AS 4021: Gravitational Dynamics

1. AS 4021: Gravitational Dynamics HongSheng Zhao hz4@st-and.ac.uk Texts: Binney and Tremaine: Galactic Dynamics (chpt 1-4) notes: star-www.st-and.ac.uk/~hz4/gravdyn/gravdyn.html

2. Gravitational Dynamics Can be applied to: • Two-body systems: Binary Stars • Planetary Systems • Stellar clusters • open and globular • Galactic structure • galactic nuclei/bulge/disk/halo • black holes • Clusters of Galaxies • The Universe: Large Scale Structure

3. Do stars hit each other? 105 to 106 stars in 10 pc Collisional timescale s ­ (2 Ro)2 n ­ 104 pc-3 v ­ 10 km / s tcoll ­ 1015 years >> age of cluster Example: Collisions in Globular Clusters

4. Importance of Gravitational Dynamics Observations: Magnitudes Spectral lines Proper Motions Distribution in ( l, b) Velocities Densities + Gravitational Dynamics 3-D mass distribution, e.g., Clues of Dark Matter/BHs, How galaxies form (merge)

5. Phase Space Fluid f(x,v) Eq. of motion Poisson’s eq. -> G Spherical equilibrium M(r) Virial theorem Jeans eq. Stellar Orbits x(t),v(t) Integrals of motion (E, J) Jeans theorem Interacting systems Tides->Satellites->Streams Relaxation = Collisions Crisis: Fin du MOND Syllabus G M(r) x(t),v(t)

6. 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.

7. 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.

8. 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

9. 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:

10. 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

11. 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

12. Why Potential φ(r) ? • Potential φ(r) is scaler, function of r only, • Easier to work with than force (vector, 3 components) • Simply relates to specific orbital energy E= φ(r) +½v2

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. 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

16. 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

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

18. 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.

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

20. Laplacian in various coordinates

21. Phase Space Density f(t,x,v) • Twinkle, twinkle, so many stars … • statistical approach • Life is too short … • Snapshot of a galaxy/cluster • Can you do sums? • a smooth and linear galaxy

22. Fluid approach:Phase Space Density PHASE SPACE DENSITY:Number of stars per unit volume per unit velocity volume f(x,v) (all called Distribution Function DF). The total number of particles per unit volume is given by:

23. E.g., air particles with Gaussian velocity (rms velocity = σ in x,y,z directions): • The distribution function is defined by: mdN=f(x,v)d3xd3v where dN is the number of particles per unit volume with a given range of velocities. • The mass distribution function is given by f(x,v).

24. The total mass is then given by the integral of the mass distribution function over space and velocity volume: • Note:in spherical coordinates d3x=4πr2dr • The total momentum is given by:

25. The mean velocity is given by:

26. Example:molecules in a room: These are gamma functions

27. Gamma Functions: