I) emission & absorption due to spin-flip transition at λ21 cm.

# (Neutral hydrogen line in Milky Way) Explain how a map of (H0 ≡) H I distribution in

Milky Way can be constructed by radio observations along Galactic Plane at different

Galactic longitudes.

# (Neutral hydrogen line & its width) A neutral hydrogen feature peaks at a frequency

100 kHz higher than the rest frequency 1420 MHz. Find its line-of-sight velocity. If its

width is 50 kHz, find the velocity dispersion within the hydrogen cloud.

# (Doppler shift) Show that a velocity difference of 1 km/s corresponds to a frequency

difference of 4.74 MHz for the λ21 cm neutral hydrogen spin-flip spectral line.

13

# (Molecular lines) Briefly describe some radio molecular lines & their main properties

& occurrence in Milky Way. [Recall that Earth’s atmosphere limits radio window at high

frequencies / short wavelengths due to presence of absorbing molecules.]

# (Spectral line) Explain the procedure for calculating optical depth & line temperature of

a line-emitting cloud by observing a discrete radio source behind the cloud.

# (Ionized hydrogen in Milky Way) Describe briefly ionized hydrogen (H+ ≡ H II)

regions in Milky Way.

# (Supernova remnants) What are supernova remnants? Describe briefly some of them.

# (Dark matter) Write a brief note on dark matter in Milky Way.

# (Milky Way differential rotation) Using V = RΩ & the definitions

A ≡ –(1/2)R0(dΩ/dR)R0 & B ≡ –(1/2)[(V0/R0)+(dV/dR)R0] of Oort A & B constants, show

that Ω0 = A – B.

# (Galactic Centre proper motion) Find the distance to Galactic Centre, with its error bar,

from its observed ℓ-direction proper motion (i.e., in Galactic longitude direction)

µℓ = –6.55±0.34 milliarcsec/yr, assuming 220 km/s for its apparent velocity.

# (Spiral structure) Write a note on spiral structure in galaxies.

# (Disk galaxy rotation curve) Find the mass within radius R of a disk galaxy, given that

the observed mean rotational velocity is V. (Use balance of gravitational & centrifugal

forces.)

# (Stellar effective temperature) Explain how the (effective) temperature of a star can be

calculated from its angular size and flux density at a specific wavelength.

# (HR diagram for stars) Write a brief note on main sequence of stars, relative to

Hertzsprung-Russell diagram.

# (Neutrinos from Sun) The energy released per helium nucleus produced in Sun’s core is

26.7 MeV. Solar power at Earth’s position is 1370 W/m2. Calculate flux of solar

neutrinos at Earth, if each helium fusion releases 2 neutrinos. (1 eV = 1.6 x 10–16

J.)

# (Particle densities) Typically, solar wind reaching Earth has 107 particles/m

3. For

comparison, estimate particle density (i) in air in the room using ideal gas law, (ii) in

solar photosphere, assuming protons of density 10–6

kg/m3, (iii) in neutron star, assuming

mass density 2 x 109 kg/m

3.

# (Star formation) Assume protosolar nebula of uniform density out to 5 AU (Jupiter’s

present orbit) & rotated with period of 12 yr. Find new rotation period after Sun collapsed

to its present size, using conservation of angular momentum.

14

# (Radio spectrum of Betelgeuse) Discuss the radio spectrum of Betelgeuse (= α Orionis)

using Fig. 11.2 & eqns (11.1) to (11.3) (pp. 164-6 in 1997 (1st) edn of book).

# (Sun & planets in radio) Discuss Sun & planets as radio sources.

# (Circumstellar masers) Write a note on circumstellar masers, especially SiO, H2O &

OH masers.

# (Novae) Describe how eqns (11.4) to (11.6), together with the simple model given,

explain nonthermal radio & infrared emission from a nova, as shown in Fig. 11.10. (Cf.

pp. 173-6, 1997 (1st) edn of book.)

# (Binaries & flare stars) Write a brief note on nonthermal radiation from binaries & flare

stars.

# (Symbiotic stars) Write a note the recurrent nova (= symbiotic star) RS Aquarii.

# (Stellar jets) Describe X-ray binaries Cyg X-3 & SS 433 as variable radio sources.

# (Pulsars in Milky Way) Name three or four aspects of pulsar research which are

important in Galactic astronomy & astrophysics.

# (Solar mass neutron star) Examine from angular momentum & magnetic energy

conservation whether our Sun can contract to become a neutron star. Assume appropriate

values for typical parameters.

# (Neutron star structure) Describe briefly the structure of a neutron star. Which parts of

Physics are needed to study this structure?

# (Neutron star magnetic field) Using eqns (12.1) to (12.3) (p.186, 1997 (1st) edn of

book), relate polar field B0 & magnetic dipole moment M. Use appropriate units for

various quantities. Relate this calculation to spherical frozen in dipolar magnet of radius

10 km as can be characterized in standard classical electromagnetic theory.

# (Neutron star braking) Derive eqn (12.4) from eqn (12.3). Further derive eqn (12.5).

(Refer to p. 186 of 1997 (1st) edn of book.)

# (Pulsar braking) Show that rate of change of pulsar frequency ν is given by

dν/dt = –ν.(dp/dt)/p, where p (= 1/ν) is the pulsar period.

# (Pulsar braking) Starting from dν/dt = –k.νn for frequency decrease of a pulsar, show

that its spin-down age is –ν/[(n–1).(dν/dt)].[1–(ν/ν0)n–1

], where ν0 is zero age

frequency. For ν<<ν0, calculate spin-down age of a pulsar of braking index n = 2.7 &

frequencies 10.000 Hz at epoch 1975.0 & 9.999 Hz at epoch 1985.0.

