Introduction to potential theory – at black board
Potentials of simple spherical systems
Point mass- keplerian potential
Homogeneous sphere  ρ = constant and M(r)=(4/3)πr3ρ
With radial size a
for r < a
3
for r > a
Isochrone potential – model a galaxy as a constant density at
the center with density decreasing at larger radii. One
potential with these properties:
where b is characteristic radius that
defines how the density decreases with r
Density pair given in BT (2-34) and yields
at center and
at r >>b
Modified Hubble profile – derived from SBs for ellipticals
where a is core radius and j is
luminosity density
Power-law density profile – many galaxies have surface
brightness profiles that approximate a power-law over
large radii
If
we can compute M(r) and Vc(r)
If α = 2, this is an isothermal sphere (density goes as 1/r2)
 Can be used to approximate galaxies with flat rotation curves;
need outer cut-off to obtain finite mass
Plummer Sphere – simple model for round galaxies/clusters
This potential “softens” force between particles in N-body simulations by avoiding
the singularity of the Newtonian potential. The density profile has finite core
density but falls as r-5 at large r (too steep for most galaxies).
Jaffe and Hernquist profiles
Both decline as r-4 at large radii which works well with galaxy models produced
from violent relaxation (i.e. stellar systems relax quickly from initial state to quasiequilibrium).
Hernquist has gentle power-law cusp at small r while Jaffe has steeper cusp.
Potential
density
Density distributions for various simple spherical potentials
Navarro, Frenk and White (NFW) profile
Good fit to dark matter haloes formed in simulations
Problem – mass diverges logarithmically with r  must be cut off at large r
Potentials for Flattened Models: Axisymmetric potential
Kuzmin Disk (cylindrical coordinates)
At points with z<0, Φk is identical with the
potential of a point mass M at (R,z) = (0,a) and
when z>0, Φk is the same as the potential
generated by a point mass at (0,-a).
Everywhere except
on plane z=0
Use divergence theorem to find the
surface density generated by Kuzmin
potential
Kuzmin (1956) or Toomre model 1 (1962)
Miyamoto & Nagai (1975) introduced a
combination Plummer sphere/Kuzmin
disk model
where b is aP in previous Plummer
notation
a=0  Plummer sphere
b=0  Kuzmin disk
b/a ~ 0.2 similar to disk galaxies
Stellar Orbits
• For a star moving through a galaxy, assume its motion does not change the
overall potential
• If the galaxy is not collapsing, colliding, etc., assume potential does not change
with time
Then, as a star moves with velocity v, the potential at its location changes as
Recall
(grad of potential is force on star)
Then,
Energy along orbit remains constant (KE always + ; PE goes to 0 at large x)
Star escapes galaxy if E > 0
Circular velocity
angular velocity
In a cluster of stars, motions of the stars can cause the potential to
change with time. The energy of each individual star is no longer
conserved, only the total for the cluster as a whole.
cluster KE
cluster PE
Stars in a cluster can change their KE and PE as long as the sum
remains constant. As they move further apart, PE increases and
their speeds must drop so that the KE can decrease.
The virial theorem tells how, on average, KE and PE are in balance
Begin with Newton’s law of gravity and add an external force F
Take the scalar product with xα and sum over all stars to get…
VT is tool for finding masses of star clusters and galaxies where the orbits are not necessarily
circular. For system in steady-state (not colliding, etc), use VT to estimate mass
Assume average motions are isotropic
<v2> ≈ 3σr2
KE ≈ (3σr2/2) (M/L) Ltot
Get PE by M = Ltot (M/L) then use galaxy SB to find volume density of stars.
Orbits in Spherical Potentials – at blackboard
Equations of motion:
In n spatial dimensions, some orbits can be decomposed into n independent
periodic motions – regular orbits
Integrals of Motion – functions of phase-space coordinates that are constant along
any orbit (not time dependent)
Regular orbits have n isolating integrals and define a surface of 2n-1 dimensions
2 independent integrals of motion are:
Each integral of motion
defines a surface in 3-d
space (R, VR, Vϕ)
Vϕ
VR
Constant E surface
revolves around R-axis
R
Constant L surface is a
hyperbola in the R, Vϕ
plane
*note that both L and J are used to denote angular momentum
Intersection is closed
curve and the orbit
travels around this
curve
The integrals of motion combine (see BT 3.1 for treatment) to produce a
differential equation
d 2u
F(1 / u)
+
u
=
df 2
L2u2
where u = 1/R
Solutions to this equation have 2 forms:
bound = orbits oscillate between finite limits in R
unbound = R  ∞ or u  0
Each bound orbit is associated with a periodic solution to this equation. Star in
this orbit also has a periodic azimuthal motion as it orbits potential center.
