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AY202a Galaxies & Dynamics
Lecture 7: Jeans’ Law, Virial Theorem
Structure of E Galaxies
Jean’s Law
Star/Galaxy Formation is most simply defined as the process of going from hydrostatic equilibrium to gravitational collapse.
There are a host of complicating factors --- left for a graduate course:
Rotation Cooling Magnetic Fields Fragmentation ……………
The Simple Model
Assume a spherical,
isothermal gas cloud
that starts near Rc
hydrostatic
equlibrium:
2K + U = 0
(constant density)
Rc
Mc
ρo
Spherical Gas Cloud
Tc
U = ∫ -4πG M(r) ρ(r) r dr
~
Mc = Cloud Mass
Rc = Cloud Radius
ρ0 = constant density =
0
Rc
35
GMc2
Rc
Mc
4/3 π Rc3
Potential Energy
The Kinetic Energy, K, is just
K = 3/2 N k T where N is the total number of particles,
N = MC /(μ mH)
where μ is the mean molecular weight and
mH is the mass of Hydrogen
The condition for collapse from the Virial theorem (more later) is
2 K < |U|
So collapse occurs if
and substituting for the cloud radius,
We can find the critical mass for collapse:
MC > MJ ~ ( ) ( )
3 MC kT 3G MC2
μ mH 5 RC<
RC = ( )3 MC 4πρ0
1/3
5 k T 3
G μ mH 4 πρ0
3/2 1/2
If the cloud’s mass is greater than MJ it will
collapse. Similarly, we can define a critical radius, RJ, such that if a cloud is larger than
that radius it will collapse:
RC > RJ ~ ( )
and note that these are of course for ideal
conditions. Rotation, B, etc. count.
15 k T
4 π G μ mH ρ0
1/2
Mass Estimators:The simplest case = zero energy bound orbit.
Test particle in orbit, mass m, velocity v, radius R, around a body of mass M
E = K + U = 1/2 mv2 - GmM/R = 0
1/2 mv2 = GmM/R
M = 1/2 v2 R /GThis formula gets modified for other orbits (i.e. not
zero energy) e.g. for circular orbits 2K + U = 0
so M = v2 R /G
What about complex systems of particles?
The Virial Theorem
Consider a moment of inertia for a system of N particles and its derivatives:
I = ½ Σ mi ri . ri (moment of inertia)
I = dI/dt = Σ mi ri . ri
I = d2I/dt2 = Σ mi (ri . ri + ri
. ri )
i=1
N
..
.. . . ..
Assume that the N particles have mi and ri and
are self gravitating --- their mass forms the overall potential.
We can use the equation of motion to elimiate
ri :
miri = Σ ( ri - rj )
and note that
Σ miri . ri = 2T (twice the Kinetic Energy)
..
|ri –rj| 3
j = i
Gmimj..
. .
Then we can write (after substitution)
I – 2T = Σ Σ ri . (ri – rj)
= Σ Σ rj . (rj – ri)
= ½ Σ Σ (ri - rj).(ri – rj)
= ½ Σ Σ = U the potential energy
.. i j=i
Gmi mj
|ri - rj|3
Gmi mjj i=j |rj - ri|
3
reversing labels
Gmi mj|ri - rj|
3i j=iadding
Gmi mj
|ri - rj|
I = 2T + U
If we have a relaxed (or statistically steady) system which is not changing shape or size, d2I/dt2 = I = 0
2T + U = 0; U = -2T; E = T+U = ½ U
conversely, for a slowly changing or periodic
system 2 <T> + <U> = 0
..
..
Virial Equilibrium
Virial Mass EstimatorWe use the Virial Theorem to estimate masses
of astrophysical systems (e.g. Zwicky and Smith and the discovery of Dark Matter)
Go back to:
Σ mi<vi2> = ΣΣ Gmimj < >
where < > denotes the time average, and we have N point masses of mass mi, position ri
and velocity vi
N
i=1
N
i=1 j<i
1
|ri – rj|
Assume the system is spherical. The observables are (1) the l.o.s. time average velocity:
< v2R,i> Ω = 1/3 vi
2
projected radial v averaged over solid angle
i.e. we only see the radial component of motion &
vi ~ √3 vr
Ditto for position, we see projected radii R,
R = θ d , d = distance, θ = angular separation
So taking the average projection,
< >Ω = < >Ω
and
< >Ω = = = π/2
Remember we only see 2 of the 3 dimensions with R
1
|Ri – Rj| |ri – rj|
1 1
sin θij
1
sin ij
∫(sinθ)-1dΩ
dΩ
∫0
π dθ
∫π
0sinθ dθ
Thus after taking into account all the projection effects, and if we assume masses are the same so that Msys = Σ mi = N mi we have
MVT = N
this is the Virial Theorem Mass Estimator
Σ vi2 = Velocity dispersion
[ Σ (1/Rij)]-1 = Harmonic Radius
3π2G Σ (1/Rij)i<j
i<j
Σ vi2
This is a good estimator but it is unstable if there exist objects in the system with very small projected separations:
x x
x x x xx
x x x x x
x x x x
x x x
x x
all the potential energy is in this pair!
