The Distribution Function I

23 Pages·2012·0.12 MB·English

The Distribution Function I
We have seen that the dynamics of our discrete system of N point masses is
given by 6N equations of motion, which allow us to compute 6N unknowns
(~x; ~v) as function of time t.
The system is completely specified by 6N initial conditions (~x ; ~v )
0 0
We can specify these initial conditions by defining the distribution function
(DF), also called the phase-space density
N N
f(~x; ~v; t ) = (cid:14)(~x x~ ) (cid:14)(~v ~v )
0 i;0 i;0
i=1 i=1
(cid:0) (cid:0)
Once f(~x; ~v; t) is specifiePd at any time t, we cPan infer f(~x; ~v; t ) at any
0
other time t
0
The DF f(~x; ~v; t) completely specifies a collisionless system
In the case of our smooth density distribution we define the 6 dimensional
phase-space density :
f(~x; ~v; t)d3~xd3~v
NOTE: A necessary, physical condition is that f 0
(cid:21)
The Distribution Function II
The density (cid:26)(~x) follows from f(~x; ~v) by integrating over velocity space:
(cid:26)(~x; t) = f(~x; ~v; t)d3~v
while the total mass follows from R R R
M(t) = (cid:26)(~x; t)d3~x = d3x~ d3~vf(~x; ~v; t)
R R R R R
It is useful to think about the DF as a probability function (once normalized
by M), which expresses the probability of finding a star in a phase-space
volume d3x~d3~v. This means we can compute the expectation value for any
quantity Q as follows:
Q(~x; t) = 1 d3~v Q(~x; ~v) f(~x; ~v; t)
h i (cid:26)(~x)
Q(t) = 1 d3~x d3~v Q(~x; ~v) f(~x; ~v; t)
R
h i M
R R
EXAMPLES:
RMS velocity: v2 = 1 d3~x d3~v v2 f(~x; ~v; t)
h i i M i
Velocity Profile: (x; y; v ) = R f(~x;R~v; t) dz dv dv
z x y
L
Surface Brightness: (cid:6)(x; y) = R R R f(~x; ~v; t) dz dv dv dv
x y z
R R R R
The Distribution Function III
Each particle (star) follows a trajectory in the 6D phase-space (~x; ~v), which
is completely governed by Newtonian Dynamics (for a collisionless system).
This trajectory projected in the 3D space ~x is called the orbit of the particle.
As we will see later, the Lagrangian time-derivative of the DF, i.e. the
time-derivative of f(~x; ~v; t) as seen when travelling through phase-space
along the particle’s trajectory, is
df
= 0
dt
This simple equation is the single most important equation for collisionless
dynamics. It completely specifies the evolution of a collisionless system, and
is called the Collisionless Boltzmann Equation (C.B.E.) or Vlasov equation.
The flow in phase-space is incompressible.
@f
NOTE: Don’t confuse this with = 0!!!!
@t
This is the Eulerian time-derivative as seen from a fixed phase-space
location, which is only equal to zero for a system in steady-state equilibrium.
Collisionless Dynamics in a Nutshell
(cid:26)(~x) = f(~x; ~v) d3~v
2(cid:8)(~x) = 4(cid:25)G(cid:26)(~x)
R
r
df
= 0
dt
The self-consistency problem of finding the orbits that reproduce (cid:26)(~x) is
equivalent to finding the DF f(~x; ~v) which yields (cid:26)(~x).
Problem: For most systems we only have constraints on a 3D projection of
the 6D distribution function.
Recall: (x; y; v ) = f(~x; ~v; t) dz dv dv
z x y
L
R R R
Circular & Escape Velocities I
Consider a spherical density distribution (cid:26)(r) for which the Poisson
Equation reads
1 @ r2 @(cid:8) = 4(cid:25)G(cid:26)(r)
r2 @r @r
from which we obtain that
(cid:0) (cid:1)
r2 @(cid:8) = 4(cid:25)G r (cid:26)(r) r2 dr = GM(r)
@r 0
with M(r) the enclosed mass. TRhis allows us to write
~ ~ d(cid:8) GM(r)
F (~r) = (cid:8)(~r) = ~e = ~e
grav r r
(cid:0)r (cid:0) dr (cid:0) r2
Because gravity is a central, conservative force, both the energy and angular
momentum are conserved, and a particle’s orbit is confined to a plane.
Introducing the polar coordinates (r; (cid:18)) we write
E = 1(r_2 + r2(cid:18)_2) + (cid:8)(r)
2
J = r2(cid:18)_
_
Eliminating (cid:18) we obtain the Radial Energy Equation:
2
1r_2 + J + (cid:8)(r) = E
2 2r2
Circular & Escape Velocities II
In the co-rotating frame, the equation of motion reduces to a
one-dimensional radial motion under influence of the effective potential
2
J
U(r) = + (cid:8)(r). The ‘extra’ term arises due to the non-inertial nature
2r2
of the reference frame, and corresponds to the centrifugal force
~ d J2 J2 v 2
F = ~e = ~e = (cid:18) ~e
cen r r r
(cid:0)dr 2r2 r3 r
(cid:16) (cid:17)
~ ~
For a circular orbit we have that F = F , so that we obtain the
cen grav
(cid:0)
circular speed.
d(cid:8) GM(r)
v (r) = r =
c
dr r
q q
Thus, rv2(r) measures the mass enclosed within radius r (in spherical
c
symmetry). Note that for a point mass v (r) r 1=2, which is called a
c (cid:0)
/
Keplerian rotation curve
Escape Speed: The speed a particle needs in order to ‘escape’ to infinity
v (r) = 2 (cid:8)(r)
esc
j j
Recall: The energy per unit mass is E =p 1v2 + (cid:8)(r). In order to escape to
2
infinity we need E 0, which translates into v2 2 (cid:8)(r)
(cid:21) (cid:21) j j
Projected Surface Density
Consider a spherical system with intrinsic, 3D luminosity distribution (cid:23)(r).
