AY202aGalaxies & Dynamics

Lecture 7:Jeans’ Law, Virial Theorem

Structure of E Galaxies

Jean’s Law

Star/Galaxy Formation is most simplydefined as the process of going fromhydrostatic equilibrium to gravitationalcollapse.

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

PotentialEnergy

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 3G µ mH 4 πρ0

3/2 1/2

If the cloud’s mass is greater than MJ it willcollapse. Similarly, we can define a critical

radius, RJ, such that if a cloud is larger thanthat radius it will collapse:

RC > RJ ~ ( )

and note that these are of course for idealconditions. Rotation, B, etc. count.

15 k T4 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 = 0so M = v2 R /G

What about complex systems of particles?

The Virial TheoremConsider 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 andare self gravitating --- their mass forms theoverall 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| 3j = 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|3Gmi mj

j i=j |rj - ri|3

reversinglabels

Gmi mj|ri - rj|

3i j=i

adding

Gmi mj|ri - rj|

I = 2T + UIf we have a relaxed (or statistically

steady) system which is not changingshape or size, d2I/dt2 = I = 0

2T + U = 0; U = -2T; E = T+U = ½ U

conversely, for a slowly changing or periodicsystem 2 <T> + <U> = 0

..

..

VirialEquilibrium

Virial Mass EstimatorWe use the Virial Theorem to estimate masses

of astrophysical systems (e.g. Zwicky andSmith and the discovery of Dark Matter)

Go back to:

Σ mi<vi2> = ΣΣ Gmimj < >

where < > denotes the time average, and wehave 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 observablesare (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 vrDitto 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

1sin θij

∫ (sinθ)-1dΩ

dΩ

∫ 0π dθ

∫ π0

sinθ dθ

Thus after taking into account all the projectioneffects, and if we assume masses are the sameso that Msys = Σ mi = N mi we have

MVT = N

this is the Virial Theorem Mass Estimator Σ vi

2 = 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 ifthere exist objects in the system with verysmall projected separations:

x x x x x xx x x x x x x x x x x x x x x

all the potentialenergy is in thispair!

Projected Mass Estimator

In the 1980’s, the search for a stable massestimator led Bahcall & Tremaine andeventually Heisler, Bahcall & Tremaine toposit a new estimator with the form

~ [dispersion x size ]

Derived PM Mass estimator checked againstsimulations:

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

fpGN

The projection factor depends fairly strongly on theaverage 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 projectedl.o.s. sigmas

Applications:Coma Cluster (PS2)

M31 Globular Cluster System σ ~ 155 km/s MPM = 3.1+/−0.5 x 1011 MSun

Virgo Cluster (core only!) σ ~ 620 km/s MVT = 7.9 x 1014 MSun MPM = 8.9 x 1014 MSun

Etc.

M31G1=

MayallII

M31 GlobularClusters

(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, θ, ϕ) 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,θ,ϕ)

The Mass M = ∫ ρ dV

The Gravitational Φ(x) = -G ∫ d3x’ PotentialForce on unit mass at x F(x) = - ∇Φ(x) plus Energy Conservation Angular Momentum Conservation Mass Conservation (orthogonally)

ρ(x’)

|x’-x|

Plus Poisson’s Equation: ∇2Φ = 4 πGρ (divergence of the gradient)Gauss’s Theorem 4 π G ∫ ρ dV = ∫ ∇Φ d2S enclose mass surface integral

For spherical systems we also have Newton’stheorems:

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 Φ = -GM/r

The circular speed is then vc

2 = r dΦ/dr =

and the escape velocity from such a potential is ve = √ 2 | Φ(r) | ~ √ 2 vcFor 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 fromthe surface to the center.

Consider the equation of motion = - = - r

Which is a harmonic oscillator with frequency 2π/Twhere 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 systemsreduces to describing the mass densitydistribution 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

3π16 G ρ

A distributions like this is called a Plummermodel --- density roughly constant near thecenter and falling to zero at large radii

For this model Φ = -

By definition, there are many other possiblespherical potentials, one that is nicely

integrable is the isochrone potential

GM√r2 + a2

Φ(r) = - GMb + √b2 +r2

Today there are a variety of “two power”density distributions in use

ρ(r) =

With β = 4 these are called Dehnen models β = 4, α = 1 is the Hernquist model β = 4, α = 2 is the Jaffe model β = 3, α = 1 is the NFW model

ρ0

(r/a)α (1 – r/a)β-α

Circular velocities versus radius

Mod Hubble law Dehnen like laws

Theory’s End

There is a theory which states that if everanyone discovers exactly what the Universeis for and why it is here, it will instantlydisappear and be replaced by somethingeven more bizzare and inexplicable. Thereis another theory which states that this hasalready happened.

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