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Dark Matter
-----------
The Virial Theorem
------------------
The Virial theorem relates the total kinetic energy
of a self-gravitating body due to the motions of its
constituent parts, K, to the gravitational potential
energy, U, of the body.
Consider N point paricles in a system.
We define the VIRIAL, G, as:
G := Σ_{i}p_{i}.r_{i}
Where p_{i} and r_{i} are the momentum and position of
particle i.
Then:
dG/dt = Σ_{i}(dp_{i}/dt).r_{i} + Σ_{i}p_{i}.(dr_{i}/dt)
Now, the force on the particle, F_{i}, is:
F_{i} = m_{i}a_{i} = dp_{i}/dt and p_{i} = m_{i}(dr_{i}/dt)
Therefore we can write:
dG/dt = Σ_{i}F_{i}.r_{i} + Σ_{i}m_{i}(dr_{i}/dt)^{2}
= Σ_{i}F_{i}.r_{i} + Σ_{i}m_{i}v_{i}^{2}
= Σ_{i}F_{i}.r_{i} + 2K (K = mv^{2}/2)
The time average is given by:
_{τ}
<dG/dt> = (1/τ)∫(Σ_{i}F_{i}.r_{i} + 2K)dt
^{0}
= <F_{i}.r_{i}> + 2<K>
If the system is cyclical such that it returns
to its initial state after an interval, τ, then τ
can be chosen to be equal to the cycle period and,
therefore, dG/dt reduces to 0. In this case:
<K> = (-1/2)Σ_{i}<F_{i}.r_{i}>
Now F_{i} = -∂U/∂r_{i} (U is PE)
<K> = (-1/2)<U>
or,
2<K> + <U> = 0 ... 1.
Astronomy has made great use of the Virial Theorem
as a way of measuring gravitational mass.
The gravitational potential, U, for a cluster of N
galaxies each separated by a distance, r, is:
U = -GNm^{2}/r
= -GM^{2}/r
Likewise, the KE is:
K = Mv^{2}/2
Using 1. we get:
Mv^{2}/2 = (1/2)GM^{2}/r
∴ v^{2} = GM/r
Therefore, the total mass of the cluster is:
M = 2v^{2}/G
The left-hand side of the equation can be estimated
if we can measure the motions of bodies in the system.
Thanks to the Doppler shift, we can determine the
motion in the radial direction very easily.
Mass-Luminosity Relationship
----------------------------
The massâ€“luminosity equation gives the relationship
between a star's mass and its luminosity. A rough
approximation can be obtained by modelling stars as
ideal gases (Eddington).
L/L_{O} = (M/M_{O})^{α}
Where L_{O} and M_{O} are the luminosity and mass of the
Sun respectively and, α, is number dependent on the
mass range of the star.
L/L_{O} ~ 0.23(M/M_{O})^{2.3}^{ } M < 0.43M_{O}
L/L_{O} = (M/M_{O})^{4.0} 0.43M_{O} < M < 2M_{O}
L/L_{O} ~ 1.5(M/M_{O})^{3.5} 2M_{O} < M < 20M_{O}
L/L_{O} ~ 3200(M/M_{O}) M > 20M_{O}
Dark Matter
-----------
The existence of Dark Matter was first proposed by
the Swiss astronomer, Fritz Zwicky in 1933.
Zwicky focussed on the COMA CLUSTER of galaxies and
measured the Doppler shift of their spectra to find
their velocities. He then used the Virial theorem to
determine their masses. In parallel he also determined
their masses using the mass-luminosity relationship
described above. He discovered that the 2 methods
yielded masses that differed greatly. The dynamical
mass from the Virial theorem turned out to be about
400 times the luminosity mass. Because of this Zwicky
believed that there must be a large amount of invisible
matter within the cluster that he termed "dark matter".
Galactic Rotation Curves
------------------------
Four decades after Zwicky's initial observations new
techniques were used to analyze the rotation speeds
of galaxies. The new data provided the first pieces
of evidence that large quantities of non-luminous mass
might exist outside the visible region of most galaxies.
The velocity of a mass, m, orbiting at radius r around
a mass M is given by:
v = √(MG/r) (F = mv^{2}/r = GMm/r^{2})
Stars in tentacles of spinning galaxies don't show
this - v doesn't change with √(1/r) but is roughly
constant. The implication of this is that the mass
is not concentrated in the center of the galaxy but
is distributed out to where the star is located. In
other words the M(r), is a constant * r. Furthermore,
whatever this mass is it does not radiate energy.
Note: If we know r and v then we can compute M(r).
v = √(M(r)G/r )
∴ M(r) ~ r
Cosmologists speculate that this dark matter may be
made of particles produced shortly after the Big Bang.
These particles would be very different from ordinary
"baryonic matter" and hypothetically referred to as
WIMPs (Weakly Interacting Massive Particles) or "non-
baryonic matter". A WIMP is thought to be a new
elementary particle which interacts via gravity and
potentially other weak forces that are not part of
the current Standard Model.
The DENSITY PARAMETER, Ω, is comprised of the following:
Ω = Ω_{mass,0} + Ω_{rad,0} + Ω_{Λ,0}
Data from the CMB indicates that Ω_{mass,0} = 0.27 ± 0.04.
But the amount of ordinary or baryonic matter is only
calculated to be 0.044 ± 0.004. Therefore, observable
baryonic matter constitutes only about 17% of the
matter of the Universe, with balance being dark
matter.