PROBLEM SHEET

MATHEMATICS PART III — GALACTIC ASTRONOMY AND DYNAMICS IProf N.W. Evans (nwe@ast.cam.ac.uk), Lent 2015

WARMING UP

(1) PRETTY PICTURES

Find images on the www of the following galaxies and classify them according to Hubble’sclassification scheme (E0, . . . , E7, S0, SB0, Sa, . . . , Sc, SBa, . . . , SBc, Irr). The galaxiesare:

NGC 2403, NGC 2683, NGC 3031, NGC 3184, NGC 3344, NGC 3379, NGC 3810, NGC4242, NGC 4406, NGC 4449, NGC 4501

(Hint: The course website (http://www.ast.cam.ac.uk/∼nwe/astrodyn.html) has links inthe section On-Line Resources. Click the galaxy catalogue and the NASA/IPAC extra-galactic database links to find images).

(2) VIZIER

Plot the colour-magnitude diagram for the 5000 brightest (V < 6.0) and the 5000 nearest( parallax > 23.4 mas) stars from the Hipparcos catalogue. A color-magnitude diagram isa plot of absolute magnitude MV versus color B − V . Interpret your diagrams.

A useful interface to the Hipparcos catalogue is the VizieR service. This is linked from thecourse website in On-Line Resources. VisieR enables you to extract ASCII tables, whichcan be plotted with your favourite plotting program.

(3) A FEW QUICK QUESTIONS

What is the nearest star visible from Cambridge other than the Sun? What is the brighteststar other then the Sun? What is the star that moves fastest on the Sky? Explain why thereare locations on Mercury where the sun rises twice and sets twice on the same day? Doesthe shadow of a solar eclipse move from east to west or west to east ? Is the accelerationof the Sun in the Galaxy due mostly to the nearest stars or the most distant stars? Whatis the inclination between the plane of the Galaxy and the plane of the Solar system (theecliptic)?

POTENTIAL THEORY

(4) BREAKFAST AT DUNKIN’S

Newton’s law of gravity states that the gravitational force between two point masses in-creases as they are brought closer together. Find an example of two bodies for which thegravitational force decreases as they are brought closer together

(5) HERNQUIST’S MODEL

Hernquist introduced a simple model of a galaxy halo. His model has gravitational poten-tial

φ = −GM

r + a

where r is the spherical polar radius, and M and a are constants Find the density ofHernquist’s model, and the radius that encloses half the mass. Plot the rotation curve ofthe model.

(6) NO CALCULATION NEEDED

A theorist writes down the gravitational potential of a galaxy model as

φ = −K

(r4 + a2r2 + b2)1/4

Without doing any detailed calculations, find the total mass of the galaxy model in termsof the constants K, a and b, and Newton’s gravitational constant G.

(7) KELVIN’S PARADOX

Newton’s Theorem on the gravitational force due to a uniform spherical shell states thatthe interior force vanishes, and that the force outside is the same as that due to an equalpoint mass at the centre of the shell. Newton’s Cosmology posits an infinite space filledwith a uniform matter distribution. Show that Newton’s Theorem is inconsistent withNewton’s Cosmology. Hint: Show that the force at a point due to an infinite uniformmedium can take any value we please.

(8) THE PUZZLED PROFESSOR**

Prove that the surface density Σ and potential φ of an infinitesimally thin disk occupyingthe plane z = 0 are related by

Σ(x) =1

4π2G

∫ ∫

dx′dy′

|x − x′|

( ∂2φ

∂x′2+

∂2φ

∂y′2

)

Hint (or Gloat): When I was a grad student, this defeated the then Professor of TheoreticalAstronomy at Cambridge University. Use Laplace Transforms.

(9) EXPONENTIAL SPHEROIDS*

Consider the exponential sphere, with density distribution

ρ = ρ0 exp(−αr),

where r is the spherical polar radius and α is a constant with dimensions of inverse length.What is the total mass M of the model? Show that the rotation curve is

v2

circ =GM

r−

GM exp(−αr)

2r

(

α2r2 + 2αr + 2)

.