# (Pulsar glitches) What are pulsar glitches? Give example.

15

# (JDK) (Dispersion measure) Using dispersion measure formula, find the mean electron

density in the direction of a pulsar 2 kpc away if a pulse delay of 2 sec is observed

between 450 MHz & 350 MHz. At which frequency does the pulse arrive first?

# (Polar caps pulsar model) Describe briefly a polar caps model of pulsar emission.

# (Polar caps model) Derive eqn (12.8) from eqn (12.7). (Cf. p.189 of 1997 (1st) edn of

book.)

# (Light cylinder for pulsar) What is light cylinder for a pulsar? Show on a diagram.

Calculate light cylinder radius for a pulsar of period 1 sec.

# (Pulsar P-Pdot diagram) Write a note on the population & evolution of pulsars in Milky

Way Galaxy.

# (Binary pulsars) Write a note on pulsar binary orbits, variation in pulse period, &

possible of interaction between binary members.

# (Binary millisecond pulsar) Describe briefly how the binary millisecond pulsar is a very

good lab for testing generl relativity in many ways.

# (Synchrotron spectrum) A radio source has flux densities 1 Jy at 300 MHz and 1 mJy at

3 GHz. Find its spectral index. For synchrotron radiation from optically thin relativistic

plasma, the spectral index is directly related to the energy index of emitting electrons.

Find this energy index for the radio source.

# (Radio galaxies & quasars) Describe different types of structures found among radio

galaxies & quasars. Draw sketches. Illustrate a typical continuum spectrum from radio to

ultraviolet.

# (Active galaxy) Describe a simple model of an active galaxy, illustrating accretion disk,

torus, core & jets.

# (Synchrotron spectrum & size) If 3C 345 (Fig. 13.3) is synchrotron-self-absorbed,

estimate model peak frequency & peak flux density, & use these to estimate minimum

angular size of emitting region. (Cf. pp. 207 & 219 of 1997 (1st) edn of book.)

# (Superluminal motion) Write a note on apparent superluminal proper motion in radio

source cores. Find angle to line of sight at which maximum superluminal speed occurs.

Verify numbers given after eqn (13.2). (Cf. pp. 221-2 in 1997 (1st) edn of book.)

# (AGN unified model) Explain terms NLR & BLR. Draw a schematic of unified AGN

model.

# (Hubble expansion) Describe briefly Hubble’s discovery & its interpretation.

# (Cosmodynamics) Integrate eqn (14.4) to get eqn (14.5), & further derive eqn (14.6).

(Cf. pp.227-8 in 1997 (1st) edn of book.)

16

# (Cosmodynamics) Describe simple Newtonian model of Universe.

# (Hubble & deceleration parameters) Expand scale factor of Universe R(t) in Taylor

series around t0, to second order, & express the coefficients in terms of H0 & q0.

# (Hubble law) Show that each of eqns (14.16) to (14.18) reduces to eqn (14.1) for small

z. (Cf. pp. 230 & 226 in 1997 (1st) edn of book.)

# (Cosmodynamics) Describe relaticistic cosmology in terms of simplest eqns.

# (Big bang & early Universe) Write a note on hot big bang cosmology & phase

transitions expected in early Universe in this model.

# (CMBR & COBE) Describe cosmic microwave background radiation as observed by

COBE satellite observatory.

# (Matter & radiation in hot big bang) How do matter & radiation densities in hot big

bang model vary relative to scale factor of Universe? Why? Express these variations

relative to redshift. At what redshift are these two densities equal? What is implied?

# (CMBR structure on sky) Describe anisotropy & distortions of CMBR on sky.

# (Inflation) Describe inflation in big bang cosmology.

# (Source counts) Assuming uniform volume distribution of radio sources, find integral

& differential counts as a function of flux density in Euclidean space.

# (Source counts) Assuming uniform volume distribution of galaxies, find integral &

differential count of galaxies as a function of apparent magnitude in flat space.

# (Differential relative count) Describe with a rough diagram ratio of observed

differential count of radio sources vs that expected in flat space as a function of flux

density. What are the implications? Take care to consider appropriate normalization.

# (Angular diameter vs redshift) Derive angular diameter vs redshift relations for world

models given by eqns (14.16) to (14.18). Plot these on log-log scale & compare with Fig.

15.3. (Cf. pp. 230 & 243 in 1997 (1st) edn of book.)

# (Gravitational lensing) Using Fig. 15.4 as an aid, integrate eqn (15.1) to get eqn (15.2).

(Cf. p. 244 in 1997 (1st) edn of book.)

# (Gravitational lensing) Using geometry specified, & drawing diagram if needed, use

eqn (15.2) to get eqn (15.3) & hence eqn (15.4). (Cf. pp. 244-5 in 1997 (1st) edn of book.)

17

# (Gravitational lensing) Show that generalization of 1D eqn (15.3) to 2D leads to eqn

(15.5). Calculate further to get eqns (15.6) & (15.7). (Cf. pp. 245-7 in 1997 (1st) edn of

book.)

# (Gravitational lensing) What is the condition for getting an Einstein ring in

gravitational lensing?

# (Hubble constant from gravitational lensing delay) Describe qualitatively how

observation of time delay between variations of images of a gravitationally lensed

variable source leads to precise determination of Hubble constant.

# (Quantum delayed choice experiment on cosmological scale) Explain how observing

time delay between variations two of images of a gravitationally lensed variable source

can be considered equivalent to a quantum mechanical delayed choice experiment as

envisaged by John A Wheeler.