MATHEMATICAL TRIPOS
Part III
Wednesday, 3 June, 2015
1:30 pm to 4:30 pm
PAPER 63
PLANETARY SYSTEM DYNAMICS
Attempt no more than THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
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Cover sheet
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Treasury Tag
Script paper
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
2
1
A
nearly
edgeon
axisymmetric
circumstellar
disk
is
observed
with
a
midplane
inclined Id
≈ 87◦
to the skyplane. An orthonormal astrocentric reference
frame [ X , Ŷ , Z]
is defined with
X and
Ŷ in the skyplane,
X pointing toward North and Z
toward Earth.
The disk’s ascending node through the skyplane (in which disk material
approaches us)
lies at a position angle Ωd
= 45◦
from North. Another orthonormal astrocentric
reference
frame [x, ŷ, z] is defined with x and ŷ in the disk midplane with x in the sky
plane in the
direction of the disk’s ascending node, at which location material moves in the
ŷ direction.
Provide an annotated sketch of this geometry, and determine the transformation
matrix
T that converts between locations in the two reference frames X and x,
respectively, such
that X = T x.
A planet is also observed to orbit the star with an inclination to the skyplane
of
Ip
≈
88◦ .
The planet’s ascending node through the skyplane lies at a
position angle
Ωp
= 49◦
from North.
The planet’s ascending node through the disk midplane
(that in
which motion is in the positive z direction) is in the direction
x′
at an
angle Ωm
from
the
x direction.
The orthonormal astrocentric reference frame
[x′, ŷ′ , z′ ]
has
ŷ′
in the
planet’s orbital plane oriented in the direction of motion at the ascending
node. Provide
an updated sketch of this geometry.
There are two ways of applying successive rotations to convert between the [ X ,
Ŷ , Z]
and [x′ , ŷ′ , z′ ] reference frames. Describe how this can be used to determine
both Ωm
and
the inclination of the planet’s orbit to the disk midplane Im
as a function of
Id, Ip, Ωp
and Ωd, defining any additional angles needed for the transformation.
Hence, or otherwise, show that
cos Im
=
cos Id cos Ip + sin Id sin Ip cos (Ωp − Ωd),
sin Ip sin (Ωp − Ωd)
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tan Ωm
=
cos Id sin Ip cos (Ωp − Ωd) − sin Id cos Ip .
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 Fall '09

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