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The instructions and questions for paper 69 of the mathematical tripos part iii exam, focusing on astrophysical dynamics and self-gravitating systems. The paper includes questions on integrals of motion, axisymmetric density, keplerian ellipses, and self-gravitating systems. Students are required to define terms, use theorems, and derive relations. Essential for those preparing for the mathematical tripos part iii exam.
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Wednesday 6 June 2007 9.00 to 12.
Attempt THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
1 Define the term integral of motion. Show that in a steady-state spherical system the binding energy E and total angular momentum L
E = ψ(r) − 12 v^2 , L = r × v
are integrals of the motion. Show that the distribution function of a collisionless steady- state stellar system depends on the integrals of motion only (Jeans Theorem).
Consider a distribution function of form
f (E, L) =
ρ 1 (2πσ^2 )^3 /^2
exp
σ^2
exp
raσ
where σ and ra are constants, and L = |L|. By introducing spherical polar coordinates in velocity space, show that the density is
ρ =
ρ(0) (1 + r^2 /r^2 a)
exp
ψ(r) − ψ(0) σ^2
and deduce that the velocity second moments are 〈v^2 r 〉 = σ^2 and 〈v φ^2 〉 = 〈v^2 θ 〉 = σ^2 r a^2 /(r a^2 + r^2 )
Show that for r << ra, the density ρ ∝ r−^2 , while for r >> ra, ρ ∝ r−^2 (log r)−^1. What is the corresponding behaviour of the circular speed in these two regimes? Give a physical interpretation of the structure of the model.
Hint: You may assume the following integrals without proof
∫ (^) π
0
sin ηdη (1 + k^2 sin^2 η)^3 /^2
1 + k^2
∫ (^) π
0
sin η cos^2 ηdη (1 + k^2 sin^2 η)^5 /^2
3(1 + k^2 )
∫ (^) ∞
0
x^4 exp(−k^2 x^2 )dx =
π 8 k^5
Paper 69
3 Using a clear diagram, define the orbital elements of a Keplerian ellipse, namely the semimajor axis a, the eccentricity e, the longitude of the ascending node Ω, the argument of pericenter ω and the inclination i.
The Hamiltonian of the Kepler problem can be written as
HK (Jr , Jθ , Jφ) = −
2(Jr + Jθ + Jφ)^2
where Jr , Jθ and Jφ are the actions referred to spherical polar coordinates. Find a canonical transformation to a new set of actions (J 1 , J 2 , J 3 ) and angles (w 1 , w 2 , w 3 ) such that H becomes a function of J 1 alone. Explain how the new actions and angles (the Delaunay variables) are related to the orbital elements.
Consider the case of a test particle orbiting very close to a planet with mass M. Let the gravitational potential consist of a monopole term and a quadrupole describing the planet’s flattening
φ = −
r
(3z^2 − r^2 ) r^5
where α is a constant. Find the radial and vertical frequencies of motion. Show that the the line of nodes of the test particle precesses backwards, while the perihelion position precesses forwards.
Paper 69
4 Show from the virial theorem that a self-gravitating system has negative heat capacity. Explain what is meant by the gravothermal catastrophe.
Consider a system of N self-gravitating infinite rods of equal mass per unit length m confined to an infinite cylinder of radius Re. The rods are restricted to move in two dimensions, with their axes parallel to that of the cylinder. Let the gravitational potential be normalised so that the potential at the boundary is
ψ(Re) = −GM log(V /V 0 )
where M = N m, V = πR^2 e and V 0 = πR 02. Anywhere else between the axis of the cylinder and the boundary, the potential ψ(R) must satisfy Poisson’s equation
1 R
d dR
dψ dR
= − 4 πGρ.
Assume that the distribution of the velocities of the rods is of Maxwellian form
f ∝ exp
α − β(v^2 / 2 − ψ)
where α is a constant and β = m/(kT ). Verify that there exist regular solutions of the form
ψ(R) = −GM log(V /V 0 ) − 2 β−^1 log
1 − 12 GM β(1 − R^2 /R^2 e )
Hence, show that there exists a lower bound to the temperature Tmin given by
kTmin = 12 GM m.
Show that the total energy E of the system is
E = N kTmin
log(V /V 0 ) + 2Θ + Θ^2 log(1 − Θ−^1 )
where Θ = T /Tmin. Use this expression to demonstrate that there is no gravothermal catastrophe in two dimensions.
Paper 69
