Dynamics
Stellar Stellar systems vs. gases
Gravitational potential
Spherical and disk potentials
Orbits in the stellar neighborhood
Orbits of single stars
Orbits of stars in clusters
The virial theorem
Measuring masses from motions
Eective potentials and epicycles
Relaxation of orbits and encounters
The Boltzmann equation
J.M. Lattimer
AST 346, Galaxies, Part 4
Stellar Systems vs. Gases
Similarities
Comprise many
interacting point-like
objects
Can be described by
distribution functions of
position and velocity
Obey continuity
equations (are not
created or destroyed)
Interactions and the
systems as a whole obey
conservation laws of
energy and momentum
Concepts like pressure
and temperature apply
Dierences
Relative importance of short (gas) and
long-range (stellar systems) forces
Stars interact continuously with entire
ensemble via long-range force of gravity
Gases interct continuously via frequent,
short-range, strong, elastic, repulsive
collisions
Stellar pairwise encounters are very rare
Pressures in stellar systems can be
anisotropic
Stellar systems have negative specic heat
and evolve away from uniform temperature
Gases evolve toward uniform temperature
and have positive specic heats
J.M. Lattimer
AST 346, Galaxies, Part 4
Potential Theory
Gravitational potential is a scalar eld whose gradient gives the net
gravitational force (per unit mass), a vector eld.
G M(r )
G M(r )
(r ) 3
dr=
+ 4 G
(r )r dr =
dr
|r r|
r
r2
r
r
V
r r 3
G M(r )
Vc (r )2
F(r) d v
=
= (r) = G
( r )
d r =
=
m
dt
|r r|3
r2
r
V
(r) =G
By convention, (r) 0 as r . Outside a spherically symmetric
object, (r ) = G M/r . Inside a spherically symmetric uniform density
shell, (r ) = 0. The divergence of F gives Poissons equation:
1
F(r) = 2 (r) = 4 G (r).
m
Using Gauss Theorem,
1
1
F(r)d 3 r =
F(r) d 2 S
4 G M = 4 G
(r)d 3 r =
mV
mA
V
Gravitational potential energy (last equality for spherical symmetry)
1
1
M(r )
W=
(r)(r)d 3 r =
| |2 d 3 r = G
d M.
2V
8 G V
r
M
J.M. Lattimer
AST 346, Galaxies, Part 4
Analytic Density-Potential Pairs in Spherical Symmetry
Homogenous sphere (radius R , (r < R ) = C )
Inside: (r ) = 2 G (R 2 r 2 /3), F (r ) = G M(r )/r 2
Vc2 = G M(r )/r = 4 G r 2 /3 ( (r ) = constant).
2
Singular isothermal sphere ((r ) = o ro /r 2 )
2
2
2
(r ) = 4 G o ro ln r + C , M(r ) = 4o ro r , Vc2 = 4 G 0 ro .
Power law ((r ) = o (r /ro ) , 2 < < 3)
2
(r ) = 4 G o ro (r /ro )2 /[(3 )( 2)],
3
M(r ) = 4 G o ro (r /ro )3 /(3 ).
Hernquist ((r ) = Ma/[2 r (r + a)3 ])
(r ) = G M/(r + a), M(r ) = Mr 2 /(r + a)2 .
Jae ((r ) = Ma/[4 r 2 (r + a)2 ])
(r ) = (G M/a) ln(1 + a/r ), M(r ) = Mr /(r + a).
Plummer ((r ) = 3a2 M/[4 (r 2 + a2 )5/2 ])
(r ) = G M/ r 2 + a2 , M(r ) = Mr 3 /(r 2 + a2 )3/2 .
Navarro-Frenk-White ((r ) = N a3 /[r (r + a)2 ])
(r ) = 4 G N a3 r 1 ln(1 + r /a),
M(r ) = 4N a3 [ln(1 + r /a) r /(r + a)].
J.M. Lattimer
AST 346, Galaxies, Part 4
Density Laws
J.M. Lattimer
AST 346, Galaxies, Part 4
Orbits of Single Stars
v
dv
+
dt
(r) = 0 =
d
dt
12
v + (r) .
2
Stars energy E is therefore constant, where
E=
m2
v + m(r) = KE + PE .
2
A star can escape only with E > 0 since KE> 0, thus
2
Vesc (r) = 2(r).