An observer, at large distance, observes the projected, 2D surface brightness
distribution (cid:6)(R)
To observer
z
R
r
z = distance along line−of−sight
R = projected radius from center
1 1 r dr
(cid:6)(R) = 2 (cid:23)(r)dz = 2 (cid:23)(r)
p
r2 R2
0 R (cid:0)
R R
This is a so-called Abel Integral, for which the inverse is:
1 1 d(cid:6) dR
(cid:23)(r) =
(cid:0)(cid:25) dR pR2 r2
r
(cid:0)
R
Thus, an observed surface brightness distribution (cid:6)(R) of a spherical
system can be deprojected to obtain the 3D light distribution (cid:23)(r). However,
because it requires the determination of a derivative, it can be fairly noisy.
Spherical Potential-Density Pairs
To compute the potential of a spherical density distribution (cid:26)(r) we can
make use of Newton’s Theorems
First Theorem A body inside a spherical shell of matter experiences
no net gravitational force from that shell.
Second Theorem The gravitational force on a body that lies outside a
closed spherical shell of mass M is the same as that
of a point mass M at the center of the shell.
Based on these two Theorems, we can compute (cid:8)(r) by splitting (cid:26)(r) in
spherical shells, and adding the potentials of all these shells:
r
(cid:8)(r) = 4(cid:25)G 1 (cid:26)(r ) r 2 dr + 1(cid:26)(r ) r dr
0 0 0 0 0 0
(cid:0) r
0 r
(cid:20) (cid:21)
R R
r
Using the definition of the enclosed mass M(r) = 4(cid:25) (cid:26)(r ) r 2 dr
0 0 0
0
this can be rewritten as
R
GM(r) 1
(cid:8)(r) = 4(cid:25)G (cid:26)(r ) r dr
0 0 0
(cid:0) r (cid:0)
r
R
Power-law Density Profiles I
Consider a spherical system with a simple power-law density distribution
(cid:11)
r (cid:0)
(cid:26)(r) = (cid:26)
0
r
0
(cid:16) (cid:17)
(cid:6)(R) = 2 1(cid:26)(r) r dr = (cid:26) r(cid:11) B((cid:11) 1; 1) R1 (cid:11)
0 (cid:0)
pr2 R2 0 2 (cid:0) 2 2
R (cid:0)
R
r
(cid:11)
M(< r) = 4(cid:25) (cid:26)(r ) r 2 dr = 4(cid:25)(cid:26)0r0 r3 (cid:11) ((cid:11) < 3)
0 0 0 (cid:0)
3 (cid:11)
0 (cid:0)
R
M(> r) = 4(cid:25) 1(cid:26)(r ) r 2 dr = 4(cid:25)(cid:26)0r0(cid:11) r3 (cid:11) ((cid:11) > 3)
0 0 0 (cid:0)
(cid:11) 3
r (cid:0)
R
NOTE: For (cid:11) 3 the enclosed mass is infinite, while for (cid:11) 3 the total
(cid:21) (cid:20)
mass (r ) is infinite: A pure power-law system can not exist in nature!
! 1
A more realistic density distribution consists of a double power-law:
At small radii: (cid:26) r (cid:11) with (cid:11) < 3
(cid:0)
/
At large radii: (cid:26) r (cid:12) with (cid:12) > 3
(cid:0)
/
B(x; y) is the so-called Beta-Function, which is related to the Gamma Function (cid:0)(x)
(cid:0)(x) (cid:0)(y)
B(x; y) = = B(y; x)
(cid:0)(x+y)
answer: the circular velocity is defined via the gradient of the potential. As shown above,
(cid:8) is only defined for 2 < (cid:11) < 3, and therefore so does v
c
Power-law Density Profiles II
The potential of a power-law density distribution is:
(cid:11)
4(cid:25)G(cid:26)0r0 r2 (cid:11) if 2 < (cid:11) < 3
(cid:0)
(cid:8)(r) = ((cid:11) 3)((cid:11) 2)
(cid:0) (cid:0)
otherwise
(
1
The circular and escape velocities of a power-law density distribution are:
(cid:11)
v2(r) = r d(cid:8) = G M(r) = 4(cid:25)G(cid:26)0r0 r2 (cid:11)
(cid:0)
c dr r 3 (cid:11)
(cid:0)
v2 (r) = 2 v2(r)
esc (cid:11) 2 c
(cid:0)
(cid:11) = 2: Singular Isothermal Sphere v = constant (flat rotation curve)
c
(cid:11) = 0: Homogeneous Sphere v r (solid body rotation)
c
/
NOTE: For (cid:11) > 3 you find that v (r) falls off more rapidly than Keplerian.
c
How can this be? After all, a Keplerian RC corresponds to a delta-function
density distribution (point mass), which is the most concentrated mass
distribution possible....