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Their Acquisition of Rotation in a Planetary Sweep of the Solar Ring About the Sun |
Our Solar System evolved out of a rotating gaseous cloud that first collected into a central solar mass and a planetary disk. The planetary disk was like a planar ring around the central mass which became the Sun. Before the central mass ignited into the Sun there was rotation of the planetary disk around that central mass. The rotation would not have been that of a rotating solid disk but instead a rotation in which the speed of rotation varied with the distance from the central mass. For details see Keplerian Disk.
In that initial formation there was no differentiation between dust, small pieces and large pieces. The dust in an orbit traveled at the same speed as the larger pieces or even a protoplanet. Everything was held in orbit by the gravitational attraction of the pieces for the other pieces, but that effectively was the same as if all the other mass was concentrated at the center of mass of the system.
When the central mass reached a critical level thermonuclear fusion was ignited and the central mass became the Sun. Now the pieces of the planetary disk were subject not only to gravitation but also to radiation pressure. The gravitational attraction was proportional to the masses of the pieces but the radiation pressure was proportional to their cross-sectional area. The impact of the radiation pressure was greater for the small particles, particularly the dust, than for the larger pieces. Thus the radiation drove the small dust-like particles away of the Sun relative to the larger pieces. As the particles moved away from the Sun they lost speed in preserving angular momentum. The small pieces thus moved into an outer orbit where they were traveling slower than the larger pieces. The larger pieces ran into the slower, smaller pieces and captured them. This built up the mass of the larger pieces.
After the larger pieces plowed through the dust and small pieces in one pass around the Sun thereafter the small pieces would tend to captured on the side of the larger pieces closer to the Sun. This would have increased the spin of the larger pieces. Over time the larger pieces would sweep the planetary disk clean and in the process accumulate angular momentum.
Once larger bodies were formed another process came into operation: resonance oscillation. Resonance occurs when a system is subject to a disturbing force with a frequency near or at its natural frequency or some harmonic of its natural frequency. Consider the planet Jupiter. A protoplanet having an orbit period one half of the orbit period of Jupiter would be subject to an oscillation that would continue until that protoplanet moved to an orbit that involved a period sufficiently different from that of Jupiter that the resonance was broken. Resonance occurs not only for 0.5 the period but also 0.4 and 2 and 2.5. If the protoplanet is nudged from an orbit with 0.4 period to one that is 0.45 of the period of the larger planet then the protoplanet will be traveling slower than the material in the new orbit and that material will crash into the protoplanet. If the protoplanet move from an orbit with a 0.5 period to one with a 0.45 orbit it will be traveling faster than the material in that orbit and crashes into the material. It sweeps up the material. For more details on this process see Bode.
The more material swept up by a protoplanet the closer its orbit gets that of the material in the new orbit. There could be more than one protoplanet in the new orbit. For protoplanets that moved outward the one that acquired the most material would have the larger speed. It would therefore sweep up the smaller protoplanets that were in its orbit. For the protoplanets that moved inward the one that acquired the most material would have a slower speed and the small protoplanets would crash into it. Thus there would end up being only one planet in each orbit unless one of the smaller protoplanets did not crash into the larger one but was acquired as a satellite.
Angular momentum is mass times velocity times radius. Consider a body of radius r acquiring a parcel of mass dm by overtaking it. That is to say the body has a greater velocity than the planetary material at its edge. Let Δv be the difference in velocity and let the distance from the Sun of the body be R. Thus the body at orbit R is acquiring mass in the solar ring at (R−r).
In a Keplerian ring the relationship between orbit velocity and orbit radius is
where α is a parameter that depends upon the mass of the Sun. More generally v will also depend upon the area and mass of the particles and the luminosity of the Sun. For simplicity of the explanation the Keplerian distribution of velocity will be presumed.
Thus
Thus the increment in angular momentum dL is given by
Since m=(4/3)πr³ρ where ρ is the density of the material then r=(3m/(4πρ))^{1/3}. Let γ=β(3/(4πρ))^{2/3}. Then
Now consider the mass and moment of inertia I of a ball (solid sphere) of radius r and density ρ. These are:
Therefore
Let ω be the angular velocity of a spinning spherical body and let T be its period of rotation. From the definitions
From the previously derived expressions for L and I it follows that
That is to say, the periods of rotation of the planets should be independent of their masses. At this point the question of what the rotation period does depend upon will be deferred. However let it be noted that the parameters ζ and γ depend upon the density of the planetary material from which the planets are formed. The parameter γ also depends upon a planet's distance from the Sun and the mass of the Sun.
Most major satellites such as Earth's Moon are subject to tidal locking which establishes as rotation period equal to their revolution period. Mercury and Venus are satellites of the Sun whose periods of rotation are anomalous. This leaves Earth, Mars, Jupiter, Saturn, Uranus and Neptune as test cases. Pluto might also be included but many characteristics of Pluto are anomalous.
Planet | Relative Mass | Sidereal Rotation Period (hours) |
Earth | 1.000 | 23.93444 |
Mars | 0.107 | 24.62278 |
Jupiter | 317.8 | 9.93333 |
Saturn | 95.2 | 10.65000 |
Uranus | 14.5 | 17.23333 |
Neptune | 17.1 | 16.11667 |
Pluto | 0.001674 | 153.3 |
It is notable that the planets come in pairs with respect to rotation period and mass.