Changes to the stars angular momentum L = mr v:
dL
d
dv
= mr v = mr
= m r
dt
dt
dt
(r).
For spherical symmetry, L is therefore conserved.
In a stellar system, individual stellar energies or angular momenta are not
conserved, but their sums are.
J.M. Lattimer
AST 346, Galaxies, Part 4
The Virial Theorem
Newtons Law of Gravity d (mv)/dt = GmMr/r 3 .
d
(mi vi ) =
dt
i
j
i
d
(mi vi rj ) =
dt
d
(mj vj rj ) =
dt
d
1
(mi vi rj ) =
dt
2
i
i
i
2
j =i
j =i
j =i
Gmi mj
(ri rj ).
|ri rj |3
Gmi mj
(ri rj ) ri +
|ri rj |3
Gmi mj
(rj ri ) ri +
|ri rj |3
d
(m i r i r i )
dt 2
mi vi vi =
i
Fi ri .
i
Fj rj .
j
1 d 2I
2 KE.
2 dt 2
2
Add, divide by 2: 1 d I 2 KE = PE +
Fi ri
2 dt 2
1 dI
dI
( ) (0) = 2KE + PE +
Fi ri 0
Average:
2 dt
dt
i
Compare to 2KE + PE 3Po V = 0.
J.M. Lattimer
AST 346, Galaxies, Part 4
Virial Theorem
Validity:
Self-gravitating
Steady state
Time-averaged (or many objects)
Isolated (or slowly varying potential)
J.M. Lattimer
AST 346, Galaxies, Part 4
Virial Theorem and Energy Changes
J.M. Lattimer
AST 346, Galaxies, Part 4
Measuring Masses
Assume uniformity of the mass-to-light ratio M/L in the system. The
surface brightness I (x) = L/D 2 is the surface luminosity density.
The
surface mass density is the projection of along the line-of-sight
z = r 2 R 2 , with R the impact parameter. Using the Plummer model,
(R ) =
(r (z ))dz = 2
0
Ma2
3a2 Mdz
=
.
(a2 + R 2 )2
4 (a2 + z 2 + R 2 )5/2
The parameter a can be inferred from I (x ):
1
(rc )
rc
I (rc )
2
==
= a4 /(a2 + rc )2 , a =
1.55rc .
I (0)
2
(0)
21
1
1
3G M2
Kinetic energy
KE = 2 M = PE =
2
2
64 a
for the Plummer model. One measures velocity dispersion 2 averaging
2
radial velocities vr relative to the systems mean motion : r =< vr2 >.
Tangential motions are undetectable. Typically r 10 km s1 and vr
2
errors are about 0.5 km s1 . For isotropy, 2 = i i = 3r . Thus
M=
J.M. Lattimer
2
32ar
.
G
AST 346, Galaxies, Part 4
Measuring Masses
An alternate method makes use of a measurement of the total luminosity
Ltot = 2
I (R )RdR = 2
0
= 2 I (0)a4
0
I (0)
(0)
(R )RdR
0
RdR
= I (0)a2 ,
(a2 + R 2 )2
giving a = Ltot /( I (0)).
To nd the average motion of one star within (x), the gravity of all
other stars gives a net external force. Then
v2 =
(x) x .
Assuming the Galaxys mass is spherically symmetrically distributed
within the location of an object located far from the Galactic center and
the Sun, so r d , one may compute MG = v 2 r /G where
v = vr , + V0 sin cos b converts the radial velocity relative to the Sun to
the velocity relative to the Galactic center.
J.M. Lattimer
AST 346, Galaxies, Part 4
Circular Motion Reprise
Oorts A (shear) and B (vorticity) constants are dened in terms of
circular rotation:
A=
1
2
Vc
dVc
R
dR
1
B =
2
A
R0
Vc
dVc
+
R
dR
15 km s1 kpc1 , B
A+B =
=
R0
R
2
d
dR
R
=
2
R0
d 2
+
dR
R
R0
12 km s1 kpc1 . Note that
dVc
dR
,
R0
AB =
Vc
dR
= 0
R0
Rotation curve is fairly at, the Suns orbital period is P0 = 2/0 = 230
Myr, and its circular velocity is V0 = 0 R0 = 220 km s1 . These results
assume circular orbits, which is not actually the case in detail.
