Introduction to potential theory - PowerPoint PPT Presentation

About This Presentation
Title:

Introduction to potential theory

Description:

Introduction to potential theory at black board Potentials of simple spherical systems Point mass- keplerian potential Homogeneous sphere = constant and M(r ... – PowerPoint PPT presentation

Number of Views:147
Avg rating:3.0/5.0
Slides: 41
Provided by: Vicki140
Category:

less

Transcript and Presenter's Notes

Title: Introduction to potential theory


1
Introduction to potential theory at black
board Potentials of simple spherical
systems Point mass- keplerian potential Homoge
neous sphere ? ? constant and
M(r)(4/3)pr3? With radial size a for
r lt a
for r gt a
3
2
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 gtgtb
Modified Hubble profile derived from SBs for
ellipticals where a is core radius and j
is luminosity density
3
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 a 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
4
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 quasi-equilibrium).
Hernquist has gentle power-law cusp at small r
while Jaffe has steeper cusp.
Potential
density
5
Density distributions for various simple
spherical potentials
6
Navarro, Frenk and White (NFW) profile Good
fit to dark matter halos 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 zlt0, Fk is identical with the
potential of a point mass M at (R,z) (0,a) and
when zgt0, Fk is the same as the potential
generated by a point mass at (0,-a).
Everywhere except on plane z0
7
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 a0 ? Plummer
sphere b0 ? Kuzmin disk b/a 0.2 similar to
disk galaxies
8
  • 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 gt 0

9
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.
10
The virial theorem tells how, on average, KE and
PE are in balance Begin with Newtons law of
gravity and add an external force F Take the
scalar product with xa 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 ltv2gt 3sr2 KE
(3sr2/2) (M/L) Ltot Get PE by M Ltot (M/L)
then use galaxy SB to find volume density of
stars.
11
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
12
V?
Each integral of motion defines a surface in 3-d
space (R, VR, V?) Constant E surface revolves
around R-axis Constant L surface is a hyperbola
in the R, V? plane Intersection is closed curve
and the orbit travels around this curve
VR
R
note that both L and J are used to denote
angular momentum
The integrals of motion combine (see BT 3.1 for
treatment) to produce a differential
equation where u 1/R
13
Solutions to this equation have 2 forms bound
orbits oscillate between finite limits in
R unbound R ? 8 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 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

14
Two special potentials where all bound orbits are
closed
  • Keplerian potential point mass
  • - radial and azimuthal periods are equal
  • - all stars advance in azimuth by
    between successive pericenters
  • 2) Harmonic potential homogeneous sphere
  • - radial period is ½ azimuthal period
  • - stars advance in azimuth by between
    successive pericenters
  • Real galaxies are somewhere between the two, so
    most orbits are rosettes advancing by
  • ? Stars oscillate from apocenter to pericenter
    and back in a shorter time than is required for
    one complete azimuthal cycle about center

15
Orbits in Axisymmetric Potentials at blackboard
Feff ½ Vo2 ln (R2 z2/q2) Lz2/(2R2)
F(R,z)
  • q axial ratio
  • Resembles F of star in oblate spheroid with
    constant Vc Vo
  • Feff rises steeply toward z-axis

16
  • If only E and Lz constrain motion of star on R,z
    plane, star should travel everywhere within
    closed contour of constant Feff
  • But, stars launched with different initial
    conditions with same Feff follow distinct orbits
  • Implies 3rd isolating integral of motion no
    analytically form

17
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 ? and ?
18
Integrals of Motion are then
Now relate back to the potentialrecall 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
19
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
  • Oo A B
  • ?2 -4B(A-B) -4BOo 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
  • ?o 36 10 km/s/kpc
  • Sun makes 1.3 oscillations in radial direction in
    the time to complete one orbit around GC
  • Does not close (rosette)

A measures shear in the disk would be zero
for solid body rotation B measures rotation of
Galaxy or local L gradient
20
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 ?/(2O) Pt mass (keplarian rotation curve)
?O and X/Y1/2 Homogeneous sphere ?2O and
X/Y1
21
Orbits in Non-Axisymmetric Potentials
Produce a richer variety of orbits F F (x,y)
or F (x,y,z) cartesian coordinates Only 1
classical integral of motion E ½ v2
F ?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)

22
2-D orbits in non-axisymmetric potential
  • For larger R gt 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 q1 at large R
  • For small RltltRc, orbits become box orbits
  • potential approximates that of homogeneous sphere
  • orbits are like harmonic oscillator

