1 : Preliminaries | 6 : Dynamics I | 11 : Star Formation | 16 : Cosmology |
2 : Morphology | 7 : Ellipticals | 12 : Interactions | 17 : Structure Growth |
3 : Surveys | 8 : Dynamics II | 13 : Groups & Clusters | 18 : Galaxy Formation |
4 : Lum. Functions | 9 : Gas & Dust | 14 : Nuclei & BHs | 19 : Reionization & IGM |
5 : Spirals | 10 : Populations | 15 : AGNs & Quasars | 20 : Dark Matter |
(8.1a)
(8.1b) |
(8.2a) (8.2b) (8.2c) |
8.2b is Poisson's equation, for locations within the mass distribution
8.2c is Laplace's equation, for locations outside the mass distribution
(8.3a)
(8.3b) |
(8.4) |
Note that, with this definition, potential energy is always negative
(8.5) |
(8.6a)
(8.6b) |
(8.7) |
(8.8a) (8.8b) |
Here are two examples :
(8.9) |
where y = R / 2Rd, and In Kn are Bessel functions
or the 1st and 2nd kind
see [Topic 5.6a] for an analytic approximation and rotation curve.
(8.11a)
(8.11b)
(8.11c) |
We begin by looking at two illustrative cases and then deal with the general case.
(8.12) |
(8.13) |
where the five tensors are :
(8.14) (a,b,c,d,e) |
where i,j arises from the expansion: <vi vj> = <vi><vj> + i,j2
(8.15a) |
the Kinetic and potential energies are related for each tensor element
for example, they are related separately along each axis
(8.15b) |
(8.15c) |
So the total energy is negative : the system is bound !
its value is equal to either
Knowing Rg and measuring <v2> allows us to
determine M, the system mass.
What to use for Rg isn't obvious for most stellar systems
with no clear "edge" or "size"
However, we can make use of the median radius : Rm which
encloses half the mass
For many stellar systems, it turns out that Rg Rm / 0.4
(note Rm is written rh in B&T)
We then have :
(8.16) |
which resembles the circular orbit relation: M = V2 R / G, but applies to a general self-gravitating system.
Here are diagrams to illustrate the situation : [image]
(8.17a) |
B&T-1 fig 4.5 shows this relation for several ,
including projection corrections [ images ]
for isotropic velocities, = 0,
and we get, for small :
(8.17b) |
However, there are other constraints :
(8.18a) |
the net flow due to the velocity gradient is
(8.18b) |
the sum of these equals the net change to f in the region, ie at x, vx of size dx dvx
(8.18c) |
or, dividing by dx dvx dt, we get
(8.19a) |
but since
(8.19b) |
we have
(8.19c) |
adding the y and z dimensions, which are independent, we finally have
| (8.19d) |
This is the collisionless Boltzmann equation (CBE)
|
(8.20) |
Clearly, the phase space density (f) along the star's orbit is constant
ie the flow is "incompressible" in phase-space
for example
If we take moments of the CBE, we transform it into equations in these
new variables.
Lets look in more detail at these first two moments in v (see B&T-2 §4.8) :
(8.21) |
where n n(x,t) is the space density and
<vx> is the mean drift velocity along x
This is a simple continuity equation for the number of stars along the
x axis.
(8.22a) |
where
x2 is the velocity
dispersion about the mean velocity,
it arises from <vx2>
= <vx>2 + x2
(8.22b) |
where the summation convention applies (sum over repeated indices)
here, i=1,2,3 and j=1,2,3 refer to x,y,z, eg x2
y and v2
vy
(8.23) |
which is clearly analogous.
Here we look briefly at the first and second :
(8.24a) |
Introducing anisotropy parameters :
=
1 -
2 / r2
and =
1 -
2 / r2
and writing 2 for +
and Vrot for
<v > this becomes
(8.24b) |
which is equivalent to the equation of hydrostatic support :
dp /dr + anisotropic correction + centrifugal correction = Fgrav
(8.24c) |
This parallels the equation for hydrostatic support of an ideal gas, where
p = nkT
the equivalences are :
The answer is yes, by introducing two new powerful constraints :
demand that the system is in steady
state ( in equilibrium)
demand that the DF generate the
full potential (not just act as a tracer population)
|
(8.25) |
Any function of integrals of motion f (I1, I2, I3, ..... ) is also a solution of the steady state CBE |
(8.26a) (8.26b) |
where f here is the mass DF (ie we've multiplied f by the mean stellar mass)
(8.27) |
This is now a fundamental equation describing spherical equilibrium systems.
Solutions not only have self consistent and f, but
f also satisfies the steady state CBE.
Such a solution now describes a self-consistent, physically plausible stellar
dynamical system.
(8.28a)
(8.28b) |
These now describe a spherical, non-rotating, isotropic velocity dispersion
system.
They will be our starting point in constructing specific spherical models in
§ 8.8
(8.29) |
(8.30) |
where rm = largest radius out to which a star with Er can be found i.e. v=0 at (rm) = Er
Integrate f(Er) over velocity to find the density in terms of (eq 8.26a) :
(8.31) |
after substituing v = (2)½cos,
we find () = cn
n ( > 0)
where cn is a constant depending on n and F.
(8.32) |
(8.33) |
Recall, more +ve & Er means more bound.
Also, note f(Er) > 0 for Er < 0: there are unbound stars! .... we anticipate problems at large radii.
OK, substituting - ½v2 for Er and integrating f(Er) over v gives = 1 exp ( / 2)
(8.34) |
This is, in fact, the equation for a hydrostatic
sphere of isothermal gas, with 2 =
kT/m
Why is this ?
At every point, N(v) exp(-½v2/2), for both
the stellar system and a gas of atoms
it is irrelevant, therefore, whether the stars are collisionless or not, they
mimic a gas of atoms.
This method is called "core fitting" or "King's method"
Typical values for ellipticals cores are 10-20 h M/ L
suggesting minimal/no dark matter
To rectify this problem, we attempt to modify things slightly by removing the unbound stars:
(8.35) |
where o is a (dispersion like) parameter.
(8.36) |
Solve this by integration, choosing boundary conditions at r = 0 :
If the initial distribution is hotter
less concentrated
If the initial distribution is rotating slowly less concentrated & rotating oblate figure
If the initial distribution is rotating faster even less concentrated & prolate/bar figure
If the initial distribution is ellipsoidal
rotating ellipsoid, anisotropic everywhere
(8.37a)
(8.37b) |
where = bmax / bmin
(8.38a)
(8.38b) |
where 10 has units 10 km/s, m has
units of M, and
3 has units 103 M / pc3
Substituing, we get
(8.38c) |
System | N | R (pc) | V (km/s) | tcross | trelax | tage | age/relax |
Open Cluster | 102 | 2 | 0.5 | 106 | 107 | 108 | 10 |
Globular Cluster | 105 | 4 | 10 | 5 ×105 | 4 ×108 | 1010 | 20 |
Dwarf Galaxy | 109 | 103 | 50 | 2 ×107 | 1014 | 1010 | 10-4 |
Elliptical | 1011 | 104.5 | 250 | 108 | 4 ×1016 | 1010 | 10-7 |
Spiral Disk | 1011 | 104.5 | 20 | 1.5 ×109 | 6 ×1017 | 1010 | 10-8 |
MW Nucleus | 106 | 1 | 150 | 104 | 108 | 1010 | 100 |
Luminous Nucleus | 108 | 10 | 500 | 2 ×104 | 1010 | 1010 | 1 |
(Galaxy Cluster) | 102 | 5 ×105 | 500 | 109 | (3 ×109) | 1010 | (3) |