J.M. Lattimer
AST 346, Galaxies, Part 4
Circular Motion Reprise
Objects close to the Sun have radial and tangential velocities
vr = Ar sin 2 ,
J.M. Lattimer
vt = r (B + A cos 2 ).
AST 346, Galaxies, Part 4
Epicycles
Stellar orbits in a rotating
galaxy can be described by
superposition of a background
circular motion (guiding center
at Rg with g ) and an
elliptical epicycle with angular
velocity g .
Consider the motion in a
rotating frame. For a
Keplerian potential
3/2
(g Rg
), the orbit and
epcicyclic frequencies are the
same, g = g . The orbit is
closed, an o-centered ellipse.
In general g = g so orbits
dont close unless viewed from
a frame rotating at g g /2.
J.M. Lattimer
phases same
AST 346, Galaxies, Part 4
phases advance with radius
Axisymmetric Geometry
Radial (R ) motions:
Equations of Motion:
= (R , z )
r
R
Lz = R 2 = constant
R = R 2 / r ,
= z
z
z = / z
z
z =0
2
z 2
3
( e / R )Rg = ( / R )Rg L2 /Rg
z
= Rg 2 Vc2 /Rg = 0
g
2
z 2
=
x
z =0
= 2 z
z =0
2 =
g
z (t ) = Z cos( t + 0 )
=
From Poissons equation:
4 G (R , 0)
Rg
L2
e
+ z2
R=
R
2R
r
Vertical (z ) motions:
( / z )z =0 = 0
z =
Vc2
= R g 2
g
Rg
=
(dVc2 /dR )/R + 2
2
J.M. Lattimer
e
R
2 e
R 2
z
Rg
=
Rg
d 2
R
+ 42
dR
R = Rg + x ,
2 e
R 2
2
R 2
= 2 z
g
Rg
+
Rg
3L2
z
4
Rg
Rg
x (t ) = X cos(g t + 0 )
AST 346, Galaxies, Part 4
Axisymmetric Geometry
Azimuthal motions:
Lz
Lz
= 2 =
R
(Rg + x )2
2x
Lz
2x
= g 1
1
2
Rg
Rg
Rg
2g X
(t ) = g t
sin(g t + 0 )
g Rg
2g
y (t ) =
X sin(g t + 0 )
g
x (t ) = X cos(g t )
2g
X sin(g t )
y (t ) =
g
Motion is retrograde. For Keplerian
potential, g = g . For at rotation,
g Rg 1 , g = 2g . For solid
rotation, g constant, g = 2g
(circular and closed).
J.M. Lattimer
AST 346, Galaxies, Part 4
Values in the Solar Neighborhood
Epicycle size X R /, Z z / .
In terms of Oorts constants:
2 = 4B (A B ) = 4B 0
0
0
37 km s1 kpc1 = 0.037 Myr 1
0
96 km s1 kpc1 = 0.096 Myr 1
0 = A B 27 km s1 kpc1
Since 0 /0 1.4, solar
neighborhood stars make 1.4
epicyclic rotations per orbit; the
orbit appears to regress.
R 30 km s1 implies X 1 kpc.
z km s1 and
30
= 4 G 0 0.1 Myr1 30
implies Z 300 pc.
The Sun is at z = 40 pc with
vz , = 7 km s1 , suggesting that
Z
80 pc.
The azimuthal/radial extent of
epicycles is 20 /0 1.46.
The mean-square azimuthal/radial
velocities at Rg : y 2 /x 2 = 42 /2 .
0
0
But the azimuthal/radial velocity
dispersion near the Sun is actually
2
2
,0 /R ,0 = 2 /42 0.47
0
0
because this is measured at R0 .
J.M. Lattimer
AST 346, Galaxies, Part 4
Velocity Dispersion Near the Sun
Epicyclic trajectories in rest frame at Rg :
y (t ) =
x (t ) = X cos(g t )
2g
X sin(g t )
g
At R0 , the azimuth obeys = g t + y (t )/Rg . Relative to circular
motion at R0 :
d
vy = v vc = R0 ( 0 ) = R0 g x (t )
dR Rg
R0 x (t )
2 d
+
R
dR
We also have vx = vR = x (t ). Then
2
2
vy
y
=2
2
vx
x
= 2Bx (t ) =
R0
2
0
x (t )
20
2
0
42
0
We ignored that the density of stars decreases with R . There should be
more stars in the solar neighborhood on the outer parts of their epicycles,
with x > 0, than the inner, with x < 0. Therefore vy < 0, which is
called asymmetric drift. The eect is enhanced in older stars, those with
velocities further removed from circular motion.
