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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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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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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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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Poissons Equation

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

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Laplacian in various coordinates
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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 uniform distribution of mass of
  density ?.

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

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From Density to Mass
M(rdr)
M(r)
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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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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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ExampleSingle 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

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Proof Density
Log(?)
??r-2 n-2
Log(r)
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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
r
ro
ro
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• Beyond ro
• We choose the constant so that the potential is
  continuous at rro.

r
? r-1
logarithmic
?
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So
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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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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.

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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 of angular momentum.

Keplers 3rd law
pericentre
apocentre
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?eff
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• at the PERICENTRE and APOCENTRE
• There are two roots for
• One of them is the pericentre and the other is
  the apocentre.
• The RADIAL PERIOD Tr is the time required for the
  star to travel from apocentre to pericentre and
  back.
• To determine Tr we use

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• The two possible signs arise because the star
  moves alternately in and out.
• In travelling from apocentre to pericentre and
  back, the azimuthal angle ? increases by an
  amount

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• The AZIMUTHAL PERIOD is
• In general will not be a rational number.
  Hence the orbit will not be closed.
• A typical orbit resembles a rosette and
  eventually passes through every point in the
  annulus between the circle of radius rp and ra.
• Orbits will only be closed if is an integer.

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?????????????Graphs

• The velocity of the star is slower at apocentre
  due to the conservation of angular momentum.
• ??????????????????????????

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Fluid approachPhase Space Density

• PHASE SPACE DENSITYNumber 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

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• E.g., air particles with Gaussian velocity (rms
  velocity s in x,y,z directions)
• The distribution function is defined by
• mdNf(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).

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• The total mass is then given by the integral of
  the mass distribution function over space and
  velocity volume
• Notein spherical coordinates d3x4pr2dr
• The total momentum is given by

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• The mean velocity is given by

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• Examplemolecules in a room
• These are gamma functions

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• Gamma Functions

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Liouvilles Theorem

• We previously introduced the concept of phase
  space density. The concept of phase space
  density is useful because it has the nice
  property that it is incompessible for
  collisionless systems.
• A COLLISIONLESS SYSTEM is one where there
  are no collisions. All the constituent particles
  move under the influence of the mean potential
  generated by all the other particles.
• INCOMPRESSIBLE means that the phase-space
  density doesnt change with time.

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• Consider Nstar identical particles moving in
  a small bundle through spacetime on neighbouring
  paths. If you measure the bundles volume in
  phase space (Vol?x ? p) as a function of a
  parameter ? (e.g., time t) along the central path
  of the bundle. It can be shown that
• It can be seen that the region of phase space
  occupied by the particle deforms but maintains
  its area. The same is true for y-py and z-pz.
  This is equivalent to saying that the phase space
  density fNstars/Vol is constant.? df/dt0!

px
px
x
x
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DF Integrals of motion

• If some quantity I(x,v) is conserved i.e.
• We know that the phase space density is conserved
  i.e
• Therefore it is likely that f(x,v) depends on
  (x,v) through the function I(x,v), so ff(I(x,v)).

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Jeans theorem

• For most stellar systems the DF depends on (x,v)
  through generally three integrals of motion
  (conserved quantities), Ii(x,v),i1..3 ? f(x,v)
  f(I1(x,v), I2(x,v), I3(x,v))
• E.g., in Spherical Equilibrium, f is a function
  of energy E(x,v) and ang. mom. vector L(x,v).s
  amplitude and z-component

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EoM?Jeans eq.
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Spherical Equilibrium System

• Described by potential f(r)
• SPHERICAL density ?(r) depends on modulus of r.
• EQUILIBRIUMProperties do not evolve with time.

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Anisotropic DF f(E,L,Lz).

• Energy is conserved as
• Angular Momentum Vector is conserved as
• DF depends on Velocity Direction through Lr X v

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Proof Angular Momentum is Conserved

• ?

Since
then the force is in the r direction.
?both cross products on the RHS 0.
So Angular Momentum L is Conserved in Spherical
Isotropic Self Gravitating Equilibrium Systems.
Alternatively ?rF F only has components in
the r direction??0 so
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Proof Energy is conserved

• The total energy of an orbit is given by

0 since and
0 for equilibrium systems.
So Energy is Conserved.
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Linear Momentum

• The change in linear momentum with time is given
  by
• Since P is not conserved.
• However, when you sum up over all stars, linear
  momentum is conserved.

