Radprod Exam Papers

School of Physics & Astronomy
Radiation Processes in Astrophysics

PHYS11067 (SCQF Level 11)
Monday 12th December, 2022 13:00 – 16:00

(December Diet)
Please read full instructions before commencing writing.
Examination Paper Information
Answer ALL questions
Special Instructions
• Only authorised Electronic Calculators may be used during this examination.

• A sheet of physical constants is supplied for use in this examination.
• Attach supplied anonymous barcodes to each script book used.
Special Items
• School supplied Constant Sheets

• School supplied barcodes
Chairman of Examiners: Prof K Rice

External Examiner: Prof I McCarthy
Anonymity of the candidate will be maintained during the marking of this

examination.
Printed: Friday 3rd February, 2023 PHYS11067

Radiation Processes in Astrophysics (PHYS11067)
1. (a) The Bremsstrahlung emissivity from a region of ionised gas can be approximated as
51 2 1/2 h⌫/kT 3 1
✏⌫ ⇡ 7 ⇥ 10 ne T e Wm Hz
where ne is the number density of electrons and T is the temperature. A cluster of galaxies
is observed to contain 2⇥1013 solar masses of intracluster gas, at a temperature of 3⇥107 K,
within the central 150 kpc radius (the cluster core). Assuming that the density is constant
within the cluster core, calculate ne . Hence calculate the Bremsstrahlung luminosity of
the cluster core. [5]
(b) What is meant by the cooling time of the gas? Write down an expression for this,
explaining why this is only an approximation. Show that the cooling time in the cluster
core is less than 1 Gyr. [4]
(c) Why is 1 Gyr an interesting number to compare the cooling time against? In the
absence of any heating source, what will be the fate of the intracluster gas, and why? [3]
(d) The galaxy at the centre of the cluster hosts a radio source, emitting synchrotron
radiation throughout a region which can be roughly approximated as having a spherical
shape of radius 65 kpc. Considering both the work done to inflate the cavities, and the
internal energy density of the relativistic particles (using equipartition, or otherwise),
show that the total energy that has been fed in to the cluster by the radio jets can
be approximated as 4P V , where P is the pressure and V is the volume of the source.
Calculate this energy, assuming that the radio lobes are in pressure balance with the
surrounding intracluster gas. [4]
(e) Radio spectral ageing techniques suggest that the radio source has an age of 2⇥107 yrs.
Without the use of equations, briefly explain how the technique of radio spectral ageing
works. [2]
(f) Using this age, derive the jet power of the radio source. Comment on the implications
of this for the fate of the intracluster gas, discussed in part (c). [2]
Printed: Friday 3rd February, 2023 Page 1 Continued overleaf. . .

2. (a) Larmor’s formula for the energy loss rate from an accelerating electron, moving with
relativistic velocity, is:
dE e2
= a⌫ a⌫
dt 6⇡✏0 c3
where a⌫ is the 4-acceleration. Write down, or derive, an expression for a⌫ . Use this
to evaluate the 4-vector product a⌫ a⌫ for the specific case where the acceleration (v̇) is
perpendicular to the velocity. Hence, show that under these circumstances,
dE e2 4
= 3
|v̇|2
dt 6⇡✏0 c
where is the Lorentz factor of the electron. [5]

(b) Show that the average energy loss rate from synchrotron radiation for an electron
moving in a magnetic field of strength B is given by
dE e4 B 2 E 2
=
dt 9⇡✏0 c5 m40
where m0 is the rest-frame mass of the electron and ✓ is the angle between the velocity
and the magnetic field. [6]
(c) Derive an expression for the time taken for an electron of energy E0 to radiate away
90% of its energy. Comment on why, in astrophysical situations, this estimate of the
lifetime is no longer valid when the magnetic field strength is very low. [4]
(d) Relativistic electrons in an expanding radio source also lose energy through adiabatic
expansion losses, according to
dE 1 dR
= E
dt R dt
where R is the size of the source. A radio source has a size of 10 kpc, is expanding at
a rate of 0.03c, and has a magnetic field strength of 10 nT. Determine the energy of an
electron that would lose energy at the same rate from synchrotron radiation as it does
from adiabatic expansion. At what frequency does such an electron emit most of its
synchrotron radiation? You can assume that an electron with a Lorentz factor radiates
most of its energy at a frequency ⌫ ⇡ 2 !g where !g = eB/m0 . [5]

3. (a) Explain what is meant by the terms ‘classical’, ‘semi-classical’ and ‘quantum’ when
referring to the interaction of light and matter. Highlight some of the limitations of
the classical and semi-classical description of this interaction which require a quantum
treatment. [4]
(b) The quantum radiation field for free photons in a box can be expressed as
s
X h̄ ⇣ ⌘
A= ek,↵ ak,↵ eik.r + a†k,↵ e ik.r
.
k,↵ 2✏0 !V
Define the terms in this expression. [4]
(c) Given the interaction Hamiltonian of radiation with a charged particle of mass m,
charge q and momentum p, is ✓ ◆
0 q
H = A.p,
m
derive the quantum interaction matrix, Hf i = hf |H 0 |ii, for a particle with two states,
X
p and Y , in the† electricp dipole approximation. You may assume the operators a|ni =
n|n 1i and a |ni = n + 1|n + 1i, and that the expectation of the particle momentum
is hf |p|ii = im!0 hf |r|ii. [7]
(d) A two-state hydrogen atom is collisionally excited in a gas at temperature 4350 K and
with density n = 106 m 3 . The energy di↵erence between states is 1.38 ⇥ 10 19 J, and the
ratio of upper to lower level populations is 0.01. What is the critical density of the cloud?
The downward collision rate in the cloud is h viul = 40 m3 s 1 , and the spontaneous dipole
transition rate is given by
!
4 c |r Y X |2
Spon = (2⇡)3 ↵ 3
,
3
where is the radiation wavelength, ↵ is the fine-structure constant, and r Y X is the dipole
matrix. What is the magnitude of the dipole matrix in this transition? [5]

4.
(a) The specific intensity of light is related to the photon occupation number by
2h⌫ 3
I⌫ = Nk,↵ .
c2
Define the specific intensity and photon occupation number. By considering the energy
flux of radiation, derive this relation. [5]
(b) The equation of radiative transfer for a spectral line from a two-state atom is
dI⌫
= ⌫ I⌫ + E⌫ ,
dl
where ⌫ = (nl nu ) Spon ( 2 /8⇡) ⌫ . Explain what the terms in both of these expressions
represent. Explain why the density nu appears in ⌫ , and why nu > nl may occur. [3]
(c) Explain the terms Natural Broadening, Thermal Broadening, and Thermal Limit,
illustrating your answer. [3]
(d) Define the optical depth and source function, and find a solution to the radiative
transfer equation as function of the optical depth and source function. Explain how an
absorption feature can be used to estimate the properties of a gas cloud. [4]
(e) An astronomer observes a collisionally excited molecular line emitted by a spherical

gas cloud. The line is broad and has a double peak. Give two plausible explanations for
what this might imply about the internal structure of the cloud. [5]
Printed: Friday 3rd February, 2023 Page 4 End of Paper

Radiation Processes in Astrophysics

Wednesday 15th December, 2021 13:00 – 16:00

(December Diet)
Answer ALL questions

• This is an open book examination.
• You may use books, notes, approved electronic calculators, and passive internet
resources.
• Your answers must be entirely your own work.
• You must not seek the assistance of any other person, organisation, or service.
• You must not use responsive internet tools or software resources such as pro-
grammable calculators or computer algebra packages.
Special Items
Chairman of Examiners: Prof P Best

External Examiner: Prof I McCarthy

examination.
Printed: Monday 6th December, 2021 PHYS11067

1. (a) By considering the 4-momentum of a photon travelling at an angle ✓ to the x-axis in

frame S, or otherwise, derive the aberration formula
cos ✓ vc
cos ✓0 =
1 vc cos ✓
which describes the transformation of angles between two inertial frames S and S 0 , moving
at relative velocity v. Use the aberration formula to explain why, for relativistic electrons
of Lorentz factor , half of the radiated energy is emitted into an angle |✓| < 1/ in frame
S. Show also that the characteristic frequency of synchrotron radiation emitted by such
electrons is ⌫ ⇡ 2 !g , where !g is the non-relativistic gryo-frequency. [7]
(b) The average rate of energy loss of the relativistic electrons due to synchrotron radiation
is given by
* +
2 4
dE e B2
=
dt 9⇡✏0 cm20
where B is the magnetic field strength and m0 is the electron rest-mass. Using this, derive
an expression for the time taken for an electron emitting at frequency ⌫ to radiate away
half of its energy. Calculate this time for frequencies of both 5 GHz and 150 MHz, for
electrons gyrating in a magnetic field of strength 10 nT. What will be the e↵ect on the
observed synchotron spectrum of these energy losses? [6]
(c) Explain what is meant by the term synchrotron self-absorption and what e↵ect this
process has on the observed synchrotron spectrum of radio sources. To what extent would
you expect to see the e↵ects of both synchrotron self-absorption and energy losses in the
spectrum of the same radio source? Explain your answer. [4]
(d) Remnant radio sources are a class of radio sources in which the fuelling of the central
black hole has recently ceased. What would you expect the appearance of remnant radio
sources to be, at radio wavelengths? What would their radio spectrum look like, and how
would it subsequently evolve? [3]
Printed: Monday 6th December, 2021 Page 1 Continued overleaf. . .

2. (a) Show that there is an upper limit (the Eddington limit) to the rate of accretion on to
a black hole of mass MBH , corresponding to a luminosity of
4⇡Gmp c
LEdd = MBH
T
where T is the Thomson scattering cross-section and mp is the proton mass. Assuming
an accretion efficiency of 10%, calculate the accretion rate of a 108 M black hole at the
Eddington luminosity. Give your answer in units of M /year. [4]
(b) Consider a black hole accreting with 10% accretion efficiency. Derive an expression
for how the mass of the black hole increases with time, if it accretes continually at its
Eddington limit. Hence determine the time taken for the black hole to grow from a mass
of 107 M to 109 M . [4]
(c) The black hole powers a radio source. The two jets are intrinsically identical, emit-
ted in precisely opposite directions, and can be assumed to be driving into an identical
interstellar medium. The hotspots at the end of the two radio jets are observed to be
at angular distances of D1 and D2 , respectively, from the black hole. By considering the
light travel time e↵ects across the source, explain why the two jets will be observed to
have di↵erent lengths, and show that the fractional di↵erence in jet lengths is given by
D1 D2 v
x⌘ = cos ✓
D1 + D2 c
where v is the velocity at which each hotspot is advancing, and ✓ is the angle between the
jet direction and the line of sight. Comment on why one of the jets might be observed to
be brighter than the other, and whether you would expect this to be on the longer or the
shorter side of the source. [5]
(d) Show that if a population of radio sources is randomly oriented within a maximum
angle of ✓max to the line of sight, then the average value of cos ✓ is
sin2 ✓max
hcos ✓i =
2(1 cos ✓max )
The population of radio sources consists of both Type 1 (face-on) and Type 2 (edge-on,
obscured) quasars. The value of hxi determined for the Type 1 quasars is 1.7 times larger
than that of the full (Type 1 plus Type 2) population. What does this imply for the
opening angle of the obscuring torus? (The opening angle is the maximum angle from the
polar axis out to which the central engine of quasar is unobscured). How would asymme-
tries in the interstellar medium intercepted by the radio jets influence this interpretation?
[7]

3. (a) The rate of the transition of a system from an initial quantum state, |ii, to a final
state, |f i, due to a perturbation to the system is given by Fermi’s Golden Rule,
2⇡ 0 2
= |H | D (! !0 ).
h̄ f i
Derive this expression, and explain each term. [5]
(b) From gauge invariance, or otherwise, derive the interaction Hamiltonian for radiation
and electron transitions. [5]
(c) Using the example of the interaction between radiation and hydrogen, where hydro-
gen goes from the ground state, 1s, to the excited orbital state, 2p, outline, without
mathematical detail, a physical description of the possible transition processes. [3]
(d) The spontaneous transition rate in the electric dipole (ED) approximation is
4! 3 e2
ED = |rY X |2 .
3h̄c3 4⇡✏0
Assuming the dipole moment, |rY X |, for the Lyman-alpha transition in hydrogen can be
approximated by the Bohr radius, a0 ⇡ 5.3 ⇥ 10 11 m, estimate the transition rate for
radiation emitted at 1216Å. By scaling from the electric dipole transition rate, estimate
the electric quadrupole transition rate at the same wavelength. Finally, estimate the
magnetic dipole transition rate for radiation at 21cm. [7]
[You may assume the Fine Structure Constant is ↵ = e2 /(4⇡✏0 h̄c) = 1/137, the Compton
wavelength is c = h̄/mc = 2.4 ⇥ 10 12 m, and the angular momentum of the electron in
hydrogen is L = h̄.]

