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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