Relationship between azimuthal and radial periods is:
Tf =
2p
TR
Df
2p
Df is usually not a rational number so orbit is not closed in most spherical
potentials
• star never returns to starting point in
phase-space
• typical orbit is a rosette and eventually
passes every point in annulus
between pericenter and apocenter
Two special potentials where all bound orbits are closed
1) Keplerian potential – point mass
F=-
GM
R
- radial and azimuthal periods are equal
- all stars advance in azimuth by Df = 2p between successive pericenters
2) Harmonic potential – homogeneous sphere
1
F = W2 R 2 + const
2
- radial period is ½ azimuthal period
- stars advance in azimuth by Df = p between successive pericenters
Real galaxies are somewhere between the two, so most orbits are rosettes
advancing by
p < Df < 2p
 Stars oscillate from apocenter to pericenter and back in a shorter time
than is required for one complete azimuthal cycle about center
Orbits in Axisymmetric Potentials – at blackboard
Φeff = ½ Vo2 ln (R2 + z2/q2) + Lz2/(2R2)
Φ(R,z)
q= axial ratio
•Resembles Φ of star in oblate
spheroid with constant Vc = Vo
•Φeff rises steeply toward z-axis
•If only E and Lz constrain motion of star on R,z plane, star should travel
everywhere within closed contour of constant Φeff
•But, stars launched with different initial conditions with same Φeff follow
distinct orbits
•Implies 3rd isolating integral of motion – no analytically form
Nearly Circular Orbits in
Axisymmetric Potentials
– epicyclic approximation
In disk galaxies, many stars are on nearly circular orbits
derive approximate valid solutions to d2R/dt2 and d2z/dt2.
Taylor expansion series around (Rg,0) or (x,z) = (0,0)
(Ignore higher order terms)
Note: x = R – Rg
 yields harmonic potential
Define two new quantities:
Epicyclic and Vertical frequencies
Then equations of motion become
x and z evolve like the displacements of 2 harmonic oscillators with frequency κ and ν
Integrals of Motion are then
Now relate back to the potential…recall
Then the equations become
Since the angular speed is related to the potential as
We can now write kappa in terms of the angular speed
This is related to the (well known) Oort constant B
The Oort constants, first derived by Jan Oort in 1927, characterize the
angular velocity of the Galactic disk near the Sun using observationally
determined quantities.
It can be shown that
Ωo = A – B
κ2 = -4B(A-B) = -4BΩo at the Sun
Hipparchos proper motions of
nearby stars yield (Feast &
Whitlock 1997):
A = 14.8 ± 0.8 km/s/kpc
B = -12.4 ± 0.6 km/s/kpc
κo = 36 ± 10 km/s/kpc
A measures “shear” in the disk – would be zero for
solid body rotation
B measures rotation of Galaxy or local L gradient
•Sun makes 1.3 oscillations in radial
direction in the time to complete one
orbit around GC
•Does not close (rosette)
Continue with Nearly Circular orbit approximation on the board….
Integrals of motion
Equations of motion in x , z and y directions
Derive epicycle shapes
X/Y = κ/(2Ω)
Pt mass (keplarian rotation curve)
κ=Ω and X/Y=1/2
Homogeneous sphere
κ=2Ω and X/Y=1
Orbits in Non-Axisymmetric Potentials
Produce a richer variety of orbits – Φ = Φ (x,y) or Φ (x,y,z) cartesian coordinates
Only 1 classical integral of motion – E = ½ v2 + Φ
though other integrals of motion may exist for certain potentials
which cannot be represented in analytical form
Orbits in non-axisymmetric potential can be grouped into Orbit Families.
Examples can be found in two types of NAPs.
Separable Potentials
- All orbits are regular (i.e. the orbits can be decomposed into 2 or 3
independent period motions (in 2 or 3-d)
- All integrals of motion can be written analytically
- These are mathematically special and therefore not likely to describe real
galaxies. However, numerical simulations for NA galaxy models with
central cores have many similarities with separable potentials.
Distinct families are associated with a set of closed, stable orbits. In 2-d:
• Oscillates back and forth along major axis (box orbits)
• Loops around the center (loop orbits)
2-D orbits in non-axisymmetric potential
For larger R > Rc, orbits are mostly loop orbits
• initial tangential velocity of star
determines width of elliptical annulus
(similar to way in which width of
annulus in AP varies with Lz)
• Rotation curve is flat with q=1 at large R
For small R<<Rc, orbits become box orbits
• potential approximates that of homogeneous
sphere
• orbits are like harmonic oscillator
In 3-d (triaxial potential), there are four families of orbits:
box orbit: move
along longest
(major) axis,
parent of family
Intermediate and
short axis orbits
are unstable!
outer long-axis
tube orbit: loop
around major
axis
Intermediate axis
loop orbits are
unstable!