4. (a) Derive the equation of radiative transfer

dI⌫
⌫ I⌫ + E⌫ ,
d`
and define the functions ⌫ and E⌫ . Explain what these two functions represent. [5]
(b) The line profile of a gas, ⌫ , is Doppler Broadened. What is the thermal velocity
dispersion for the gas of CO molecules at a temperature T = 30K, given that the mass of
the CO molecule is m = 6.65mH ?
What is the central value of the line profile for a gas of CO molecules at T = 30K which
is rotationally excited and emitting at = 2.6mm? [5]
(c) The number densities for CO molecules in the J = 0 ground state and J = 1 ex-
cited state are n0 and n1 , respectively. Assuming the states are thermally populated by
collisions, and the total number density of CO molecules in a cloud is nCO , what is the
number density for each state as a function of rotation energy and temperature? What
is the ratio of the number densities of the states when the rotational excitation energy is
much lower and much higher than the thermal collision energy? Explain why this is. [5]
(d) Estimate the depth of a cloud of CO molecules which is just optically thick (⌧ = 1 at
the line centre) in CO J = 0 to J = 1 radiation. The kinetic temperature of the CO is
T = 30K and its density is nCO = 106 m 3 . [5]
[You can assume energy levels are thermally populated, the J = 1 rotational excitation
energy for CO is E1 = 2.7K, the spontaneous transition rate for emission is Spon (J =
1 ! 0) = 7.4 ⇥ 10 8 s 1 and it emits at = 2.6mm.]
Printed: Monday 6th December, 2021 Page 4 End of Paper

High Energy Astrophysics

Wednesday 12th May, 2021 13:00 - 15:00

(May Diet)
Answer TWO questions

resources.
Special Items

External Examiner: Prof L Miller

examination.
Printed: Wednesday 21st April, 2021 PHYS11013

High Energy Astrophysics (PHYS11013)
51 2 1/2 h⌫/kT 3 1
✏⌫ ⇡ 7 ⇥ 10 ne T e Wm Hz
where ne is the number density of electrons and T is the temperature. Without using
equations, explain briefly why the emissivity scales as n2e , and why it shows an exponential
cut-o↵ at high frequencies. [2]
(b) The density profile of the gas in a galaxy cluster can be approximated as being
constant, n0 , within the core radius of the cluster, Rc , and then falling as n / r 2 at radii
r > Rc . Assuming a constant temperature T throughout the cluster, show that the total
luminosity of the cluster gas is given by
39 2 0.5 3
L ⇡ 2.4 ⇥ 10 n 0 T RC W
and determine the fraction of the cluster luminosity that arises from inside the core radius.
[8]
(c) What is meant by the cooling time of the cluster gas? Write down an equation for
this, and explain why it is only a useful approximation. [2]
(d) The cluster in part (b) has a temperature of 5 ⇥ 107 K, a core radius of 100 kpc and a
total Bremsstrahlung luminosity of 1038 W. Calculate the cooling time of the gas in years,
assuming that the intracluster gas is entirely hydrogen. Comment on whether this cluster
would be expected to develop a cooling flow, giving a justification for your answer. [6]
(e) The central galaxy of the cluster contains an Active Galactic Nucleus. The temper-
ature of the accretion disk surrounding the black hole has the following dependence on
radius, r:
!1/4
GMBH Ṁ
T (r) =
8⇡ s r3
where MBH = 108 M is the mass of the black hole and Ṁ = 1M yr 1 is the accretion
rate. Calculate the temperature of the inner accretion disk, at 5 Schwarzschild radii, and
of the outer accretion disk at 500 Schwarzschild radii [the Schwarzschild radius is given
by 2GMBH /c2 ]. In which part of the electromagnetic spectrum does the emission peak at
each of these radii? Describe how you would expect the spectrum of direct emission from
the accretion disk to appear. What is the origin of the X-ray emission from this AGN? [5]
(f) In what two ways could an astronomer distinguish between X-ray emission from the
AGN and X-ray emission from the intracluster gas? [2]
Printed: Wednesday 21st April, 2021 Page 1 Continued overleaf. . .

2. (a) The average energy loss rate due to synchrotron radiation, for an electron with a
Lorentz factor moving in a magnetic field of strength B, is given by
2 4
dE e B2
h i=
dt 9⇡✏0 cm2e
where me is the electron rest-mass. The electron radiates most of its energy close to the
critical frequency, ⌫c ⇡ 2 ⌫g , where
eB
⌫g =
2⇡me
is the non-relativistic gyro-frequency. Using these results, calculate the time taken for an
electron radiating primarily at 5 GHz to radiate away half of its energy in a magnetic field
of strength B = 10 nT. [6]
(b) A radio source has a spectral shape of S⌫ / ⌫ ↵ at high frequencies, with ↵ = 0.7. At
low frequencies, the spectrum of the source shows synchrotron self-absorption such that
2me ⌦
S⌫ = 1/2
⌫ 5/2
3⌫g
where ⌦ is the solid angle subtended by the source. Explain briefly why self-absorption
occurs. [2]
(c) The source has a spectrum which peaks at a frequency of 1 GHz, with a peak flux
density of S⌫ = 1 Jy = 10 26 W Hz 1 m 2 . If the magnetic field strength is 10 nT, calculate
the angular radius of the source in units of arcseconds, assuming the source to be spherical
in shape. [3]
(d) If the source increases in radius by a factor f , estimate the new peak frequency and
new peak flux density of the source in terms of f , assuming no change in either the
magnetic field or in the (high-frequency) unabsorbed flux density of the source. [5]
(e) If the source expansion in part (d) occurs adiabatically with no input of new relativistic
particles, show that the energy of the relativistic particles in the source would be expected
to fall as E / R 1 , where R is the radius of the source. How would you then expect the
high-frequency (unabsorbed) luminosity of the source to vary with f ? [5]
(f) The radio source expands with a velocity v / t 3/5 . Taking into account both the
adiabatic expansion losses and the synchrotron self-absorption, derive an expression for
how the peak frequency of the radio source evolves with time. [4]

3. (a) In the rest-frame of a shock, the equations of conservation of mass, momentum and
energy for material passing through the shock can be written as:
⇢0 u0 = ⇢1 u1
⇢0 u20 + P0 = ⇢1 u21 + P1
✓ ◆ ✓ ◆
1 1
u0 ⇢0 u20 + ✏0 + P0 u0 = u1 ⇢1 u21 + ✏1 + P1 u1
2 2
where ⇢, u, P and ✏ are the density, velocity, pressure and internal energy density of
the gas respectively, and the sub-scripts 0 and 1 refer to the unshocked and shocked gas
respectively.
Define clearly what is meant by a strong shock and how this helps to solve these equations.
In the case of a strong shock, show that u1 = u0 /4. [4]
(b) A shock is passing through the intergalactic medium. Consider a highly relativistic
particle which is travelling perpendicular to the shock and crosses from the unshocked
to the shocked gas. It is then scattered by 180 degrees in a collision and re-crosses the
shock, where it is once again scattered by 180 degrees so that it is travelling in the same
direction as it began. By using Lorentz Transformations between relevant frames, or
otherwise, calculate the fractional increase in momentum of the particle on this double-
crossing of the shock. [4]
(c) Consider relativistic particles crossing the shock at all angles. By considering the rate
at which particles are swept away downstream, show that the probability of a particle not
returning to re-cross the shock is P (no return) ' 4uc 1 . [3]
(d) Averaging over all angles that particles might cross the shock, the fractional gain in
momentum from part (b) can be shown to be
p 4 u0 u 1
=
p 3 c
Combining this with your result from part (c), derive an expression for the number of
double-crossings that particles will make before 50% of them have been swept downstream,
and hence determine the median factor by which a shock moving at 1200 km s 1 increases
the momentum of relativistic particles. [4]
(e) A radio source has a flux density S⌫ = 1 mJy at 150 MHz and a flux density of
S⌫ = 0.37 mJy at 610 MHz. Define and calculate the spectral index of the source. A radio
survey is carried out at 5 GHz with an rms noise level of 0.01 mJy. Would you expect the
source to be detected? Give two possible explanations for why your expectation might be
proven wrong. [5]
(f) An emitting region within a jet in the radio source is observed repeatedly, and shows
an apparent speed on the sky of 7 times the speed of light. Show how this is possible and
calculate the minimum speed at which the emitting region must actually be moving. [5]
Printed: Wednesday 21st April, 2021 Page 3 End of Paper


Friday 15th May, 2020 13:00 – 15:00 BST

(May Diet)
• This is an open-book examination.

• Electronic Calculators may be used during this examination.
Special Items
Chairman of Examiners: Prof. P Best

External Examiner: Prof. L Miller

examination.
Printed: Friday 10th April, 2020 PHYS11013

1. (a) Describe how observations at X-ray wavelengths can be used to determine the metal-
licity of the intracluster gas in galaxy clusters. This metallicity is found to be about 30%
of Solar metallicity: what implications does this have? [4]
(b) The Bremsstrahlung emissivity from a region of ionised gas can be approximated as
51 2 1/2 h⌫/kT 3 1
✏⌫ ⇡ 7 ⇥ 10 ne T e Wm Hz
where ne is the number density of electrons and T is the temperature. Assuming that
the intracluster gas has uniform density and temperature, argue that its Bremsstrahlung
luminosity will scale as
L / M 2 T 0.5 R 3
where M is the total cluster mass, T is the temperature, and R is the radius. [5]
(c) Show that if a spherical gas cloud of mass M and radius R has uniform density then
its total gravitational potential energy is given by
3 GM 2
Egrav =
5 R
[3]
(d) The Virial Theorem states that the thermal kinetic energy (Ekin ) of a system in
hydrostatic equilibrium is related to its gravitational potential energy by
2Ekin + Egrav = 0
What is meant by ‘hydrostatic equilibrium’ ? What does the Virial Theorem imply about
the energy radiated from a system as it slowly collapses under virialised conditions? [3]
(e) Using the Virial Theorem and your results from part (b), or otherwise, show that the
Bremsstrahlung luminosity of the intracluster gas is expected to scale as
T 5/2
L/
R
In practice this luminosity-temperature relation holds at high cluster masses, but the
observed relation at lower cluster masses is steeper. Suggest where the above analysis
breaks down, and why. [5]
(f) Assume that the intracluster gas collapses over time in a virialised manner (retaining
uniform density). If the intracluster gas collapses to half of its original radius, by what
factor would the luminosity increase? Relate the luminosity of the gas cloud to its rate
of collapse to show that
dR 3/2
/R
dt
[5]
Printed: Friday 10th April, 2020 Page 1 Continued overleaf. . .

2. (a) A supernova expands into the interstellar medium, driving a shock. In the rest-frame
of the shock, the following three equations hold:
⇢0 u0 = ⇢1 u1
⇢0 u20 + P0 = ⇢1 u21 + P1
✓ ◆ ✓ ◆
1 1
u0 ⇢0 u20 + ✏0 + P0 u0 = u1 ⇢1 u21 + ✏1 + P1 u1
2 2
Briefly explain what each of these three equations represents, defining the terms within
them and outlining why the equations have the form that they do. Explain how u0 relates
to the shock speed vs . [4]
(b) In a strong shock, P0 ⌧ ⇢0 u20 . The solution to the shock equations for a strong
shock gives u1 = u0 /4. Using this result, determine the dependence of the post-shock gas
temperature on the shock speed, and argue that the total energy of the supernova
ESN / MSN vs2
where MSN is the mass of swept-up gas. [4]
(c) Hence, if the supernova remnant expands into a uniform density medium, show that
the radius of the supernova remnant (RSN ) will initially grow with time as
RSN / t2/5
[3]
(d) The shock accelerates relativistic electrons, which emit synchrotron radiation accord-
ing to:
2 4 2
dE eB
h i=
dt 9⇡✏0 cm20
where is the Lorentz factor of the electrons, B is the strength of the magnetic field, and
m0 is the electron rest-mass. Calculate the time taken for a 1 GeV electron to radiate
away 90% of its energy through synchrotron radiation, if B = 10 nT. [4]
(e) Consider a sphere filled with relativistic electrons. Show that if the radius of the
sphere, R(t), increases with time, then the energy loss rate of the electrons due to adiabatic
expansion is:
!
dE 1 dR
= E
dt R dt
What explicit assumption is made in this calculation that is not valid in the case of the
expanding supernova remnant? [5]
(f) Explain why, for an expanding sphere of relativistic electrons, adiabatic expansion
losses may be relatively unimportant (compared to synchrotron losses) for electrons radi-
ating primarily at frequencies of 1400 MHz, and yet be the dominant energy loss mecha-
nism for electrons radiating at 30 MHz. [3]
(g) Two massive elliptical galaxies each host a radio AGN, which launch equally powerful
radio jets. One of the radio AGN is hosted by a galaxy located in a cluster, and the other
by a galaxy located in the field. After the radio sources have aged for 107 years they are
observed at low radio frequencies. Would you expect to observe any di↵erences in their
synchrotron emission? Justify your answer. [2]

3. (a) Write down or derive the expression for the 4-acceleration, a⌫ . Hence show that the
4-vector product a⌫ a⌫ is given by:
2 !2 3
2
24c d d
a⌫ a⌫ = 2
2 v·a 2 25
a
dt dt
where v is the 3-velocity, a is the 3-acceleration (whose modulus is a), and is the Lorentz
3
factor. Noting that ddt = c2 v · a, go on to show that
" #
⌫ 4 2 2 (v · a)2
a a⌫ = a +
c2 [5]
(b) Using this result, and expressing a as a combination of components parallel to (ak )
and perpendicular to (a? ) the velocity, show that
h i
a⌫ a⌫ = 4
a2? + 2 2
ak
[3]
(c) Larmor’s formula for the rate of energy loss from an accelerating relativistic electron
is
dE e2
= a⌫ a⌫
dt 6⇡✏0 c3
Show that an electron of energy E moving in a magnetic field of strength B has an average
rate of energy loss due to synchrotron radiation given by:
dE E 2 e4 B 2
h i=
dt 9⇡✏0 c5 m40
where m0 is the electron rest-mass. [5]

(d) By considering the 4-momentum of a photon travelling at an angle ✓ to the x-axis in
frame S, or otherwise, derive the aberration formula describing the transformation of an-
gles between frames S and S 0 , moving at relative velocity v in the standard configuration:
cos ✓ vc
cos ✓0 =
1 vc cos ✓ [3]
(e) Synchrotron emission is observed from the radio jet in a quasar. The jet is oriented
at an angle ✓ to the line of sight. A blob of emitting plasma within the jet is moving
along the jet at velocity v, and is emitting isotropically in its own rest-frame, emitting an
energy dW 0 into each solid angle d⌦0 . Show that in the observer’s frame the emission per
solid angle is given by
dW 1 dW 0
= v
d⌦ 3 (1
c
cos ✓)3 d⌦0 [5]
(f) A radio survey observes a radio source whose jet travels with a velocity of 0.98c at an
angle ✓ = 10 to the line of sight. The jet is observed to have a flux density of 2 Jy, and
a flat spectral index (↵ = 0). The survey has a sensitivity of 20µJy. Will the counter-jet
be detected? [4]
Printed: Friday 10th April, 2020 Page 3 End of Paper


