11 th lec phase space. collisionless systems we showed collisions or deflections are rare...
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11th Lec
• Phase Space
Collisionless Systems
• We showed collisions or deflections are rare• Collisionless: stellar motions under influence of
mean gravitational potential!• Rational:• Gravity is a long-distance force, decreases as r-2
– as opposed to the statistical mechanics of molecules in a box
Collisionless Systems
• stars move under influence of a smooth gravitational potential– determined by overall structure of system
• Statistical treatment of motions– collisionless Boltzman equation– Jeans equations
• provide link between theoretical models (potentials) and observable quantities.
• instead of following individual orbits
• study motions as a function of position in system
• Use CBE, Jeans eqs. to determine mass distributions and total masses
Fluid approach:Phase Space Density
PHASE SPACE DENSITY:Number of stars per unit volume per unit velocity volume f(x,v) (all called Distribution Function DF).
The total number of particles per unit volume is given by:
313 )(kmspc
mN
volumevelocityvolumespace
mstarsofnumberv)f(x,
• E.g., air particles with Gaussian velocity (rms velocity = σ in x,y,z directions):
• The distribution function f(x,v) is defined by:
mdN=f(x,v)d3xd3v
where dN is the number of particles per unit volume with a given range of velocities.
3
2
222
o
)2(
2expnm
v)f(x,
xyx vvv
• The total mass is then given by the integral of the mass distribution function over space and velocity volume:
• Note:in spherical symmetry d3x=4πr2dr,
• for isotropic systems d3v=4πv2dv
• The total momentum is given by:
xdvdvxfxdxM total
333 ),()(
vdxdvvxfmdNvPtotal
33),(
• Example:mean speed of air molecules in a box of dx3 :
These are gamma functions
3
2
222
o
)2(
2expnm
v)f(x,
xyx vvv
• Gamma Functions:
2
1
)1()1()(
)( 1
0
nnn
dxxen nx
How to calculate dx3 and dv3
22 2 2
0 0
3 2 2
3 2 2
2 2 2
sin 2 2 ,
4 (if spherical)
4 (if isotropic)
2[ ( )]
x y z
x y z
d d d r x x y z
d dxdydz r drdΩ πr dr
d dv dv dv v dvdΩ πv dv
V v v v v E x
x
v
DF and its moments
mass-weighted average,
,
AdMA A
dM
3 3 3
3 3 3
2 2 2
( ) ( , )
( , )
1: A( , ) Vx , ( ) ,
2
d A dM A d Af x v d
d dM d f x v d
For Vx Vy Vz x
x x x v
x x v
x v ,VxVy x v
Additive: subcomponents add upto the total gravitational mass
1 2
1 2
A B A B
f f f
Full Notes online
• http://www-star.st-and.ac.uk/~hz4/gravdyn/GraviDynFinal3.ppt GraviDynFinal3.pdf
Liouvilles Theorem
We previously introduced the concept of phase space density. The concept of phase space density is useful because it has the nice property that it is incompessible for collisionless systems.
A COLLISIONLESS SYSTEM is one where there are no collisions. All the constituent particles move under the influence of the mean potential generated by all the other particles.
INCOMPRESSIBLE means that the phase-space density doesn’t change with time.
Consider Nstar identical particles moving in a small bundle through spacetime on neighbouring paths. If you measure the bundles volume in phase space (Vol=Δx Δ p) as a function of a parameter λ (e.g., time t) along the central path of the bundle. It can be shown that:
It can be seen that the region of phase space occupied by the particle deforms but maintains its area. The same is true for y-py and z-pz. This is equivalent to saying that the phase space density f=Nstars/Vol is constant. df/dt=0!df/dt=0!
THEOREM'LIOUVILLES',0,0 d
dNstar
d
dVol
px
x
px
x
motions in phase-space
• Flow of points in phase space corresponding to stars moving along their orbits.
