San José State University |
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applet-magic.comThayer WatkinsSilicon Valley & Tornado Alley USA |
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Nonequilibrium Systemsand Dark Matter |
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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

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

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

and thus

M = 2v²s/G

The supposed dark matter for the system would be

and its share

of total matter

m

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

and thus

m

and

m

or, equivalently

m

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.

Angular momentum is conserved so

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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