Triaxial potentials with
cores have orbit
families like those in
separable potentials.
short axis tube
orbit: loop around
minor axis
(resemble
annular orbit of
axisymmetric
potential
inner long-axis
tube orbit: loop
around major
axis
Scale Free Potentials
All properties have either a power-law or logarithmic dependence on radius
(i.e. ρ ~ r-2)
These density distributions are similar to central regions of E’s and halos of
galaxies in general
If density falls as r-2 or faster, box orbits are replaced by boxlets
box orbits about minor-axis arising from resonance between motion in
x and y directions (Miralda-Escude & Schwarzchild 1989)
Some irregular orbits exist as well (i.e. stochastic motions which wander
anywhere permitted by conservation of energy).
Stellar Dynamical Systems
Unlike molecules in a gas, where collisions distribute and average out their
motions, stellar systems are governed strictly by gravitation forces.
For stars, the cumulative effect of small pulls of distant stars is more important
than large pulls caused as one star passes close to another. But we will see that
even these have little effect over a galaxy’s lifetime of randomizing or relaxing
the stellar motions. Therefore,
The smooth Galactic potential of the Milky Way almost entirely dominates the
motion of the Sun.
Consider a system where physical collision are rare. This can be idealized as N
point-sized bodies with masses Mi, positions ri and velocities vi
Potential runs over all pairs
twice, hence the 1/2
Equations of motion are
A general result of the equations of motions is the scalar virial theorem
where T is KE and U is PE
since E = T + U
Total mass M and energy E of N-body system define a characteristic velocity
and size  the virial velocity and virial radius
The crossing time is a system can be then be defined
tc is constant even for systems far from
equilibrium
tc is time scale over which system evolves toward equilibrium
For systems near equilibrium, Vv2 = GM/Rv
density
For systems w/galaxy-like profiles
tc ~ 1.36 (Gρh)-0.5
radius
Rv = 2.5 Rh (half-mass radius)
where density is defined within the half-mass
Since crossing time is supposed to be just the typical time scale for
orbital motion, we can define the crossing time as
tc = (Gρh)-0.5
Under virial assumptions, crossing time depends only on density and
increases as density decreases to the square power.
How does the crossing time relate to the relaxation time, or time it
would take the small pulls of distant stars to randomize the stellar
orbits?
In a distant encounter, the force of one star on another is so weak that stars
hardly deviate. We can use the impulse approximation to calculate the forces
that a star would feel as it moves along an undisturbed path
m
ΔVt
where ΔVt = 2Gm/(bV)
But, when is it a close enough encounter to matter??
A strong encounter occurs when, at closest approach, the change in the PE
is as great as the initial KE
Gm2/r ≥ 1/2mV2
so
r ≤ rs = 2Gm/V2 this is the strong encounter radius
Near the Sun, V ~ 30 km/s, m ~ 0.5 M then, rs ~ 1 AU  pretty close!
How often does this occur?
Assume the Sun is moving with speed V for a time t through a cylinder with
radius rs and volume π rs2 V t. What is time ts such that n π rs2 V t = 1?
= 1015 yrs with typical solar values
Back to considering effects of distant encounters…
Using impulse approximation, a star will have dnenc encounters during a single
passage through a system
Surface density of stars
area of annulus with
radius b and width db
The star receives many deflections due to dn encounters, each with random
direction, so expected tangential velocity after time t is obtained by adding
perturbations in quadrature
Total velocity perturbation acquired in one crossing time
So a single distant encounter may barely effect the star, but the cumulative effects
are important!
Now estimate V as the virial velocity Vv where Vv ~ sqrt(GNm/Rv) and let
Then…
….is the total change in a star’s velocity per crossing time tc
Relaxation time is the time over which the cumulative effects of stellar
encounters become comparable to a star’s initial velocity
For galaxies N~1011 stars and tr = 5x108 tc (relaxation important after ~100 million
crossings)
But, galaxies in general are only ~100 crossing times old  cumulative effects of
encounters between stars are pretty insignificant!
For globular clusters, N~106 or 105 stars and tc~105 yrs
tr=5000tc  stellar encounters important after ~109-1010 yrs
In denser cores, encounters play a key role…
Collisionless Dynamics – In the continuum limit, stars move in the
smooth gravitational field Φ(x,t) of the galaxy. So instead of thinking
about motion in 6N dimensions, we can simplify to just 6 dimensions.