Wednesday 22nd May, 2019 14:30 – 16:30

(May Diet)

• Attach supplied anonymous bar codes to each script book used.
Special Items



examination.
Printed: Monday 22nd April, 2019 PHYS11013

1. (a) Show that a relativistic charged particle of charge e, rest mass m0 and Lorentz factor
moves in a magnetic field of strength B with a velocity whose component parallel to
the magnetic field is constant and whose component perpendicular to the magnetic field
undergoes circular motion. Show that the gyro-frequency, !B , is given by
eB
!B =
m0
[4]
(b) Larmor’s formula for the energy radiation rate from a relativistic electron accelerating
in a direction perpendicular to its velocity can be written as
e2 4
P = |v̇|2
6⇡✏0 c3
Using this equation, show that the average energy loss rate from an electron due to
synchrotron radiation is given by
2 4
dE e B2
=
dt 9⇡✏0 cm20
[4]
(c) By considering the 4-momentum of a photon travelling at an angle ✓ to the x-axis in
0 cos ✓ vc
cos ✓ =
1 vc cos ✓
[3]
(d) Show that for relativistic electrons half of the radiated energy is emitted into an angle
|✓| < 1/ in frame S. Building on this result, show that the characteristic frequency of the
synchrotron radiation is ⌫ ⇡ 2 !g , where !g = !B is the non-relativistic gyro-frequency.
[6]
(e) A population of electrons has a distribution of energies that follows a power-law,
N (E) / E x . Using your results from parts (b) and (d), show that the observed frequency
spectrum of emission from the electron population is also a power-law, P⌫ / ⌫ ↵ , with
↵ = (x 1)/2. What value of ↵ is typically observed for galactic supernovae and for
extragalactic radio galaxies? [5]
(f) A radio source is detected in a radio survey carried out at 150 MHz, with a measured
flux density of 2 mJy. A survey of the same region of sky is then carried out at 1.4 GHz,
with a noise level of 0.15 mJy. For what range of spectral indices would the radio source
be detectable with signal-to-noise ratio greater than 3 in the 1.4 GHz survey? [3]
Printed: Monday 22nd April, 2019 Page 1 Continued overleaf. . .

2. (a) What is meant by an astrophysical shock? In the rest-frame of a shock, the following
three equations hold:
⇢0 u0 = ⇢1 u1
⇢0 u20 + P0 = ⇢1 u21 + P1
✓ ◆ ✓ ◆
1 1
u0 ⇢0 u20 + ✏0 + P0 u0 = u1 ⇢1 u21 + ✏1 + P1 u1
2 2
Briefly explain what each of these three equations represents, defining the terms within
them and outlining why the equations have the form that they do. [5]
(b) For the case where P0 ⌧ ⇢0 u20 , use these equations to show that
u0 3
u1 = and P1 = ⇢0 vs2
4 4
where vs is the shock speed. [4]

(c) Assuming that the post-shock gas is composed of pure ionised hydrogen, show that
the post-shock gas temperature is
3 mH 2
T = v
32 k s
where mH is the hydrogen mass. Hence, estimate the minimum shock velocity required
to ionise hydrogen (ionisation energy of 13.6 eV). [5]
(d) Explain qualitatively why shocks lead to the Fermi acceleration of mildly relativistic
particles up to much higher energies. [3]
(e) The Bremsstrahlung emissivity from a region of ionised gas can be approximated as
51 2 1/2 h⌫/kT 3 1
✏⌫ ⇡ 7 ⇥ 10 ne T e Wm Hz
where ne is the number density of electrons and T is the temperature. A cluster of

galaxies contains approximately 2 ⇥ 1013 solar masses of intracluster gas, within a radius
of 1 Mpc. It is observed to have a line-of-sight velocity dispersion of 600 km s 1 . Estimate
the temperature of the gas. Hence, assuming a uniform density for the gas, estimate the
total Bremsstrahlung luminosity of the cluster. [5]
(f) A second (similar mass) cluster collides with the first cluster, with a relative velocity
of 5000 km s 1 . Describe how you might expect this cluster merger to appear at radio
wavelengths, and why. [3]
Printed: Monday 22nd April, 2019 Page 2 Continued overleaf. . .

3. (a) Larmor’s formula for the energy loss rate from a non-relativistic accelerating charge
is:
e2 |v̇|2
P =
6⇡✏0 c3
Use this to calculate the energy loss rate of an electron oscillating in a sinusoidal electric
field of strength E = E0 ei!t (you can assume that the electron remains non-relativistic
at all times). The electron will intercept the passing energy density of the radiation field
over a cross-sectional area T (the Thomson scattering cross-section). By considering the
balance between the energy the electron intercepts and the energy it radiates, show that
e4
T ⌘
6⇡✏20 m2e c4
[5]
(b) Using the Thomson cross-section, estimate the probability of a photon being scat-
tered by an electron in the intracluster medium of a galaxy cluster, while passing on a
line through the centre of the cluster. You can approximate the intracluster medium as
comprising 3 ⇥ 1013 M of pure ionised hydrogen gas uniformly distributed over a sphere
of radius 1 Mpc. [3]
(c) Show that if a photon of frequency ⌫ is inverse-Compton scattered by an electron of
Lorentz factor , then its frequency will be increased to approximately 2 ⌫. [5]
(d) A more precise calculation shows that during inverse-Compton scattering the photon
frequency is increased to 43 2 ⌫. Use this result to show that the rate at which a highly
relativistic ( 1) electron loses energy through inverse Compton scattering is given by
dE 4 2
= Uphoton Tc
dt 3
where Uphoton is the energy density of the photon field. [3]

(e) Hence, calculate the time taken for a 1 GeV electron to lose half of its energy in a loca-
tion where the only significant photon field is that of the Cosmic Microwave Background
(CMB; Uphoton,CMB = 4 ⇥ 10 14 J m 3 ). [4]
(f) Explain briefly why the Thomson scattering cross-section is also relevant for the upper
limit to the rate of accretion on to a black hole (the Eddington limit). Show that the
Eddington luminosity limit is given by:
4⇡Gmp c
LEdd = MBH
T
where mp is the proton mass and MBH is the black hole mass. A quasar has a bolometric
luminosity of 4 ⇥ 1039 W. What information does this provide about the mass of the black
hole powering the quasar? [5]
Printed: Monday 22nd April, 2019 Page 3 End of Paper


Monday 14th May, 2018 14:30 – 16:30

(May Diet)

• Attach supplied anonymous bar codes to each script book.
Special Items

Chairman of Examiners: Prof. J Dunlop


examination.
Printed: Thursday 19th April, 2018 PHYS11013

1. (a) What is a 4-vector? Show that since electric charge is conserved, then it must obey
the continuity equation, which in 4-vector formalism is
r⌫ J ⌫ = 0
where J ⌫ = (c⇢, j) is the 4-current.

R H
(You may wish to make use of the Divergence Theorem, V r· dV = S · dS). [5]
(b) Maxwell’s four equations are:
r · E = ⇢/✏0 rÊ= Ḃ
1
r·B=0 r ^ B = µ0 j + Ė
c2
Define the terms in these equations. If the magnetic vector potential A is defined such
that B = r ^ A, argue that E = Ȧ r for some scalar field (x, t). [3]
(c) Show that obeys the wave equation
!
1 @2 ⇢ @ 1@
r2 = r·A+ 2
c2 @t2 ✏0 @t c @t
[3]
(d) Explain what a Gauge transformation is and how one can be used to simplify the
wave equation for . Write down the Gauge transformation in 4-vector format, using
the electromagnetic 4-potential Aµ = ( /c, A). Also write down the 4-vector form of the
Lorentz Gauge which simplifies the wave equation. [4]
(e) The physical field tensor can be written as
F µ⌫ = rµ A⌫ r ⌫ Aµ
Show that this definition of the fields ensures that they are Gauge invariant. [2]
(f) Taking advantage of the continuity equation from part (a), show that
rµ F µ⌫ = µ0 J ⌫
is a solution to this equation. Using the Lorentz Gauge, show that the wave equations
can therefore be written as
rµ rµ A⌫ = µ0 J ⌫
[4]
(g) The solution to the wave equations leads to Larmor’s formula for the energy loss rate
from a non-relativistic accelerating charge:
e2 |v̇|2
P =
6⇡✏0 c3
Use this to derive an expression for the total energy radiated by a non-relativistic electron
accelerated in an electric field whose strength decays exponentially as |E| = E0 e t/t0 . [4]
Printed: Thursday 19th April, 2018 Page 1 Continued overleaf. . .

2. (a) Larmor’s formula for the rate of energy loss from an accelerating relativistic electron
is
dE e2 4 ⇣ 2 2 2
⌘
= a + ak
dt 6⇡✏0 c3 ?
where a? and ak are the components of acceleration of the electron perpendicular to and
parallel to its velocity, respectively.
Using this formula, show that an electron with a Lorentz factor moving in a magnetic
field of strength B has an average rate of energy loss due to synchrotron radiation given
by
2 4
dE e B2
h i=
dt 9⇡✏0 cm20

(b) The aberration formula, describing the transformation of angles between frames S
and S 0 moving at relative velocity v in the standard configuration, is given by:
cos ✓ vc
cos ✓0 =
1 vc cos ✓
Using this, or otherwise, show that for highly relativistic electrons half of the radiated
energy is emitted into an angle |✓| < 1/ in frame S. Building on this result, show that
the characteristic frequency of the synchrotron radiation is ⌫ ⇡ 2 !g , where !g = eB/m0
is the non-relativistic gyro-frequency. [6]
(c) Without using equations, describe two methods of estimating the magnetic field
strength in an extragalactic radio source. [4]
(d) An extragalactic radio source has a magnetic field strength of 20 nT. Estimate the
Lorentz factor of the electrons that are seen in observations at a frequency of 1.4 GHz,
and hence estimate their half-life. [4]
(e) The spectral index ↵ (defined as S⌫ / ⌫ ↵ , where S⌫ is the flux density at frequency ⌫)
of the radio source in part (d) is ↵ ⇡ 0.7 at a frequency of 150 MHz, but significantly
steeper at 1.4 GHz. Comment on what this implies for the age of the radio source. Ap-
proximately what physical size would you expect an extragalactic radio source of such an
age to have? [4]
(f) The core of the radio source has a much flatter spectral index (↵ ⇡ 0). Using your
knowledge of other processes that a↵ect the synchrotron spectrum, comment on what
might give rise to this. [2]
Printed: Thursday 19th April, 2018 Page 2 Continued overleaf. . .

3. (a) The Eddington luminosity, LEdd , is given by

4⇡Gmp c
LEdd = M
T
Define precisely what is meant by the Eddington luminosity, and clarify what the other
terms in the equation are. What assumption goes into the derivation of this equation? [3]
(b) The innermost stable orbit of a non-rotating black hole occurs at R = 3Rs , where
Rs = 2GMBH /c2 is the Schwarzschild radius of the black hole. Use this to show that
matter falling into a black hole through an accretion disk radiates a fraction ✏ ⇡ 0.1 of
its rest-mass energy. [4]
(c) A quasar is observed to have a bolometric luminosity of L = 1039 W. Assuming that
the quasar is radiating at the Eddington luminosity, calculate the mass of the black hole
powering it. Derive an expression for the rate of growth of the black hole. Use this
to calculate the mass that the black hole will have grown to after the quasar has been
switched on for a period of 108 years, assuming it continues to radiate at the Eddington
limit with an accretion efficiency ✏ = 0.1. [6]
(d) The cosmic X-ray background is observed to peak in the hard X-ray band (around
30 keV), whereas quasars typically show X-ray spectra which rise towards softer X-rays
(0.5-10 keV). Briefly outline how this X-ray ‘spectral paradox’ can be explained. [3]
(e) Consider a population of AGN of fixed luminosity L, distributed at constant density
throughout Euclidean space. Show that the number of AGN observable at a given flux
density S scales as
dN
/ S 5/2
dS
Hence, argue that, even if AGN have a distribution of luminosities (provided that lumi-
nosity distribution is the same at all locations), a plot of Euclidean-scaled source counts
(i.e. S 5/2 dN
dS
versus S) should be flat. [5]
(f) At radio flux densities above ⇡ 3 Jy the distribution of Euclidean-scaled radio source
counts increases with decreasing flux density, as shown in the Figure below. What infor-
mation does this provide about the most luminous radio sources? Towards fainter flux
densities, the radio source counts then fall (see Figure). What can be inferred about the
global population of radio sources from these data? [4]
Printed: Thursday 19th April, 2018 Page 3 End of Paper

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Monday 15th May, 2017 09:30 – 11:30

(May Diet)

Special Items


External Examiner: Pro. C Tadhunter

examination.

1. (a) Maxwell’s equations are:

r · E = ⇢/✏0
r·B = 0
rÊ = Ḃ
1
r ^ B = µ0 j + Ė
c2
Explain why a magnetic vector potential, A, defined as B = r ^ A, can be introduced.
Show that E = Ȧ r for some scalar field (x, t). [3]
(b) The equation for the Poynting vector is:
1
S= |Ȧ? |2 n
µ0 c
Explain precisely what the terms S, Ȧ? and n are.

