ay202a galaxies & dynamics lecture 7
TRANSCRIPT
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:
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.
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