GALACTIC ASTRONOMY and DYNAMICS I Prof NW Evans

PROBLEM SHEET

MATHEMATICS PART III — GALACTIC ASTRONOMY AND DYNAMICS I Prof N.W. Evans ([email protected]), Lent 2015

WARMING UP

(1) PRETTY PICTURES

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

NGC 2403, NGC 2683, NGC 3031, NGC 3184, NGC 3344, NGC 3379, NGC 3810, NGC 4242, NGC 4406, NGC 4449, NGC 4501 (Hint: The course website (http://www.ast.cam.ac.uk/∼nwe/astrodyn.html) has links in the 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 is a 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 the course website in On-Line Resources. VisieR enables you to extract ASCII tables, which can 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 brightest star other then the Sun? What is the star that moves fastest on the Sky? Explain why there are locations on Mercury where the sun rises twice and sets twice on the same day? Does the shadow of a solar eclipse move from east to west or west to east ? Is the acceleration of the Sun in the Galaxy due mostly to the nearest stars or the most distant stars? What is the inclination between the plane of the Galaxy and the plane of the Solar system (the ecliptic)?

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 the gravitational 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 potential GM φ = − r + a where r is the spherical polar radius, and M and a are constants Find the density of Hernquist’s model, and the radius that encloses half the mass. Plot the rotation curve of the model. (6) NO CALCULATION NEEDED A theorist writes down the gravitational potential of a galaxy model as K φ = − (r4 + a2r2 + b2)1/4

Without doing any detailed calculations, find the total mass of the galaxy model in terms of 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 that the interior force vanishes, and that the force outside is the same as that due to an equal point mass at the centre of the shell. Newton’s Cosmology posits an infinite space filled with a uniform matter distribution. Show that Newton’s Theorem is inconsistent with Newton’s Cosmology. Hint: Show that the force at a point due to an infinite uniform medium can take any value we please. (8) THE PUZZLED PROFESSOR** Prove that the surface density Σ and potential φ of an infinitesimally thin disk occupying the plane z = 0 are related by

′ ′ 2 2 x 1 dx dy ∂ φ ∂ φ Σ( ) = 2 ′ 2 + 2 4π G ZZ |x − x |∂x′ ∂y′

Hint (or Gloat): When I was a grad student, this defeated the then Professor of Theoretical Astronomy 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

2 GM GM exp(−αr) 2 2 vcirc = − α r + 2αr + 2 . r 2r Now consider the ﬂattened analogues of the exponential sphere, which we call the exponential 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 the constant 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 that the rotation curve is

2 3 1 2 2 GMR α du u exp(−αRu) vcirc = 2 2 1 2 . 2 Z0 (1 − u (1 − q )) /

Find an approximation for vcirc when q is near unity (i.e., the model is only slightly ﬂattened). Does the circular speed at the same distance R required for centrifugal balance increase or decrease as we ﬂatten the model? Why? (10) LOGARITHMIC SPIRALS** Let (r, θ, φ) be spherical polar coordinates. Consider the density distribution (known as the log–spiral) b 5/2 ρ(r, θ) = ρ0 Pn(θ) cos(α log r), r where Pn is a Legendre polynomial of degree n and α, b and ρ0 are all constants. What is the gravitational potential corresponding to this density law? Such potential–density pairs are used in the studies of the axisymmetric stability of axisymmetric stellar systems. Why? MATHEMATICS PART III — GALACTIC ASTRONOMY AND DYNAMICS II Prof N.W. Evans ([email protected]), Lent 2015

ORBITS (1) THE ISOCHRONE! A galaxy has the gravitational potential GM φ(r) = − √ 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 2πGM T = r (−2E)3/2

(2) NIELS HENRIK ABEL AGAIN! Find a one-dimensional potential for which the action J and energy E are related by J ∝ E2. Is your solution unique? If not, can it be continuously deformed through a set of potentials 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). Introducing coordinates x = r − r0 and y = r0(θ − Ω0t), derive the linearised equations of motion of a particle as x¨ − 2Ω0y˙ = 4Ω0A0x,

y¨ + 2Ω0x˙ = 0, where r0 dΩ A0 = − 2 h dr ir=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 and describe 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 orbits remain closed when the potential is slowly and adiabatically deformed. An initially circular orbit in a spherical potential ψ does not remain closed when ψ is squashed along any line that is not parallel to the orbit’s original angular momentum vector. Why not? Is there any way the squashing could be done leaving some orbits closed? Paul Ehrenfest, the great discoverer of adiabatic invariance, shot himself. (5) ONE, TWO, THREE, FOUR OR FIVE? A particle moves in the Keplerian potential

k V = − kep r

By separating the Hamilton-Jacobi equation in spherical polar and rotational parabolic coordinates, 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 the potential k k k k V = − + 1 + 2 + 3 kep r x2 y2 z2 How many integrals of motion are there now? What are the orbits ? What are the integrals of 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) 3Mb2 ρ = , 4π(r2 + b2)5/2 where b is a constant with the dimensions of length. Prove that the projected density Σ seen at radius R on the plane of the sky is