• phase space coords:
• and the velocity of the flow is then:– where wdot is the 6-D vector related to w as
the 3-D velocity vector v relates to x
),(),(
),...,,(),( 621
vvxw
wwwwvx
• stars are conserved in this flow, with no encounters, stars do not jump from one point to another in phase space.
• they drift slowly through phase space
• In the COMBINED potential of stars and dark matter
fluid analogy
• regard stars as making up a fluid in phase space with a phase space density
•
• assume that f is a smooth function, continuous and differentiable– good for N >105
f (x,v,t) f (w,t)
• as in a fluid, we have a continuity equation• fluid in box of volume V, density, and
velocity v, the change in mass is then:
– Used the divergence theorem
0
3
23
23
V
SV
SV
xdvt
SdFxdF
SdvxdtdtdM
continuity equation• must hold for any volume V, hence:
• in same manner, density of stars in phase space obeys a continuity equation:
If we integrate over a volume of phase space V, then 1st term is the rate of change of the stars in V, while 2nd term is the rate of outflow/inflow of stars from/into V.
0
0
0
3
1
3
1
6
1
6
1
i iii i
i
i
i
xvv
v
x
v
w
w
w
wf
t
f
vt
0
Collisionless Boltzmann Equation
• Hence, we can simplify the continuity equation to the CBE:
• Vector form 0
0
0
3
1
6
1
v
ffv
t
f
v
f
xx
fv
t
f
w
fw
t
f
i iiii
• in the event of stellar encounters, no longer collisionless
• require additional terms to rhs of equation
CBE cont.
• can define a Lagrangian derivative• Lagrangian flows are where the coordinates travel along with the motions
(flow)– hence x= x0 = constant for a given star
• then we have:
• and
• rate of change of phase space density seen by observer travelling with star• the flow of stellar phase points through phase space is incompressible• f around the phase point of a given star remains the same0
6
1
6
w
fw
t
f
dt
df
wtdt
d
incompressible flow
• example of incompressible flow• idealised marathon race: each runner runs at
constant speed• At start: the number density of runners is
large, but they travel at wide variety of speeds• At finish: the number density is low, but at
any given time the runners going past have nearly the same speed
DF & Integrals of motion
• If some quantity I(x,v) is conserved i.e.
• Assume f(x,v) depends on (x,v) through the function I(x,v), so f=f(I(x,v)).
• Such phase space density is incompressible, i.e
0),(
dt
vxdI
0dt
df
Jeans theorem
• For most stellar systems the DF depends on (x,v) through generally three integrals of motion (conserved quantities), Ii(x,v),i=1..3 f(x,v) = f(I1(x,v), I2(x,v), I3(x,v))
• E.g., in Spherical Equilibrium, f is a function of energy E(x,v) and ang. mom. vector L(x,v).’s amplitude and z-component
)ˆ||,||,(),( zLLEfvxf
3D Analogy of 6D Phase space
• If DF(x,v) is analogous to density(x,y,z), • Then DF(E,L,Lz) is ~ density(r,theta,phi), • Integrals analogous to spherical coordinates
– E(x,v) analogous to r(x,y,z)
• Isotropic DF(E) ~ spherical density(r)– Normalization dM=f(E)dx3dv3 ~ dM=density(r)dr3
– Have non-self-gravitating subcomponents: DF1+DF2, like rho1+rho2 to make up total gravity.
12th Lec
• Phase Space
Tensor Virial Theorem• Equation of motion:
dt
vd
0 0
1 1.
T Tdvdt r dtr
T dt T
dt
rdv
dt
vrd
)(
TTT
rdtT
dtvvTT
rv
000
.11)(
rvv
This is Tensor Virial Theorem
• E.g.
• So the time averaged value of v2 is equal to the time averaged value of the circular velocity squared.
22
2
2222
)(
.
etc
cir
cir
zyx
yyx
xxx
vv
sphericalvdr
dr
r
vvvv
xvv
xvv
Scalar Virial Theorem
• the kinetic energy of a system with mass M is just
where <v2> is the mean-squared speed of the system’s stars.
