1. Collisionless Systems
does not refer only to actual collisions
but we showed collisions are rare
Collisionless: stellar motions under influence of mean gravitational potential
can neglect stellar interactions between passing stars!
Rational:
Gravity is a long-distance force
as opposed to the statistical mechanics of molecules in a box
gravitational force decreases as r-2

2. collisionless systems
total mass at r dependes on number of stars
but number of stars at r is ~ nr r2
so the net effect is that if the density is uniform, all regions contribute equally.
force determined by gross structure of system!
use smoothed gravitational potential
collisionless no evolution

3. Stellar interactions When are interactions important?
Consider a system of N stars or mass m
evaluate deflection of star as it crosses system
consider en encounter with star of mass m at a distance b:
where x = v t x v q r b
Fperp

4. Stellar interactions cont.
the change in the velocity Dvperp is then
using s = vt / b
an easier way of estimating Dvperp is by using impulse approximation: Dvperp = Fperp Dt
where Fperp is the force at closest approach and
the duration of the interaction can be estimated as : Dt = 2 b / v
using this estimate Dvperp is :

5. Number of encounters How many encounters occur at impact parameter b?
let system diameter be: 2R
surface density of stars is ~ N/šR2
the number encountering within b b + Db is:
each encounter has effect Dvperp but each one randomly oriented
sum is zero: b
b+Db

6. change in kinetic energy
but suming over squares (Dvperp2) is > 0
hence
now consider encounters over all b
then
b=0, 1/b is infinte!
need to replace lower limit with some bmin
estimate as expected distance of closest approach such that
set by equating kinetic and potential energies

7. Relaxation time
hence v2 changes by Dv2 each time it crosses the system where Dv2 is:
the system is therefore no longer collisionless when v2 ~ Dv2
which occurs after nrelax times across the system or nrelax crossing times
and thus the relaxation time is:

8. Relaxation time cont.
can use collisionless approx. only for t < trelax !
mass segregation occurs on relaxation timescale
also referred to as equipartition
where kinetic energy is mass independent
hence
and the massive stars, with lower specific energy sink to the centre of the gravitational p otential
globular cluster, N=105, R=10 pc
tcross ~ 2 R / v ~ 105 years
trelax ~ 108 years << age of cluster: relaxed
galaxy, N=1011, R=15 kpc
tcross ~ 108 years
trelax ~ years >> age of galaxy: collisionless
cluster of galaxie: trelax ~ age

9. Collisionless Systems
stars move under influence of a smooth gravitational potential
determined by overall structure of system
Statistical treatment of motions
collisionless Boltzman equation
Jeans equations
provide link between theoretical models (potentials) and observable quantities.
instead of following individual orbits
study motions as a function of position in system
Use CBE, Jeans to determine mass distributions and total masses

10. Collisionless Boltzman Equation
for a smooth potential F(x,t)
The number of stars with positions in volume d3x centred on x and with velocities d3v centred on v is:
where is the Distribution Function (DF)
or the Phase-Space Density
and everywhere is phase space
Now, for a stellar system, if we know the phase-space density at time t0, then we can predict it at any subsequent time

11. motions in phase-space
Flow of points in phase space corresponding to stars moving along their orbits.
phase space coords:
and the velocity of the flow is then:
where is the 6-D vector related to in the same way the 3-D velocity vector relates to x
stars are conserved in this flow, with no encounters, stars do not jump from one point to another in phase space.
they drift slowly through phase space

12. fluid analogy
regard stars as making up a fluid in phase space with a phase space density
assume that is a smooth function, continuous and differentiable
good for N 105
as in a fluid, we have a continuity equation
fluid in box of volume V, density r, and velocity v, the change in mass is then:
using the divergence theorem:

13. continuity equation must hold for any volume V, hence:
in same, manner, density of stars in phase space obeys a continuity equation:
If we integrate over a volume of phase space V, then we see that the first term is the rate of change of the stars in V, while the second term is the rate of outflow/inflow of stars from/into V.
as in the fluid case
note:
ind. variables, x,v, and grad of pot ind. of v!

14. Collisionless Boltzmann Equation
Hence, we can simplify the continuity equation to the CBE: or
in vector notation
notice that in the event of stellar encounters, then differs significantly from that given by
no longer collisionless
require additional terms in equation

15. CBE cont. can define a Lagrangian derivative
Lagrangian flows are where the coordinates travel along with the motions (flow)
hence x= x0 = constant for a given star
then we have:
and
rate of change of phase space density seen by observer travelling with star
the flow of stellar phase points through phase space is incompressible
f around the phase point of a given star remains the same

16. incompressible flow example of incompressible flow
idealised marathon race
each runner runs at constant speed
At start
the number density of runners is large, but they travel at wide variety of speeds
At finish
the number density is low, but at any given time the runners going past have nearly the same speed

17. The Jeans Equations
The DF (phase space density f) is a function of 7 variables and hence generally difficult to solve
Can gain insights by taking moments of the equation.
from the CBE:
then integrate over all possible velocities
where the summation over subscripts is implicit

18. Jeans equations cont. simplify
first term: velocity range doesn’t depend on time, hence we can take partial w.r.t. time outside
second term, vi does not depend on xi, so we can take partial w.r.t xi outside
third term: apply divergence theorem so that
but at very large velocities, f , hence last term is zero
must be true for all bound systems

19. First Jeans equation define spatial density of stars n(x)
and the mean stellar velocity v(x)
then our first (zeroth) moment equation becomes
this is the first Jeans equation
analogous to the continuity equation for a fluid.