4ρ b5 Σ(R) = 0 , 3(R2 + b2)2

where ρ0 denotes the central value of the density. Use Eddington’s formula to show the distribution function is √ 7/2 32 2 −3/2 E F (E) = 2 ρ0ψ0 , 7π ψ0

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 star cluster. Why is this remarkable? (7) THE SHRINKING CLUSTER?** Every star in a spherical system loses mass slowly and isotropically. Suppose the initial distribution function is of the form f(E), where E is the binding energy. What is the distribution function fp after every star has been reduced to a fraction p of its original mass. Is fp a function of binding energy alone? How has the gravitational radius of the system changed? MATHEMATICS PART III — ASTROPHYSICAL DYNAMICS III Prof N.W. Evans ([email protected]), Lent 2015

GALAXY MODELS (1) THE ISOCHRONE, AGAIN The isochrone potential is GM ψ(r) = − √ b + b2 + r2 Find the mass density that generates the potential. Hence, ﬁnd ρ(ψ) and use Eddington’s formula 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) as a 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. Suppose that the velocity dispersion tensor of a steady-state spherical galaxy is diagonalised in the spherical polar coordinate system (i.e., vrvθ = vrvφ = vθvφ = 0). By multiplying the collisionless Boltzmann equation by vr, derive the Jeans equations in the form 2 d(ρv ) 2ρ 2 2 2 dφ r + (v − v − v ) = −ρ dr r r θ φ 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(ρv2) 2βρv2 dφ r + r = −ρ dr r dr Find an integrating factor and hence show that the equation can always be solved formally 2 for the velocity dispersion vr 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)

2 ψ = −v0 log(r/r0),

where v0 and r0 are constants. What is the velocity of cold gas or stars on circular orbits in this model? Show from Jeans’ theorem that that the distribution of velocities of the stars is Maxwellian with 1 f v2/v2 = 3/2 3 exp(− 0 ) π v0 What is the (three-dimensional) velocity dispersion? What is the escape speed? What is the 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 space distribution function of a spherical system depends on the binding energy E and the modulus 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 the radial 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

GM M a φ(r) = − , ρ(r) = . r + a 2π r(r + a)3

Show that the model has an anisotropic distribution function

E2 f(E,L) = C L

where C is a constant to be identiﬁed, and E and L are the energy and the angular momentum. What is the ratio of the radial velocity dispersion to the tangential velocity dispersion in the model? (5) A PRESENT FROM THE OLD DAMTP** Ken Freeman, working in the old DAMTP building on Silver St, discovered an exact family of galaxy models. Suppose we have an inﬁntesimally thin elliptical disk of stars with surface density 1/2 2 2 Σ(x, y) = Σ0"1 − (x/a) − (y/b) #

Show the gravitational potential has form

2 2 φ(x, y) = Ax + By + C and determine A, B and C. Write down the equations of motion and hence ﬁnd the adiabatic invariants J1 and J2. Show that the distribution function has the form

−1/2 2 2 F (J1,J2) = "1 − c1J1 − c2J2 # with constants c1 and c2. STABILITY (6) WHY DO FLAGS FLAP?** Consider an idealised disk galaxy as an inﬁnitesimally thin, homogeneous, self–gravitating sheet of inﬁnite extent. Suppose a ripple causes the vertical position of the sheet zs to oscillate like zs = ǫz0 exp i(kx − ωt) Neglect any horizontal motions or external vertical potential so that the equation of motion is just ∂2z s = F (x, t) ∂t2 v

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

∞ ′ ′ 2 ′ 2 φ = GΣ0 dx log[(x − x ) + (z − zs(x , t)) ]. Z−∞

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

2 ω = 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 approach on a hyperbolic orbit and your task is to throw in a grain of sand so as to extract enough kinetic energy to bind them into a binary. (8) COLLISIONS A test particle approaches a compact object of mass M and radius R from inﬁnity with speed v∞ and impact parameter p. Use the particle’s energy and angular momentum with respect to the object to derive expressions for the semi-major axis and eccentricity of the hyperbolic orbit followed by the test particle, and for the pericentric distance r0. Show 2 2 that the eccentricity may be written e = 1 = 2v∞/v0, 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 the vicinity of the compact object ψ is given by sin(ψ/2) = 1/e. What is the condition for the avoidance of a physical collision? (9) NEGATIVE SPECIFIC HEAT CAPACITY Consider a system in which the interparticle potential has the form

p+1 φij = C|xi − xj | 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 values of p does the system have a negative heat capacity. MATHEMATICS PART III – GALACTIC ASTRONOMY AND DYNAMICS IV Prof N.W. Evans ([email protected]), 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, the syllabus 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 can also discuss any of the remaining questions above, if desired.