23
Triaxial potentials with cores have orbit
families like those in separable potentials.
In 3-d (triaxial potential), there are four
families of orbits
box orbit move along longest (major) axis,
parent of family
short axis tube orbit loop around minor axis
(resemble annular orbit of axisymmetric potential
Intermediate and short axis orbits are unstable!
inner long-axis tube orbit loop around major axis
outer long-axis tube orbit loop around major axis
Intermediate axis loop orbits are unstable!
24
Scale Free Potentials All properties have either
a power-law or logarithmic dependence on radius
(i.e. ? r-2) These density distributions are
similar to central regions of Es 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).
25
  • 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 galaxys 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
26
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
27
For systems near equilibrium, Vv2 GM/Rv
density
For systems w/galaxy-like profiles Rv 2.5 Rh
(half-mass radius) tc 1.36 (G?h)-0.5
where density is defined within the half-mass
radius Since crossing time is supposed to be
just the typical time scale for orbital motion,
we can define the crossing time as 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?
tc (G?h)-0.5
28
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
?Vt
where ?Vt 2Gm/(bV)
m
impact parameter
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 p rs2 V t.
What is time ts such that n p rs2 V t 1?

1015 yrs with typical solar values
29
Back to considering effects of distant
encounters Using impulse approximation, a star
will have dnenc encounters during a single
passage through a system
area of annulus with radius b and width db
Surface density of stars
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
Each decade between bmin and Rv contribute
equally to total deflection Note rs2bmin
Total velocity perturbation acquired in one
crossing time So a single distant encounter
may barely effect the star, but the cumulative
effects are important!
30
Now estimate V as the virial velocity Vv where Vv
sqrt(GNm/Rv) and let Then
.is the total change in a stars velocity
per crossing time tc
Relaxation time is the time over which the
cumulative effects of stellar encounters become
comparable to a stars initial velocity For
galaxies N1011 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, N106 or 105 stars and
tc105 yrs tr5000tc ? stellar encounters
important after 109-1010 yrs In denser cores,
encounters play a key role
31
Collisionless Dynamics In the continuum limit,
stars move in the smooth gravitational field
F(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)?x?y?z?vx?vy?vz -averag
e 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 ltv(x,t)gtn(x,t) integral v f(x,v,t)
d3v Find equations relating changes in the
density and DF to the gravitational potential as
stars move about the galaxy
32
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) ?x is stars in box between x and x?x
    at time t
  • After time ?t

x x?x
?xn(x,t ?t) n(x,t) n(x,t)v(x) ?t
n(x?x,t) v(x?x) ?t
entered in ?t left in ?t
Left side is the change in between the two
times and right side is change in stars entering
and leaving which should be equivalent
33
Take limits at ?t ? 0 and ?x ? 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.
  • Assume acceleration of star dv/dt depends only on
    potential at (x,t)
  • If dv/dtgt0, after ?t all stars will be moving
    faster by ?t(dv/dt)
  • Stars with velocities between v and
  • v-?t(dv/dt) move in
  • Those with velocities below v?v have left

Then, the net of stars gained the center box
after ?t due to change in v and x
34
In the limit that all ?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
35
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
36
Integrate CBE over velocity and apply
? 0th velocity moment

? 1st velocity
moment

where i1-3 (3D) 1. 0th moment of CBE
This is the EOC no surprise since we just
integrate over velocity
2. 1st moment of CBE
37
From 2nd moment of velocity, velocity dispersion
is Combine 1 and 2 and divide by n 3.
2
acceleration
kinematic viscosity
gravity
pressure
-analogous to Eulers equation in fluid mechanics
Equations 1, 2, and 3 are known as Jeans
Equations (Sir James Jeans, 1919) - first
applied to stellar dynamics
38
2
  • 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, ltvzgtn(z) ? 0, thus Eq. 1 gives ltvzgt
    0 everywhere
  • Eq. 3 with s sz and since ltvzgt0 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 Poissons Eq., which relates that force to
mass density of the Galaxy. Assume MW is
axisymmetric so potential and density only depend
on R, z
This is equal to Vc2(R)
39
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 S than
volume density ?
Oort (1932) measured n(z) for F dwarfs and K
giants and obtained S (lt700pc)90 Msun/pc2
(assumed sz didnt vary with height) ?o(Ro,0)0.15
Msun/pc3 Bahcall (1984) gets ?o(Ro,0)0.18
Msun/pc3 averaging several tracers
40
  • More recent work with fainter K dwarfs (more
    numerous and evenly spread out) indicates sz
    increases with z
  • sz 20 km/s _at_ 250 pc and 30 km/s _at_ 1 kpc
  • Yields S (lt1100pc) 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? Srot Vc2/(2pGRo) 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)
Write a Comment
User Comments (0)
About PowerShow.com