Dark Matter



24 June 2013

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 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.

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
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, that is, the velocity stays the same with increasing radius once outside the galactic core. This is called a
flat rotation curve. 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 is maintained. 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.

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)
vt2/R2 = c3*age/(c3*age3)
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.
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, 6.12E41_kg gives 107 to 1 as the ratio of dark to visible matter. Certainly, within reason.

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 occuring 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 may be currently overwhelmed by the noise of the changing gravitational forces exerted by the planets and moons. It may be detected by large ring laser gyroscopes which detect absolute rotations.

r is the radius from the sun to any planet. vr = r/age, the radial velocity, the Hubble expansion velocity in meters per second.

The length of the orbit, when divided by, 2*pi*radians per revolution, equals the meters per radian or 2*pi*r/(2*pi*radians)= r/radians.
vr/(meters per radian) = r/age/(r/radians) = radians/age = precession
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.
radians/age = radians/(4.73E17_s) = 2.11E-18 radians/second . This is Hubble as a rotation rate.
radians/(15 billion years) = 6.66E-11 radians/year =
1.375E-5_arcs/year = 1.375E-3_arcs/century
. This is a very small angle to measure. The gravity probe B satellite is 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 and every other orbit in the solar system every 2,986.8 years. In a year Mercury does more than two orbits while the Earth does one. Both have the same Hubble precession per year. 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.

Inertial accelerations

To calculate the path of expansion of a planet we need the vector sum of three accelerations; the centrifugal, tangent and coriolis.

Centrifugal acceleration

Centrifugal force uses the tangent velocity, but here, we are looking for the precessional component or factor, of the centrifugal force which is due to the Hubble expansion. The tangent and radial velocity 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.
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

moment of inertia*angular acceleration = force*radius, the tangent acceleration using the torque formula.
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 is the radial velocity.
2*vr/age = coriolis acceleration


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.

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