Thanks to Stephen Murray and Michael Edmiston for reminding me of the
importance of angular momentum conservation in the accretion process for
the inner planets. As Stephen explained, the main thing I was missing in
my analysis was the effect of changes in the gravitational potential
energy (relative to the Sun) for the small masses that were accreted onto
the protoplanet from a radial distance far from that of the protoplanet.
I now no longer think that a prograde planetary spin is a less likely
final state for the accretion process for an inner planet. I even think
that Stephen may be correct that a prograde spin is more likely than a
retrograde spin. But I still do not think that the picture presented by
Stephen is necessarily the whole story. I am still not ready to concede
that a retrograde spin is so unnatural that a major post-accretion
collision is necessary to account for the spin of Venus.
I have spent some time this weekend considering a simple model of the
accretion process keeping track of the total energy and angular momentum
for the process in a simple model that has the initial state of the mass
widely distributed in an annular band around the Sun in independent
circular Keplerian orbits with (near) angular symmetry, and the final
state a single planet orbiting the Sun -- also in a circular orbit. I
neglected the possiblity that the dynamics of the process would result in
a significant fraction of the original mass in the band to be ejected
from the vicinity of the band or captured by another incipient planet
growing in another region. I also neglected any gravitational
interference effects that may occur between different incipient planets
growing in different regions. Although these assumptions are probably
unrealistic to some extent, not making them resulted in a problem that
was too complicated for me to want deal with it.
First, I convinced myself that if one neglects any gravitational binding
energy for the matter once it is bound up into a planet, or if one
assumes that the total amount of energy dissipation in the accretion
process is greater than this binding energy, then, indeed, as Michael and
Stephen have claimed, the conservation of angular momentum and the
non-increase in mechanical energy will result in a final state for the
planet to have a prograde spin. I checked this for a wide ranging family
of initial radial mass distribution functions, and the result holds for
them all. Essentially, (as Michael tried to say and Stephen corrected)
the fact that the angular momentum of a mass in a circular Keplerian
orbit increases with radius at a faster rate than does the energy (since
sqrt(r) tends to grow more rapidly with r than -1/r does, as the former
grows in an unbounded manner and the latter saturates at large radii).
The difference in this functional behavior causes the final orbital
radius of the planet (if it is spinless) if all the initial state angular
momentum is tied up in the orbital angular momentum of the planet to be
strictly greater than the final orbital radius the planet would have if
100% of the initial energy was conserved. In effect, in order to
conserve orbital angular momentum (without having a prograde spin) the
system needs some external source of energy to lift the planet up to a
greater orbital radius than is consistent with the initial energy. The
amount of extra energy needed is quite small however. The system
*almost* has enough energy to pull it off. So barring a *huge* amount
of dissipation of enrgy the planet would tend to be left with a slow
net prograde spin.
Now if we include the fact that as the process of binding the initial
matter into a planet results in that matter having a lower mutual
potential energy than had it had when that matter was widely distributed
around the annulus then we *have* a source of extra energy--from the
released mutual binding energy of the matter. If only a small fraction
of this released binding energy is not dissipated as heat, radiation,
interior chemistry, etc. then there *is* enough extra energy for the
final state of the planet to have a retrograde spin. For definiteness,
suppose we take the parameters of Venus as an example. If initially 100%
of Venus' mass was distributed around the Sun in an annulus whose inner
radius was 0.5998 AU and whose outer radius was 0.8505 AU, and if the
density of the matter as a function of radius was inversely proportional
to the radius (this makes the mass per unit radius a uniform distribution
once we include the extra factor of r in the differential measure), and
if only 13.46% of the energy released in binding the matter together to
form a planet was dynamically available, then the final state would
result in a planet of the mass of Venus orbiting at Venus' radius of
0.7233 AU with Venus' retrograde spin period of 243 days. (In doing this
calculation I assumed that no matter was ejected or injected to/from out
of the annulus and the incipiently forming adjacent protoplanetary
"Earth" and "Mercury" had no effect).
