San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

Nonequilibrium Systems
and Dark Matter

In the early 1930's the Swiss astronomer Fritz Zwicky applied the Virial Theorem to the estimated velocities of stars in galaxies and concluded that there would have to be more mass in galaxies to hold the stars in their orbits than could be accounted for by the measured amount of light coming from those galaxies. The required additional mass he named dark matter. Later it was concuded that such dark matter was not required to account for the radial distribution of tangential velocities in galaxies. However the same analysis based on the Virial Theorem applied to galactic clusters indicated that such dark matter was needed to hold the clusters together.

What is argued below is that presuming the galactic clusters are equilibrium systems when they are not leads to the erroneous conclusion that galactic clusters are actually being held together by dark matter. The distance involved for galactic clusters are so enormous that they can appear on a human time scale as though they are motionless in all directions.

Consider two equal bodies, such as galaxies, rotating about their center of mass. Let m be their common mass, v be their tangengial velocitis and u their radial velocities. Let s be their separation distance.

The force of gravitation attraction experienced by each is

F = −Gm²/s²

where G is the gravitational constant,

When u is small compared to v each body is roughly traveling in an orbit of radius r=s/2. Its centripetal acceleration is mv²/r. Thus the net radial force on each body is

Fr = F + mv²/r = −Gm²/s² + mv²/(s/2)
and hence
m(du/dt) = −Gm²/s² + 2mv²/s
or, equivalently
(du/dt) = −Gm/s² + 2v²/s

However if u is presumed to be zero and each body thought to be in a stable orbit of radius r then the mass M required by each body would have to be such that

GM/s² = 2v²/s
and thus
M = 2v²s/G

The supposed dark matter for the system would be

mdm = 2M − 2m
and its share
of total matter
mdm/(2M) = 1 − m/M

The radial acceleration (du/dt) can be represented as

(du/dt) = GM/s² − Gm/s² = G(M−m)/s² = Gmdm/s²
and thus
mdm = (du/dt)s²/G
mdm/(2M) = (du/dt)s²/G/(4v²s/G) = (du/dt)s/(4v²) = ½(du/dt)/(v²/(s/2))
or, equivalently
mdm/M = (du/dt)/(v²/(s/2))

But v²/(s/2) is the centripetal acceleration so the share of supposed dark matter to supposed total matter is proportional to the ratio of radial acceleration to centripetal acceleration.

The Dynamics of the System

Angular momentum is conserved so

mvr = L
and therefore
v = L/mr = 2L/ms

So the tangential velocities depend upon their original angular momentum, their mass and their separation distance. Their radial accelerations depend upon their tangential velocities.

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