For an electron moving at a non-relativistic velocity v, the magnetic vector potential is
given by A ' µ0 ev/(4⇡R). Show that the total energy radiation rate of the electron can
be expressed in terms of the acceleration, v̇, as:
e2 |v̇|2
P =
6⇡✏0 c3 [5]
(c) The energy loss rate from the accelerating electron can be written in a relativistically
correct form as:
dE e2
= 3
a⌫ a⌫
dt 6⇡✏0 c
where a⌫ is the 4-acceleration. Write down an expression for a⌫ and use this to evaluate
the 4-vector product a⌫ a⌫ for the case where the acceleration (v̇) is perpendicular to the
velocity. Hence, show that under these circumstances,
dE e2 4
= 3
|v̇|2
dt 6⇡✏0 c
where is the Lorentz factor of the electron. [5]

(d) Using this equation, show that the energy loss rate from synchrotron radiation for an
electron moving in a magnetic field of strength B is given by
dE e B 2 sin2 ✓
2 4
=
dt 6⇡✏0 cm20
where m0 is the rest-frame mass of the electron and ✓ is the pitch angle. [5]
(e) Assume that the electron is travelling in a circular orbit. Derive an expression for
the time that it takes the electron to complete one revolution. Hence, using your result
from part (d), calculate an expression for the fraction of the electron’s energy that gets
radiated in a single revolution. [4]
(f) Evaluate this fraction for a 5 GeV electron gyrating in a magnetic field of strength
20 nT. Comment on the implications of your result for the assumption (normally adopted
in the derivation of the synchrotron formula) that the electron energy is constant. [3]

51 2 1/2 h⌫/kT 3 1
✏⌫ ⇡ 7 ⇥ 10 ne T e Wm Hz
Explain what the terms in this equation are, and provide a brief justification of why the
emissivity scales as n2e , and why it shows an exponential cut-o↵ at high frequency. [3]
(b) A cluster of galaxies is observed to have a velocity dispersion of 800 km s 1 . The
cluster contains approximately 5 ⇥ 1013 solar masses of intracluster gas, within a radius
of 1 Mpc. Assuming a uniform density for the gas, estimate the total Bremsstrahlung
luminosity of the cluster. [5]
(c) As well as the Bremsstrahlung emission, what other process can give rise to emission
at X-ray wavelengths from the hot gas in the intracluster medium? Explain why this
causes the X-ray spectra of galaxy clusters at 108 K to have a very di↵erent appearance
from those of galaxy groups at 107 K. [3]
(d) The galaxy at the centre of a cluster contains an active galactic nucleus (AGN), which
also emits at X-ray wavelengths. The accretion disk surrounding the black hole radiates
as a thermal black-body. By considering the energy losses of the material as it flows
through the accretion disk, or otherwise, show that the temperature (T ) of the accretion
disk as a function of radius (r) can be approximated as:
!1/4
GMBH Ṁ
T (r) =
8⇡ s r3
where MBH is the mass of the black hole, Ṁ is the accretion rate and s is the Stefan-
Boltzmann constant. [5]
(e) Show that there is an upper limit (the Eddington limit) to the rate of accretion on to
the black hole, corresponding to a luminosity of
4⇡Gmp c
LEdd = MBH
T
where T is the Thomson scattering cross-section and mp is the proton mass. Assuming
an accretion efficiency of 10%, calculate the accretion rate of a 108 M black hole at the
Eddington luminosity. Give your answer in units of solar masses per year. [5]
(f) Calculate the temperature of the inner accretion disk, at 5 Schwarzschild radii, of the
accreting black hole from part (e). In which part of the electromagnetic spectrum does
the emission peak? What is the origin of the X-ray emission from this AGN? [4]

3. (a) The average energy loss rate through synchrotron radiation from an isotropic distri-
bution of electrons of Lorentz factor , moving in a magnetic field of strength B, is given
by
2 4 2
dE eB
h i=
dt 9⇡✏0 cm20
where m0 is the electron rest-mass. Using this, derive an expression for how the electron
energy evolves with time. Hence, calculate the half life of a 1 GeV electron emitting
synchrotron radiation in a 10 nT magnetic field. [5]
(b) Consider a distribution of electrons in which the number of electrons between energy
E and E + dE is given by N (E)dE. The electrons lose energy at a rate dE dt
, and new
particles are injected at a rate Q(E). By considering the flow of particles in energy
space, or otherwise, derive the equation for the evolution of the energy distribution of the
particles (ignoring any spatial di↵usion):
" #
@N @ dE
= N (E) + Q(E)
@t @E dt
[4]
(c) Consider the relativistic electrons to be confined to a sphere of radius R(t), which is
expanding with time. Show that the energy loss rate of the particles due to adiabatic
expansion losses is:
!
dE 1 dR
= E
dt R dt
[4]
(d) Consider a steady-state solution (ie. @N
@t
= 0), in which electrons are injected with a
power law injection spectrum Q(E) / E x . Show that in the case of adiabatic losses the
electron energy distribution follows the injected distribution (N (E) / E x ), but that for
synchrotron radiation losses the energy distribution is steepened by one power of E from
the injected distribution, to N (E) / E (x+1) . [3]
(e) A sphere of plasma travels down the jet of a radio galaxy, away from the core, at a
speed v. The jet has a conical shape, and the plasma expands with the jet, retaining
its spherical shape. The plasma also contains a magnetic field whose strength decays
⇣ ⌘ 1/2
with distance (r) from the core as B(r) = B0 rr0 , where B0 is the strength of the
magnetic field at distance r0 . Show that the expression for the energy loss rate of an
electron of energy E, combining both synchrotron and adiabatic expansion losses (but
ignoring inverse Compton losses), can be written as:
1 dE v
2
=
E + aE dt r
and give the expression for a. [5]
(f) Using this equation, show that if the electron has energy E0 at a distance r0 from the
core, then its energy at some larger distance r is given by:
✓ ◆
1 1 r
+a= +a
E(r) E0 r0
[4]

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Thursday 19th May, 2016 09:30 – 11:30

(May Diet)

Special Items

Chairman of Examiners: Professor J Dunlop

External Examiner: Professor C Tadhunter

examination.

1. (a) The relativistic Larmor formula for the rate of energy loss from a charged particle
undergoing acceleration is
dE e2 4 ⇣ 2 2 2
⌘
= a + ak
dt 6⇡✏0 c3 ?
where a? and ak are the components of particle acceleration perpendicular to and parallel
to its velocity, respectively.
Use this formula to show that the averate rate of energy loss due to synchrotron emission
from an electron with a Lorentz factor gyrating in a magnetic field of strength B is
given by
2 4
dE e B2
h i=
dt 9⇡✏0 cm20

(b) Show that if the distribution of electron energies follows a power-law, with N (E) /
E x , then the observed synchrotron spectrum also follows a power-law: P⌫ / ⌫ ↵ , with
↵ = (x 1)/2.
Note: you can assume that all of the synchrotron radiation from an electron of Lorentz
factor is emitted at the characteristic frequency ⌫ ⇡ 2 !g , where !g = eB/mo is the
non-relativistic gyro-frequency. [4]
(c) The radio jet in a quasar is oriented at an angle ✓ to the line of sight. Show that if a
blob of emitting plasma within the jet is moving at velocity v, then its apparent velocity
on the sky will be given by
v sin ✓
vapp =
1 vc cos ✓
[3]
(d) By considering the 4-momentum of a photon travelling at an angle ✓ to the x-axis in
cos ✓ vc
cos ✓0 =
1 vc cos ✓
[3]
(e) Show that if the blob of plasma in part (c) is emitting isotropically in its own rest-
frame, emitting an energy dW 0 into each solid angle d⌦0 , then the emission per solid angle
in the observer’s frame is given by
dW 1 dW 0
= v
d⌦ 3 (1
c
cos ✓)3 d⌦0
[5]
(f) The quasar radio jet is oriented at 10 to the line of sight, with an apparent velocity of
vapp = 3c. Calculate the jet velocity. Hence calculate the expected flux ratio between the
jet and an identical counter-jet pointed in the opposite direction, assuming a flat spectral
index, ↵ = 0.0. [5]

2. (a) Without detailed calculation, show that if a photon of frequency ⌫ is inverse-Compton

scattered by an electron of Lorentz factor , then its frequency will be increased to roughly
2
⌫. [5]
(b) A more precise calculation shows that the photon frequency is increased to 43 2 ⌫. Use
this result to show that the rate at which a highly relativistic ( 1) electron loses
energy through inverse Compton scattering is given by
dE 4 2
= Uphoton Tc
dt 3
where Uphoton is the energy density of the photon field and T is the Thomson scattering
cross-section.
Hence, calculate the characteristic lifetime, ⌧ , of a 5 GeV electron in a region where the
only significant photon field is that of the Cosmic Microwave Background (CMB), which
has Uphoton,CMB = 4 ⇥ 10 14 J m 3 . Express your answer in years. [6]
(c) The Bremsstrahlung emissivity from a region of ionised gas can be approximated as
51 2 1/2 h⌫/kT 3 1
✏⌫ ⇡ 7 ⇥ 10 ne T e Wm Hz
where ne is the number density of electrons and T is the temperature. Provide a brief
justification of why this emissivity scales as n2e , and why there is an exponential cut-o↵
at high frequency. [2]
(d) A cluster of galaxies is observed to have a velocity dispersion of 1000 km s 1 . The clus-
ter contains approximately 1014 solar masses of intracluster gas, within a radius of 1 Mpc.
Assuming a uniform density for the gas, estimate the total Bremsstrahlung luminosity of
the cluster. [5]
(e) Calculate the probability of a photon from the CMB undergoing a scattering event as
it passes through this cluster (on a line through the centre). You can assume the Thomson
scattering cross-section. [3]
(f) The CMB has a black-body spectrum with a temperature of 2.73K, which corresponds
to a peak frequency of about 160 GHz. Sketch the spectral shape of the CMB. Bearing
in mind the results of parts (a) and (e), on the same plot also sketch the expected shape
that the CMB would be observed to have when looking in the direction of the cluster.
Hence, comment on how the cluster might be expected to appear at frequencies of 50 GHz
and 300 GHz. [4]

3. (a) A shock is passing through the interstellar medium. In the rest-frame of the shock,
the equations of conservation of mass, momentum and energy are:
⇢0 u0 = ⇢1 u1
⇢0 u20 + P0 = ⇢1 u21 + P1
✓ ◆ ✓ ◆
1 1
u0 ⇢0 u20 + ✏0 + P0 u0 = u1 ⇢1 u21 + ✏1 + P1 u1
2 2
Explain what the terms in these equations are, and how u0 relates to the shock speed vs . [2]
(b) What is meant by a strong shock? Under the strong shock conditions, use these
equations to show that
u0
u1 =
4
and that
3
P1 = ⇢0 vs2
4 [5]
(c) Assuming that the post-shock gas is composed of pure ionised hydrogen, show that
3
the temperature of the post-shock gas can be expressed as kT = 32 mH vs2 , where mH is the
hydrogen mass. Hence, estimate the minimum shock velocity required to ionise hydrogen
(ionisation energy of 13.6eV). [5]
(d) A supernova explosion leads to an expanding supernova remnant, which drives a
strong shock into its environment. Using the result from (c), argue that the total energy
of the shocked gas scales as MSN vs2 , where MSN is the mass of swept-up gas. Hence, show
that if the supernova remnant expands into a uniform density medium then the radius of
the supernova remnant (RSN ) will initially grow with time as
RSN / t2/5
[5]
(e) The supernova remnant gives rise to radio synchrotron emission. Define the brightness
temperature (Tb ) of the synchrotron emission, and show that it is given by
c 2 S⌫
Tb ⌘
2k⌫ 2 ⌦
where S⌫ is the flux density of the source at frequency ⌫, and ⌦ is the solid angle that
the source subtends on the sky. [4]
(f) At early times, the synchrotron emission of the supernova is self-absorbed at low
frequencies. Explain briefly what this means and why it happens.
The flux density of the self-absorbed synchrotron radiation is given by
2me ⌦
S⌫ = 1/2
⌫ 5/2
3⌫g
where ⌫g is the non-relativistic gyro-frequency. Assuming a constant magnetic field

strength, calculate how the flux density of the supernova will initially evolve with time at
low frequencies. [4]

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Wednesday 14th May, 2014 2:30pm – 4:30pm

(May Diet)
• Only the supplied Electronic Calculators may be used during this examination.
Special Items
• School supplied calculators

Chairman of Examiners: Professor J Dunlop

External Examiner: Professor C Tadhunter

examination.
Printed: Thu 17th Apr, 2014 PHYS11013

1.
(a) Maxwell’s four equations are:
r · E = ⇢/✏0
r·B = 0
rÊ = Ḃ
1
r ^ B = µ0 j + Ė
c2
Justify the introduction of a magnetic vector potential A such that B = r ^ A,

and argue that E = Ȧ r for some scalar field (x, t). [3]
(b) Show that obeys the wave equation

!
2 1 @2 ⇢ @ 1@
r = r·A+ 2
c2 @t2 ✏0 @t c @t
Explain what a Gauge transformation is and how one can be used to simplify the
wave equation for . [6]
(c) Solving the wave equation leads to the Larmor formula for the energy loss rate from
an accelerating charge:
dE e2
= a⌫ a⌫
dt 6⇡✏0 c3
Write down the expression for the 4-acceleration a⌫ , and use this to evaluate the
4-vector product a⌫ a⌫ for the case where the acceleration (v̇) is perpendicular to
the velocity (it may help to note that ˙ = 2 3 v · v̇/c2 where = (1 v 2 /c2 ) 1/2 is
the Lorentz factor for velocity v). Hence show that under these circumstances
dE e2 4
= |v̇|2
dt 6⇡✏0 c3
[5]
(d) Using this equation, show that the average energy loss rate due to synchrotron
radiation from an isotropic distribution of electrons moving in a magnetic field of
strength B is given by * +
2 4 2
dE eB
=
dt 9⇡✏0 cm20
where m0 is the rest-frame mass of the electron. [6]
(e) Derive an expression for the synchrotron lifetime, ⌧ , of an electron of Lorentz factor
radiating in a magnetic field of strength B. Evaluate this lifetime for a 1 GeV
electron in a field strength of B = 5 ⇥ 10 8 T. Comment on what will cause your
derived relationship between ⌧ and B to break down at low B in astrophysical
situations. [5]
Printed: Thu 17th Apr, 2014 Page 1 Continued overleaf. . .