J.M. Lattimer
AST 346, Galaxies, Part 4
Axisymmetric, Flattened Potentials
Kuzmin disk An innitely thin sheet of mass M.
GM
(R , z ) =
,
R 2 + (a + |z |)2
1
aM
(R ) =
=
2 + a2 )3/2
2 G z z =0
2 (R
Miyamoto-Nagai
b /a = 0.2
Miyamoto-Nagai disk b = 0 is a Kuzmin disk,
a = 0 is a Plummer sphere.
GM
(R , z ) =
R 2 + (a + z 2 + b 2 )2
b 2 M aR 2 + (a + 3 z 2 + b 2 )(a + z 2 + b 2 )
( R , z ) =
4 (z 2 + b 2 )3/2 [R 2 + (a + z 2 + b 2 )2 ]5/2
Satoh disk
G M
(R , z ) =
S
(R , z ) =
G M
R 2 + (a + z 2 + b 2 )2 b 2
b2 M
a
R2 + z2
+33
.
2)
+b
S2
z 2 + b2
4 S 3 (z 2
J.M. Lattimer
AST 346, Galaxies, Part 4
b /a = 1.0
b /a = 5.0
Stellar Encounters
Although the overall galactic potential is smooth, on small scales it has
deep potential wells around each star. Encounters arent as catastrophic
as collisions, and dont aect the overall motion of a star as much as the
overall smoot potential, but are extremely important in changing an
individual stars motion and randomizing the overall velocity distribution.
We distinguish between tidal capture (b < 3rstar ), strong encounters
(b < rs , V V ), in which the potential energy at closest approach is
larger than the initial kinetic energy, and weak encounters
(b >> rs , V << V ), when it is less. The strong encounter radius is
rs = 2Gm/V 2
1 AU
where m 0.5 M is a stellar mass and V 30 km s1 is the initial
relative velocity.
Had this happened to the Sun since its formation, the orbits of the
planets would have been disrupted. The time between close encounters is
ts
( rs2 Vn)1 = V 3 /(4 G 2 m2 n)
4 1012 yr
V
10 km s1
J.M. Lattimer
3
m
M
2
AST 346, Galaxies, Part 4
n
pc3
1
.
Encounter Geometry
2
V = V 2 when t = trelax :
Distant weak encounters
Use the impulse ignoring
the approximation, deviation in the stellar paths. The
impact parameter is b . The perpindicular
pull of star m on star M is GmM/r 2
times b /r , with r 2 = b 2 + V 2 t 2 :
dV
GmMb
=M
F (t ) = 2
2 t 2 )3/2
dt
(b + V
Deection angle:
+
V
1
2Gm
=
=
F dt =
V
MV
bV 2
trelax =
=
V3
ts
=
8 nG 2 m2 ln
2 ln
R
0.3 30 kpc
bmax
=
bmin
rs
1 AU
ln = 18 22
After many encounters
bmax
2
V =
nVt
bmin
=
2Gm
bV
2
2 bdb
8 G 2 m2 nt bmax
8 G 2 m2 nt
ln
=
ln
V
bmin
V
J.M. Lattimer
AST 346, Galaxies, Part 4
Relaxation Applications
If integration is instead performed over a Maxwellian velocity distribution,
trelax increases by a factor of 8 (replace 1/8 by 0.34).
3
2
M
2 1010 yr
V
103 pc3
1
ln
m
n
10 km s
For the Sun, trelax 1012 yr.
Cen has N = 105 , trelax 0.5 Gyr and tcross 0.5 Myr. On
crossing times, stars are little aected by encounters. But over its
lifetime, Cen has been modied by relaxation.
Open clusters, have lower densities and random velocities:
N = 100, trelax 10 Myr, tcross 1 Myr. Have to include eects of
stellar evolution and mass loss to simulate evolution of open clusters.
Elliptical Galaxy: N = 1011 , trelax = 4 1016 yr, tcross = 108 yr.
