1. Gravitational Dynamics

2. 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

3. 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

4. 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.

5. 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

6. 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:

7. 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

8. 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.

9. 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

10. 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

11. 2nd Lec

12. 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.

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

14. 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.

15. 3rd Lec
Animation of GC formation

16. 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

17. 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

18. Poisson’s Equation
Relates potential with density
Proof hints:

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

20. 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.

21. Laplacian in various coordinates

22. 4th Lec
Potential,density,orbits

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

24. 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.

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

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

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

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

29. Sun escapes if Galactic potential well is made shallower

30. Solar system accelerates weakly in MW
200km/s circulation
g(R0 =8kpc)~0.8a0,
a0= 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)

31. 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

32. 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

33. 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.

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

35. 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)

36. 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.

37. Tutorial Question 2: Isochrone Potential
Prove G is approximately 4 x (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)