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Spherical Isotropic Self Gravitating Equilibrium
Systems

• ISOTROPICThe distribution function only depends
  on the modulus of the velocity rather than the
  direction.

Notethe tangential direction has ? and ?
components
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Isotropic DF f(E)

• In a static potential the energy of an particle
  is conserved.
• So,if we write f as a function of E then it will
  agree with the statement

NoteEenergy per unit mass
For incompressible fluids
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• So
• Ecst since
• For a bound equilibrium system f(E) is positive
  everywhere (can be zero) and is monotonically
  decreasing.

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• SELF GRAVITATINGThe masses are kept together by
  their mutual gravity. In non self gravitating
  systems the density that creates the potential is
  not equal to the density of stars. e.g a black
  hole with stars orbiting about it is not self
  gravitating.

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Eddington Formulae

• EDDINGTON FORMULAEThese can be used to get the
  density as a function of r from the energy
  density distribution function f(E).

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Proof of the 1st Eddington Formula
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So, from a given distribution function we can
compute the spherical density.
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Relation Between Pressure Gradient and
Gravitational Force

• Pressure is given by

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• So, this gives
• This relates the pressure gradient to the
  gravitational force. This is the JEANS EQUATION.

Note ??2P
Note-ve sign has gone since we reversed the
limits.
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So, gravity, potential, density and Mass are all
related and can be calculated from each other by
several different methods.
g
M
??2
vesc
?(E)
?(r)
?(r)
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Proof Situation where Vc2const is a Singular
Isothermal Sphere

• From Previously
• Conservation of momentum gives

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• ?

?
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• ?
• Since the circular velocity is independent of
  radius then so is the velocity dispersion?Isotherm
  al.

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Finding the normalising constant for the phase
space density

• If we assume the phase space density is given by
• ?

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• We can then find the normalizing constant so that
  ??(r) is reproduced.
• Note you want to integrate f(E) over all
  energies that the star can have I.e. only
  energies above the potential?
• We are integrating over stars of different
  velocities ranging from 0 to ?.

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• One way is to stick the velocity into the
  distribution function
• Using the substitution gives

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• Now ? can also be found from poissons equation.
  Substituting in ? from before gives
• Equating the r terms gives

as before.
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Flattened Disks

• Here the potential is of the form ?(R,z).
• No longer spherically symmetric.
• Now it is Axisymmetric

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Orbits in Axisymmetric Potentials

• We employ a cylindrical coordinate system (R,?,z)
  centred on the galactic nucleus and align the z
  axis with the galaxies axis of symmetry.
• Stars confined to the equatorial plane have no
  way of perceiving that the potential is not
  spherically symmetric.
• ?Their orbits will be identical to those in
  spherical potentials.
• R of a star on such an orbit oscillates around
  some mean value as the star revolves around the
  centre forming a rosette figure.

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Reducing the Study of Orbits to a 2D Problem

• This is done by exploiting the conservation of
  the z component of angular momentum.
• Let the potential which we assume to be symmetric
  about the plane z0, be ?(R,z).
• The general equation of motion of the star is
  then
• The acceleration in cylindrical coordinates is

Motion in the meridional plane
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• The component of angular momentum about the
  z-axis is conserved.
• If ? has no dependence on ? then the azimuthal
  angular momentum is conserved (r?F0).
• Eliminating in the energy equation using
  conservation of angular momentum gives

Specific energy density in 3D
?eff
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• Thus, the 3D motion of a star in an axisymmetric
  potential ?(R,z) can be reduced to the motion of
  a star in a plane.
• This (non uniformly) rotating plane with
  cartesian coordinates (R,z) is often called the
  MERIDIONAL PLANE.
• ?eff(R,z) is called the EFFECTIVE POTENTIAL.

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Minimum in ?eff

• The minimum in ?eff has a simple physical
  significance. It occurs where
• (2) is satisfied anywhere in the equatorial plane
  z0.
• (1) is satisfied at radius Rg where
• This is the condition for a circular orbit of
  angular speed

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• So the minimum in ?eff occurs at the radius at
  which a circular orbit has angular momentum Lz.
• The value of ?eff at the minimum is the energy of
  this circular orbit.

?eff
R
E
?
Rcir
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• The angular momentum barrier for an orbit of
  energy E is given by
• The effective potential cannot be greater than
  the energy of the orbit.
• The equations of motion in the 2D meridional
  plane then become .

