Gravitational Dynamics

Gravitational Dynamics can be applied to: Two body systems:binary stars Planetary Systems Stellar Clusters:open & globular Galactic Structure:nuclei/bulge/disk/halo Clusters of Galaxies The universe:large scale structure

Syllabus Phase Space Fluid f(x,v) Eqn of motion Poisson’s equation Stellar Orbits Integrals of motion (E,J) Jeans Theorem Spherical Equilibrium Virial Theorem Jeans Equation Interacting Systems TidesSatellitesStreams Relaxationcollisions

How to model motions of 1010stars in a galaxy? Direct N-body approach (as in simulations) At time t particles have (mi,xi,yi,zi,vxi,vyi,vzi), i=1,2,...,N (feasible for N<<106). Statistical or fluid approach (N very large) At time t particles have a spatial density distribution n(x,y,z)*m, e.g., uniform, at each point have a velocity distribution G(vx,vy,vz), e.g., a 3D Gaussian.

N-body Potential and Force In N-body system with mass m1…mN, the gravitational acceleration g(r) and potential φ(r) at position r is given by: r12 r mi Ri

Eq. of Motion in N-body Newton’s law: a point mass m at position r moving with a velocity dr/dt with Potential energy Φ(r) =mφ(r) experiences a Force F=mg , accelerates with following Eq. of Motion:

Orbits defined by EoM & Gravity Solve for a complete prescription of history of a particle r(t) E.g., if G=0  F=0, Φ(r)=cst,  dxi/dt = vxi=ci  xi(t) =ci t +x0, likewise for yi,zi(t) E.g., relativistic neutrinos in universe go straight lines Repeat for all N particles.  N-body system fully described

Example: 5-body rectangle problem Four point masses m=3,4,5 at rest of three vertices of a P-triangle, integrate with time step=1 and ½ find the positions at time t=1.

Star clusters differ from air: Size doesn’t matter: size of stars<<distance between them stars collide far less frequently than molecules in air. Inhomogeneous In a Gravitational Potential φ(r) Spectacularly rich in structure because φ(r) is non-linear function of r

Why Potential φ(r) ? More convenient to work with force, potential per unit mass. e.g. KE½v2 Potential φ(r) is scaler, function of r only, Easier to work with than force (vector, 3 components) Simply relates to orbital energy E= φ(r) +½v2

2nd Lec

Example: energy per unit mass The orbital energy of a star is given by: 0 since and 0 for static potential. So orbital Energy is Conserved in a static potential.

Example: Force field of two-body system in Cartesian coordinates

Example: Energy is conserved The orbital energy of a star is given by: 0 since and 0 for static potential. So orbital Energy is Conserved in a static potential.

3rd Lec Animation of GC formation

A fluid element: Potential & Gravity For large N or a continuous fluid, the gravity dg and potential dφ due to a small mass element dM is calculated by replacing mi with dM: r12 dM r d3R R

Potential in a galaxy Replace a summation over all N-body particles with the integration: Remember dM=ρ(R)d3R for average density ρ(R) in small volume d3R So the equation for the gravitational force becomes: RRi

Poisson’s Equation Relates potential with density Proof hints:

Poisson’s Equation Poissons equation relates the potential to the density of matter generating the potential. It is given by:

Gauss’s Theorem Gauss’s theorem is obtained by integrating poisson’s equation: i.e. the integral ,over any closed surface, of the normal component of the gradient of the potential is equal to 4G times the Mass enclosed within that surface.

Laplacian in various coordinates

4th Lec Potential,density,orbits

From Gravitational Force to Potential From Potential to Density Use Poisson’s Equation The integrated form of Poisson’s equation is given by:

More on Spherical Systems Newton proved 2 results which enable us to calculate the potential of any spherical system very easily. NEWTONS 1st THEOREM:A body that is inside a spherical shell of matter experiences no net gravitational force from that shell NEWTONS 2nd THEOREM:The gravitational force on a body that lies outside a closed spherical shell of matter is the same as it would be if all the matter were concentrated at its centre.

From Spherical Density to Mass M(r+dr) M(r)

Poisson’s eq. in Spherical systems Poisson’s eq. in a spherical potential with no θ or Φ dependence is:

Proof of Poissons Equation Consider a spherical distribution of mass of density ρ(r). g r

Take d/dr and multiply r2  Take d/dr and divide r2

Sun escapes if Galactic potential well is made shallower

Solar system accelerates weakly in MW 200km/s circulation g(R0 =8kpc)~0.8a0, a0=1.2 10-8 cm2 s-1 Merely gn ~0.5 a0 from all stars/gas Obs. g(R=20 R0) ~20 gn ~0.02 a0 g-gn ~ (0-1)a0 “GM” ~ R if weak! Motivates M(R) dark particles G(R) (MOND)

Circular Velocity CIRCULAR VELOCITY= the speed of a test particle in a circular orbit at radius r. For a point mass: For a homogeneous sphere

Escape Velocity ESCAPE VELOCITY= velocity required in order for an object to escape from a gravitational potential well and arrive at  with zero KE. It is the velocity for which the kinetic energy balances potential. -ve

Plummer Model PLUMMER MODEL=the special case of the gravitational potential of a galaxy. This is a spherically symmetric potential of the form: Corresponding to a density: which can be proved using poisson’s equation.

The potential of the plummer model looks like this:  r 

Since, the potential is spherically symmetric g is also given by:  The density can then be obtained from: dM is found from the equation for M above and dV=4r2dr. This gives (as before from Poisson’s)

Tutorial Question 1: Singular Isothermal Sphere Has Potential Beyond ro: And Inside r<r0 Prove that the potential AND gravity is continuous at r=ro if Prove density drops sharply to 0 beyond r0, and inside r0 Integrate density to prove total mass=M0 What is circular and escape velocities at r=r0? Draw Log-log diagrams of M(r), Vesc(r), Vcir(r), Phi(r), rho(r), g(r) for V0=200km/s, r0=100kpc.

Tutorial Question 2: Isochrone Potential Prove G is approximately 4 x 10-3 (km/s)2pc/Msun. Given an ISOCHRONE POTENTIAL For M=105 Msun, b=1pc, show the central escape velocity = (GM/b)1/2 ~ 20km/s. Argue why M must be the total mass. What fraction of the total mass is inside radius r=b=1pc? Calculate the local Vcir(b) and Vesc(b) and acceleration g(b). What is your unit of g? Draw log-log diagram of Vcir(r). What is the central density in Msun pc-3? Compare with average density inside r=1pc. (Answer in BT, p38)