2.
(a) In a non-relativistic shock, viewed in the rest-frame of the shock, gas flows in to the
shock on the upstream side at velocity u0 and out of the shock on the downstream
side at velocity u1 = u0 /4. Consider a relativistic particle with momentum p which
crosses the shock, scatters in the downstream fluid, and then crosses back. Using
Lorentz transformations of the 4-momentum of the particle, or otherwise, show that
on average the particle’s gain in momentum for this double-crossing will be
4p
p= (u0 u1 )
3c
You can assume that the shock velocity is sufficiently low that aberrations of angles
can be ignored. [7]
(b) Show that the probability of the particle not re-crossing the shock from the down-
stream side is approximately 4u1 /c. What is the probability of the particle not
re-crossing the shock from the upstream side, assuming a plane shock and an infi-
nite medium? [3]
(c) Show that the distribution of particle energies following this shock acceleration is
expected to be a power-law, N (E) / E x , with x = 2. [4]
(d) The aberration formula describing the transformation of angles between frames S 0
and S moving at a relatively velocity v in the standard configuration is given by
cos ✓0 + vc
cos ✓ =
1 + vc cos ✓0
Using this, show that when v approaches c, half of the radiated energy is emitted
into an angle |✓| < 1/ in frame S, where is the Lorentz factor corresponding to
velocity v. Use this result to show that a relativistic electron gyrating in a magnetic
field of strength B will emit synchrotron radiation with a characteristic frequency
⌫ ⇡ 2 !g , where !g is the non-relativistic gyro-frequency. [7]
(e) The average energy loss rate for synchrotron-emitting electrons scales with energy
as h dE
dt
i / E 2 . Combining this with the results from parts (c) and (d), show that
the observed spectrum of emission from a population of shock-accelerated electrons
is also a power-law, S⌫ / ⌫ ↵ , with ↵ = (x 1)/2. [4]
Printed: Thu 17th Apr, 2014 Page 2 Continued overleaf. . .

3.
(a) Gas is accreting onto a magnetised star. The Alfvén radius, Rm , occurs when the
following condition is satisfied:
B2
⇢v 2 =
2µ0
Explain this condition, and describe the behaviour of the accreting gas within the
Alfvén radius. [2]
(b) Assuming that the Alfvén radius corresponds to the inner edge of an accretion disk,
show that the transfer of angular momentum from the disk to the magnetised star
will cause the rotation period P of the star to decrease as
p
dP 2 Ṁ GM Rm
= P
dt 2⇡I
where I is the moment of inertia of the star. What assumption must be made in
this derivation? [5]
(c) What two pieces of evidence can be used to show that the magnetised star in an
X-ray binary pulsar system is a neutron star rather than a white dwarf? [2]
(d) A binary X-ray pulsar has been spun up (via accretion from its companion star) to a
period of 0.1 second, after 107 years. Assuming that the accretion rate and magnetic
field strength both remain constant, what will its period be when it is 1 Gyr old? [3]
(e) Consider instead gas accretion on to a black hole, of mass MBH . Show that there is
an upper limit (the Eddington limit) to the rate of accretion on to the black hole,
corresponding to a luminosity of
4⇡Gmp c
LEdd = MBH
T
where T is the Thomson scattering cross-section. [3]
(f ) A quasar has an integrated X-ray luminosity between 0.8 and 10 keV of 1038 Watts.
Assuming that the quasar has a spectral shape of f⌫ / ⌫ 1 extending from the far-
infrared (100µm) to hard X-rays (100 keV), calculate the total bolometric luminosity
of the quasar. Hence derive the mass of the black hole powering the quasar, assuming
that it is accreting at the Eddington limit. [5]
(g) Calculate the time taken for a black hole accreting at the Eddington limit to double
its mass, assuming a radiative efficiency of ✏ = 0.1. If the black hole powering a
quasar grows from a seed mass of 1 M to 108 M in a period of 1 Gyr at a constant
fraction f of the Eddington rate, calculate f . [5]
Printed: Thu 17th Apr, 2014 Page 3 End of Paper

College of Science and Engineering
School of Physics

SCQF Level 11, U01432, PHY-5-HighEnAst
Tuesday, 13th May 2008

9.30 - 11.30 a.m.
Chairman of Examiners
Prof J A Peacock
External Examiner
Prof S Rawlings
The bracketed numbers give an indication of the value assigned

to each portion of a question.
Only the supplied Electronic Calculators may be used during this examination.
Anonymity of the candidate will be maintained during the marking of

this examination.
Printed: March 16, 2008

High Energy Astrophysics (U01432)
1.
a) In a uniform magnetic field B a relativistic electron with energy E = me c2
executes a circular orbit in a plane perpendicular to the field direction. Given
the relativistic transformation of angles from the electron rest frame S 0 to the
observer’s rest frame S
cos ✓0 + vc
cos ✓ =
1 + vc cos ✓0
where the angles are measured from the forward direction of motion, show that
the observer finds the synchrotron radiation to be concentrated near the char-
acteristic frequency ⌫c = 2 !g , where !g is the non-relativistic gyro frequency.
[7]
b) Given that the rate at which an accelerated electron loses energy is
dE e2 4 ⇣ 2 2 2
⌘
= a + ak
dt 6⇡✏0 c3 ?
where a? and ak are the components of the acceleration perpendicular and par-
allel to the electron velocity vector, show that the electron in the magnetic field
radiates energy at a rate
2 4
dE e B2
= .
dt 6⇡✏0 m2e c
[5]
c) An ensemble of electrons in a uniform magnetic field has a power-law energy

spectrum N (E) = E ⇠ (where  and ⇠ are constants) at time t = 0. Supposing
that no further particle acceleration or injection of fresh particles takes place,
find the energy Emax (t) above which there are no electrons after time t due to
synchrotron losses, and sketch the particle energy spectrum and the synchrotron
spectrum at this time. [6]
d) What is the condition for the validity of the Rayleigh-Jeans approximation to

the black-body formula? [2]
e) Given that the surface brightness S⌫ /⌦ of radiation from a black body at a

temperature Tb in the Rayleigh-Jeans part of the curve is approximately
S⌫ 2kTb
= 2
⌦
where ⌦ is the solid angle subtended by the source, show that when synchrotron
radiation from an ensemble of particles in a radio source is optically thick, the
5
spectrum is S⌫ / ⌫ 2 , whatever the spectral index of the electron energy distri-
bution. [5]
Printed: March 16, 2008 Page 1 Continued overleaf. . .

2. a) Define, carefully, the terms right-hand circular and left-hand circular as applied
to the polarization of an electromagnetic wave. [2]
b) A cold electron-proton plasma of electron density n is threaded by a uniform

magnetic field of strength B0 lying parallel to the z-axis. A right-hand circularly-
polarized, monochromatic transverse electromagnetic wave of angular frequency
! propagates along the z-axis as exp[ i!t + ikz]. Show that each electron makes
a circular orbit whose radius vector r rotates in the xy-plane along the E -vector,
with
|e|
r=+ E.
!2m e |e|!B0
[3]
c) Defining the Maxwell displacement vector as
D = ✏0 E + P = ✏✏0 E
show that the dielectric constant ✏ is given by
ne2
✏=1
me ! 2 + e!B0
and give the appropriate equation for left-hand polarization. [3]
d) Show that the plane of polarization of a linearly-polarized wave propagating

along the positive direction of the z-axis rotates at a rate
d✓ ne3 B0
=
dz 2! 2 m2e ✏0 c
[3]
e) How would this result change if the ordinary plasma were replaced, first by
the equivalent antiproton-positron plasma, and second by a cold electron-positron
ambiplasma? [2]
f) How could you make use of observations at more than one frequency to infer
a reliable intrinsic position angle for the polarization of radio emission from a
distant galaxy seen through a magnetized plasma? [3]
g) In what ways does Faraday rotation often reduce the apparent polarized in-
tensity of a distant radio source? [4]
h) A certain region of thickness L along the line of sight (the z-axis) contains a
uniform, cold electron-proton plasma of electron density n threaded by a uniform
magnetic field of strength B0 along the line of sight. It also contains a medium
which radiates linearly-polarized radio waves towards the observer, with all the
E-vectors parallel to the x-axis at emission; this emission is optically thin and has
a broad spectrum, giving a strong signal right across the radio band. There is no
emission or Faraday rotation outside this region. What is the highest frequency
at which the observer receives a completely unpolarized signal? [5]

3.
a) Without detailed calculation, show that a photon of frequency ⌫ will typically
have a frequency of order 2 ⌫ after being inverse-Compton scattered by a charged
particle whose relativistic factor is . [5]
b) The rate of loss of energy E of a charged particle (mass m, charge e) to the

inverse-Compton process is
dE 4 2 e4
= U
dt 3 6⇡✏20 m2 c3
where U is the energy density of the ambient photons . Explain in general terms,
without detailed calculation, why the dependence on the charge is as e4 and on
the ambient energy density as the first power of U . [5]
c) An electron with energy E = 10 GeV is situated in a region where the only

significant ambient photons are those of the microwave background radiation,
whose energy density is appropriate to blackbody radiation at a temperature
T = 2.7 K. Calculate the characteristic lifetime ⌧ (in years) of the electron energy
against inverse-Compton losses, where ⌧ is defined as
E
⌧ = ⇣ dE ⌘ ,
dt
and also calculate the characteristic lifetime for a 10-GeV proton in the same
circumstances. (The Stefan-Boltzmann constant = 5.7 ⇥ 10 8 W m 2 K 4 ). [5]
d) Given a population of electrons having an energy distribution such that

N (E)dE, the number with energies between E and E + dE, is given by the
power-law N (E) = E ⇠ , where  and ⇠ are constants, show that the spectrum
radiated by inverse-Compton collisions with photons of the microwave back-
ground S⌫ is also a power-law function of the frequency ⌫, and derive its spectral
index. [5]
e) A radio wave which has right-hand circular polarization propagates along

the z-axis in the negative direction and is inverse-Compton scattered by high-
energy electrons in a source region situated at the origin. What is the state of
polarization of the scattered photons as seen by an observer situated a long way
from the source on the z-axis in the positive direction? [5]
Printed: March 16, 2008 Page 3 End of paper

Radiation and Matter

Saturday 8th May, 2021 13:00 - 15:00

(May Diet)

resources.
Special Items

External Examiner: Prof L Miller

examination.
Printed: Wednesday 21st April, 2021 PHYS11020

Radiation and Matter (PHYS11020)
1.
(a) Write down Maxwell’s equations for the electric field, E, and magnetic field, B,
and give a physical interpretation for each of them. Show that they are satisfied by the
electromagnetic scalar and vector potentials defined by
.
E = −∇φ − A, and B = ∇ ∧ A,
where a dot denotes differentiation with respect to time. [5]

(b) Show that E and B are invariant under the gauge transformations
A0 = A + ∇χ, and φ0 = φ − χ,
.
where χ(r, t) is an arbitrary field. [4]

(c) What is the Lorenz Gauge? Write down the Lorenz Gauge condition and give a
physical interpretation of it. [4]
(d) The electromagnetic potential in the wavezone can be written
1
A(r, t) = F (t − r · n/c) ,
r
where r is the distance between the source and observer, and n is the unit vector pointing
from the source to the observer. Using the Lorenz Gauge
. condition show . the scalar field
is given by φ = c n · A. Hence show that E = n ∧ n ∧ A and B = −n ∧ A/c. [6]
(e) Show that the electric field can also be written as Feynman’s equation for a moving
charged particle in the wavezone,
µ0 .. 0
E= q [n ],
4π
..
where [n0 ] is the transverse acceleration of the unit vector pointing from the observer
towards the apparent (retarded) position of the particle. You may ignore terms which
drop off as v 2 /r2 or faster. [6]

2.
(a) Using the Heisenberg equation of motion for a particle, p = im[H, r]/h̄, where H
is the Hamiltonian, show that the expectation value of a particle’s momentum from an
initial state, X, to a final state, Y , is
hY |p|Xi = imωhY |r|Xi,
where ω = h̄(EY − EX ) is the frequency associated with the energy difference. Explain
the terms in this expression. [5]
(b) Fermi’s Golden Rule for the rate of transition from a state X to a state Y is
2π 2
Γ= |hY |H 0 |Xi| ρω (ω0 ),
h̄
where ρω = V /(2π)3 (ω 2 /c3 )dΩ is the density of perturbing states and H 0 is the per-
turbing Hamiltonian. Assuming an electric dipole transition for a charged particle in an
unpolarised electromagnetic field, show that the spontaneous transition rate is
4ω 3 e2
Γspon (X → Y ) = |r Y X |2 ,
3h̄c3 4π0
where r Y X = hY |r|Xi, and other terms have their usual meaning. You may consider just
emission for the photon transition. [10]
(c) A gas of two-state atoms of density n = 106 m−3 is collisionally excited at a temperature
of T = 5000 K. The energy difference between the two states is 1.38 × 10−19 J, and the
downward collisional rate is hσviUL = 50 m3 s−1 . The ratio of upper to lower population
levels is nU /nL = 0.01. What is the spontaneous dipole transition timescale for these
atoms? [10]