• Hence the virial theorem states that 2
2
1vMK
.2
2
rv
r
GM
M
Wv
g
Virial02 WK
Stress Tensor • describes a pressure which is anisotropic
– not the same in all directions
• and we can refer to a “pressure supported” system
• the tensor is symmetric.
• can chose a set of orthogonal axes such that the tensor is diagonal
• Velocity ellipsoid with semi-major axes given by
Pnx ij
i
2
ij2 ii
2 ij
11 , 22 , 33
n ij2
Subcomponents in Spherical Equilibrium Potential
• Described by spherical potential φ(r)• SPHERICAL subcomponent density ρ(r) depends on
modulus of r.
• EQUILIBRIUM:Properties do not evolve with time.
,
,0,0 (0, ,0) (0,0, )
0 0
r r
r r r
x xy
000
ttt
f
• In a spherical potential
rrxy
dr
rd
r
yx
r
r
y
rx
y
rx
)(
)()(
r
y
dy
dr
ydyrdr
ydrd
zyxr
)()( 22
2222
So <xy>=0 since the average value of xy will be zero.
<vxvy>=0
Spherical Isotropic f(E) Equilibrium Systems
• ISOTROPIC:The distribution function f(E) only depends on the modulus of the velocity rather than the direction.
2
2 2 2 2 2 2tangential
, / 2 ( )
1
20
x y z r
x y
f E E v r
v v
Note:the tangential direction has and components
Anisotropic DF f(E,L) in spherical potential.
• Energy E is conserved as:
• Angular Momentum Vector L is conserved as:
• DF depends on Velocity Direction through L=r X v• Hence anisotropic
0t
0
2 2tangential
1
2r
e.g., f(E,L) is an incompressible fluid
• The total energy of an orbit is given by:
),(2
1 2 trvE
( , ) ( , ) ( , )0 0
df E L f E L dE f E L dL
dt E dt L dt
0 for static potential, 0 for spherical potential
So f(E,L) constant along orbit or flow
drdrdrd trr //)2(/)( 222
• spherical Jeans eq. of a tracer density rho(r)
•
• Proof :
)/()(),(//
/)(),(
/)()(
///
/22
23232
223
2223
32
222
drdvdELdLEffdvrrLdrd
rvdELdLEffdv
rvdELddvvddv
rLdrddrdvv
rLEv
r
r
rrt
rr
r
Jeans eq. Proof cont.
222
422
2
222
00
222
4222
223
/2/)(
//),(
/)/(),(
/)/(*),(/)(
),(
)/(*),(
)/(),(
2
trr
r
r
rr
r
vL
r
rt
r
drrddrrd
vrLdELdLEf
vrdrddELdLEf
rdrdvdELdLEfrdrrd
vdELEfLdr
vrLdELdLEf
rvdELdLEffdv
r
• SELF GRAVITATING:The masses are kept together by their mutual gravity.
• In non-self gravitating systems the density that creates the potential is not equal to the density of stars. e.g a black hole with stars orbiting about it is NOT self gravitating.
13th Lec
• Phase Space
Velocity dispersions of a subcomponent
in spherical potential• For a spherically symmetric system we have
• a non-rotating galaxy has – and the velocity ellipsoids are spheroids with
their symmetry axes pointing towards the galactic centre
• Define anisotropy
22
2222
/1
2
r
rr
vvr
dr
dnvvv
r
nvn
dr
d
v2 v
2
Spherical mass profile from velocity dispersions.
• Get M(r) or Vcir from:
• RHS: observations of dispersion and as a function of radius r for a stellar population.
2ln
ln
ln
ln
21
222
circ
2
22
rd
vd
rd
ndv
r
rGM
dr
drv
r
rGM
dr
d
r
vvn
dr
d
n
rr
rr
• Isotropic Spherical system, β=0
• This is the isotropic JEANS EQUATION, relating the pressure gradient to the gravitational force.
dr
dr
dr
d )(
)( 2
Note: 2=P
2 | |r r
dP dr g dr
dr
Above Solution to Isotropic Jeans Eq:
negative sign has gone since we reversed the limits.
