Gravitational Dynamics Gravitational Dynamics can be applied to

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 1010 stars 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 3 D Gaussian.

N-body Potential and Force • In N-body system with mass m 1…m. N, the gravitational acceleration g(r) and potential φ(r) at position r is given by: r 12 r Ri mi

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 Eo. M & 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 +x 0, 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 ½v 2 • 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) +½v 2

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

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A fluid element: Potential & Gravity • For large N or a continuous fluid, the gravity dg and potential dφ due to a small mass element d. M is calculated by replacing mi with d. M: r 12 r R d. M d 3 R

Potential in a galaxy • Replace a summation over all N-body particles with the integration: R Ri • Remember d. M=ρ(R)d 3 R for average density ρ(R) in small volume d 3 R • So the equation for the gravitational force becomes:

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

th 4 • Potential, density, orbits Lec

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 1 st THEOREM: A body that is inside a spherical shell of matter experiences no net gravitational force from that shell • NEWTONS 2 nd 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 r 2 • Take d/dr and divide r 2

Sun escapes if Galactic potential well is made shallower

Solar system accelerates weakly in MW • 200 km/s circulation g(R 0 =8 kpc)~0. 8 a 0, a 0=1. 2 10 -8 cm 2 s-1 Merely gn ~0. 5 a 0 from all stars/gas • Obs. g(R=20 R 0) ~20 gn ~0. 02 a 0 • g-gn ~ (0 -1)a 0 • “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. -ve • It is the velocity for which the kinetic energy balances potential.

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: • d. M is found from the equation for M above and d. V=4 r 2 dr. • This gives (as before from Poisson’s)

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

Tutorial Question 2: Isochrone Potential • Prove G is approximately 4 x 10 -3 (km/s)2 pc/Msun. • Given an ISOCHRONE POTENTIAL • For M=105 Msun, b=1 pc, show the central escape velocity = (GM/b)1/2 ~ 20 km/s. • Argue why M must be the total mass. What fraction of the total mass is inside radius r=b=1 pc? 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=1 pc. (Answer in BT, p 38)