20. Second Jeans equation
multiply the CBE by vj and then integrate over all velocities
taking the integrand from the third term
but from the divergence theorem
and thus B

21. 2nd Jeans equations cont.
now, clearly the term except where i=j where it is =1
hence we replace differential with the Kroneker delta function dij (=0 when i°j, =1 when i=j)
we can also define:
and thus we can rewrite B as:
the second Jeans equation!

22. 3rd Jeans equation
now, take the last equation and subtract vj times the continuity equation (1st Jeans equation) ie
which yields:
we can separate into a part that is due to the streaming motions and a part
that arises because the stars near any given point x do not have all the same velocity. ie, a velocity dispersion
gives a measure of the velocity dispersion in various directions C

23. 3rd Jeans Equation then C becomes:
This equation, the third Jeans equation, is very similar to the Euler equation for a fluid flow:
both LHS are of the same form, as are the first terms on the RHS thus it follows that the last term must represent some sort of pressure force

24. Stress Tensor Hence if we associate:
then the quantity is called the Stress Tensor
it describes a pressure which is anisotropic
not the same in all directions
and we can refer to a “pressure supported” system
the tensor is symmetric, as seen from the definition of
therefore we can chose a set of orthogonal axes such that the tensor is diagonal
ellipsoid with these axes, and semi-major axes given by the is called the Velocity Ellipsoid

25. Jeans Equations
So, we have the three Jeans equations, first applied to stellar dynamics by Sir James Jeans
again:
the continuity equation
the (first)
Jeans eq.
the “Euler flow” eq.
these equations link observable quantities to potential gradients which describe the mass distribution in a system.
can be applied to many diverse problems of stellar dynamics

26. Jeans Equation Compact form, s=x, y, z, R, r, …
e.g., oblate spheroid, s=[R,phi,z],
Isotropic rotator, a=[-Vrot^2/R, 0, 0], sigma=sigma_s
Tangential anisotropic (b<0), a=[b*sigma^2)/R, 0, 0], sigma=sigma_R=sigma_z=sigma_phi/(1-b),
e.g., non-rotating sphere, s=[r,th,phi], a=[-2*b*sigma_r^2/r, 0, 0], sigma_th=sigma_phi=(1-b)*sigma_r

27. Applications of the Jeans Equations
I. The mass density in the solar neighbourhood
Using velocity and density distribution perpendicular to the Galactic disc
cylindrical coordinates
Multiply CBE by vz and integrate over all velocities to obtain:
compare observed and predicted number-density and velocity distributions as a function of height Z above Galactic disc
assume steady state so drop first term

28. mass desnity of solar neighbourhood cont.
2nd and 4th terms contain
For small z/R, we find that
hence these terms are (z/R)2 smaller than the first and fifth terms
in the solar neighbourhood, R~8.5 kpc, |z|< 1kpc, these terms can be neglected.
Hence, Jeans equation reduces to:

29. mass density in solar neighbourhood
Using Poisson’s equation in cylindrical coordinates:
for axisymmetric systems
and density a function of R,z, so
where the radial force,
Near z=0, the second term on the LHS is relatively small, so can be neglected.

30. mass density cont. Poisson’s equation then reduces to
and we can then replace the potential in the Jeans equation as
all quantities on the LHS are, in principle, determinable from observations.
Hence we can determine local mass density r = r0
Known as the Oort limit
difficult due to double differentiation!

31. local mass density Don’t need to calculate for all stars Procedure
just a well defined population (ie G stars, BDs etc)
test particles (don’t need all the mass to test potential)
Procedure
determine the number density n, and the mean square vertical velocity, vz2, the variance of the square of the velocity dispersion in the solar neighbourhood.
need a reliable “tracer population” of stars
whose motions do not reflect formation
hence old population that has orbited Galaxy many times
ages > several x 109 years
N.B. problems of double differentiation of the number density n derived from observations
need a large sample of stars to obtain vz as f(z)
ensure that vz is constant in time
ie stars have forgotten initial motion

32. local mass density > 1000 stars required Oort : r0 = 0.15 Msol pc-3
K dwarf stars (Kuijken and Gilmore 1989)
MNRAS 239, 651
Dynamical mass density of r0 = 0.11 Msol pc-3
also done with F stars (Knude 1994)
Observed mass density of stars plus interstellar gas within a 20 pc radius is r0 = 0.10 Msol pc-3
can get better estimate of surface density
out to 700 pc S ~ 90 Msol pc-2
from rotation curve Srot ~ 200 Msol pc-2

33. Application II Velocity dispersions and masses in spherical systems
For a spherically symmetric system we have
a non-rotating galaxy has
and the velocity ellipsoids are spheroids with their symmetry axes pointing towards the galactic centre
If we define b as
the degree of anisotropy of the velocity dispersion

34. spherical systems cont.
then the Jeans equation is:
Need observations to determine , vr and b as a function of radius r for a stellar population in a galaxy
then, with
and the circular speed

35. Mass of spherical systems
we can use this equation to establish the total mass (and as a function of radius) for a spherical galaxy
including any unseen halo
Example 1) globular clusters and satellite galaxies around major galaxies
within 150 kpc
Need n(r), vr2, b to find M(r)
Several attempts for Milky Way
all suffer from problem of small numbers
N ~ 15

36. Mass of the Milky Way
Little and Tremaine tried two extreme values of b
Isotropic orbits:
Radial orbits
If we assume a power law for the density distribution
then we obtain

37. Mass of the Milky Way Drop first term, we have solutions
E.g., Point mass a=0, Tracer gam=3.5, Isotro b = 0 :
E.g. Flat rotation a=1, Self-grav gamma=2, Radial anis. b >0 :
For the isotropic case, Little and Tremaine determined a TOTAL mass of 2.4 (+1.3, -0,7) 1011 Msol
3 times the disc and spheroidal mass