Simultaneous transits of Venus and Mercury?

In article ,
wrote:
CeeBee writes:

Andrew Goldish wrote:

Is it possible for Venus and Mercury to be transiting the sun at the
same time? Assuming the two orbits are independent of each other, it
should occur (though extremely rarely: millions of years between
double transits probably or something like that).

You're right. What should be the problem preventing the two passing the
Sun simultaneuosly as seen from Earth?

The problem that currently prevents it is the non-alignment of the
nodes of Mercury's and Venus' orbits. I do not know their precession
rates, so I'm not in a position to say how long it might be before
the nodes are aligned. But for such a thing to happen, you'd need
(a) the nodes aligned to within some tight tolerance, (b) both
planets near their nodes at approximately the same time, (c) both
planets on the same side of the Sun, (d) the Earth aligned with both
planets at that time. I wouldn't be surprised if it's never happened
in the 4.5 billion year history of the Solar System.

I tried to guess at the very-very-long-term average frequency of these events,
assuming that Mercury and Venus aren't in an orbital resonance (so their
longitudes should be uncorrelated), and that the precession of their nodes is
at least fast enough that they've coincided many times over the life
of the solar system. And I guessed that Mercury's and Venus' nodes
needed to be within a one-degree span (half degree delta) for double
transits to be possible.

During periods when the nodes do nearly coincide, it looked like there
should be about three double transits, with these 5-or-6-hour-long
events at least partly overlapping, per million years. And if the
+/- 0.5 degree tolerance is right, those periods should be about 1/180th
of the time. So the long-term-average rate would be about
one per 60 million years, or maybe 80 times during the life of
the solar system, within a factor of a few.

Of course this kind of handwaving gives no clue about the *distribution*
of these events in time. I browsed a bit but couldn't find any on-line
references suggesting specific values for rates of node advance for
Mercury or Venus. There must be people (Gerald Sussmann?) who have
done very-long-term solar system integrations -- maybe their papers
would give statistics like these.

I'm crossposting to sci.astro.research too since this seems like
a sane thread so far, and one raising a question that its readers
might have something to say about...

Stuart Levy