# The Virial Theorem - Harvard-Smithsonian Center for Astrophysics

```AY202a
Galaxies &amp; 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
hydrostatic
equlibrium:
2K + U = 0
(constant density)
Mc
Rc
ρo
Rc
Tc
Spherical Gas Cloud
Rc
U =
∫0
-4πG M(r) ρ(r) r dr
~-
3 GMc2
5
Rc
Mc = Cloud Mass
ρ0 = constant density =
Potential
Energy
Mc
4/3 π Rc3
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 &lt; |U|
So collapse occurs if
3 MC kT
3G MC2
&lt;
μ mH
5 RC
and substituting for the cloud radius,
RC =
(
3 MC
4πρ0
)
1/3
We can find the critical mass for collapse:
MC &gt; MJ ~
(
1/2
5 k T 3/2 3
)
(
)
G μ mH
4 πρ0
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
RC &gt; RJ ~
15 k T
( 4πGμm
)
ρ
1/2
H 0
and note that these are of course for ideal
conditions. Rotation, B, etc. count.
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
2
1/2 mv = GmM/R
M = 1/2 v2 R /G
This formula gets modified for other orbits (i.e. not
zero energy) e.g. for circular orbits 2K + U = 0
2
so
M = v 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:
N
I = &frac12; Σ mi ri . ri (moment of inertia)
.
i=1
..
I = dI/dt = Σ mi ri ri
..
. .
..
I = d2I/dt2 = Σ mi (ri . ri + ri . ri )
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 :
Gmimj
..
miri = -Σ |r –r | 3 ( ri - rj )
j=i
i j
and note that
. .
Σ miri . ri = 2T
(twice the Kinetic Energy)
Then we can write (after substitution)
..
I – 2T = =-
Gmi mj
Σ Σ |r - r |3 ri . (ri – rj)
i j=i
i j
Gmi mj
j i=j |r - r |3
j i
Gmi mj
3
|r
r
|
i j=i
i j
Gmi mj
ΣΣ
=- &frac12;Σ
Σ
=- &frac12;Σ
Σ
|ri - rj|
rj . (rj – ri)
(ri - rj).(ri – rj)
=U
reversing
labels
the potential energy
..
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 = &frac12; U
conversely, for a slowly changing or periodic
system 2 &lt;T&gt; + &lt;U&gt; = 0 Virial
Equilibrium
Virial Mass Estimator
We use the Virial Theorem to estimate masses
of astrophysical systems (e.g. Zwicky and
Smith and the discovery of Dark Matter)
Go back to:
N
N
Σ
2
mi&lt;vi &gt;
i=1
=
1
ΣΣ
Gm
m
&lt;
&gt;
i
j
|r
–
r
|
i=1 j&lt;i
i
j
where &lt; &gt; denotes the time average, and we
have N point masses of mass mi, position ri
and velocity vi
Assume the system is spherical. The observables
are (1) the l.o.s. time average velocity:
2
2
&lt; v R,i&gt; Ω = 1/3 vi
averaged over solid angle
i.e. we only see the radial component of motion &amp;
vi ~ √3 vr
Ditto for position, we see projected radii R,
R = θ d , d = distance, θ = angular separation
So taking the average projection,
&lt;
1
|Ri – Rj|
&gt;Ω =
and
&lt;
1
sin ij
&gt;Ω =
1
|ri – rj|
&lt;
∫(sinθ)-1dΩ
dΩ
1
&gt;Ω
sin θij
π
∫0 dθ
=
0
∫π sinθ dθ
= π/2
Remember we only see 2 of the 3 dimensions with R
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
2
Σ
v
i
3π
=
N
Σ (1/Rij)
2G
i&lt;j
this is the Virial Theorem Mass Estimator
Σ vi = Velocity dispersion
2
-1
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
all the potential
x x
x x
x
energy is in this
pair!
x
x
x
x
x
x
x
x
x
Projected Mass Estimator
In the 1980’s, the search for a stable mass
estimator led Bahcall &amp; Tremaine and
eventually Heisler, Bahcall &amp; Tremaine to
posit a new estimator with the form
~ [dispersion x size ]
Derived PM Mass estimator checked against
simulations:
fp
2
MP =
Σ vi Ri,c where
GN
Ri,c = Projected distance from the center
vi = l.o.s. difference from the center
fp = Projection factor which depends on
(includes) orbital eccentricities
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 &amp; Tremaine 1985)
Richstone and Tremaine plotted the effect of
eccentricity vs radius on the velocity dispersion
profile:
Expected projected
l.o.s. sigmas
Richstone &amp;
Tremaine
Applications:
Coma Cluster (PS2)
M31 Globular Cluster System
σ ~ 155 km/s MPM = 3.1+/-0.5 x 1011 MSun
Virgo Cluster (core only!)
14
σ ~ 620 km/s
MVT = 7.9 x 10 MSun
MPM = 8.9 x 1014 MSun
Etc.
M31
G1
=
Mayall
II
M31 Globular
Clusters
(Perrett et al.)
The Structure of Elliptical Galaxies
Main questions
1. Why do elliptical galaxies have the shapes they
do?
2. What is the connection between light &amp; mass &amp;
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 space
Generally 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
ρ(x’)
The Gravitational (x) = -G ∫
d3x’
|x’-x|
Potential
Force on unit mass at x F(x) = - (x)
plus Energy Conservation
Angular Momentum Conservation
Mass Conservation
(orthogonally)
Plus Poisson’s Equation: 2 = 4 πGρ
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 =
G M(r)
r
and the escape velocity from such a potential is
ve =  2 | (r) | ~  2 vc
For homogeneous spheres with ρ = const r  rs
= 0
r &gt; rs
vc =
(
4πGρ
3
1/2
)
r
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
d2r
GM(r) = - 4πGρ r
=
2
2
dt
r
3
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
Thus the free fall time is &frac14; of the period
td =
(
3π
16 G ρ
)
&frac12;
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
-2
2
I(r) = I0/(a + r) = I0a /(1 + r/a)2
so ρ(r)  [(1 + r/a)2]-3/2  ρ0 [(1 + r/a)2]-3/2
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  = - GM
r2 + a2
By definition, there are many other possible
spherical potentials, one that is nicely
integrable is the isochrone potential
(r) =
- GM
b + b2 +r2
Today there are a variety of “two power”
density distributions in use
ρ(r) =
ρ0
(r/a)α (1 – r/a)-α
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