3.
(a) Radiation with specific intensity, Iν , propagating through a medium varies according
to the equation of radiative transfer,
dIν
= −κν Iν + Eν .
d`
Define each of the terms in this equation. [5]
(b) Using the equation of radiative transfer, show that if the medium is in thermal equi-
librium then Kirchoff’s Law,
Eν = κν B(T ),
applies, where B(T ) is the Planck function. [4]
(c) The opacity and emissivity are given by
κν = (gu nL − gL nu )Γspon c2 φν /(8πν 2 ) and Eν = gL nu Γspon hνφν /(4π).
When the population number densities, nu and nL , are in thermal equilibrium show that
Kirchoff’s Law holds. [4]
(d) If the medium is in local thermodynamic equilibrium, we can assume Kirchoff’s Law
with the local temperature of the gas depending on depth, T = T (`). Using this in the
equation of radiative transfer, find the general integral solution for the specific intensity
as a function of optical depth, Iν (τν ). Explain why this shows that the specific intensity
will always try to approach the local Planck function as it travels through a medium.
Assuming the temperature is a constant along the light path, show that this general
solution reduces to
Iµ (τν ) = Iµ,0 e−τν + (1 − e−τν )Bν (T ).
[6]
(e) An astronomer observes an optically thick cloud which emits radiation as a blackbody,
Bν (TBG ) at temperature TBG in the Rayleigh-Jeans limit. In front of it is a foreground
layer of gas with temperature TF G . The gas is in local thermodynamic equilibrium and
the optical depth of the gas is a Gaussian function in frequency, with mean frequency ν0
and width σν . Sketch the observed specific intensity when the background cloud has a
temperature of TBG = 5000K and the foreground layer is TF G = 6000K, (i) when the peak
optical depth is unity, and (ii) when it is much greater than unity. Draw the corresponding
figure when the temperatures of the background and foreground are swapped. [6]
Printed: Wednesday 21st April, 2021 Page 3 End of Paper

College of Science and Engineering

SCQF Level 11, PHYS11020
Thursday, 16th May 2013
9:30am to 11:30am
Chairman of Examiners
Prof J A Peacock
External Examiner
Prof C Tadhunter
The bracketed numbers give an indication of the value assigned

to each portion of a question.
Only the supplied Electronic Calculators may be used during this examination.
A sheet of standard constants is available for use during this examination.
Anonymity of the candidate will be maintained during the marking of

this examination.
Printed: March 20, 2013 PHYS11020

1. Fermi’s Golden Rule is
2π 0 2
w= |H | (dN/dE)f , Ef = Ei
h̄ f i
Explain what each term in this equation is. [3]
For an atom in initial state X and final state Y the dipole matrix element of H 0
for absorption of a photon directed along n within solid angle dΩ is
s
h̄ωNk,α
q ek,α .rY X
20 V
Explain what the terms in this expression represent. [3]
By considering the number density of quantized states in k-space, derive an

expression for (dN/dE)f , and hence show that, for all polarizations and angles,
the spontaneous dipole transition rate for this system can be written
4ω 3 e2
wspon = |rY X |2
3h̄c3 4π0
[6]
By assuming both atomic states X and Y can be described by hydrogenic wave-
functions, and considering the projection of the dipole operator r in spherical
polar coordinates, derive the selection rules for dipole radiation.
m m m m
l
(l−ml +1)Pl+1l +(l+ml )Pl−1 l
Pl+1l −Pl−1
[N OT E : cos θPlml = 2l+1
; sin θPlml −1 = 2l+1
) [5]
Assuming that rY X is of order the Bohr radius, calculate the transition rate wspon
for Lyman-α emission. [3]
Explain the implications of this rate for the observability of UV continuum and
Lyman-α line emission from galaxies and quasars as viewed at increasingly high
redshift back into the reionization epoch. How does the situation differ for the
Balmer-α line and why. [5]

2. Given the transition rate for photon absorption
ωdΩ e2
w(Ω)abs dΩ = N k,α |MY X (k, α)|2
2πh̄c3 m2 4π0
write down the analogous expression for photon emission, and explain clearly
what Nk,α means. [2]
Hence write down the corresponding expression for the rate of spontaneous emis-
sion, wspon , in the absence of a radiation field. [1]
Define specific intensity Iν , and show that the relationship between Iν and Nk,α
describing the same field is
2hν 3
Iν = Nk,α
c2 [5]
Hence, derive the equation of radiative transfer for a spectral line from a two-state
atom
dIν
= −κν Iν + Eν
dl
where κν = (nl − nu )wspon (λ2 /8π)φν , and explain clearly what each term in this
equation represents. [5]
What happens if i) nu = nl , and ii) nu > nl ? [2]
Define the optical depth τν and source function Sν , and show that if κν and Sν
are independent of position, the solution to the equation of radiative transfer is
Iν = Iν (0)e−τν + Sν (1 − e−τν )
[3]
Show that, if absorption and emission are due to the same process, and the source
is in thermal equilibrium and optically thick, that Iν is given by Planck’s Black
Body formula. [3]
A simplified galaxy model consists of an infinitely extended slab of finite thickness

with a uniform mixture of stars and dust. Assuming no background emission,
calculate the surface brightness observed as a function of inclination and show
that it is independent of inclination in the limit that the amount of dust becomes
very large. Specifically, let the slab have a thickness L, and let τ denote the
optical depth for a light ray passing perpendicularly through the slab. Assume
the emissivity of starlight Eν and the density of dust to be constant throughout
the slab. [3]
How does the observed flux density from the galaxy depend on inclination in this
optically thick situation? [1]

3. Show that, if one insists that Schrödinger’s equation for a free particle is invariant
to local phase changes in the wave function of the form ψ → eiα ψ (where α =
α(r, t)), one has to introduce a 4-vector potential which obeys the same local
gauge invariance transformations as the electromagnetic potential (A, φ). [4]
Hence show that, adopting the gauge where φ = 0, ∇. A = 0, the interaction

Hamiltonian for radiation and matter has the form (to first order)
H 0 = −(q/m)A.p
where A is the electromagnetic vector potential, and p is the particle momentum. [3]
Outline how, starting from the source-free classical wave equation for A, the
vector potential A can be quantized and ultimately written in the form
s
h̄ h i
ek,α ak,α (t)eik.r + a∗k,α (t)e−ik.r
X
A(r, t) =
k,α 20 V ω
and explain what the operators a and a∗ do when operating on a radiation state
with photon occupation number Nk,α . [5]
Hence justify why, when the interaction Hamiltonian is inserted into the matrix
element connecting an initial atomic state X and a final atomic state Y , the
result can be expressed as a sum of two terms, one of which is
s
h̄ q Z
Hf0 i =q Nk,α ek,α . ψY∗ eik.r v ψX dV.
20 V ω
[2]
By expanding the exponential term in this equation show that, if electric dipole
transitions are forbidden, transitions can still be allowed by either the magnetic
dipole or electric quadrupole operators, and estimate how the rates for these
“higher order” transitions compare with those produced by electric dipole tran-
sitions. [6]
What kind of transition produces an emission or absorption line at λ = 21cm

from neutral Hydrogen atomic gas? The transition rate wspon (21cm) is only
2.85 × 10−15 s−1 . Give two reasons why this transition is so unlikely (compared
with, say, Lyman-α emission) and explain why the 21cm line is still an important
and useful spectral feature in modern astrophysics. [5]
Printed: March 20, 2013 Page 3 End of paper

N I V ER
U S
IT
TH
Y
0131 668 8477
O F
H
G
E
R
D I U
N B

Friday 15th May, 2015 14:30 – 16:30

(May Diet)
• Only the supplied Electronic Calculators may be used during this examination.
Special Items
• School supplied calculators


External Examiner: Prof. C Tadhunter

examination.
Printed: Thursday 23rd April, 2015 PHYS11020

1. Outline how, starting from the source-free classical wave equation, the vector potential,
A, can be quantized and ultimately written in the form
s
h̄ h i
ek,α ak,α (t)eik.r + a∗k,α (t)e−ik.r
X
A(r, t) =
k,α 20 V ω
[4]
It can be shown that the operators a and a∗ are analogous to the “ladder” operators c, c† ,
which can be defined in terms of q (the displacement) and p (the corresponding canonical
momentum) as follows:
c = (2h̄ω)−1/2 (ωq + ip) c† = (2h̄ω)−1/2 (ωq − ip).
Given the standard value for the commutator between position and momentum, [q, p] =
ih̄, and the simple harmonic oscillator Hamiltonian H = (1/2)(p2 + ω 2 q 2 ), show that
[c, c† ] = 1, and H = (h̄ω/2)(cc† + c† c). [4]
Thus show that c is an “annihilation” operator which, operating
√ on state N , produces
a new state of energy h̄ω lower, which is amplified by N . Show, similarly, that c†
is
√ a “creation” operator which produces a new state of energy h̄ω higher, amplified by
N + 1. [6]
Hence, given the transition rate for photon absorption
ωdΩ e2
w(Ω)abs dΩ = Nk,α |MY X (k, α)|2
2πh̄c3 m2 4π0
write down the analogous expression for photon emission, and explain clearly what Nk,α
means. [2]
Define specific intensity Iν , and show that the relationship between Iν and Nk,α describing
the same field is
2hν 3
Iν = 2 Nk,α
c
[5]
The wavelength of Lyman-α emission is 1216Å, and the rate of spontaneous emission is
wspon = 6.3 × 108 s−1 . Calculate the specific intensity of the radiation field at which the
rate of Lyman-α absorption equals the rate of spontaneous Lyman-α emission. [4]
Printed: Thursday 23rd April, 2015 Page 1 Continued overleaf. . .

2. Outline, with mathematical illustration, the steps to obtain Fermi’s Golden Rule
2π 0 2
w= |H | (dN/dE)f , Ef = Ei
h̄ f i
explaining clearly the meaning of (dN/dE)f , and how the requirement for energy conser-
vation arises. [8]
When the interaction Hamiltonian for radiation and matter is inserted into the matrix
element, the result can be expressed as a sum of two terms, one of which is
s
h̄ q Z
Hf0 i = q Nk,α ek,α . ψY∗ eik.r v ψX dV.
20 ωV
Explain how this expression can be simplified further by the dipole approximation, stating
clearly when this approximation can be used. [2]
Outline why the next “higher-order” term can be subdivided into electric quadrupole and
magnetic dipole transitions, and estimate how the rates of such transitions compare with
those produced by electric dipole transitions. [5]
Explain why, for a single electron transition in a hydrogen atom, the dipole approximation
leads to the selection rules δl = ±1, δml = 0, ±1. [4]
Hence explain and write down the selection rules for the quantum number J describing a
multi-electron atom in the case of i) dipole radiation, and ii) electric quadrupole radiation. [2]
Why are dipole-forbidden transitions rarely seen in the lab, but so prevalent and important
in astrophysics? Give an example of such a transition, and briefly discuss its astrophysical
importance. [4]
Printed: Thursday 23rd April, 2015 Page 2 Continued overleaf. . .

3. The rotational energy levels of the CO molecule have energies proportional to J(J + 1)
where J is the angular momentum quantum number.
Briefly explain why CO emission is generally only observed for transitions between ad-
jacent rotational energy levels (i.e. CO 1–0, CO 2–1, CO 3–2, etc) and, given that the
emission wavelength of the CO 1–0 transition is 2.6 mm, calculate the wavelengths of the
CO 2–1 and CO 3–2 transitions. [3]
Define collision cross section σ, collision rate coefficent hσvi (m3 s−1 ), and critical density
ncrit . [3]
Prove that, in a gas of two-level atoms, with a Maxwellian velocity distribution, the rate
coefficient for upward collisions is equal to that for downward collisions multiplied by the
Boltzmann ratio of level occupancy. [3]
Hence show that the upper level occupancy can be written as
nBoltz
u
nu =
1 + ncrit /n
where
Eu −El
nBoltz
u = nl (gu /gl )e− kT
explaining clearly the meaning of all the terms in this equation. [4]
The emission from the J → J − 1 transition of CO is proportional to the number of
molecules in the upper state. Show that in a cloud where the levels are excited by
collisions with molecular hydrogen, the critical density increases monotonically with the
level being excited. You may assume the dipole of CO is approximately independent of
J, that the cross section is proportional to the dipole squared, and that the spontaneous
transition rate is
4ω 3 e2
wspon = 3
|r|2 .
3h̄c 4π0
[5]
One of the CO emission lines from a molecular cloud is observed to have a full width at
half maximum of 1 kms−1 and a peak brightness temperature of 30 K. Assuming the line
has a Doppler broadened profile, work out its expected line width if it was optically thin,
and hence estimate the optical depth at the centre of the line. [5]
What problems does this present for estimating the gas mass of a molecular cloud or a
galaxy from CO line emission? [2]
Printed: Thursday 23rd April, 2015 Page 3 End of Paper

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Monday 2nd May, 2016 2:30pm – 4:30pm

(May Diet)

Special Items

Chairman of Examiners: Prof J Dunlop

External Examiner: Prof C Tadhunter

examination.

