Gravitational Dynamics

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 energyΦ(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) =cit +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: Force field of two-body system in Cartesian coordinates

9. Example: 4-body problem • Four point masses Gm=1 at rest (x,y,z)=(0,1,0),(0,-1,0),(-1,0,0),(1,0,0). What is the initial total energy? • Integrate EoM by brutal force with time step=1 to find the positions/velocities at time t=1. i.e., use straight-orbit V=V0+gt, R=R0+V0t+gt2/2. What is the new total energy?

10. 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

11. 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

12. 2nd Lec

13. 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.

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. 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) Solar system accelerates weakly in MW

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. 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.

34. 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)

35. Example:Single Isothermal Sphere Model • For a SINGLE ISOTHERMAL SPHERE (SIS) the line of sight velocity dispersion is constant. This also results in the circular velocity being constant (proof later). • The potential and density are given by:

36. Proof: Density Log() r-2 n=-2 Log(r)

37. Proof: Potential We redefine the zero of potential If the SIS extends to a radius ro then the mass and density distribution look like this: M  r ro r ro

38. Beyond ro: • We choose the constant so that the potential is continuous at r=ro. r  r-1 logarithmic 

39. So:

40. 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.

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

42. 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)

43. Isochrone Potential • We might expect that a spherical galaxy has roughly constant  near its centre and it falls to 0 at sufficiently large radii. • i.e. • A potential of this form is the ISOCHRONE POTENTIAL.

44. 5th Lec • orbits

45. Stellar Orbits • Once we have solved for the gravitational potential (Poisson’s eq.) of a system we want to know: How do stars move in gravitational potentials? • Neglect stellar encounters • use smoothed potential due to system or galaxy as a whole

46. Motions in spherical potential

47. Proof: Angular Momentum is Conserved  Since then the force is in the r direction. both cross products on the RHS = 0. So Angular MomentumL is Conserved in Spherical Isotropic Self Gravitating Equilibrium Systems. Alternatively: =r×F & F only has components in the r direction=0 so

48. In static spherical potentials: star moves in a plane (r,q) • central force field • angular momentum • equations of motion are • radial acceleration: • tangential acceleration:

49. Orbits in Spherical Potentials • The motion of a star in a centrally directed field of force is greatly simplified by the familiar law of conservation (WHY?) of angular momentum. Keplers 3rd law pericentre apocentre

50. Energy Conservation (WHY?) eff