In order to do the calculation I needed to make a model for the interior
of "Venus" to find its gravitational binding energy and its moment of
inertia about its spin axis. In doing this I assumed that it had a
spherical mass distribution whose density [rho] was a simple quadratic
function of radius, i.e. [rho] = A - B*R^2 for 0 <= R <= R_o where R is
the radial distance inside the planet from the center, R_o is the overall
radius of the planet, and constants A and B were chosen so that the
overall mass of the planet came out right and so the density at the
surface (2800 kg/m^3) was typical of terrestrial crustal rock.
Of course there is the question of whether or not it is realistic to
expect any significant amount of the binding energy to be made available
as orbital energy. I doubt that a very large fraction of it would not be
be dissipated and reinjected into the orbital dynamics. But since only a
small percentage of it is needed I can't say that amount would not be
made available. For instance I can imagine that near-miss multi-body
interactions could easily result in some bodies being ejected at a
higher kinetic energy then they were incident with while other bodies
coalesce to provide the kick. Also, at a later stage of the process
some collisions could be great enough for some ejected debris to acquire
more than escape velocity and the energy for its ejection would have come
from the binding energy of the material that remained and coalesced.
I should also point out that apparently the expectation of at least
some experts in the planetary science community is that (barring any
catastrophic post-accretion collisions) the inner planets would be
expected to have a net slow spin either prograde or possibly even
retrograde. As an example of this there is an article I came across
in Scientific American (G. Jeffrey Taylor, "The Scientific Legacy of
Apollo", Sci. Am., Jul 94, pp. 40-47) which tends to support this
claim--although in the context of the spin of the Earth since the
article is concerned with explaining why the "giant impact theory" is
the preferred theory for the origin of the Moon, and why the
earlier "fission theory", "capture theory", and the "in situ double
planet theory" of the origin of the moon all have major problems. In
the section about the problems with the "fission" theory/hypothesis on
p. 41 Taylor writes:
"Subsequent calculations showed that the earth would have to have been
rotating once every 2.5 hours in order to have spun off the material
that became the moon. This short day is among the chief problems
with the hypothesis: no one can figure out how the earth would have
been spinning so fast in the first place. The models that describe
planetary formation as an accumulation of dust grains indicated that
the earth would end up spinning rather slowly, if at all.
Incorporating events that add angular momentum--most notably, impacts
of planetesimals up to a few hundred kilometers across--did not help.
Computer simulations showed that for every object that struck the
earth to add clockwise spin, another impact would cause the planet to
spin counterclockwise. Even if there were a mechanism for imparting
enough angular momentum into the earth, advocates of the fission
hypothesis had to find a way to eliminate much of the rotational
energy. The earth-moon system of today does not have nearly the
amount of momentum needed to initiate separation of the two bodies
from one another."
Also, later on p. 42 while discussing the problems with the "in situ
double planet" theory/hypothesis Taylor writes:
"Most important, it runs into the angular momentum problem. That is,
it does not explain how the earth's rotation came to be 24 hours,
which is faster than predicted by simple accretion models, and how
the ring could have acquired enough circular motion to stay in
orbit."
And on pp. 43-4 while discussing some of the advantages of the "giant
impact" theory Taylor writes:
"Finally, the hypothesis explains the most difficult problem: the
angular momentum of the earth-moon system. The projectile must have
struck the earth off-center, away from the central axis. This type
of blow would have sped up the earth's rotation to its current value.
"The most enticing aspect of the giant impact theory is that such
a collision is a natural consequence of planet formation. No unusual
or ad hoc circumstances need to be invoked. Such catastrophes, while
enormous, are not unlikely. Indeed, planetary scientists now appeal
to giant impacts to explain the composition of Mercury and the large
tilt of Uranus. Without this colossal event early in the history of
the solar system, there would be no moon in the sky. The earth would
not be rotating as fast as it does, nor would it have such strong
tides. Days might even last a year, as they do on Venus. But then,
we probably would not be here to notice."
Although Taylor doesn't come right out and say that Venus had not
suffered a giant collision, he does suggest that if the earth had not
suffered one its days would probably be like those on Venus. He also
did not include Venus in is list of other planetary bodies that were
thought to have suffered such collisions (i.e. Mercury and Uranus).
So this, along with my calculation, discussed above, suggest to me that
it is probably quite possible that Venus, indeed, acquired its slow
back-spin by a non-catastrophic accretion process.