Download Images

Topic 8: Stellar Dynamics
Homework Solutions
6. Conceptual Question on the DF.
a) Systems of stars can be described by a 7-dimensional distribution function, DF or just f.
What are those 7 dimensions and what, exactly, does the DF describe? For a coordinate
system centered on the MW center, with the x-axis towards the sun and z-axis
perpendicular to the plane, describe qualitative terms the form of the velocity portion of
the DF for (i) stars at the galaxy center; (ii) stars in the solar neighborhood?
The 7 dimensions are three of space, r (x,y,z), three of velocity v (vx vy vz), and
time, t. The DF specifies the density of stars at each location in this phase space,
i.e. the relative number at each location, r, with each velocity, v, within a small
range dx dy dz dvx dvy dvz at time t.
(i) The velocity part of the DF at the center of a galaxy would be a spherically
symmetric function peaking at the origin, possibly in the form of a 3-D Gaussian.
(i.e. isotropic dispersion with Gaussian velocity field).
(ii) For stars in the solar neighborhood, the velocity part of the DF would be a
peaked function centered at a velocity corresponding to our circular velocity (e.g.
vx = 220 km/s, vy = 0, vz = 0) and with ellipsoidal form, with dispersions of about
40 km/s.
b) Write down the collisionless Boltzmann equation (CBE) for f, and briefly discuss the origin
of the three terms and why they sum to zero. What qualities of galaxies means that the DF for
stars must satisfy the CBE?
The first term is the time derivative of the number of stars in a small volume
element at a given location in phase space. The second and third terms track the
net flow of stars (into minus out) into this volume element. The second term
tracks the flow of stars across the spatial coordinates arising from the local
velocity field. The third term tracks the flow of stars across the velocity
coordinates arising from the local acceleration (force). Since stars move smoothly
through the phase space (no collisions) and are neither created nor destroyed, then
the sum of all three terms is zero. This continuity equation is also independently
true in three orthogonal directions.
This equation describes a system of particles that is collisionless (no jumps across
the phase space), and whose numbers are conserved (no creation or destruction of
particles). Since these are also conditions for stars in a galaxy, then the DF for
stars in a galaxy must be such that it is a solution to the CBE.
c) Consider following a star as it moves through the 6-D phase space. Why does the (phase
space) density of stars near this star remain constant along its orbit? Try to answer this in both
intuitive terms and with reference to the Lagrangian derivative, Df/Dt.
In intuitive terms: imagine you are at a location in the 6-D phase space,
surrounded by a few neighbors. Because you are all located at the same position
and share the same velocity, then you will ALL travel along the same path
through the phase space (recall Galileo’s discovery that gravitational trajectories
are independent of mass). So you will always be surrounded by your original
cohort. So your local density within the phase space will not change as you move
through it. I.e. f is constant along the trajectory.
Mathematically, the Lagrangian derivative Df/Dt, expresses the change in f
moving along the trajectory. Expanding this, we have:
Df
¶f ¶t
¶f ¶x
¶f ¶vx
=
+
+
Dt
¶t ¶t
¶x ¶t
¶vx ¶t
which in 3-D is the LHS of the CBE, which equals zero. Hence the derivative of f
along the trajectory through phase space is zero, so f is constant.
d) For a static potential, why is a distribution function with simple form f(E) automatically a
solution of the CBE, where E is the energy at a particular point in position-velocity phase
space? For a static spherical potential, what can you say about the velocity parts of the DF
when the DF has the following form: f(E), f(E,|L|)?
In a static potential, the energy of a star is constant as it moves along its orbit. But
we’ve just learned that any function that is constant along its trajectory through
phase space is a solution to the CBE. Hence the DF defined by E(r,v) will be a
solution to the CBE. But since the CBE is a linear equation, then functions of E
will also be solutions to the CBE. Hence any DF with form f(E) will be a solution
to the CBE.
For spherical systems, a DF with form f(E) must have isotropic velocities – i.e. the
velocity part of the DF is spherically symmetric for all r. For a spherical system
with DF with form f(E,|L|) must have anisotropic velocities, i.e. the velocity part
of the DF cannot be symmetric for any r.
e) How is the CBE "processed" to yield an observationally more accessible equation: the Jeans
equation? What does the Jeans equation express, in physical terms?
If we multiply each term in the CBE by velocity, and then integrate over that
velocity, we have “taken a moment of the equation” – meaning, since f gives the
relative number of stars at each location in the phase space, then we are
effectively taking the weighted mean of the velocities. This transformation
generates a new equation whose variables are much closer to the kinds of things
we can measure (n, <v>, <v2> etc). The equation reads (summation convention
over repeated indices):
¶áv j ñ
¶áv j ñ
¶F
1 ¶ ( ns i, j )
+ ávi ñ
= ¶t
¶xi
¶x j
n ¶xi
2
This is akin to Newton’s second law of motion, with acceleration terms on the left,
and force terms on the right. The RHS clearly has the gravitational force term, but
also a “pressure” term, in the form of a velocity dispersion. So the Jeans equation
is equivalent to an equation of “fluid flow”, where the stars are the fluid.
Compare it with Euler’s equation for fluid flow:
In the case of fluid flow, the pressure is always isotropic, but this need not be true
for the stellar system, since i,j is a tensor not a scalar.
f) Write down the Jeans equation for a spherical galaxy or star cluster. Describe in physical
terms what this equation means. What must astronomers measure, and how must How do
astronomers use this equation to derive the mass distribution in a spherical galaxy? Answer in
both What basic observations and assumptions must be made, and how can higher quality
observations help inform those assumptions?
2
1 d ( ns r )
s r2
Vrot2
dF
+ 2b
= n dr
r
r
dr
Clearly, the RHS gives the mass distribution, via d / dr = GM(<r) / r2 . What
needs to be measured are the quantities on the left. The light profile (after
deprojection) gives n, and stellar spectra yield Vrot and r (again, after appropriate
deprojection). The most challenging parameter is , the anisotropy parameter.
For a non-rotating galaxy, the third term is zero (Vrot = 0). Normally, one starts by
assuming an isotropic velocity field, so the second term is also zero ( = 0), in
which case the simple negative gradients in light (n) and dispersion (r) give the
mass profile. However, if we acknowledge the galaxy may have an anisotropic
velocity field, then the positive second term (with positive ) can partly cancel the
first term, and the mass profile is reduced.
To measure  one needs higher quality spectra that yield additional parameters
from the LOSVD, such as skewness, h3 and kurtosis, h4.