Chapter 4 - Dark matter modified 20110926
Introduction
Dark matter can be explained by matter present farther out from the galaxy than is seen in the optical spectrum. The orbiting atomic hydrogen clouds are also in this region. Each meter of radius in moving out from the center of a galaxy adds a fixed amount of mass to a bigger volume so the density decrease with radius. The galaxy extends to the radius at which the galaxy is the same density as the Cosmos.

Go Back to Chapter 3 - Rotation of the Cosmos
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Measured tangent velocity within our galaxy
We have a tangent velocity vt of 210,000_m/s at a radius within the galaxy of 30_kpc or 98,000 light years = 9.26E20_m. The mass contained within this orbit is
M = vt2*radius/G = 6.12E41_kg or 308 billion solar masses.
The mass/radius ratio, m/r = 6.12E41_kg/9.26E20_m = 6.61E20_kg/m or vt2/G.
The mass added by the next meter of radius, 6.61E20_kg when divided by the volume added by the next meter of radius, is the density at this radius, or the mass/radius ratio, m/r = vt2/G, divided by the surface area of a sphere at this radius.
(m/r)/(4*pi*r2) or (vt2/G)/(4*pi*r2) = density = 6.61E20_kg/m/(4*pi*(9.26E20_m)2) = 6.13E-23_kg/m3.
I suspect that this low density of matter or dark matter, would usually be hard to detect for example with 21_cm radiation, but it is obviously still probably matter not some mysterious stuff. Radio telescopes can detect the atomic hydrogen at 21_cm if it is dense enough along their line of sight. Cold molecular hydrogen which is more stable, is probably much more common but is unfortunately invisible at radio wavelengths. It may be detected in the future as the unseen dark matter.

Flat rotation curves of galaxies
m*vt2/r = G*m*M/r2, centrifugal equals gravitational force.
In reference to galaxies, vt2 *r = G*M, as r the radius within a galaxy increases, vt2 the tangent velocity of stars, at that radius within the galaxy should decrease with G*M taken as a constant. However, what is seen is that the tangent velocity vt is largely flat. Vera Rubin's made the discovery that the velocity stays the same with increasing radius once outside the galactic core. This is called a
flat rotation curve, an unsolved problem of physics for which we provide an answer. A way this can be written is vt2/G = M/r. The vt will stay constant with increasing radius and mass if the mass/radius ratio M/r is maintained in vt2/G = M/r. This is similar to the mass/radius ratio seen as a defining condition for a black hole and the Cosmos with c2/G = M/r, another clue we are on the right track.
If the galaxy increases with a constant m/r then the density m/r*R/(4/3*pi*R3) decreases as 1/R2. A good upper limit for the radius of the galaxy would be the radius at which the density of the galaxy equals the average density of the Cosmos.
m/r*R/(4/3*pi*R3) = Mc/(4/3*pi*c3*age3) , but m/r = vt2/G so
vt2/G*R/(4/3*pi*R3) = Mc/ (4/3*pi*c3*age3)
vt2/R2 = Mc*G/(c3*age3)
, substitute for Mc*G = c3*age
vt2/R2 = c3*age/(c3*age3)
, collect terms
vt2*age2 = R2
vt*age = R
The tangent velocity vt, which is seen in the flat rotation curves of galaxies, times the age of the Cosmos equals the radius of the galaxy. The outer extent of the galaxy which also orbit at vt is too low in density to be seen with telescopes except for occasional higher density gas clouds, which may produce stars and containing atomic hydrogen, which are seen with radio telescopes. The more stable low density and low temperature molecular hydrogen remains invisible.
210,000_m/s *4.73E17_ s = R = 9.94E22_m = 3.22_Mpc = 10.5 million light years to where the galaxy is as low in density as is the Cosmos. This is the perimeter of the galaxy.
m/r times the galactic radius gives a total galactic mass = 6.61E20_kg/m * 9.94E22_m = 6.57E43_kg
The total galactic mass when divided by the visible mass within the 30_kpc radius of 6.12E41_kg gives 107 to 1 as the ratio of dark to visible matter. This explains the anomaly between observed galactic dynamics and estimated masses of the galaxies.
For vt2 and M/r to remain constant, both G = c3*age/Mc and r = vt*age vary with age.
vt2/G = Mg/r
vt2*Mc/(c3*age) = Mg/(vt*age)
vt3/c3*Mc = Mg,
the mass of the galaxy
vt*age = R, suggest that there is a Hubble expansion occurring within the galaxy. Hubble's constant is about 65_km/(s*Mpc) so we have vr = 3.22_Mpc * 65_km/(s*Mpc) = 209,000_m/s or vr = vt. The radial velocity equals the tangent velocity at the perimeter of the galaxy. When the tangent velocity of something equals its radial velocity, it spirals out at a constant angle of 45 degrees. This is the same spiral as the Cosmos.

Torque of the spinning galaxy
Here mass is the mass of the galaxy. r is the radius of the galaxy. vr is the radial velocity of expansion at the perimeter of the galaxy or vr = vt the characteristic tangent velocity of the flat rotation curve of the galaxy.
moment of inertia *angular acceleration = torque =
mass *r2 *angular acceleration =
mass *vr2 *age2 *(1/age2) = mass *vr2

We see that the square of the radius in the moment of inertia for the galaxy vr2*age2 increases at the same rate the angular acceleration of the galaxy 1/(age2) decreases so that the age2 in each cancels and the energy stays constant.