1. (a) Maxwell’s equations may be written as

∇ ∧ E = −Ḃ ∇·B=0
∇ · E = ρ/0 ∇ ∧ B = µ0 (j + 0 Ė).
Show that the electric and magnetic fields can be derived from a vector and a scalar
potential as follows:
B = ∇∧A E = −∂A/∂t − ∇φ.
[5]
(b) Explain what is meant by a gauge transformation of the potentials, and say why such
a transformation is possible. Show that a given pair of potentials (φ0 , A0 ) can always be
transformed into new potentials that obey ∇ · A + (1/c2 )∂φ/∂t = 0. [5]
(c) In this gauge, the solution for the vector potential can be written as
µ0 Z [j(r0 , t)] 3 0

A(r, t) = d r.
4π |r − r0 |
Explain the meaning of the square brackets inside the integral, and show that the field
measured by a sufficiently distant observer satisfies
n ∂
∇∧A = − ∧ A,
c ∂t
where n is a unit vector pointing from source to observer. Using one of Maxwell’s equa-
tions, show that the fields for an oscillating source are transverse, with E, B, n all mutually
perpendicular. [5]
(d) For a charge q with velocity v, the volume integral of j is qv. Hence determine the
E and B fields from an accelerated charge as measured by a sufficiently distant observer,
and show that the rate of loss of energy is α|v̇|2 , where α is a constant that need not be
explicitly evaluated. [5]
(e) A charge is constrained to undergo one-dimensional harmonic motion with angular
frequency ω. Discuss conservation of energy and show that the time-averaged radiation
emitted can be accounted for if the particle experiences an additional ‘radiation reaction’
force F = α d2 v/dt2 . [5]

2. (a) The classical vector potential can be written as a Fourier superposition:

C(ω) ek,α ak,α (t)eik·r + a∗k,α (t)e−ik·r ,
X
A(r, t) =
k,α
where C(ω) is a normalization factor. Explain the meaning of all quantities in this ex-
pression. [5]
(b) If the electromagnetic field is described with zero scalar potential, compute the con-
tribution of the electric field to the Hamiltonian within a box of volume V , and show that
it is
1 X
HE = ak,α a∗k,α + a∗k,α ak,α h̄ω,
4 k,α
provided |C(ω)|2 = h̄/(20 V ω). Show that the electric and magnetic energy densities are
equal for each mode, so that the total Hamiltonian is 2HE . [5]
(c) In quantum electrodynamics, the coefficients a, a∗ become operators a, a† , with the
commutator [a, a† ] = 1. Show that H = modes (N +1/2)h̄ω, where the operator N = a† a.
P
Given eigenstates N |ni = n |ni, show that a and a† act as lowering and raising operators
and derive the effect of the operators on the normalization of the states. Prove that n
cannot be negative and hence that it must take integer values, n = 0, 1, 2, · · · [5]
(d) According to the Golden Rule, the semiclassical transition rate between two states
can be written as
ω dΩ e2
Γ= 3 2
Nk,α |MYX (k, α)|2 .
2πh̄c m 4π0
Explain the meaning of the terms in this expression, and say how it is modified in the
fully quantum case, especially considering spontaneous transitions. [3]
(e) An electron is confined within an infinitely high cubical potential barrier of side L,
and is placed in one of the first excited states. Use the dipole approximation to calculate
the rate at which the electron makes spontaneous radiative transitions to the ground state
(you may assume that 0π θ sin θ sin 2θ dθ = −8/9).
R
[7]

3. (a) Define specific intensity of radiation, Iν . For unpolarized radiation, show that specific
intensity is related to photon occupation number, N , via
Iν = (4πh̄ν 3 /c2 ) N.
[5]
(b) In the case where radiation passes through material with two non-degenerate energy
levels, derive the equation of radiative transfer in the form dIν /d` = −κν Iν + Eν . Discuss
the contribution of stimulated and spontaneous transitions to the opacity and emissivity.
Under what circumstances will Iν be constant along light rays? [7]
(c) In intergalactic space, clouds of Hydrogen are inferred to have a high degree of ioniza-
tion. Explain why the ionization arises, and why the temperature is therefore expected
to be of order T = 104 K. Calculate the column density of neutral Hydrogen that would
yield an optical depth of unity at the line centre for gas at this temperature (assume a
Gaussian line profile; central wavelength of Ly α is 121.5 nm; the spontaneous transition
rate associated with the Lyman α line is 6.3 × 108 s−1 ). [8]
(d) The spontaneous transition rate associated with the 21-cm line is 2.85 × 10−15 s−1 .
Calculate the expected peak brightness temperature of 21-cm emission from the cloud in
part (c). [5]

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Tuesday 2nd May, 2017 14:30 – 16:30

(May Diet)

Special Items


External Examiner: Prof. C Tadhunter

examination.
Printed: Tuesday 18th April, 2017 PHYS11020

1. (a) Write down Maxwell’s equations, and explain how two of these equations lead to a
re-expression of the electric and magnetic fields in terms of vector and scalar potentials,
A and φ. [3]
(b) Explain what is meant by gauge invariance in electromagnetism, and show how it can
be used to enforce the Lorentz condition between A and φ. [3]
(c) Suppose that the electromagnetic fields are to be described in terms of a superpotential,
Z, where A = (1/c2 )Ż and φ = −∇ · Z; show that the Lorentz condition is automatically
satisfied in this case. If Z is assumed to satisfy the wave equation Z = −(1/0 ) P, show
that the usual wave equations for A and φ can be obtained, and derive the necessary
relations between the vector P and the charge and current densities. [6]
(d) Write down the general solutions to the electromagnetic wave equations for the po-
tentials, and explain how these apply in the case that the source is a charge q, with
non-relativistic velocity v. Consider only a distant source in the radiation zone. Explain
carefully what is meant by [ρ] and [j], the retarded values of the charge density and cur-
rent density, and hence explain why the volume integral of [ρ] only reduces to q in the
nonrelativistic limit. [5]
(e) A quadrupole source consists of two charges, q, separated by a vector d, with equal and
opposite velocities, oscillating with angular frequency ω. Calculate the vector potential in
the radiation zone at large radii and show that its amplitude is ωn · d/c times the dipole
field of a single charge. Hence obtain the angular distribution of the radiated power from
the quadrupole. [8]
Printed: Tuesday 18th April, 2017 Page 1 Continued overleaf. . .

2. (a) The quantum vector potential can be written as

s
h̄
ek,α ak,α (t)eik·r + a†k,α (t)e−ik·r ;
X
A(r, t) =
k,α 20 V ω
write down the meaning of the terms in this expression. [3]

(b) If the operators a and a† for a given mode have the commutator [a, a† ] = 1 and
the Hamiltonian is i (a†i ai + 1/2)h̄ωi summed over all modes, prove that the energy
P
P
eigenvalues are E = i (ni + 1/2)h̄ωi , where the ni are independent non-negative integers.
[6]
(c) The theory of supersymmetry proposes that the photon has a Fermion counterpart
that is identical in properties, except that commutators would be replaced by anti-
commutators: aa† + a† a = 1. If eigenstates |ni are defined via a† a |ni = n |ni, show
that n is an integer that obeys the exclusion principle, so that n = 0 or n = 1 are the
only allowed values. [6]
(d) If phase in quantum mechanics is to be locally unobservable, show that this requires
a first-order perturbation to the Hamiltonian of a charged particle: ∆H = −(q/m)A · p
in the gauge where φ = 0. Explain why the full perturbation allows the simultaneous
emission or absorption of up to two photons. [4]
(e) According to the Golden Rule, the spontaneous decay rate from an excited state |Yi
to a lower state |Xi is
ω dΩ e2
Γ= 3 2
|MYX |2 .
2πh̄c m 4π0
Write down the matrix element MYX in the dipole approximation and give an expression for
the lifetime of the m = 0 2p level in the Hydrogen atom [wavefunction (32πa5 )−1/2 z exp(−r/2a)]
decaying
R∞ 4
to the ground state [wavefunction (πa3 )−1/2 exp(−r/a)]. You may assume that
0 y exp(−3y/2) dy = 256/81. [6]

3. (a) The twice-ionized OIII ion has two outer-shell electrons in the 2p state. By factorising
the wave function into a function of total spin and a function of total angular momentum,
show that the three possible spectroscopic terms are 3 P , 1 D and 1 S. [4]
(b) According to Hund’s rule, which of these is the ground state, and why? Spectroscopic
selection rules for permitted transitions in this case require zero change in spin; thus
explain why the transitions between 1 D & 1 S (4363Å) and 3 P & 1 D (4959/5007Å) are
both forbidden lines. [4]
(c) The above transitions are observed in nebulae as a result of collisional excitation.
Discuss the operation of this mechanism, explaining in particular the reciprocity relation
obeyed by collisional cross-sections, and the concept of a critical density. Explain why
the line emissivity scales in proportion to ion density at high densities, but in proportion
to the square of the density at low densities. [8]
(d) The de-excitation cross-section affecting the 4959/5007Å transition is 8 times larger
than the de-excitation cross-section affecting the 4363Å transition. Hence derive an ex-
pression for the ratio of emissivities in these two lines in the limit of low densities. At
what temperature are they in the ratio 100:1? [6]
(e) Explain how the calculation of part (d) would differ in the case of high densities. In
particular, how does the line emissivity ratio now depend on temperature? [3]
Printed: Tuesday 18th April, 2017 Page 3 End of Paper


Wednesday 16th May, 2018 14:30 – 16:30

(May Diet)

Special Items

Chairman of Examiners: Prof A. Trew

External Examiner: Prof L. Miller

examination.

1. (a) The electric field in a plane polarized electromagnetic wave of angular frequency ω
may be assumed to be transverse; use a Maxwell equation to derive the relation between
the electric and magnetic fields in the wave. The wave accelerates a free electron, of
charge e and mass m: show that the acceleration is normally dominated by the electric
field. Under what circumstances would this not be true? [5]
(b) The radiated electromagnetic fields from an accelerated charge scale in proportion to
a sin α/R, where R is distance from the charge, a is the acceleration, and α is the angle
between the direction of acceleration and the direction to the observer. Hence explain
how the electron can be said to scatter radiation, and argue that the direction-dependent
scattering cross-section is
dσ
= γ σT sin2 α,
dΩ
where Ω denotes solid angle and σT is the total (Thomson) cross-section. Evaluate the
dimensionless constant γ. [5]
(c) Now consider an electron placed in a damped harmonic potential, obeying the equation
of motion
e
ẍ + Γẋ + ω02 x = E.
m
Show that the total scattering cross-section for this case is
ω4
σ(ω) = σT .
(ω 2 − ω02 )2 + Γ2 ω 2
Discuss the form of this expression at (i) very high frequencies; (ii) very low frequencies;
(iii) ω close to ω0 (assume Γ ω0 ). [6]
(d) The Lyman-α transition (121.5 nm) has an absorption cross-section identical to that
of a damped oscillator, but lower by a factor 0.42. Explain how the damping parameter
Γ relates to the lifetime of the Lyman-α excited state (1.59 × 10−9 s). Hence deduce
the column density of neutral Hydrogen that would yield optical depth unity at the line
centre (σT = 6.65 × 10−29 m2 ). A high-redshift quasar shows a broad Lyman-α absorption
feature with rest-frame width 3 nm FWHM: deduce the foreground column density of
neutral Hydrogen. You may neglect Doppler broadening. [9]

2. (a) For the case of two spins, j1 = j2 = 1/2, what are the allowed values of the total spin
quantum number, S? Assuming that the (S, m) = (1, 1) state must factorise as | ↑1 i| ↑2 i,
use the total lowering operator to deduce the other m states, and show that they are
all symmetric. The total raising and lowering operators must act on the S = 0 state to
give zero. Hence prove that the singlet state is antisymmetric. The raising and lowering
operators j± = jx ±ij
q y
can be assumed to have the following effect on angular momentum
states: j± |j, mi = (j ∓ m)(j ± m + 1)h̄|j, m ± 1i. [5]
(b) The 21-cm hyperfine transition of neutral Hydrogen is between an excited S = 1 state
and the ground S = 0 state of the combined electron and proton spins. The spontaneous
decay rate is Γspon = 2.85 × 10−15 s−1 . Radiation propagates through Hydrogen gas
containing this 2-level system. Derive the equation of radiation transfer, and show that
it involves an opacity
κ = (gU nL − gL nU )Γspon λ2 φν /8π
and an emissivity
E = gL nU Γspon hνφν /4π.
Explain the meaning of all terms in these expressions. [9]
(c) Assume that the level populations are in equilibrium at a gas temperature Tg and
that the incident radiation is black-body radiation at temperature Tr . Use the above
expressions for κ and E to show that the change in radiation specific intensity vanishes if
the gas and radiation are at the same temperature. [5]
(d) It the gas temperature is Tg 0.07 K, show that the optical depth of an absorbing
Hydrogen cloud is inversely proportional to its temperature:
3hc2 Γspon
τ= NH φν .
32πνkTg
[3]
(e) An HI cloud has a line-of-sight thickness of 10 pc. At what density would the cloud
be optically thick to 21-cm radiation if its temperature were 50 K? [3]

3. (a) According to the Golden Rule, the semiclassical transition rate between two non-
degenerate energy levels can be written as
ω dΩ e2
Γ= Nk,α |MYX (k, α)|2 .
2πh̄c3 m2 4π0
Explain the meaning of the terms in this expression, and say how it is modified if the
levels are degenerate. In the fully quantum case, Nk,α would be replaced by Nk,α + 1 for
an emission process. Explain how Einstein was able to derive this result without using
quantum electrodynamics. [6]
(b) Explain the concept of collisional excitation, with reference to a 2-level atom. Define
the two collisional cross-sections, and also say what is meant by the collision time and
mean free path. [5]
(c) By considering statistical equilibrium of transition rates, derive the reciprocity relation
between the two collisional cross-sections, and hence show that the relation between the
number densities of particles in the upper and lower levels can be written in the following
form:
nU (gU /gL )e−x EU − EL
= , x= ;
nL 1 + ncrit /n kT
give the definition of the critical density ncrit . [5]
(d) Once-ionized Sulphur, S + , has three 2p electrons in its outer shell. The resulting
electronic structure produces a doublet at 671.6 and 673.2 nm, between a 2 D excited state
and a 4 S ground state. The upper level splits according to the total angular momentum
quantum number, with the larger J giving a lower energy. Explain briefly how this result
relates to the phenomenon of spin-orbit coupling. [4]
(e) If the spontaneous decay rate of the 671.6 nm transition is three times that of the
673.2 nm transition, show that the intensity ratio in the lines is I671.6 /I673.2 = 2 above
the critical density. What is the ratio below the critical density? [Hint: the collisional
excitation rate may be assumed not to depend on energy of the upper level]. [5]


Wednesday 8th May, 2019 09:30 - 11:30

(May Diet)