Now consider the flattened analogues of the exponential sphere, which we call the expo-nential spheroids. These have density law

ρ = ρ0 exp(−αm)

where m2 = R2 + z2q−2. Here, R and z are cylindrical polar coordinates and q is theconstant axis ratio of the isodensity contours. What is the total mass M of the model?Find an expression for the gravitational potential of the exponential spheroid. Show thatthe rotation curve is

v2

circ =GMR2α3

2

∫ 1

0

du u2 exp(−αRu)

(1 − u2(1 − q2))1/2.

Find an approximation for vcirc when q is near unity (i.e., the model is only slightlyflattened). Does the circular speed at the same distance R required for centrifugal balanceincrease or decrease as we flatten the model? Why?

(10) LOGARITHMIC SPIRALS**

Let (r, θ, φ) be spherical polar coordinates. Consider the density distribution (known asthe log–spiral)

ρ(r, θ) = ρ0

( b

r

)5/2

Pn(θ) cos(α log r),

where Pn is a Legendre polynomial of degree n and α, b and ρ0 are all constants. Whatis the gravitational potential corresponding to this density law? Such potential–densitypairs are used in the studies of the axisymmetric stability of axisymmetric stellar systems.Why?

MATHEMATICS PART III — GALACTIC ASTRONOMY AND DYNAMICS IIProf N.W. Evans (nwe@ast.cam.ac.uk), Lent 2015

ORBITS

(1) THE ISOCHRONE!

A galaxy has the gravitational potential

φ(r) = −GM

b +√r2 + b2

Explain why the energy E and the angular momentum L of a star are conserved quantities.Show that the radial period Tr of a bound orbit is

Tr =2πGM

(−2E)3/2

(2) NIELS HENRIK ABEL AGAIN!

Find a one-dimensional potential for which the action J and energy E are related byJ ∝ E2. Is your solution unique? If not, can it be continuously deformed through a set ofpotentials with the same property?

(3) THE SHEARING SHEET**

Consider an idealisation of a disk (whether planetary or galactic) as a shearing two–dimensional sheet and let the angular frequency of rotation vary with radius r like Ω(r).Consider a patch centred on radius r0 with angular frequency Ω0 = Ω(r0). Introducingcoordinates x = r− r0 and y = r0(θ −Ω0t), derive the linearised equations of motion of aparticle as

x− 2Ω0y = 4Ω0A0x,

y + 2Ω0x = 0,

where

A0 = −r0

2

[dΩ

dr

]

r=r0

Explain carefully the physical interpretation of every term in the equations of motion.Estimate values of A appropriate to the dynamics of (1) a disk star in the Milky Way and(2) a lump of breccia orbiting in the A ring of Saturn. Solve the equations of motion anddescribe the orbits qualitatively.(These are Hill’s equations).

(4) EHRENFEST’S LEGACY

Show that the adiabatic invariance of the actions implies (in general) that closed orbitsremain closed when the potential is slowly and adiabatically deformed. An initially circularorbit in a spherical potential ψ does not remain closed when ψ is squashed along any linethat is not parallel to the orbit’s original angular momentum vector. Why not? Is thereany way the squashing could be done leaving some orbits closed? Paul Ehrenfest, the greatdiscoverer of adiabatic invariance, shot himself.

(5) ONE, TWO, THREE, FOUR OR FIVE?

A particle moves in the Keplerian potential

Vkep = −k

r

By separating the Hamilton-Jacobi equation in spherical polar and rotational paraboliccoordinates, show that there are five functionally independent isolating integrals of motion?What implication does this have for the orbits? Now suppose a particle moves in thepotential

Vkep = −k

r+k1

x2+k2

y2+k3

z2

How many integrals of motion are there now? What are the orbits ? What are the integralsof motion (research problem)? Hint: These potentials are super-integrable.

GALAXY MODELS

(6) HYPERVIRIALITY

Suppose a star cluster of total mass M has the spherically symmetric density distribution(the Plummer model)

ρ =3Mb2

4π(r2 + b2)5/2,

where b is a constant with the dimensions of length. Prove that the projected density Σseen at radius R on the plane of the sky is

Σ(R) =4ρ0b

5

3(R2 + b2)2,

where ρ0 denotes the central value of the density. Use Eddington’s formula to show thedistribution function is

F (E) =32

√2

7π2ρ0ψ

−3/2

0

( E

ψ0

)7/2

,

where E is the binding energy per unit mass and ψ0 is the central value of the potential.Find the velocity dispersion, kinetic energy and potential energy at any point in the starcluster. Why is this remarkable?

