The thickness h of the solar disk was roughly a linear function of the distance R from the center of the system; i.e.,

h(R) = α − βR
 

Let ρ be the volume density of the solar material and assume that it is independent of R. The areal density A(R) is then ρh(R). The radial density Π(R) is the amount of material per unit length at a distance of R. Since the circumference of a circle of radius R is 2πR the radial density is given by

Π(R) = 2πρ(αR − βR²)
 

It is thus a parabolic relationship.

The mass M of a planet at distance R would the sum total of material between some R_min and some R_max. These limits would be determined by the phenomenon of resonance and would be connected with the period of revolution.

By Kepler's Law the period of revolution T and orbit radius R are connected by the equation

R = γT^2/3
 

where the parameter γ depends upon the mass of the Sun.

Thus R_min might be the distance such that the period of revolution is one half of its value at R. However rather than specifying the periods of revolution at R_min and R_max let them be expressed as some ratios to T; i.e.,

R_min = γ(εT)^2/3
and
R_max = γ(δT)^2/3
 

These equations however reduce to

R_min = ε^2/3γT^2/3 = ε^2/3R
and
R_max = δT^2/3γT^2/3 = δ^2/3R
 

For convenience the coefficients of R are just renamed μ and ν.

Thus the mass M of a planet at R would be

M(R) = ∫_Rmin^R_maxΠ(r)dr
which reduces to
M(R) = ∫_μR^νR[2πρ(αr − βr²)dr
which evaluates to
M(R) = 2πρ[α(μ²−ν²)R²/2 − β(μ³−ν³)R³/3]
 

This function is not a parabola but it can have the general shape of a parabola. One notable aspect of his function is that it does not have a constant term.

The actual masses of the planets and their distances from the center of the Sun are

The graph of the data is as follows.

In the above graph distance is expressed in terms of astromical units (a.u.); i.e. relative to the Earth's orbit.

There is no way that a parabolic type curve can fit this data. However if we look at the inner planets (Mercury through Mars) the picture is different.

A regression of the form

M(R) = c₁R² + c₂R³
 

yields the following results

M(R) = 2.708R² − 1.743R³
    (0.418)    (0.293)   
R² = 0.9
 

The coefficients are of the right sign and they are significantly different from zero at the 95% level of condidence. Unfortunately there are only 2 degrees of freedom for the regression.

The graph of the mass versus distance for the outer planets is entirely different from the one for the inner planets.

What this suggests is that the proto-solar system had two rings in the planetary disk. The inner one fit the assumptions of the analysis, but the outer one had a diffferent structure. Perhaps the volume density of the disk material fell off with distance as well as the thickness.

Some notion of the areal density of the planetary rings can be obtained by supposing a planet at distance R is the collection of all the mass between &nuR and μR. The planetary annulus would have an area A(R) of

A(R) = π(νR)² − π(μR)² = π[ν²−μ²]R²
 

Thus the areal density would be proportional to M(R)/R². The values of this quantity for the planets is shown below.

Because of the wide range of orbit radii it is more instructive to view the relation between orbit distance and relative areal density in terms of the logarithms of the variables as shown below:

Again the data suggest that there were two planetary rings in the original proto-solar system.

(To be continued.)