For a virialized system of size R with N stars moving with an average V :
trelax
G (Nm)2
R
GmN V 2
N
N
mV 2 =
, =
=
=
2
2
2R
rs
V
2Gm
2
With tcross = R /V and 4 n = 3N /R 3
trelax
V3
V
V 4R 2
N
=
=
=
2 m2 ln R
2 m2 ln
tcross
8 nG
6NG
6 ln(N /2)
J.M. Lattimer
AST 346, Galaxies, Part 4
Evaporation
Without collisions, (x) does not change. But encounters alter the
energies of individual stars, preferentially removing energy from massive
stars. On average, encounters shue velocities toward a Maxwellian
distribution
mv 2
/kT
f (E ) exp m(X) +
2
for equal mass stars. The eective temperature is
m v 2 (x) /2 = 3kT /2.
More massive stars move less rapidly. At the upper end of the velocity
distribution, stars achieve escape velocity:
12
1
2
4
mve (x) =
mi (xi ) = PE =
KE.
2
N
N
N
i
2
This means escaping stars satisfy ve 12kT /m. Note that the fraction,
at any given time, of stars capable of escaping is
ve
0
f (E )v 2 dv
f
(E )v 2 dv
= 0.0074
1
.
136
Thus tevap = 136trelax .
J.M. Lattimer
AST 346, Galaxies, Part 4
Mass Segregation
As massive stars (and binaries) lose
energy, they sink to the center; light
stars migrate outwards. In addition,
the stars near the center gain velocity,
so stars near the center tend to lose
energy even faster.
Pleiades
M<M
M>M
Mass segregation is a runaway process,
leading to core collapse after
12 20trelax . Note the
too-small-to-see dense core in M15.
Encounters with binaries lead to
energy losses from binaries; they
become tighter. Release of energy
from binaries (binary burning) can
halt or reverse core contraction.
J.M. Lattimer
AST 346, Galaxies, Part 4
X-ray Sources
J.M. Lattimer
AST 346, Galaxies, Part 4
Collisionless Flows
Assume all stars have the same mass m and ignore encounters
(collisions). The distribution function f (x, v, t ) is the probability density
in phase space, so that the number density at position x and time t is
n(x, t ) =
f (x, v, t )dvx dvy dvz .
Begin with 1-D, and the concepts that no stars are created or destroyed
in the ow and stars dont jump across phase space (no deective
encounters). The net ow in x:
dx
f
dtdvx [f (x , vx , t ) f (x + dx , vx , t )] = dtdvx vx dx .
dt
x
The net ow due to the velocity gradient:
dvx
dvx f
dxdt
[f (x , vx , t ) f (x , vx + dvx , t )] = dtdx
dvx .
dt
dt vx
Adding:
f
f
dvx f
dxdvx dt
= dtdxdvx vx
+
.
t
x
dt vx
f
f
dvx f
f
f
f
0=
+ vx
+
=
+ vx
t
x
dt vx
t
x
x vx
J.M. Lattimer
AST 346, Galaxies, Part 4
Collisionless Boltzmann Equation
Extending this to 3-D (other dimensions are independent) gives the CBE
f
+v
t
f
f
= 0.
v
This has followed from: 1. conservation of stars; 2. smooth orbits; 3.
ow through r implicitly denes v; 4. ow through v given by - .
It can also be written with a convective (or total or Lagrangian)
derivative instead of an Eulerian one:
f
f d x f d v
df
=
+
+
= 0.
dt
t
x dt
v dt
This is incompressible ow. Think of a trac jam: in a dense region,
increases; in a rareed region, decreases. It also applies to all
sub-populations of stars (e.g., spectral classes) even though no one class
determines . A self-consistent eld can be introduced which itself
generates .
J.M. Lattimer
AST 346, Galaxies, Part 4
Jeans Equations
The CBE is of limited use; what we observe are averages (e.g., v 2 ).