Hydrostatic equilibrium Isotropic spherical Jeans equation
• Conservation of momentum gives:
dr
dr
dr
d
dr
dP )(
)( 2
0
1
P g
g P
Tutorial
g
(r) (r)
2
(E)vesc
M
Tutorial Question 3• Question: Show dispersion sigma is constant in potential Phi=V0
2ln(r). What might be the reason that this model is called Singular Isothermal Sphere?
24
24
0,rrAt
1
4
1
2
2
2
2
222
2o
2
2
222
2
2
cc
cc
cr
cr
c
oc
r
r
v
Gr
v
r
v
G
v
P
drr
v
rG
vdr
r
v
rrr
v
dr
d
• Since the circular velocity is independent of radius then so is the velocity dispersionIsothermal.
2
2
22
2
22
c
c
c
v
v
v
Flattened Disks
• Here the potential is of the form (R,z).• No longer spherically symmetric.• Now it is Axisymmetric
zg
Rg
zRR
RR
GzRzR
zr
2
2
4
1),(),(
Question 4: Oblate Log. potential
• oblate galaxy with Vcirc ~ V0 =100km/s
• Draw contours of the corresponding Self-gravitating Density to show it is unphysical.
• Let Lz=1kpc*V0 , E=0.55*V02 +C0, Plot
effective potential contours in RZ plane to show it is an epicycle orbit.
• Taylor expand the potential near (R,z)=(1,0) to find epicycle frequencies and the approximate z-height and peri-apo range.
2 2 210 02( , ) ln 2R z v R z C
Orbits in Axisymmetric Potentials (disk galaxies)
• cylindrical (R,,z) symmetry z-axis• stars in equatorial plane: same motions as in
spherically symmetric potential– non-closed rosette orbits
• stars moving out of plane– can be reduced to 2-D problem in (R,z)
– conservation of z-angular momentum, Lz
z
R
yx
R2=x2+y2
• Angular momentum about the z-axis is conserved, toque(rF=0) if no dependence on .
• Energy is also conserved (no time-dependence)
• Eliminating in the energy equation using conservation of angular momentum gives:
0)( 2222 Rdt
dRLZ
constzRzRR ),(2
1 2222
ER
JzRzR z
2
222
2),()(
2
1
Specific energy density in 3D
eff
Total Angular momentum almost conserved
• These orbits can be thought of as being planar with more or less fixed eccentricity.
• The approximate orbital planes have a fixed inclination to the z axis but they process about this axis.
• star picks up angular momentum as it goes towards the plane and returns it as it leaves.
Orbital energy• Energy of orbit is (per unit mass)
• effective potential is the gravitational potential energy plus the specific kinetic energy associated with motion in direction
• orbit bound within
2
eff
eff22
21
2
222
21
22221
E
zR
R
LzR
zRRE
z
• The angular momentum barrier for an orbit of energy E is given by
• The effective potential cannot be greater than the energy of the orbit.
• The equations of motion in the 2D meridional (RZ)plane then become: .
EzReff ),(
0
),(2222
zREzR eff
z
eff
eff
JR
zz
RR
2
• Thus, the 3D motion of a star in an axisymmetric potential (R,z) can be reduced to the motion of a star in a plane (Rz).
• This (non uniformly) rotating plane with cartesian coordinates (R,z) is often called the MERIDIONAL PLANE.
eff(R,z) is called the EFFECTIVE POTENTIAL.
• The orbits are bound between two radii (where the effective potential equals the total energy) and oscillates in the z direction.
• The minimum in eff occurs at the radius at which a circular orbit has angular momentum Lz.
• The value of eff at the minimum is the energy of this circular orbit.
R
eff
E
2
2
2R
J z
Rcir
• The effective potential is the sum of the gravitational potential energy of the orbiting star and the kinetic energy associated with its motion in the direction (rotation).
• Any difference between eff and E is simply kinetic energy of the motion in the (R,z) plane.
