🚀 go-pugleaf

<*MUNCH*>

In article <1183@cit-vax.Caltech.Edu> jon@cit-vax.UUCP (Jon Leech) writes:
>In article <13@spar.SPAR.SLB.COM> freeman@spar.UUCP (Jay Freeman) writes:
>>    Each planet should see transits of all planets inferior to itself.  Proof:
>>The orbital planes of every pair of planets necessarily intersect in at least a
>>line.  By happenstance, sooner or later both planets will both be on the same
>>side of the Sun, "on" that line.  At that time, the superior planet will see
>>a transit of the inferior.
>>
>	It seems to me that with the right sort of resonant orbits
>this would not always be true (nitpicking admittedly). I'd be interested
>in hearing more about this from someone better informed.

Bill Jefferys also made a similar comment in private mail,
which I was unable to answer directly due to mailer problems:

Subject: Re: Transits of Planets

> Interestingly enough, your argument fails for Pluto. Neptune and
> Pluto are locked in an exact 3:2 resonance, so it is possible that
> the configuration you propose would never come about for these
> two planets. (You implicitly recognize this possibility when you 
> mention "stroboscopic effects", of course.)

> Cheers, Bill Jefferys

    I indeed intended to include resonance in "stroboscopic effects",
but now I am curious.  Let's see, now.  If Neptune and Pluto
are in 3:2 resonance, that means that whenever Pluto is at ecliptic
longitude (say) zero, then Neptune is at either ecliptic longitude
theta or ecliptic longitude theta plus pi, for some particular value
of theta.  My question is, how strong is the resonance?  In specific,
in what manner does theta vary over the long term?  A range of theta
of pi radians would be sufficient to make the proposed transit
possible.  (Did I get that all right?)
    A truly constant value of theta is only likely in an undergraduate
classical mechanics course.  A quite strong resonance might have theta
oscillating quasiperiodically about a constant value, in which case
transits could happen only if the range of oscillation were greater
than pi or in case of fortuitous near coincidence of theta-nought and
the line of intersecion of the orbital planes.  A still weaker
resonance might have theta not "bound" (in the sense of an oscillator)
but still changing at a very slow rate compared to the rates of
revolution of the planets.  (Possible opportunity to quibble about the
precise definition of "resonance".)  Do you happen to know what
condition obtains with Pluto / Neptune?

     I wonder if anyone has attempted numerical searches for times of
planetary transits as seen from other planets than Earth?

     Are transits of Neptune in fact visible from Pluto, or not?

     Hmnn.  With a lot of luck on the orbital elements of the two
planets, it might be possible that instead of a transit one would
have an eclipse of the Sun, by Neptune, as seen from Pluto.  (The
perihelion of Pluto is inside the orbit of Neptune -- the issue is
where the orbit of Pluto crosses the orbital plane of Neptune.  I
know the orbits themselves do not intersect.)  (At the distance
of Neptune from the Sun, Neptune's umbra is on the order of one
astronomical unit long.)


                                    Cordially,
                                    Jay Freeman