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• The effective potential is the sum of the
  gravitational potential energy of the orbiting
  star and the kinetic energy associated with its
  motion in the ? direction.
• Any difference between ?eff and E is simply
  kinetic energy of the motion in the (R,z) plane.
• Since the kinetic energy is non negative, the
  orbit is restricted to the area of the meridional
  plane satisfying .
• The curve bounding this area is called the ZERO
  VELOCITY CURVE since the orbit can only reach
  this curve if its velocity is instantaneously
  zero.

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• When the star is close to z0 the effective
  potential can be expanded to give

Can generally be ignored since the gravitational
force is only in the z direction close to the
equatorial plane.
?2
So, the orbit is oscillating in the z direction.
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• The orbits are bound between two radii (where the
  effective potential equals the total energy) and
  oscillates in the z direction.
• ????????graph of z vs R??????????

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• If the energy of the orbit is reduced the two
  points between which the orbit is bound
  eventually become one.
• You then get no radial oscillation.
• You have circular orbits in the plane of the
  galaxy.
• This is one of the closed orbits in an
  axisymmetric potential and has the property that.

(Minimum in effective potential.)
Centrifugal Force
Gravitational force in radial direction
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• The following figure shows the total angular
  momentum along the orbit.
• ??????????????????????????
• Angular momentum is not quite conserved but
  almost.
• This is because the star picks up angular
  momentum as it goes towards the plane and vice
  versa.
• These orbits can be thought of as being planar
  with more or less fixed eccentricity.
• The approximate orbital planes have a fixed
  inclination to the z axis but they precess about
  this axis.

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Orbits in Planar Non-Axisymmetrical Potentials

• Here the angular momentum is not exactly
  conserved.
• There are two main types of orbit
• BOX ORBITS
• LOOP ORBITS

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LOOP ORBITS

• Star rotates in a fixed sense about the centre of
  the potential while oscillating in radius
• Star orbits between allowed radii given by its
  energy.
• There are two periods associated with the orbit
• Period of the radial oscillation
• Period of the star going around 2?
• The energy is generally conserved and determines
  the outer radius of the orbit.
• The inner radius is determined by the angular
  momentum.

apocentre
pericentre
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Box Orbits

• Have no particular sense of circulation about the
  centre.
• They are the sum of independent harmonic motions
  parallel to the x and y axes.
• In a logarithmic potential of the form
• box orbits will occur when RltltRc and loop
  orbits will occur when RgtgtRc.

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JEANS EQUATION

• The jeans equation for oblate rotator
• a steady-state axisymmetrical system in which ?2
  is isotropic and the only streaming motion is
  azimuthal rotation

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• The velocity dispersions in this case are given
  by
• If we know the forms of ?(R,z) and ?(R,z) then at
  any radius R we may integrate the Jeans equation
  in the z direction to obtain ?2.

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Obtaining ?2

• Inserting this into the jeans equation in the
  R
• direction gives

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• Example suppose

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Tidal Stripping

• TIDAL RADIUSRadius within which a particle is
  bound to the satellite rather than the host
  system.
• Consider a satellite of mass profile ms(R) moving
  in a spherical potential ?(r) made from a mass
  profile M(r).

R
r
V(r)
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• The condition for a particle to be bound to the
  satellite rather than the host system is

Differential (tidal) force on the particle due to
the host galaxy
Force on particle due to satellite
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• So,
• Therefore the formula for Tidal Radius is
• The tidal radius is smallest at pericentre where
  r is smallest.
• The inequality can be written in terms of the
  mean densities.
• The satellite mass is not constant as the tidal
  radius changes.

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Lagrange Points

• There is a point between two bodies where a
  particle can belong to either one of them.
• This point is known as the LAGRANGE POINT.
• A small body orbiting at this point would remain
  in the orbital plane of the two massive bodies.
• The Lagrange points mark positions where the
  gravitational pull of the two large masses
  precisely cancels the centripetal acceleration
  required to rotate with them.
• At the lagrange points

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Effective Force of Gravity

• A particle will experience gravity due to the
  galaxy and the satellite along with a centrifugal
  force and a coriolis term.
• The effective force of gravity is given by
• The acceleration of the particle is given by

Coriolis term
Centrifugal term
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Jakobis Energy

• The JAKOBIS ENERGY is given by
• Jakobis energy is conserved because

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• For an orbit in the plane (r perpendicular to ?)