• Since the kinetic energy is non negative, the orbit is restricted to the area of the meridional plane satisfying .
• The curve bounding this area is called the ZERO VELOCITY CURVE since the orbit can only reach this curve if its velocity is instantaneously zero.
E - eff (R,z)>= 0
Nearly circular orbits: epicycles• In disk galaxies, many stars (disk stars) are
on nearly-circular orbits
• EoM:
• x=R-Rg
– expand in Taylor series about (x,z)=(0,0)
– then2/ 2/
0,at 0
;
2222
2
)0,(
2eff
2
212
)0,(
2eff
2
21
eff
effeff
effeff
zx
zz
xR
zRRzR
zz
RR
gg RR
g
• When the star is close to z=0 the effective potential can be expanded to give
22
2
2
1)0,(),( z
zz
zRzR eff
effeff
Zero, changes sign above/below z=0 equatorial plane. 2
.......2
1)0,(),( 22 zRzR effeff
zz 2
So, the orbit is oscillating in the z direction.
epicyclic approximation
• ignore all higher / cross terms:
• EoM: harmonic oscillators– epiclyclic frequency :– vertical frequency :
– with
2 2
2
2eff
222
2 4
( ,0)
R , and
( , )2
3;
g
z
z
gR
R z z
LR zR
L
R R
epicycles cont.
• using the circular frequency , given by
– so that
disk galaxy: ~ constant near centre– so ~ 2
~ declines with R, » slower than Keplerian R-3/2
» lower limit is ~
in general < 2
gR
z
z
R
RR
RRR
L
RR
R
L
RRR
22
224
22
4
2
)0,(
2
4
33
1)(
R
Vrot
Example:Oort’s constants near Sun
– where R0 is the galacto-centric distance
• then 2 = -4A(A-B) + 4(A-B)2 = -4B(A-B) = -4B0
• Obs. A = 14.5 km/s /kpc and B=-12 km/s /kpc
00
21
21 ;
RR RRB
RRA
2.03.12 0
0
BA
B
the sun makes 1.3 oscillations in the radial direction per azimuthal (2) orbit– epicyclic approximation not valid for z-motions
when |z|>300 pc
General Jeans Equations
• CBE of the phase space density f is eq. of 7 variables and hence generally difficult to solve
• Gain insights by taking moments of the CBE
• where integrate over all possible velocities– U=1, vj, vjvk
3
1
3 3 3
0
0
ii i i i
ii i i
f f fv
t x x v
f f fUd v v Ud v Ud v
t x x v
1st Jeans (continuity) equation
• define spatial density of stars n(x)
• and the mean stellar velocity v(x)
• then our zeroth moment equation becomes
n fd 3v
v i 1
nfvi d 3v
nt nv i xi
0
3rd Jeans Equation
similar to the Euler equation for a fluid flow:– last term of RHS represents pressure force
Pvvt
v
x
n
xn
x
vvn
t
vn
i
ij
ii
j
ij
1
2
JEANS EQUATION for oblate rotator
: a steady-state axisymmetrical system in which ASSUME ij
2 is isotropic and the only streaming motion is azimuthal rotation:
RR
v
R
zz
rot
22
2
)(1
)(1
• The velocity dispersions in this case are given by:
• If we know the forms of (R,z) and (R,z) then at any radius R we may integrate the Jeans equation in the z direction to obtain 2.
222
rotrot
2222
222222
isotropic sit' vfromapart since vvbut
2)(
since
),(
rot
rotrotrot
rotzr
vv
vvvvvv
vvvvzR
Obtaining 2
Inserting this into the jeans equation in the R
direction gives:
dzz
zRz
1
),(2
z
rot dzzR
R
RRv
2
20th Lec
• orbits
Applications of the Jeans Equations
• I. The mass density in the solar neighbourhood
• Using velocity and density distribution perpendicular to the Galactic disc– cylindrical coordinates.– Ignore R dependence
E.g.: Total Mass of spherical Milky WAY
• Motions of globular clusters and satellite galaxies around 100kpc of MW– Need n(r), vr
2, to find M(r), including dark halo
• Several attempts all suffer from problem of small numbers N ~ 15
• For the isotropic case, Little and Tremaine TOTAL mass of 2.4 (+1.3, -0,7) 1011 Msol
• 3 times the disc need DM
Power-law model of Milky Way
• Isotropic orbits: • Radial orbits• If we assume a power law for the density
distribution
– E.g. Flat rotation a=1, Self-grav gamma=2, Radial anis.