Galaxy may be described by a one-body distribution function (probability
density in phase-space):
f(x,v,t)ΔxΔyΔzΔvxΔvyΔvz
-average number of stars in phase-space volume at (x,v) and time t
Number density (at position x) is then
n(x,t) = integral f(x,v,t) d3v
Average velocity
<v(x,t)>n(x,t) = integral v f(x,v,t) d3v
Find equations relating changes in the density and DF as stars move
about the galaxy…
As stars move through a galaxy, how do changes in the density
and DF of stars relate to the potential?
vx
Simplify to one direction x
•n(x,t) Δx is # stars in “box”
between x and x+Δx at time t
•After time Δt:
x
x+Δx
Δx[n(x,t + Δt) – n(x,t)] = n(x,t)v(x) Δt – n(x+Δx,t) v(x+Δx) Δt
entered in Δt
left in Δt
Left side is the change in # between the two times and right side is
change in stars entering and leaving which should be equivalent
Take limits at Δt  0 and Δx  0
Equation of continuity – stars are not destroyed or added
Rate of stars flowing in + rate of stars flowing out is zero.
The Collisionless Boltzmann equation is like the EOC, but allows for
changes in velocity and relates changes in DF to forces on the stars.
(x+Δx,v+Δv)
v + Δv
v
(x,v)
x
x+Δx
 Assume acceleration of star dv/dt
depends only on potential at (x,t)
 If dv/dt>0, after Δt all stars will be
moving faster by Δt(dv/dt)
 Stars with velocities between v and vΔt(dv/dt) move in
 Those with velocities below v+Δv
have left
Then, the net # of stars that enter the center box after Δt due to change in v and x
In the limit that all Δ’s are small
EOC in phase-space space
Under gravity, stars’ acceleration depends only on position
So, I-D CBE
And in 3D
Collisionless Boltzmann Equation – the fundamental equation of stellar
dynamics
Equation holds if stars are neither created or destroyed and change
position & velocity smoothly.
BT describe CBE this way = “The flow of stellar phase points through phase
space is incompressible; the phase-space density f around the phase point
of a given star always remains the same.”
If there are close encounters between stars, these can alter the position
and velocity much faster than a smoothed potential. In this case, the
effects are given as an extra “collisional” term on the right side.
Since f is a function of seven variables (phase-space and time), the complete
solution of CBE is usually too difficult – but, velocity moments of CBE can
be used to answer specific questions in stellar dynamics
Integrate CBE over velocity and apply
 0th velocity moment
 1st velocity moment
where i=1-3 (3D)
1.
0th
moment of CBE
This is the EOC – no
surprise since we just
integrate over velocity
Now multiply CBE by vj and integrate over velocity and apply
 2nd velocity moment
2. 1st moment of CBE
From 2nd moment of velocity, velocity dispersion is
Combine 1 and 2 and divide by n
3.
acceleration
kinematic viscosity
gravity
pressure
-analogous to Euler’s equation in fluid mechanics
Equations 1, 2, and 3 are known as Jeans Equations
(Sir James Jeans, 1919) - first applied to stellar dynamics
Applying Jeans Equations and CBE – Mass Density in the Galactic Disk
•Select tracer stellar type (K dwarfs) and measure density n(z) at height z
above disk (coordinates (z, vz) instead of (x, v))
•Assume potential, DF and number density n do not change with time
•At large z, <vz>n(z)  0, thus Eq. 1 gives <vz> = 0 everywhere
•Eq. 3 with σ = σz and since <vz>=0 we lose the 1st and 2nd term
If we measure how density and sigma changes with z, we get vertical force at
any height z.
Now use Poisson’s Eq., which relates that force to mass density of the Galaxy.
Assume MW is axisymmetric so potential and density only depend on R, z
Since V(R) is ~constant at Sun, let the last term = 0
Then, if we know # density wrt z and the velocity dispersion in z, we get density!
More accurate to determine mass surface density Σ than volume density ρ
Oort (1932) measured n(z) for F dwarfs and K giants and obtained
Σ (<700pc)=90 Msun/pc2 (assumed σz didn’t vary with height)
ρo(Ro,0)=0.15 Msun/pc3
Bahcall (1984) gets ρo(Ro,0)=0.18 Msun/pc3 averaging several tracers
More recent work with fainter K dwarfs (more numerous and evenly spread out)
indicates σz increases with z:
σz ~ 20 km/s @ 250 pc and 30 km/s @ 1 kpc
Yields Σ (<1100pc) = 71 +/- 6 Msun/pc2
If some of this is in the halo, disk is ~50 to 60 Msun/pc2
Compare this dynamical estimate to mass summed up in gas and stars in the disk
 40 to 55 Msun/pc2
Not much DM in the disk
What is disk surface density required to maintain a flat rotation curve?
Σrot = Vc2/(2πGRo) ~ 210 Msun/pc2
- way too big!!
Also tells us that most of this mass must be distributed in a halo component
well out of the disk (scale height above 1 kpc)