San José State University

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Thayer Watkins
Silicon Valley
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 The Enhancement of the Rotation Speed ofGaseous Planets Under Gravitational Contraction

A theoretical investigation of the formation of planets in a planetary ring about the Sun indicated that the period of rotation of planets should be independent of the mass of the planets. While empirically this is not precisely true it is nearly so. Jupiter has a mass nearly three thousand times larger than that of Mars yet Jupiter's period of rotation is only about sixty percent smaller than that of Mars; ten hours for Jupiter and 24 hours for Mars.

The second order differences in the periods of rotation can be accounted for by the gravitational contractions of the planets. A larger gaseous planet contracts more than a smaller rocky planet and thus its rotation speed increases more.

In a gravitational field of constant intensity such as for Earth's atmosphere near the surface the profiles of pressure and density are negative exponential functions. It is much more difficult to characterize the pressure and density profiles for the gravitational fields generated in general. However some important insights can be gained by examining some highly simplified models.

## Analysis for a Single Proto-Planet

In the formation of a planet the process of gravitational contraction would take place along with its acquisition of mass and angular momentum. However it is convenient to consider the planet forming first and the gravitational contraction taking place afterwards.

Suppose a proto-planet has formed of radius R0 and of radially constant mass density ρ0 rotating at a speed of ω0. Its mass M and angular momentum L would then be

#### M = (4/3)πR0³ρ0 L = I0ω0 = (2/5)MR0²ω

where I0 is the initial moment of inertia of the planet.

Under contraction both mass and angular momentum are preserved. Therefore mass and angular momentum are not subscripted because their values are constant.

The initial condition is not an equilibrium. Under gravitational contraction a new density profile would be established. The functional form is not simple to express. Suppose the gaseous sphere of the planet contracts until an equilibrium is established at its surface between the gravitational force, pressure and centrifugal force at a new radius R1. The gravitational force on a unit mass at the surface is GM/R1² and the centrifugal force is ω1²R1. where G is the gravitational constant and ω1 is the new rate of rotation.

The pressure force depends upon the gradient of pressure which in turn depends upon the density gradient. If the density gradient is presumed to be zero anywhere below the surface a balance of forces just below the surface would require

#### GM/R1² = ω1²R1and hence ω1²R1³ = GM, a constant

Since M=(4/3)πR1³ρ1 this means

#### GM/R1² = ω1²R1implies G(4/3)πR1ρ1 = ω1²R1and hence (4/3)Gπρ1 = ω1² and thus ω1 = [(4/3)Gπρ1]½

This is one of the three conditions that prevail among the three unknowns. The other two conditions that prevail between the initial and final values of R, ρ and ω are

#### R1³ρ1 = R0³ρ0 (Conservation of Mass) and ω1R1² = ω0R0² (Conservation of Angular Momentum)

There are thus three equations in three unknowns. The variable ω1 may be replaced by αρ½ in the third equation, where α=[(4/3)πG]½.

The result is two equations in two unknowns; i.e.,

#### R1³ρ1 = R0³ρ0and αρ1½R1² = ω0R0²

Squaring this last expression gives

#### α²ρ1R14 = ω0²R04

Dividing this expression by the first of the above two equations gives a solution for R1; i.e.,

#### α²R1 = ω0²R0/ρ0or R1 = ω0²R0/(α²ρ0)

Since ρ1 = ρ0R0³/R0³

Thus

#### ω1 = αρ11/2 = α4ρ02/ω03

Thus the higher the initial density ρ0 the higher the final density ρ1 and the faster the final rotation speed ω1. The dependences are nonlinear so a small change in initial density results in a larger change in final density and rotation speed and period. It is notable that the final density and rotation speed is independent of the initial radius of the protoplanet.

## Relationships Among a Collection of Planets

The previous analysis indicated that for the assumptions made the final rate of rotation and hence the period of rotation is independent of the initial radius. Thus the relationship between final density and the rotation period should be

#### T1 = 2π/ω1 = (2π/α)ρ1-1/2

For the planets of the solar system there is a relationship between the rotation period and the planet density, as shown below.

The magnitude of the slope is about the 0.5 the model implies but the slope is positive rather than negative. However the initial density affects the final density and hence the rotation period. The above data is consistent with there having been three bands or rings of planetary material. Earth and Mars would have been formed from the inner ring and have approximately the same rotation period. Jupiter and Saturn would have been formed from the second ring and have the same rotation period. And from the third ring Uranus and Neptune would have been formed. (Mercury is left out of the analysis because it is subject to tidal locking. Venus is left out because something catastrophic occurred to it that left it with a retrograde rotation of extremely long period.)

A close observation of the density and rotation within the rings shows the relationships to be slightly inverse. The exponents for T=λρε for the three cases are: Earth-Mars ε=−0.08, Jupiter-Saturn ε=−0.1, Uranus-Neptune ε=−0.09. Thus all three indicate an inverse relation between density and rotation period and they are approximately the same.

(To be continued.)