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and 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.
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. The parameters ζ and γ depend upon the density of the planets and possibly upon their distance from 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.
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 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.
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 material being accumulated. From the previous analysis it is seen that
Thus the rotation period should be independent of the density as well as the mass.
There is 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 millions 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 will be faster for a bigger planet than a smaller planet. For more on this effect see Direction of Rotation of the Planets.
The data for limited sample of planets is consistent with the model of the formation of the planets in which the radiation pressure from the Sun upset the equilibrium of the planetary ring resulting in the larger particles sweeping up the smaller particles until essentially only the planets were left. 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. 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 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.
(To be continued.)
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