• Attach supplied anonymous bar codes to each script book used.
Special Items

Chairman of Examiners: Prof. P N Best

External Examiner: Prof L. Miller

examination.
Printed: Monday 15th April, 2019 PHYS11020

1. (a) Write down Maxwell’s equations for the electric and magnetic fields. Combine these
to derive the continuity equation, and hence show that charge conservation is guaranteed
in Maxwell’s electrodynamics. [4]
(b) Assuming the vector identity ∇ · (E ∧ B) = B · (∇ ∧ E) − E · (∇ ∧ B), show that
electromagnetic energy is conserved in free space (ρ = 0; j = 0), and that the Poynting
vector S = (E ∧ B)/µ0 represents the energy flux density. [4]
(c) Consider a free electron interacting with a beam of linearly polarized radiation of
angular frequency ω. Use a Maxwell equation to establish the relation between E and
B fields in the radiation (which can be assumed to be transverse), and hence show that
acceleration due to the magnetic field can be neglected provided the electron motion is
nonrelativistic. The vector potential from the electron at a large distance, r, is
A = −µ0 e [v] /4πr.
Explain the distinction between [v] and v, and show how it allows the magnetic field to
be expressed in terms of the electron acceleration. [6]
(d) Hence show that the power emitted by the accelerated electron can be expressed in
terms of a angle-dependent cross-section: dσ/dΩ = (3/8π) σT (1 − (n · e)2 ), where e is
the direction of polarization and n is the direction of the emitted radiation. Express σT
in terms of fundamental constants. If the incident radiation is unpolarized, show that the
angular dependence of the cross-section is now proportional to (1 + cos2 θ), where θ is the
angle between incident and emitted radiation. [6]
(e) Now consider polarized radiation with angular frequency ω that propagates through a
plasma where the number density of free electrons is n. Show that the radiation induces
a current density of
e2 n ∂E
j=− 2 .
ω m ∂t
Consider how this modifies the Maxwell equation for ∇ ∧ B and hence show that the phase
velocity of radiation in the plasma is c (1 − ωP2 /ω 2 )−1/2 . Give an expression for the plasma
frequency, ωP . [5]
Printed: Monday 15th April, 2019 Page 1 Continued overleaf. . .

2. (a) The quantum vector potential can be written as

s
h̄
ek,α ak,α (t)eik·r + a†k,α (t)e−ik·r .
X
A(r, t) =
k,α 20 V ω
Write down the meaning of the terms in this expression. [3]

(b) The first-order perturbation to the Hamiltonian of an electron in an electromagnetic
field is H 0 = −(e/m)A · p, where p is the momentum operator. Explain how this pertur-
bation arises as a consequence of gauge invariance. [6]
(c) The golden rule states that the radiative transition rate between a pair of levels with
energy separation h̄ω0 is Γ = (2π/h̄2 )|Hf0 i |2 δ(ω − ω0 ). The wave function for the combined
system of radiation+electron can be written as a product state, |N i ψ(x). What are the
effects of the operators a and a† on the radiation part of the state? Hence show that the
matrix element for photon absorption is
s
−e h̄ q
Hf0 i = Nk,α Mf i (k, α),
m 20 ωV
where Z
Mf i (k, α) ≡ ek,α · ψf∗ eik·r p ψi dV.
What is the corresponding matrix element for photon emission? [5]
(d) The form of the golden rule in part (c) considers exactly monochromatic radiation.
Explain how to calculate the transition rate for broad-band radiation, which for the case
of absorption is
ω dΩ e2
Γ= 3 2
Nk,α |Mf i (k, α)|2 .
2πh̄c m 4π0
[5]
(e) Explain what is meant by the dipole approximation. Now consider the sequence of
hydrogen-like ions consisting of one electron and a nuclear charge of Ze. Show that the
dipole approximation is not valid if Z is beyond a critical value [you may use Bohr-atom
arguments to estimate the characteristic size and energy levels of the ions]. [6]
Printed: Monday 15th April, 2019 Page 2 Continued overleaf. . .

3. (a) Define the specific intensity of radiation, Iν , in terms of the rate of transport of energy
through a measuring aperture. Use the concept of density of states to show that, for
unpolarised radiation, the specific intensity is related to the photon occupation number,
N , via Iν = (2hν 3 /c2 ) N . [4]
(b) Consider an atom with non-degenerate lower and upper energy levels L and U, and let
Γs dΩ/4π be the spontaneous decay rate of the upper level into solid angle dΩ. Consider
an element of length d` along a beam of radiation with this solid angle, and show that
spontaneous decays within the beam cause a change in specific intensity of
dIν /hν = nU d` (Γs /4π) φν ,
where nU is the number density of atoms in the upper state. The function φν represents the
line profile of the transition: say how it is normalised, and give two physical mechanisms
that can contribute to the shape of the function. [8]
(c) If the photon occupation number is N , stimulated processes proceed at a rate N times
the spontaneous transition rate. Use this fact to include stimulated processes in the result
of part (b) and hence derive the equation of radiative transfer, showing that the opacity
and emissivity are respectively
κν = (nL − nU )Γs (λ2 /8π)φν
and
Eν = nU Γs (hν/4π)φν ,
where λ is the wavelength of the radiation. [6]
(d) Black-body radiation with temperature T shines through a partially ionized hydrogen
gas with the same temperature. Consider the following possible cross-sections: bound-free
scattering, free-free scattering, and Thomson scattering. The respective cross-sections are
approximately
σbf = 2.0 × 10−18 (h̄ω/eV)−3 m2
σff = 2.6 × 10−45 (h̄ω/eV)−3 (n/m−3 ) (T /K)−1/2 m2
σT = 6.7 × 10−29 m2 ,
where n is the number density of free electrons. Which of the three processes dominates
the opacity when the gas is neutral? If, instead, the gas is 50 percent ionized, determine
the temperature and density for which all three processes make an approximately equal
contribution to the opacity [hint: consider a typical photon energy, and neglect any
detailed averaging over the spectral shape of the radiation]. [7]
Printed: Monday 15th April, 2019 Page 3 End of Paper


Thursday 7th May, 2020 13:00 - 15:00 BST

(May Diet)
• This is an open-book examination.

• Electronic Calculators may be used during this examination.
Special Items


examination.

1.
(a) Write down Maxwell’s equations for the electric field, E and magnetic field B. Show
that they are satisfied by the electromagnetic scalar and vector potentials defined by
.
E = −∇φ − A, and B = ∇ ∧ A,
where a dot denotes differentiation with respect to time. [5]

(b) Derive the wave equations for the potentials φ and A, and define the Lorenz gauge.
You may use the vector identity ∇ ∧ ∇ ∧ A = ∇(∇.A) − ∇2 A. [5]
(c) The solution for the vector potential wave equation in the Lorenz gauge is
µ Z 3 0 j(r 0 , tret )
A(r, t) = dr
4π |r − r 0 |
where tret = t − R/c and R = |r − r 0 |. Explain why for a distant source we can express
A in the form
1 r.n

A(r, t) = F t − ,
R c
where n is a unit vector pointing from the source to observer. [5]
(d) Show that for a distant source
1 . .
B = − n ∧ A, and E = n ∧ n ∧ A.
c
[5]
(e) A particle with charge e, travelling at a velocity v is irradiated by a distant source
behind it. Given the particle feels a Lorentz force, F = e(E + v ∧ B) from the radiation,
and the particle is in the wavezone, show that the radiated power from the particle is
e2
2
v⊥

P = n − E 2,
6π0 c3 c
where v ⊥ is the velocity component of the particle perpendicular to B. You may assume
the emitted power is P = (e2 /6π0 c3 )a2 , where a is the acceleration of the particle. [5]

2.
(a) In quantum electrodynamics the electromagnetic vector potential can be written as
s
h̄
ek,α ak,α (t)eik·r + a†k,α (t)e−ik·r .
X
A(r , t) =
20 V ω
k,α
Describe the physical set-up of the field this expression describes, and explain the meaning
of the terms in this expression. [5]
(b) Assuming a gauge choice where the scalar potential vanishes, φ = 0, show that the
electric field contributes a term
1X
HE = h̄ω(an a†n + a† an )
4 n
to the total Hamiltonian of the field, where n = (k, α). [4]

(c) The quantum operators a and a† for each mode have the commutator relation [a, a† ] =
1. Show that the total Hamiltonian can be written
h̄ω(a†n an + 1/2).
X
H=
n
†
Show that
√ a and a lower† and raise
√ the number eigenstate of the system by one, so that
a|ni = n|n − 1i and a |ni = n + 1|n + 1i, where n is a positive integer. Show that
P
H = n h̄ω(n + 1/2). Why is the term with a factor 1/2 problematic? [5]
(d) Derive Fermi’s Golden Rule for the transition rate,
2π
Γ= |huf |H 0 |ui i|2 δD (ω − ω0 )
h̄2
between two quantum states when the initial state undergoes a small perturbation de-
scribed by a change in Hamiltonian, H = H0 + H 0 , and |ui is the time-independent part
of the wavevector. [6]
(e) The spontaneous decay rate from an excited state, |Xi, to a lower state, |Y i, in the
dipole approximation, is
4ω 3 e2
Γspon (X → Y ) = |r Y X |2 ,
3h̄c3 4π0
where r Y X = hY |r|Xi. The rotationally excited CO molecule spontaneously decays from
the J = 1 state (X) to the J = 0 state (Y ). Derive the expression for this decay rate.
Estimate the decay time if the emitted radiation has wavelength λ = 2.6mm and the
average separation
q of the molecules
q is 0.023Å. (The lowest
q order spherical harmonics are
±iφ
Y0,0 (Ω) = 1/4π, Y1,±1 = ± 3/8π sin θe and Y1,0 = 3/4π cos θ.) [5]

3.
(a) Define specific intensity of radiation, Iν . Show that specific intensity is related to
photon occupation number, Nk,α , via
4πh̄ν 3
Iν = Nk,α
c2
for unpolarized radiation. [5]
(b) Show that for black-body radiation in thermal equilibrium with matter at temperature
T , the photon occupation number is Nk,α = (ehν/kT − 1)−1 , and hence derive Planck’s
formula for the specific intensity, Bν (T ). [4]
(c) Derive the equation of radiative transfer
dIν
= −κν Iν + Eν ,
d`
clearly defining κν and Eν . Consider radiation propagating through a medium with two
non-degenerate states. Under what conditions will the medium be transparent? Explain
what maser emission is, and under what conditions it will occur. [6]
(d) Explain what the collisional cross-section, σ, the collision coefficient, hσvi, and the
critical density, ncrit , are for a two-level atom. [5]
(e) By considering thermal equilibrium of transition rates, when n ncrit , derive the
reciprocity relation between the two collisional cross-sections, and hence show that the
number density of particles in an upper level is:
nL (gU /gL )e−∆E/kT

nU = ,
1 + ncrit /n
where ∆E = EU − EL . What is the Source Function for this medium? [5]

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School of Physics & Astronomy

Radiation Processes in Astrophysics

Monday 12th December, 2022 13:00 – 16:00

Please read full instructions before commencing writing.

Examination Paper Information

Answer ALL questions

• Only authorised Electronic Calculators may be used during this examination.

• School supplied Constant Sheets

Chairman of Examiners: Prof K Rice

Anonymity of the candidate will be maintained during the marking of this

Printed: Friday 3rd February, 2023 PHYS11067

Printed: Friday 3rd February, 2023 Page 1 Continued overleaf. . .

where is the Lorentz factor of the electron. [5]

Printed: Friday 3rd February, 2023 Page 2 Continued overleaf. . .

Define the terms in this expression. [4]

Printed: Friday 3rd February, 2023 Page 3 Continued overleaf. . .

(e) An astronomer observes a collisionally excited molecular line emitted by a spherical

Printed: Friday 3rd February, 2023 Page 4 End of Paper

Radiation Processes in Astrophysics

Wednesday 15th December, 2021 13:00 – 16:00

Please read full instructions before commencing writing.

Examination Paper Information

Answer ALL questions

• A sheet of physical constants is supplied for use in this examination.

• School supplied Constant Sheets

Chairman of Examiners: Prof P Best

Anonymity of the candidate will be maintained during the marking of this

Printed: Monday 6th December, 2021 PHYS11067

1. (a) By considering the 4-momentum of a photon travelling at an angle ✓ to the x-axis in

Printed: Monday 6th December, 2021 Page 1 Continued overleaf. . .

Printed: Monday 6th December, 2021 Page 2 Continued overleaf. . .

Printed: Monday 6th December, 2021 Page 3 Continued overleaf. . .

4. (a) Derive the equation of radiative transfer

Printed: Monday 6th December, 2021 Page 4 End of Paper

High Energy Astrophysics

Wednesday 12th May, 2021 13:00 - 15:00

Please read full instructions before commencing writing.

Examination Paper Information

Answer TWO questions

• A sheet of physical constants is supplied for use in this examination.

• School supplied Constant Sheets

Chairman of Examiners: Prof P Best

Anonymity of the candidate will be maintained during the marking of this

Printed: Wednesday 21st April, 2021 PHYS11013

Printed: Wednesday 21st April, 2021 Page 1 Continued overleaf. . .

Printed: Wednesday 21st April, 2021 Page 2 Continued overleaf. . .

Printed: Wednesday 21st April, 2021 Page 3 End of Paper

High Energy Astrophysics

Friday 15th May, 2020 13:00 – 15:00 BST

Please read full instructions before commencing writing.

Examination Paper Information

Answer TWO questions

• This is an open-book examination.

• School supplied Constant Sheets

Chairman of Examiners: Prof. P Best

Anonymity of the candidate will be maintained during the marking of this

Printed: Friday 10th April, 2020 PHYS11013

Printed: Friday 10th April, 2020 Page 1 Continued overleaf. . .

Printed: Friday 10th April, 2020 Page 2 Continued overleaf. . .

where m0 is the electron rest-mass. [5]