These can be extracted using moments. The number density is the
zeroth moment, the mean velocity is the rst moment:
1
vi f (r, v, t )d 3 v .
n(r, t ) = f (r, v, t )d 3 v ,
vi (r, t ) =
n
0th moment CBE in 1-D:
n (n vx )
+
= 0.
t
x
1st moment CBE in 1-D:
2
vx
vx
1 (nx )
+ vx
=
t
x
x
n x
2
2
where x = vx vx 2 . You can show in 3-D (i ,j is the stress tensor,
representing an anisotropic pressure):
vj
vj
1 (ni2,j )
+ vi
=
.
t
i
xj
n xi
Compare to the Euler Equation for uid ow, which has, however, p ():
v
1
+ ( v )v =
p
t
J.M. Lattimer
AST 346, Galaxies, Part 4
Applications of the Jeans Equations
Deriving M/L proles in spherical galaxies
Determining of the surface and volume densities of the Galactic disc
Deriving the attening of a rotating spheroid with isotropic velocity
dispersion
Analysis of asymmetric drift
Analysis of the local velocity ellipsoid in terms of Oorts constants
In spherical symmetry, vr = v = 0, vi2 = i2 (i , j , k = r , , ).
2
v
1 d (nr ) 1
2
2
2
+
2r
n dr
r
r
2
=
d
dt
2
2
2
Dene = 1 ( + )/(2r ), Vrot = v ,
2
2
V2
d
1 d (nr )
+ 2 r rot =
n dr
r
r
dr
2
d ( n r )
n 2
G M(r )n n 2
n2
+ 2 r =
+ Vrot = (Vrot Vc2 )
2
dr
r
r
r
r
2
2
r looks like T , nr looks like p : equation of hydrostatic equilibrium.
Measuring I (x), r , Vrot , and assuming , can nd M(r ) and M/L (r ).
J.M. Lattimer
AST 346, Galaxies, Part 4
Mass of the Galactic disc
Select a tracer population of stars (e.g., K dwarfs) and measure n(z ) and
z (z ). Assuming is time-independent and stars are well-mixed, then f
and n are also time-independent. At large heights, vz n(z ) 0, so
vz = 0. The CBE for z is
1d
2
[n(z )z (z )] =
.
n(z ) dz
z
Take a derivative:
d
1d
2
2
[n(z )z (z )] = 2 .
dz n(z ) dz
z
The Poisson equation in cylindrical coordinates with axisymmetry is
4 G (R , z ) =
2
(R , z ) =
2 1
+
z 2 R R
R
R
=
2 1 d
+
[V 2 (R )].
z 2 R dR
For uniform rotation, the last term is small. Integrating along z :
z
2 G (R , z )dz 2 G (< z ) =
z
1
2
z
d
z
1d
2
[n(z )z (z )]
n(z ) dz
1d
2
[n(z )z (z )]
=
n(z ) dz
J.M. Lattimer
AST 346, Galaxies, Part 4
Integrals of Motion
Functions I (x, v) that remain constant along an orbit are integrals of
motion.
The energy per mass E (x, v ) = v2 /2 + (x) if is independent of
time.
Lz in an axisymmetric potential (R , z , t ).
L in a spherically symmetric potential (r , t ).
An integral of motion satises
d
I (x, v) = v
dt
I+
d v I
= 0.
dt v
Any function f (x, v) which is a time-independent solution of the CBE is
an integral of motion. Conversely, the function f (I1 , I2 , . . . ) is a
steady-state solution of the equations of motion: the Jeans Theorem.
The strong Jeans Theorem states that steady state distribution functions
are functions only of 3 (or less) independent integrals of motion.
For spherical systems, f = f (E , |L|)
If f = f (E ), velocity dispersions are isotropic r = = .
If f = f (E , |L|), velocity dispersions are anisotropic r = =
J.M. Lattimer
AST 346, Galaxies, Part 4
Integrals of Motion
Motion of disk stars on circular orbits perpindicular to the plane is
independent of motion in the plane, so the energy of vertical motion Ez is
an integral of motion. Select a tracer population of stars that are easy to
nd and measure and which are well-mixed (f is time-independent). Then
2
f (z , vz ) = f (Ez ) = f ((R0 , z ) + vz /2).
If we knew f (Ez ) and (R0 , z ) we could integrate f (vz ) to nd n(z )
and z .
If we measured n(z ) and guessed f (Ez ) we could determine
(R0 , z ). Suppose stars with Ez > 0 escape:
2
n0
e Ez /z ,
Ez < 0;
f (Ez ) = 0,
Ez > 0.
f (Ez ) =
2
2z
2
n(z ) = n0 e (R0 ,z )/ , z =
if ve = 2(R0 , z ) >>
However, note that ve 2 .