Projected Mass Estimator
In the 1980’s, the search for a stable mass estimator led Bahcall & Tremaine and eventually Heisler, Bahcall & Tremaine to posit a new estimator with the form
~ [dispersion x size ]
Derived PM Mass estimator checked against simulations:
MP = Σ vi2 Ri,c where
Ri,c = Projected distance from the center
vi = l.o.s. difference from the center
fp = Projection factor which depends on
(includes) orbital eccentricities
fp
GN
The projection factor depends fairly strongly on the average eccentricities of the orbits of the objects (galaxies, stars, clusters) in the system:
fp = 64/π for primarily Radial Orbits
= 32/π for primarily Isotropic Orbits = 16/π for primarily Circular Orbits (Heisler, Bahcall & Tremaine 1985)
Richstone and Tremaine plotted the effect ofeccentricity vs radius on the velocity dispersion
profile:
Richstone & Tremaine
Expected projected l.o.s. sigmas
Applications:Coma Cluster (PS2)
M31 Globular Cluster System
σ ~ 155 km/s MPM = 3.10.5 x 1011 MSun
Virgo Cluster (core only!)
σ ~ 620 km/s MVT = 7.9 x 1014 MSun
MPM = 8.9 x 1014 MSun
Etc.
M31 G1=
Mayall II
M31 Globular Clusters
(Perrett et al.)
The Structure of Elliptical GalaxiesMain questions1. Why do elliptical galaxies have the shapes they
do?2. What is the connection between light & mass &
kinematics? = How do stars move in galaxies?Basic physical description: star piles.For each star we have (r, , ) or (x,y,z)
and (dx/dt, dy/dt, dz/dt) = (vx,vy,vz)
the six dimensional kinematical phase spaceGenerally treat this problem as the motion of stars
(test particles) in smooth gravitational potentials
For the system as a whole, we have the density, ρ(x,y,z) or ρ(r,,)
The Mass M = ∫ ρ dV
The Gravitational (x) = -G ∫ d3x’
Potential Force on unit mass at x F(x) = - (x) plus Energy Conservation Angular Momentum Conservation Mass Conservation (orthogonally)
ρ(x’)
|x’-x|
Plus Poisson’s Equation: 2 = 4 πGρ
(divergence of the gradient)
Gauss’s Theorem 4 π G ∫ ρ dV = ∫ d2S
enclose mass surface integral
For spherical systems we also have Newton’s theorems:
1. A body inside a spherical shell sees no net force
2. A body outside a closed spherical shell sees a force = all the mass at a point in the center.
The potential = -GM/r
The circular speed is then
vc2 = r d/dr =
and the escape velocity from such a potential is
ve = 2 | (r) | ~ 2 vc
For homogeneous spheres with ρ = const r rs
= 0 r > rs
vc = ( )1/2 r
G M(r)
r
4πGρ3
We can also ask what is the “dynamical time” of such a system the Free Fall Time from the surface to the center.
Consider the equation of motion
= - = - r
Which is a harmonic oscillator with frequency 2π/T
where T is the orbital peiod of a mass on a circular orbit T = 2πr/vc = (3π/Gρ)1/2
d2r GM(r) 4πGρ dt2 r2 3
Thus the free fall time is ¼ of the period
td = ( ) ½
The problem for most astrophysical systems reduces to describing the mass density distribution which defines the potential.
E.g. for a Hubble Law, if M/L is constant
I(r) = I0/(a + r)2 = I0a-2/(1 + r/a)2
so ρ(r) [(1 + r/a)2]-3/2 ρ0 [(1 + r/a)2]-3/2
3π
16 G ρ
A distributions like this is called a Plummer model --- density roughly constant near the center and falling to zero at large radii
For this model = -
By definition, there are many other possible
spherical potentials, one that is nicely integrable is the isochrone potential
GM
r2 + a2
(r) = - GM
b + b2 +r2
Today there are a variety of “two power”
density distributions in use
ρ(r) =
With = 4 these are called Dehnen models
= 4, α = 1 is the Hernquist model
= 4, α = 2 is the Jaffe model
= 3, α = 1 is the NFW model
ρ0
(r/a)α (1 – r/a)-α
Circular velocities versus radius
Mod Hubble law Dehnen like laws
Theory’s End
There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizzare and inexplicable. There is another theory which states that this has already happened.
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