(7) THE SHRINKING CLUSTER?**

Every star in a spherical system loses mass slowly and isotropically. Suppose the initialdistribution function is of the form f(E), where E is the binding energy. What is thedistribution function fp after every star has been reduced to a fraction p of its originalmass. Is fp a function of binding energy alone? How has the gravitational radius of thesystem changed?

MATHEMATICS PART III — ASTROPHYSICAL DYNAMICS IIIProf N.W. Evans (nwe@ast.cam.ac.uk), Lent 2015

GALAXY MODELS

(1) THE ISOCHRONE, AGAIN

The isochrone potential is

ψ(r) = −GM

b+√b2 + r2

Find the mass density that generates the potential. Hence, find ρ(ψ) and use Eddington’sformula to find the isotropic distribution function f(E). If a DF only depends on energy,how are the equipotential and isodensity surfaces related ? ( Hint: Try expanding ρ(ψ) asa power series in ψ, and leave your answer as a hypergeometric function. Or see Binney& Tremaine, Section 4.4)

(2) JEANS EQUATIONS

Write down the collisionless Boltzmann equation in spherical polar coordinates. Supposethat the velocity dispersion tensor of a steady-state spherical galaxy is diagonalised in thespherical polar coordinate system (i.e., 〈vrvθ〉 = 〈vrvφ〉 = 〈vθvφ〉 = 0).

By multiplying the collisionless Boltzmann equation by vr, derive the Jeans equationsin the form

d(ρ〈v2

r〉)

dr+

2ρ

r(〈v2

r〉 − 〈v2

θ〉 − 〈v2

φ〉) = −ρdφ

dr

where ρ is the mass density and φ the gravitational potential.Suppose now that the galaxy has a distribution function of the form F (E,L) =

L−2βf(E), where β is a constant. Obtain the Jeans equation in the form

d(ρ〈v2

r〉)

dr+

2βρ〈v2

r 〉

r= −ρ

dφ

dr

Find an integrating factor and hence show that the equation can always be solved formallyfor the velocity dispersion 〈v2

r 〉 if the density and potential are known.

(3) THE ISOTHERMAL SPHERE

A widely used and simple model of a Galactic potential is (the isothermal sphere)

ψ = −v2

0log(r/r0),

where v0 and r0 are constants. What is the velocity of cold gas or stars on circular orbitsin this model? Show from Jeans’ theorem that that the distribution of velocities of thestars is Maxwellian with

f =1

π3/2v30

exp(−v2/v2

0)

What is the (three-dimensional) velocity dispersion? What is the escape speed? What isthe rms speed?

(4) ANISOTROPY

Suppose the phase space distribution function f depends on binding energy E alone.Show that the velocity dispersion tensor is isotropic. Now suppose that the phase spacedistribution function of a spherical system depends on the binding energy E and themodulus of the angular momentum vector L via

f(E,L) = L−2βg(E),

where β is a constant and g is an arbitrary function. Demonstrate that the ratio of theradial velocity dispersion to the tangential velocity dispersion is constant.

We already met Hernquist’s model in problem sheet 1. It has the potential-density pair

φ(r) = −GM

r + a, ρ(r) =

M

2π

a

r(r + a)3.

Show that the model has an anisotropic distribution function

f(E,L) = CE2

L

where C is a constant to be identified, and E and L are the energy and the angularmomentum. What is the ratio of the radial velocity dispersion to the tangential velocitydispersion in the model?

(5) A PRESENT FROM THE OLD DAMTP**

Ken Freeman, working in the old DAMTP building on Silver St, discovered an exact familyof galaxy models. Suppose we have an infintesimally thin elliptical disk of stars with surfacedensity

Σ(x, y) = Σ0

[

1 − (x/a)2 − (y/b)2

]1/2

Show the gravitational potential has form

φ(x, y) = Ax2 +By2 + C

and determine A,B and C. Write down the equations of motion and hence find theadiabatic invariants J1 and J2. Show that the distribution function has the form

F (J1, J2) =

[

1 − c1J2

1− c2J

2

2

]

−1/2

with constants c1 and c2.

