UNIVERSITY COLLEGE LONDON
EXAMINATION FOR INTERNAL STUDENTS
MODULE CODE : SPCE0005
ASSESSMENT : SPCE0005A7PD
PATTERN
MODULE NAME : SPCE0005 – Space Plasma and Magnetospheric
Physics (SS109)
LEVEL: : Postgraduate
DATE : 16-May-2022
TIME : 10:00
Controlled Condition Exam: 2 Hours exam
You cannot submit your work after the date and time shown on
AssessmentUCL – you must ensure to allow sufficient time to upload and
hand in your work
This paper is suitable for candidates who attended classes for this
module in the following academic year(s):
Year
2021/22
Additional material
Special instructions
Exam paper word
count
TURN OVER
Answer ALL questions.
The numbers in square brackets show the provisional allocation of
maximum marks per question or part of question.
Ideal magnetohydrodynamics (MHD)
u: bulk velocity, B: magnetic field
Induction equation:
@B
@t = r (u B)
Useful vector identities in spherical coordinates
The vector A denotes any vector function of the spherical coordinates r, and φ.
r · A = 1
r2
@
@r
!
r2
Ar
” +
1
r sin
@
@ (A sin ) + 1
r sin
@Aφ
@φ
(r A) = 1
r sin
@Ar
@φ 1
r @
@
r (rAφ)
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
Solar radius R” 6.96 108 m
Solar angular rotation frequency ” 2.972 10 6 s 1
Earth radius RE 6.371 106 m
Astronomical unit au 1.496 1011 m
SPCE0005/2021-22 2/5 CONTINUED
Question 1.
(a) Using a clearly labelled diagram, show the three different types of motion that a
charged particle undergoes in a planetary dipole magnetic loop. [4]
(b) The L-shell value of a particle within Earth’s ring current can be found from the
average drift speed vd of a particle, assuming a pitch angle of 90 , using
L2 = vdqBERE
3W .
Derive this equation, starting with the equation for the average drift period of a particle. [2]
(c) In the Earth’s inner magnetosphere, charged particles experience a co-rotation
electric field drift such that low-energy particles rotate about the Earth at the same
rate that the Earth spins. Ions in the ring current under the influence of gradient and
curvature drifts only would be directed in the opposite direction to the co-rotation drift.
Determine the L-shell at which a 10 keV ion is stationary due to having equal and
opposite drift from co-rotation and the ring current. You may assume that the ion has an
equatorial pitch angle of 90 . Assume also that the Earth’s magnetic field is a perfect
dipole with its axis aligned to the Earth’s spin axis. [4]
(d) Consider a situation in which an intense auroral electric field suddenly turns on.
Instantaneously, a 10 keV proton with a pitch angle of 90 and an L-shell value of 4.1
experiences an acceleration in the direction of the local magnetic field. In this process,
the proton gains an energy of 2 keV. Calculate the new pitch angle of the particle. [3]
(e) Using the concepts of the equatorial loss cone, determine whether the proton from
part (d) will be lost to the atmosphere or not. [2]
[Total: 15]
SPCE0005/2021-22 3/5 CONTINUED
Question 2.
(a) Explain the significance of the magnetohydrodynamics (MHD) frozen-in flow ap_x005f proximation. Draw a diagram and discuss how this concept can be used to understand
the spiral form of the interplanetary magnetic field (IMF). [3]
(b) Assume that the solar wind is a constant and highly-conducting plasma outflow
with a velocity profile of the form v(r)=(vr, 0, vφ) in spherical coordinates. The IMF is
frozen into this flow and has the form B(r)=(Br, 0, Bφ). Show that, under steady-state
conditions, the induction equation requires that
r (uφBr urBφ) = C,
where C is a constant for all r. You can assume that the system is azimuthally sym metric, which means that all partial derivatives with respect to φ vanish. [3]
(c) Assume that the solar wind transitions from rigid co-rotation with the Sun to a bal listic outflow at a critical distance r = rc from the Sun’s centre. At r = rc, the magnetic
field is purely radial. Furthermore, assume that the radial solar-wind speed ur is con stant at r ≥ rc, where rc 10R” . Using the result from part (b), show that the constant
C has a value of
C = rc
2
” Bc sin ,
where Bc is the value of Br at r = rc. [2]
(d) Using the condition that r · B = 0 and the results given in parts (b) and (c), show
that
Bφ = Bc
r2
c ” sin
rur
at large distances from the Sun where r ” sin & uφ. [4]
(e) The expression derived in part (d) describes the Parker magnetic field. Calculate
the angle between B and the radial direction in the Sun’s equatorial plane ( = 90 ) at
r = 1 au. [3]
[Total: 15]
SPCE0005/2021-22 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. What are the conditions of the plasma particles in the quiescent solar atmo sphere
ii. How do Solar Energetic Particles (SEPs) travel through the solar wind
iii. What is the mirror-mode instability, and what role does it play in the Earth’s mag netosheath
iv. How does the solar wind interact with unmagnetized bodies in the solar system
[15]
[Total: 15]
SPCE0005/2021-22 5/5 END OF PAPER