Note that the ratio of the largest mass (Jupiter) to the smallest (Pluto) is about 190,000 to 1 but the ratio of rotation periods is only 1 to 15.4. If Pluto is left out of the comparison the ratio of the mass of Jupiter to that of Mars is 2970 yet the ratio of their rotation periods is only 0.4. While the rotation period is not precisely constant with respect to mass it is nearly so. It is virtually a quantization of rotation period.
The relationship sought is of the form
The graph of the data is shown below.
A linear regression of log(T) on log(M) yields
The t-ratio for the regression coefficient of log(M) is 6.47 and therefore it is significantly different from zero at the 99 percent level of confidence. If Pluto is left out of the analysis the regression equation is
It is notable and perhaps significant that the regression coefficient for log(M) is almost exactly (1/8). If there is a theoretical explanation for such an exponent it will be a simple fraction of this sort.
Although the theory sketched out above indicated that the rotation period should be independent of mass the theory could be incomplete. The other explanation for the observed correlation of mass and rotation period could be due to a correlation of planet mass with other variables which do affect the rotation period. The theory allowed for a dependence upon a planet's density and its distance from the Sun.
The relationships of density with rotation period and distance with rotation period are given below.
Analysis elsewhere of the relationship between rotation period and density as a result of gravitational contraction indicates that there should be an inverse relationship between rotation period and the planet density. The above empirical relationship shows a positive relationship. However suppose the rotation period depends upon the density of the material that went to form the planets. Suppose further that there were three rings of differing densities. Earth and Mars could have been formed from one ring; Jupiter and Saturn from a second ring and Uranus and Neptune from a third ring. If one observes the relationship between density and planet rotation period within the supposed ring the relationship is slightly inverse in all three cases.
The relationship between rotation period and distance from the Sun is neither direct nor inverse but instead a quadratic relationship.
As can be seen below planet mass is correlated with planet density;
The way to handle such relationships among explanatory variables is to include all of the variables in the regression analysis. A multiple regression of log(T) on log(M), log(density) and log(distance) yields.
The t-ratio for the coefficient of log(M) is only 1.89 and thus not significantly different from zero at the 95 percent level of confidence.
The inclusion of Pluto definitely had a strong influence upon the results. However if Pluto is left out there are not enough data points to obtain a standard deviation of the estimates of the regression coefficients. The regression estimates with Pluto left out are
It is appropriate at this point to consider explicitly the dependence of the rotation period on the density of the primodial material being accumulated. From the previous analysis it is seen that
Thus the rotation period should be independent of the mass density in the solar ring as well as the mass.
There is however another process that would contribute to the rotation of the planets. It is the gravitational coalescence of the material in a planet. Two bodies with some angular velocity with respect to their center of mass will rotate faster with respect to that center as they move closer to each other.
This effect can be illustrated using data for the Earth's orbit. If the Earth's center is 93.5 million miles from the Sun's center and it travels in a circular orbit then it travels 67,018 miles per hour. Material at 94 million miles from the sun would be traveling at a speed of 67,018/(94/93.5)^{1/2}=66,839 mph. Material at 93 million miles would be traveling at a speed of 67,018/(93/93.5)^{1/2}=67,198 mph. If the material at 94 million miles moved to 93.5 million its speed would increase to 66,839(94/93.5)=67,196. If material at 93 million miles moved out to 93.5 million miles its speed would decrease to 66,839 mph. Thus the material from 0.5 million farther out would be traveling (67,196-66,839)=357 miles per hour faster than the material 0.5 million miles farther in. This would give a body composed of material farther out with material farther in a spin in the same direction as the spin of the planetary disk; in this case counterclockwise. This is shown in the diagram below.
Since this enhancement of rotation will be greater when the distance over which the material coalesces the rotation period might be faster for a bigger planet than a smaller planet. For more on this effect see Direction of Rotation of the Planets.
In addition to the above effect there would be an enhance in rotation speed due to the gravitational contraction of a planet. For an investigation of this effect see Rotation speed enhancement under gravitational contraction.
The data for limited sample of planets is consistent with the model of the formation of the planets in which some factor moving material out of its previous orbit such that it is traveling at a different speed than the material in its new orbit. The material in the new orbit is swept up or swept into the disturbed material until essentially only the planets are left. One such factor that could disturb the previous balance is radiation pressure from an ignited Sun. Thus the larger particles sweep up the smaller particles. Once some planets begin to form resonance effects would nudge material out of its previously equilibrium orbits in the planetary ring(s).
As the proto-planets acquire mass they also acquire angular momenta. The mechanism for the acquisition of angular momentum in the planetary sweep of the ring resulted in rotation periods for the planets that are largely independent of their masses. Jupiter is nearly three thousand times more massive than Mars but its rotation speed is only about sixty percent faster.
The small level of statistical dependence of rotation period on mass is apparently not due to the correlation of mass with other factors affecting the rotation period. There is an effect of mass on rotation period that arises from the gravitational coalescence and contraction of the material of the planets which could account for the second order level of dependence of rotation period on mass. See Revolution and Rotation and Rotation Enhancement.
It is now worthwhile to examine what does the period of rotation depend upon, according to the above analysis.
The definition of ζ was
The definition of γ was
Thus the ratio of ζ to γ reduces to
The definition of β was
where R is the orbit distance. The parameter α was defined as
It arises from the balancing of gravitational and centrifugal force. If G is the gravitational constant and M_{sun} is the mass of the Sun then
The implication is that the planetary rotation periods should be inversely related to the mass of the central star. Specifically it should be inversely related to the square root of the mass of the star.
(To be continued.)
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