Gravitational Dynamics

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Gravitational Dynamics
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Gravitational Dynamics can be applied to

• Two body systemsbinary stars
• Planetary Systems
• Stellar Clustersopen globular
• Galactic Structurenuclei/bulge/disk/halo
• Clusters of Galaxies
• The universelarge scale structure

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Syllabus

• Phase Space Fluid f(x,v)
• Eqn of motion
• Poissons equation
• Stellar Orbits
• Integrals of motion (E,J)
• Jeans Theorem
• Spherical Equilibrium
• Virial Theorem
• Jeans Equation
• Interacting Systems
• Tides?Satellites?Streams
• Relaxation?collisions

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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)
  , i1,2,...,N (feasible for Nltlt106).
• 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.

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N-body Potential and Force

• In N-body system with mass m1mN, the
  gravitational acceleration g(r) and potential
  f(r) at position r is given by

r12
r
mi
Ri
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Eq. of Motion in N-body

• Newtons law a point mass m at position r moving
  with a velocity dr/dt with Potential energy F(r)
  mf(r) experiences a Force Fmg , accelerates
  with following Eq. of Motion

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Orbits defined by EoM Gravity

• Solve for a complete prescription of history of a
  particle r(t)
• E.g., if G0 ? F0, F(r)cst, ? dxi/dt vxici
  ? 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

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Example 5-body rectangle problem

• Four point masses m3,4,5 at rest of three
  vertices of a P-triangle, integrate with time
  step1 and ½ find the positions at time t1.

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Star clusters differ from air

• Size doesnt matter
• size of starsltltdistance between them
• ?stars collide far less frequently than molecules
  in air.
• Inhomogeneous
• In a Gravitational Potential f(r)
• Spectacularly rich in structure because f(r) is
  non-linear function of r

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Why Potential f(r) ?

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

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2nd Lec
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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.
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Example Force field of two-body system in
Cartesian coordinates
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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.
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3rd Lec

• Animation of GC formation

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A fluid element Potential Gravity

• For large N or a continuous fluid, the gravity dg
  and potential df due to a small mass element dM
  is calculated by replacing mi with dM

r12
dM
r
d3R
R
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Potential in a galaxy

• Replace a summation over all N-body particles
  with the integration
• Remember dM?(R)d3R for average density ?(R) in
  small volume d3R
• So the equation for the gravitational force
  becomes

R?Ri
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Poissons Equation

• Relates potential with density
• Proof hints

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Poissons Equation

• Poissons equation relates the potential to the
  density of matter generating the potential.
• It is given by

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Gausss Theorem

• Gausss theorem is obtained by integrating
  poissons 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.

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Laplacian in various coordinates
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4th Lec

• Potential,density,orbits

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• From Gravitational Force to Potential

From Potential to Density
Use Poissons Equation
The integrated form of Poissons equation is
given by
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More on Spherical Systems

• Newton proved 2 results which enable us to
  calculate the potential of any spherical system
  very easily.
• NEWTONS 1st THEOREMA body that is inside a
  spherical shell of matter experiences no net
  gravitational force from that shell
• NEWTONS 2nd THEOREMThe 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.

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From Spherical Density to Mass
M(rdr)
M(r)
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Poissons eq. in Spherical systems

• Poissons eq. in a spherical potential with no ?
  or F dependence is

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Proof of Poissons Equation

• Consider a spherical distribution of mass of
  density ?(r).

g
r
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• Take d/dr and multiply r2 ?
• Take d/dr and divide r2?

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Sun escapes if Galactic potential well is made
shallower
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Solar system accelerates weakly in MW

• 200km/s circulation
• g(R0 8kpc)0.8a0,
• a01.2 10-8 cm2 s-1
• Merely gn 0.5 a0 from all stars/gas
• Obs. g(R20 R0)
• 20 gn
• 0.02 a0

• g-gn (0-1)a0
• GM R if weak!
• Motivates
• M(R) dark particles
• G(R) (MOND)

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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
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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
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Plummer Model

• PLUMMER MODELthe 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 poissons equation.
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• The potential of the plummer model looks like
  this

?
r
?
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• 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
  dV4?r2dr.
• This gives

(as before from Poissons)
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Tutorial Question 1 Singular Isothermal Sphere

• Has Potential Beyond ro
• And Inside rltr0
• Prove that the potential AND gravity is
  continuous at rro if
• Prove density drops sharply to 0 beyond r0, and
  inside r0
• Integrate density to prove total massM0
• What is circular and escape velocities at rr0?
• Draw Log-log diagrams of M(r), Vesc(r), Vcir(r),
  Phi(r), rho(r), g(r) for V0200km/s, r0100kpc.

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Tutorial Question 2 Isochrone Potential

• Prove G is approximately 4 x 10-3
  (km/s)2pc/Msun.
• Given an ISOCHRONE POTENTIAL
• For M105 Msun, b1pc, 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
  rb1pc? 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 r1pc.
  (Answer in BT, p38)