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Dark Matter
-----------
The Virial Theorem
------------------
Consider N point paricles in a system.
We define the VIRIAL, G, as:
G = Σ_{i}p_{i}.r_{i}
Then:
dG/dt = Σ_{i}(dp_{i}/dt).r_{i} + Σ_{i}p_{i}.(dr_{i}/dt)
Now, 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
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 then τ can be chosen equal to the
cycle period and dG/dt reduces to 0. In this case:
<K> = (-1/2)Σ_{i}<F_{i}.r_{i}>
Now F_{i} = -∂V/∂r_{i}
<K> = (-1/2)<V>
Astronomy has made great use of the Virial Theorem as a way
of measuring gravitational mass.
The gravitational potential for two galaxies separated by a
distance, r, is:
V = -Gm^{2}/r
For N galaxies we get:
Nmv^{2}/2 = (1/2)N(N - 1)Gm^{2}/r
∴ m = 2v^{2}R/G(N - 1)
Total mass of the cluster is:
nm = 2nv^{2}R/G(N - 1)
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. An approximate formula can be
obtained by modelling stars as ideal gases (Eddington).
L/L_{O} = (M/M_{O})^{a}
Where L_{O} and M_{O} are the luminosity and mass of the Sun respectively
and a 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 determine their velocities. He
then used the Virial theorem to determine their masses. In
parallel he also determined their masses using the mass-luminosity
relationship. He discovered that the masses determined by these
2 methods 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)
Stars in tentacles of spinning galaxies don't show this - v doesn't
change with 1/r but is approximately 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 the dark matter may be made of particles
produced shortly after the Big Bang. These particles would be very
different from ordinary "baryonic matter". These hypothetical particles
are referred to as WIMPs (Weakly Interacting Massive Particles) or
"non-baryonic matter".
The DENSITY PARAMETER, Ω, is comprised of the following:
Ω = Ω_{mass,0} + Ω_{rad,0} + Ω_{Γ,0}
Data from the CMB indicates that Ω_{mass,0} = 0.27 ± 0.04. But the amount
of ordinary or baryonic matter is only 0.044 ± 0.004. Therefore,
baryonic matter constitutes only about 17% of the matter of the
universe with balance being dark matter.