Transcript L7-Potentials-orbits

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
then
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 is only + and 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 – terms to know
In n space dimensions, some orbits can be decomposed in n independent periodic
motions – regular orbits (winding paths on a n-dimensional torus)
Constants of Motion – functions of phase-space coordinates and time which are
constant along the orbit
C (x,v,t) = const where v = dx/dt
In phase-space of 2n dimensions, there are always 2n independent constants of
motion. We will see in spherical potentials, there are 4 constants of motion (2
dimensions) relating to the 4 equations of motion.
Example: ϕ = Ωt + ϕ0  C = t – ϕ/Ω for a circular orbit where ϕ is the only dimension
Integrals of Motion – functions of phase-space coordinates alone that are constant
along any orbit
I (x,v) = const
Regular orbits have n isolating integrals and define a surface of 2n-1 dimensions
Example: E(x,v) = ½ v2 + PE  conservation of energy along an orbit
Orbits in Spherical Potentials – at blackboard
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
hyperbolas 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 found to be:
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 – at blackboard
Orbits in Non-Axisymmetric Potentials
Produce a richer variety of orbits – Φ = Φ (x,y,z) cartesian coordiates
Only 1 classical integral of motion – E = ½ v2 + Φ
though other integrals of motion may exist for certain potential
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 close 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
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 – at blackboard
Collisionless Dynamics – at blackboard