– E.g., Point mass a=0, Tracer gamma=3.5, Isotro
)(, rrMrn
0, v2 vr
2
1, v2 0
Grvvr /)(5.4M 22
Mass of the Milky Way: point-mass potential model
We find
2 2circ
2
2
2 2
1/
lnln2 1 2
ln ln
For =3.5, and isotropic tracer =0, we have
4.5
/ /
r
r
r
r
GM dr v v p
r dr
v r
d vd np
d r d r
p
M pv r G pv r G
Vertical Jeans equation
• Small z/R in the solar neighbourhood, R~8.5 kpc, |z|< 1kpc, R-dependence neglected.
• Hence, reduces to vertical hydrostatic eq.:
z
nvnz
z
2
mass density in solar neighbourhood
• Drop R, theta in Poisson’s equation in cylindrical coordinates:
GzRR
RRR
411
2
2
2
22
Gz
42
2
local mass density = 0
Finally
• all quantities on the LHS are, in principle, determinable from observations. RHS Known as the Oort limit.
• Uncertain due to double differentiation!
Gvnznz z 4/
1- 2
local mass density
• Don’t need to calculate for all stars– just a well defined population (ie G stars, BDs etc)
– test particles (don’t need all the mass to test potential)
• Procedure– determine the number density n, and the mean square vertical velocity, v z
2, the variance of the square of the velocity dispersion in the solar neighbourhood.
– need a reliable “tracer population” of stars• whose motions do not reflect formation• hence old population that has orbited Galaxy many times• ages > several x 109 years
• N.B. problems of double differentiation of the number density n derived from observations
• need a large sample of stars to obtain vz as f(z)
– ensure that vz is constant in time
– ie stars have forgotten initial motion
local mass density
• > 1000 stars required
• Oort : 0 = 0.15 Msol pc-3
• K dwarf stars (Kuijken and Gilmore 1989)– MNRAS 239, 651
• Dynamical mass density of 0 = 0.11 Msol pc-3
• also done with F stars (Knude 1994)
• Observed mass density of stars plus interstellar gas within a 20 pc radius is 0 = 0.10 Msol pc-3
• can get better estimate of surface density
• out to 700 pc ~ 90 Msol pc-2
• from rotation curve rot ~ 200 Msol pc-2
• Question 5:
).,(density star and
stars of mass totalcalculate ./1000V assume
term),(2nd stars and (1st term) halodark todue
,)1/)(1(
)2ln(5.0),( potentialIn 2/12222
0
2220
zR
skm
kpczRv
zRvzR
s
)0,1( unphysical haverotator isotropic Show
),1()0,1(stellar Calculate
(1kpc,0)?z)(R,equator on density halodark theisWhat
2
02
kpcv
dzz
zkpckpc
rot
ss
Helpful Math/Approximations(To be shown at AS4021 exam)
• Convenient Units
• Gravitational Constant
• Laplacian operator in various coordinates
• Phase Space Density f(x,v) relation with the mass in a small position cube and velocity cube
3dv3dx),(dM
)(spherical 2sin2r
2
sin2r
)(sin
2r
)2(
al)(cylindric 22-R2)(1-R
ar)(rectangul 222
1-sun
M2(km/s) kpc6104
1-sun
M2(km/s) pc3104
Gyr1
kpc1
1Myr
1pc 1km/s
vxf
rr
r
zRR
R
zyx
G
G
21th Lec: MOND
• orbits
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