The dynamics of the galaxy
The radius of the galaxy increases while the rotation of the galaxy slows down without a change in energy or use of power. Orbits spiral out as the gravitational force decreases with the age of the Cosmos.

Hubble in the solar system
If the Hubble expansion extends to the solar system, then all the planets share the same very small precession rate, of their major axis within the planet's elliptic plane. An expanding ring slows in its rotation. We calculate a change in angular velocity, which is due to a radial Hubble velocity. This effect is quite small.
r is the radius from the sun to any planet. vr = r/age, the radial velocity, the Hubble expansion velocity in meters per second.
We see that the r's cancel so that this rate of precession is universally true for the entire solar system and is not tied to any radius.
1.375E-5_arcs/year = 1.375E-3_arcs/century
. This is a very small angle to measure. The Gravity Probe B satellite was seeking to measure a Lense-Thirring frame dragging of 4,200E-5_arcs/year.
31,556,925.9747_seconds/year/1,296,000_arcs/year = 24.3495_s/arcs for the earth to cover one arc second of circumference in its orbit.
24.3495_s/arcs *1.375E-5_arcs/year = 3.348E-5 _s/year which is the amount added yearly to the orbital period by the Hubble expansion. This is one leap second being added to our orbit every 2,986.8 years. The entire solar system is slowing in its rotation while it expands, like a dynamic unit, like the galaxy and like the Cosmos. This rate of precession is universally true. It is a consequence of the slowing rotation of the Cosmos and is tied by dynamics to the expansion of the Cosmos. The precession is proportional to 1/age. The rate of change of 1/age is -1/(age2). It may be detected by large ring laser gyroscopes, a form of interferometer which detect absolute rotations by interference fringes between light beams traveling in opposite directions around a loop. The Michelson-Morley interferometer did not find fringes when their interferometer was parallel to the surface of the Earth. This was used as evidence that the aether does not exist and that the velocity of light is not additive, increased or decreased by the velocity of the light source. These erroneous beliefs have misguided physics for 120 years. When the interferometer is rotated to a vertical position fringes are detected. What does this mean? I found this in the NPA world science database under experiments.
Inertial accelerations
To calculate the path of expansion of a planet we need the vector sum of three accelerations; the centrifugal, tangent and Coriolis accelerations.

Centrifugal acceleration
Centrifugal force uses the tangent velocity, but here, we are looking for the recessional component or factor, of the centrifugal force which is due to the Hubble expansion. The tangent and radial velocity vt = vr due to precession are equal.
m*vr2/r equals the centrifugal force and vr2/r is the acceleration.
vr = r/age.
For the earth, vr = 149E9_m/4.73E17_s = 3.16E-7_m/s or 9.97_m/year.
For the moon, vr = 379737123_m/4.73E17_s = 8.018E-10_m/s or 25.3_mm/year. The moon's slightly larger recession from us is usually attributed to tidal friction.
vr2/r = vr2/(vr*age) = vr/age or
vr2/r = r2/(age2*r) = r/age2

This is the radius divided by the angular acceleration of the Cosmos. This is a clue that the ultimate source of the centrifugal force is the Cosmos.

Tangent acceleration
We can calculate the tangent acceleration using the torque formula.
moment of inertia*angular acceleration = force*radius.
m*r2*angular acceleration = m*a*r.
a = r *angular acceleration =
a = vr*age *(1/age2) = vr/age

The direction of deceleration is opposite of rotation.

Coriolis acceleration
2*angular velocity *vr = Coriolis acceleration
vr
2*vr/age = Coriolis acceleration

Spirals
Now that we have calculated the inertial accelerations we can look at the way the solar system expands. We have the centrifugal acceleration of vr/age directed radially out. We have the Coriolis acceleration of 2*vr/age in the direction of rotation and the deceleration of vr/age in the direction opposite of rotation. The resultant of these accelerations is 45 degrees between the direction of rotation and the outward directed radius. It has a value of 21/2*vr/age. A planet moving in this way traces out a logarithmic spiral.
Torque in the solar system
Here mass is the mass of a planet. r is the distance from the sun. vr = r/age is the radial velocity.
moment of inertia*angular acceleration = torque.
mass *r2 *angular acceleration =
mass *vr2 *age2 *1/age2 = mass *vr2

We see that the square of the radius in the moment of inertia for the planet, vr2*age2 increases at the same rate the angular acceleration of the planet 1/age2 decreases so that the age2 in each cancels and the energy stays constant.

The dynamics of the solar system
Expansion and rotation rates are linked. The radius from the sun, to the planets, increases while the orbital periods of the planets slow down, without a change in energy or use of power. Orbits spiral out as the gravitational force decreases with the age of the Cosmos.
References
http://blackholeformulas.com/files/GSJDarkMatter.docx
1. Galaxy tangent velocity @ http://www.astronomynotes.com/ismnotes/s7.htm
2. Atomic hydrogen 21 cm radiation @ http://en.wikipedia.org/wiki/Hydrogen_line
3. Molecular hydrogen @ http://www.newtonphysics.on.ca/hydrogen/index.html
4. Galactic flat rotation curves @ http://en.wikipedia.org/wiki/Galaxy_rotation_curve
5. Precession @ http://en.wikipedia.org/wiki/Precession
6. Large laser gyroscopes @ http://www.wettzell.ifag.de/LKREISEL/G/LaserGyros.html
7. Interferometer detects the aether drift @ http://www.21stcenturyideas.com/interferometer.html
8. Natural Philosophy Alliance world science data base @ http://www.worldsci.org/php/index.php?tab0=Experiments&tab1=Display&id=3
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