STABILITY

(6) WHY DO FLAGS FLAP?**

Consider an idealised disk galaxy as an infinitesimally thin, homogeneous, self–gravitatingsheet of infinite extent. Suppose a ripple causes the vertical position of the sheet zs tooscillate like

zs = ǫz0 exp i(kx− ωt)

Neglect any horizontal motions or external vertical potential so that the equation of motionis just

∂2zs

∂t2= Fv(x, t)

where Fv is the vertical force due to the self–gravity of the perturbed sheet. Show thatthe potential of the sheet is

φ = GΣ0

∫

∞

−∞

dx′ log[(x− x′)2 + (z − zs(x′, t))2].

Show that the vertical force per unit mass on an element of the sheet is

Fv(x, t) = −2πGΣ0|k|zs(x, t).

Prove that the dispersion relation for bending modes is

ω2 = 2πGΣ0|k|.

Is the galaxy stable to the ripple? Why might this answer be misleading?

COLLISIONS AND STATISTICAL PHYSICS

(7) A GRAIN OF SAND

How do you make a binary star with a grain of sand? In other words, two stars approachon a hyperbolic orbit and your task is to throw in a grain of sand so as to extract enoughkinetic energy to bind them into a binary.

(8) COLLISIONS

A test particle approaches a compact object of mass M and radius R from infinity withspeed v∞ and impact parameter p. Use the particle’s energy and angular momentum withrespect to the object to derive expressions for the semi-major axis and eccentricity of thehyperbolic orbit followed by the test particle, and for the pericentric distance r0. Showthat the eccentricity may be written e = 1 = 2v2

∞/v2

0, where v0 is the escape velocity at r0.

Use the expression for the true anomaly corresponding to the asymptote of the hyperbola(r → ∞) to show that the overall deflection of the test particle’s orbit after it leaves thevicinity of the compact object ψ is given by sin(ψ/2) = 1/e. What is the condition for theavoidance of a physical collision?

(9) NEGATIVE SPECIFIC HEAT CAPACITY

Consider a system in which the interparticle potential has the form

φij = C|xi − xj |p+1

where C is a constant. Show that the scalar virial theorem takes the form

2T − (p+ 1)W = 0

where T is the kinetic energy and W the potential energy of the system. For which valuesof p does the system have a negative heat capacity.

MATHEMATICS PART III – GALACTIC ASTRONOMY AND DYNAMICS IVProf N.W. Evans (nwe@ast.cam.ac.uk), Lent 2015

Past Questions

The syllabuses of the various courses in this area have changed over the years. Don’t worry,therefore, if you think questions have not been covered in lectures. In all likelihood, thesyllabus has changed!

Part III past papers are at“http://www.maths.cam.ac.uk/postgrad/mathiii/pastpapers/”

The following questions are in the present syllabus of the Galactic Astronomy and Dynam-

ics course, and would be reasonable questions for the 2015 exam!

2014: “Galactic Astronomy and Dynamics”, Questions 1, 2, 3 and 4

2012: “Galactic Astronomy and Dynamics”, Questions 1, 2, 3 and 4

2011: “Galaxies”, Questions 1, 2 (NOT 3,4,5)

2010: “Astrophysical Dynamics”, Questions 1, 2, 3 and 4

2009: “Astrophysical Dynamics”, Questions 1, 2 and 3 (NOT 4)

2007: “Astrophysical Dynamics”, Questions 1, 2 and 4 (NOT 3)

2006: “Galaxies and Dark Matter”, Questions 2 and 4

2005: “Galaxies and Dark Matter”, Questions 1, 3 and 4

2004: “Galaxies and Dark Matter”, Questions 1, 2 and 4

2003: “Galaxies”, Questions 1, 2 and 3

2002: “Galaxies”, Questions 1, 2 and 3

2001: “Galaxies”, Questions 2 and 3

For the final examples class, Please attempt the 8 questions from the 2012 and 2014 papers

beforehand.

In the examples class, I will go through the questions on the 2012 and 2014 papers. I canalso discuss any of the remaining questions above, if desired.