2
If n(z ) and z measured, (R0 , z ) can be found from
d
(R0 , z )
2
n(z )z = n(z )
.
dz
z
J.M. Lattimer
AST 346, Galaxies, Part 4
Consistency
If the stars described by f provide all the gravitational force, then the
density n(x, v) found by integrating f (x, v, t ) over v is equivalent to the
density (x, t ) in Poissons Equation. Many forms of f can give rise to
the same (x, t ): all give the same n(x, t ) but dierent v(x, t ).
In a spherically symmetric potential, any function f (E , L) not including
unbound stars will be a solution. If f = f (E ), velocity dispersions are
isotropic.
Example: f (E ) = k (E )N 3/2 for E < 0, N > 3/2.
ve
k (r )
n(r ) = 4
0
v2
2
N 3/2
v 2 dv
/2
sin2N 2 cos2 d = kcN ((r ))N ,
= 4 k 23/2 ((r ))N
0
after substituting cos() = v / 2(r ). Compare to Plummer sphere
3a 2
3a2
M
(r ) =
5 (r ) =
4 G 5 M4
4 (r 2 + a2 )5/2
suggesting N = 5 and f (E ) = k (E )7/2 .
Total mass M k , a = [G M/(0)]2 .
J.M. Lattimer
AST 346, Galaxies, Part 4
Isothermal Models
Consider a Boltzmann-like distribution function:
f (E ) =
2
2
2
no
no
e E / =
e (+v /2)/
(2 2 )3/2
(2 2 )3/2
f (E (v ))v 2 dv = no e /
n(r ) = 4
2
0
Poissons equation
d
dr
r2
d ln n
dr
=
4 G 2
rn
2
which is the isothermal spherical solution.
(i) Singular isothermal sphere
2
, Vc = 2,
v 2 = 3 2
2
2 Gr
But has innite central density and M as r .
n(r ) =
J.M. Lattimer
AST 346, Galaxies, Part 4
Isothermal Models
General isothermal sphere
n(0) = n0 , (dn/dr )r =0 = 0.
Measure I (R ) and determine ro
and I (0)
The density varies slowly near the
center, out to ro = 3/ 4 G 0 .
Also measure 2 .
ro is the core (King) radius, and is also
the scale length of the envelope.
I (ro ) = 0.5013I (0).
Vc2 = 2 d ln n/d ln r .
Then M/L = 9 2 /(2 GI (0)ro ).
But this still has an innite total mass.
The problem is f (E ) > 0 even when
E is positive, i.e., the model includes
unbound stars.
At small radii,
n(r ) = n0 (1 + (r /ro )2 )3/2 .
At large radii, n(r ) (r /ro )2
2 = 4 Gno ro /9
A good t to the centers of elliptical
galaxies can be used to estimate
central M/L.
J.M. Lattimer
AST 346, Galaxies, Part 4
Isothermal Models
Lowered isothermal sphere
Suppress stars at large radii; f (E ) 0
when E 0, v ve .
It is convenient to dene = and
Er = E = v 2 /2.
f (Er ) =
d
dr
r2
d
dr
2
no
e Er /o 1
(2o )3/2
= 4 Gno r 2
e
2
/o
erf
o
4
2
o
1+
2
2
3o
2
Inner regions: core radius ro , 2 o
2
2
Outer regions: truncated at rt , << o
2
q /4
If (0) = q o , rt ro 10 .
J.M. Lattimer
AST 346, Galaxies, Part 4
Consistency
In general, we nd a single equation to be satised for consistency with
the steady state CBE and Poissons equation:
1d
r 2 dr
d
r2
dr
= 16 2 G
2
f ( v 2 /2)v 2 dv
0
4 G () = 16 2 G
f (Er ) 2( Er )dEr
0
1 d
()
=2
f (Er ) Er dEr ,
=
8
8 d
0
This is an Abel integral equation with solution
Er
1d
d
d
f ( Er ) =
8 dEr 0 d Er
=
1
2
Er
8
0
d 2
d
1
+
2
d Er
Er
0
d
d
f (E )dEr
r
.
Er
=0
This is an alternate method, begininning with measuring (r ) from
surface photometry. Find (r ) = (r ) = G M(< r )/r from (r ), then
eliminate r to nd ().
J.M. Lattimer
AST 346, Galaxies, Part 4