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UNIVERSITY COLLEGE LONDON
EXAMINATION FOR INTERNAL STUDENTS
MODULE CODE : SPCE0005
ASSESSMENT : SPCE0005
PATTERN
MODULE NAME : Space plasma and magnetospheric physics
LEVEL : Postgraduate
DATE : 18 May 2021
TIME : 14:30
TIME ALLOWED : 3 hrs
This paper is suitable for candidates who attended classes for this mod_xfffe_ule in the following academic year(s):
Year
2020/21
EXAMINATION PAPER CANNOT BE REMOVED FROM THE EXAM HALL. PLACE EXAM
PAPER AND ALL COMPLETED SCRIPTS INSIDE THE EXAMINATION ENVELOPE
Hall Instructions .
Standard Calculators Y
Non-Standard Calculators N
SPCE0005/2020-21 1/5 CONTINUED
Answer ALL questions.
The numbers in square brackets show the provisional allocation of
maximum marks per question or part of question.
Equations of magnetohydrodynamics (MHD)
ρ: mass density, u: bulk velocity, P: thermal pressure, B: magnetic field
Continuity equation:
ρ
t + · (ρu) = 0
Momentum equation:
ρ
u
t + ρ(u · )u = P +
(B · )B
μ0
_x0012_ B2
2μ0

Useful vector identities in spherical coordinates
In the following vector identities, f denotes any scalar function, and A and B denote
any vector functions of the spherical coordinates r, θ and φ.
( f)r =
f
r ; ( f)θ =
1
r
f
θ ; ( f)φ =
1
r sin θ
f
φ
(A · B)φ = Ar
Bφ
r +
Aθ
r
Bφ
θ +
Aφ
r sin θ
Bφ
φ +
AφBr
r
+
cotθAφBθ
r
Table of constants
Proton mass mp 1.673 × 10 27 kg
Electron mass me 9.109 × 10 31 kg
Electron charge e 1.602 × 10 19 C
Electron volt eV 1.602 × 10 19 J
Boltzmann constant kb 1.381 × 10 23 J K 1
Vacuum permittivity 0 8.854 × 10 12 F m 1
Magnetic permeability μ0 4π × 10 7 H m 1
Solar radius R 6.96 × 108 m
Earth radius RE 6.371 × 106 m
Mercury radius RM 2.440 × 106 m
Mercury’s mean distance from the Sun rM 0.39 au
Saturn’s mean distance from the Sun rS 9.54 au
Astronomical unit au 1.496 × 1011 m
SPCE0005/2020-21 2/5 CONTINUED
Question 1.
(a) The Debye length in an electron-proton plasma is given by the expression
λ
2
D =

0kbTe
nee
2
,
where the electron density and temperature are given by ne and Te respectively, and e
is the electronic charge. Explain the two key concepts of quasi-neutrality and collec tive behaviour and provide expressions involving the Debye length that describe these
concepts mathematically. [4]
(b) To measure space plasma, micro-satellites (with typical dimensions around 1 m)
are being used more frequently. To measure the electric field, long wires are used that
are electrically insulated from the rest of the spacecraft and in direct contact with the
ambient plasma. You may assume that the micro-satellites become positively charged
and treat them as point sources. What lengths of wires are required for the wire tips to
be shielded from 90% of the spacecraft potential if the micro-satellites are
i. in the ring current, where the temperature is 3 000 eV and the density is 106 m 3
.
ii. in the solar wind at 1 au, where the temperature is 5 eV and the density is 107 m 3
Discuss whether this is a practical approach for each of these space plasmas. [6]
(c) Assume that the micro-satellites are exactly located at the Sun–Mercury Lagrange
point L2, at a distance of 110 Mm farther away from the Sun than Mercury. Demon strate that the micro-satellites experience significantly different environmental condi tions at this point from the rest of their orbit around Mercury. Discuss how this might
modify the potential of the spacecraft and the potential measured at the ends of the
wires. [5]
[Total: 15]
SPCE0005/2020-21 3/5 CONTINUED
Question 2.
(a) The interplanetary magnetic field’s (IMF’s) spiral configuration is equivalent to flow parallel field lines in the reference frame that is co-rotating with the Sun’s rotation. This
reference frame introduces an additional azimuthal component of the solar-wind speed,
uφ = r, where is the Sun’s angular rotation frequency and r is the heliocentric
distance. The frozen-in flux condition then implies that
Bφ
Br
=
uφ
ur
=
r
ur
Use Maxwells equations to determine the variation of the radial component of the mag netic field Br with radial distance r, and thus use the above condition to determine the
azimuthal component Bφ as a function of r. [3]
(b) Spacecraft near Earth observe an interplanetary magnetic field (IMF) of |Br| =
|Bφ| ～ 5 nT on average. What is the average field strength of the two components, and
thus the Parker spiral angle, of the IMF expected at Saturn’s orbit Assume that the
radial component of the solar wind velocity is constant between the Earth and Saturn. [3]
(c) We now work in the non-rotating, heliocentric reference frame. Even in this frame,
the solar wind has a finite azimuthal velocity component. Re-write the MHD continuity
equation and the azimuthal component of the momentum equation in spherical coor dinates. Assuming steady-state conditions and spherical symmetry, show that MHD
fulfills angular-momentum conservation of the solar wind in the form

r r
3
ρuruφ r
3BrBφ
μ0

= 0.
Briefly explain the terms in this equation. [5]
(d) On a particular day, the solar wind just upstream of the subsolar terrestrial bow
shock has an ion number density of 1.2 × 107 m 3 and flows in a direction normal to
the shock surface at a speed of 450 km s 1
. It carries a magnetic field of strength 7 nT,
which is directed exactly perpendicular to the shock surface normal. Across the shock,
the plasma density is observed to increase by a factor of 3. What is the plasma flow
speed and magnetic field strength downstream from the shock What is the expected
increase in the plasma pressure P [4]
[Total: 15]
SPCE0005/2020-21 4/5 CONTINUED
Question 3.
Briefly answer THREE of the following four questions. These questions are based
on the group research projects in the lecture course. There is a maximum of 5 marks
for each answer. As a guideline, the answers require two or three sentences per avail able mark.
i. How is the ionosphere formed and how do we measure its density profile
ii. Under what conditions is MHD valid, and which plasmas in the solar system can
be accurately described with MHD
iii. What are the characteristics of tangential and rotational discontinuities, and what
examples of each can be found in the solar system
iv. How does the Earth’s magnetosphere react to a coronal mass ejection, and how
does this relate to space weather effects
[15]
[Total: 15]
SPCE0005/2020-21 5/5 END OF PAPER