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• EJ is also known as the Jakobi Integral.
• Since v2 is always positive a star whose Jakobi
  integral takes the value EJ will never tresspass
  into a region where ?eff(x)gtEJ.
• Consequently the surface ?eff(x)gtEJ, which we
  call the zero velocity surface for stars of
  Jakobi Integral EJ, forms an impenetrable wall
  for such stars.

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• By taking a Taylor Expansion rrs, you end up
  with
• Where we call the radius rJ the JAKOBI LIMIT of
  the mass m.
• This provides a crude estimate of the tidal
  radius rt.

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Dynamical Friction

• DYNAMICAL FRICTION slows a satellite on its orbit
  causing it to spiral towards the centre of the
  parent galaxy.
• As the satellite moves through a sea of stars
  I.e. the individual stars in the parent galaxy
  the satellites gravity alters the trajectory of
  the stars, building up a slight density
  enhancement of stars behind the satellite
• The gravity from the wake pulls backwards on the
  satellites motion, slowing it down a little

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• The satellite loses angular momentum and slowly
  spirals inwards.
• This effect is referred to as dynamical
  friction because it acts like a frictional or
  viscous force, but its pure gravity.

109

• More massive satellites feel a greater friction
  since they can alter trajectories more and build
  up a more massive wake behind them.
• Dynamical friction is stronger in higher density
  regions since there are more stars to contribute
  to the wake so the wake is more massive.
• For low v the dynamical friction increases as v
  increases since the build up of a wake depends on
  the speed of the satellite being large enough so
  that it can scatter stars preferentially behind
  it (if its not moving, it scatters as many stars
  in front as it does behind).
• However, at high speeds the frictional force
  ?v-2, since the ability to scatter drops as the
  velocity increases.
• Note both stars and dark matter contribute to
  dynamical friction

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• The dynamical friction acting on a satellite of
  mass ms moving at vs kms-1 in a sea of particles
  with gaussian velocity distribution
• is

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• For an isotropic distribution of stellar
  velocities this is
• i.e. only stars moving slower than M contribute
  to the force. This is usually called the
  Chandrasekhar Dynamical Friction Formula.
• For sufficiently small vM we may replace f(vM) by
  f(0) to give
• For a sufficiently large vM, the integral
  converges to a definite limit and the frictional
  force therefore falls like
• vM-2.

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• When there is dynamical friction there is a drag
  force which dissipates angular momentum. The
  decay is faster at pericentre resulting in the
  staircase like appearance.
• ??????????????????????????????????

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• As the satellite moves inward the tidal force
  becomes greater so the tidal radius decreases and
  the mass will decay.
• ???????????????????????????????????????

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Scalar Virial Theorem

• The Scalar Virial Theorem states that the kinetic
  energy of a system with mass M is just
• where ltv2gt is the mean-squared speed of the
  systems stars.
• Hence the virial theorem states that

Virial
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• Equation of motion

This is Tensor Virial Theorem
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• E.g.
• So the time averaged value of v2 is equal to the
  time averaged value of the circular velocity
  squared.

117

• In a spherical potential

So ltxygt0 since the average value of xy will be
zero. ?ltvxvygt0
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Adiabatic Invariance

• Suppose we have a sequence of potentials ?p(r)
  that depends continuously on the parameter P(t).
• P(t) varies slowly with time.
• For each fixed P we would assume that the orbits
  supported by ?p(r) are regular and thus phase
  space is filled by arrays of nested tori on which
  phase points of individual stars move.
• Suppose

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• The orbit energy of a test particle will change.
• Suppose
• The angular momentum J is still conserved because
  r?F0.
• E.g. circular orbit in a flat potential

Initial Final
Let P(t)Vcir
?
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• So, if Vcir increases by a factor of 2 then the
  radius of the orbit would decrease by a factor of
  2 and the period would decrease by a factor of 4.
• E.g. in a potential
• In general the orbital eccentricity is conserved
  e.g. a circular orbit will stay circular.

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• In general, two stellar phase points that started
  out on the same torus will move onto two
  different tori.
• However, if P is changed very slowly compared to
  all characteristic times associated with the
  motion on each torus, all phase points that are
  initially on a given torus will be equally
  affected by the variation in P
• Any two stars that are on a common orbit will
  still be on a common orbit after the variation in
  P is complete. This is ADIABATIC INVARIANCE.