Re: mars and venus (long)

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Stephen Murray's questions about my orbital calculation for a
simple model proto-Venus:

> This still leaves the question of the different final orbital radii for the
> planet derived from conservation of angular momentum and conservation of
> energy.  David's results do depend a bit on the values that he assumed.
> For example, constant surface density is probably more appropriate in the
> inner Solar System,

I've looked at this case too.  See below.

>                    and I would have expected that the "feeding zone" of
> Venus would extend more than halfway to Mercury (not less than half) and
> out to about halfway to Earth (how did you get 0.5998 AU and 0.8505 AU,
> David?).

I *chose* the outer radius of the annular "feeding zone" to be 0.8505 AU
since that value happens to be the geometric mean between the mean
orbital radii of Venus and Earth.  It somehow seemed that taking the
geometric mean would be more appropriate (considering the geometric
nature of Bode's law) than using the arithmetic mean.  The inner radius
of the Venusian "feeding zone" was *solved for* by finding the inner
radius for the annular region with the assumed form for the radial
dependence for the initial mass distribution which would give the correct
final orbital radius for "Venus" assuming that all the mass and all the
angular momentum were conserved and that the final state of the model
"Venus" had the correct final orbital radius and final retrograde spin.
It ended up that an inner annular radius of 0.5998 AU was what did the
trick when the initial mass distribution was assumed uniform in radius
but had a mass per unit disk area which was inversely proportional to
radius.  Once both the inner and outer radii of the annulus were
determined then the difference was calculated between the initial orbital
energy of the annulus and the final orbital energy of the planet
(including its rotational kinetic energy) so that the extra orbital
energy need could be found.  This deficit amount was compared to the
total gravitational binding energy of the planet, and it was found that
just 13.46% of the released binding energy could fund the deficit in the
orbital energy.

>          A uniform density with the feeding zone defined by the
> Mercury-Venus and Venus-Earth midpoints seems to lead to about twice the
> required change in Venus' orbit as found for the 1/R law.

I also looked at the model mass distribution you propose here where the
mass distribution is uniform over disk area but had a mass per unit
orbital radius which was proportional to the radius due to a factor of r
in the differential measure.  (Actually, I also looked at the family of
all power law mass distributions as a function of radius.)  In the case
for your preferred model a choice of an outer annular radius of 0.8505 AU
resulted in a value of 0.5837 AU for the inner annular radius, and the
orbital energy deficit could be made up with 15.34% of the gravitational
binding energy of the planet.

> The main problem, I think, is how you could systematically boost a planet
> like Venus to a larger orbit during accretion in something like the "swarm
> accretion" scenario.

I doubt it would be very systematic.

>                     It seems to be generally assumed that the collisions
> will lead to net dissipation of kinetic energy, shrinking the orbit.

Certainly there would be net dissipation.  It seems the real question
essentially is whether or not the total amount of dissipation would
exceed or consume just about all of the binding energy for the assembly
of the planet or not.

> To cope with the radius discrepancy, the terrestrial planets are thought to
> have transferred excess angular momentum to objects at larger orbital radii
> via gravitational interactions.

That's a possibility too--one which I did not account for in my simple
model. The mechanism would tend to obviate the need for a source of extra
orbital energy if the system could shed enough of its initial angular
momentum to end up with a net retrograde spin.

David Bowman
David_Bowman@georgetowncollege.edu