[meteorite-list] There IS an Earth Trojan Asteroid (probably)!

[Sterling K. Webb]

Hi, Ron, List,

    The web page I cited was put up by: Paul Wiegert, UWO Physics Dept,
Astronomy Group, The University of Western Ontario, London, Ontario
Canada; Kimmo Innanen, Dept. of Physics and Astronomy, York University,
Toronto, Ontario, Canada; and Seppo Mikkola, Tuorla Observatory,
University of Turku, Finland, on 3763 Cruithne.
    <http://www.astro.uwo.ca/~wiegert/3753/3753.html>
    This web site is apparently based on their announcement paper in
(Nature, 1997, 387, 685-686.
    I chose the single word "share" from the following early
introductory paragraph:
    "The near-Earth asteroid 3753 Cruithne is in an unusual orbit about
that of the Earth, one which is known in the lingo of celestial
mechanics as being co-orbital with the Earth (meaning it SHARES [my
emphasis] the Earth's orbit with it) and, more particularly, as being of
the "horseshoe" type. Horseshoe orbits are named because of their shape
in a reference frame which corotates with their accompanying planet, and
have been known theoretically for many years. A corotating frame just
means one in which the viewer orbits along with one of the planets, in
this case the Earth. It is analogous to imagining oneself floating above
the north pole of the Earth as our planet goes around its orbit, and
watching the path the asteroid from this vantage point. By looking at
the asteroid in this way its unusual behaviour becomes apparent: it is
much more difficult to distinguish when viewed in a non-rotating frame."

    When Cruithne is viewed in the ROTATING frame of reference:
    "Cruithne is on an spiraling horseshoe orbit... but its behaviour is
even stranger. First, it is an even more tightly-wound spiral, with
kidney-bean shaped loops. Secondly, one part of the horseshoe actually
overlaps the position of the Earth when viewed from above. A series of
four frames (to be read left to right) showing a time lapse of
Cruithne's horseshoe is shown below. Note again that Cruithne's
horseshoe is composed of tightly wound kidney-bean shaped spirals which
are almost impossible to distinguish individually in the picture.
Remember, there are two types of motion going on: 1) every year, the
asteroid traces out a kidney bean, 2) over time, this kidney bean drifts
along the Earth's orbit, tracing out a spiral which, when complete
(after 385 yrs) fills in an overlapping horseshoe."
    They do not claim the Cruithne "orbits" the Earth in the direct
sense, and indeed their multi-colored orbital diagrams show the same
elements of eccentricity and inclination, etc. (see below) as the NASA
page you cite.  Unfortunately, my computer's MicroSoft mind has
difficulties with  JAVA applets for some reason, and won't execute the
one on the page you cite, so I can not get it to visualize and therefore
cannot compare the two directly.
    Even Wiegert and company regard Cruithne as a very strange duck.
They have an MPEG movie showing it in the heliocentric frame of
reference, about which they say: "Note the highly eccentric and
inclined... orbit of Cruithne, and the lack of obvious connection to the
Earth, except that they both go around the Sun in about the same amount
of time."
    This is then followed by another MPEG movie in the rotating
Earth-centric frame of reference: "After a few orbits, the lines of
sight connecting Cruithne to the Earth are drawn in green. Note that,
though they both go around the Sun, both are always on the same side of
the Sun. Then, the Earth will begin "picking up" the lines of sight,
constructing the apparent path of Cruithne as seen from the rotating
Earth. Their motions are such that Cruithne appears to perform a
kidney-shaped loop around the Earth. Though the asteroid thus appears to
orbit the Earth, this is a result of the frame of reference we are
using. Also note that the presence of the Earth at the centre of the
kidney is simply a result of their mutual synchronization: the Earth is
lined up with the asteroid and Sun when the asteroid is at perihelion
(its closest approach to the Sun)... Because the asteroid and the Earth
do not go around in exactly the same amount of time (the asteroid
currently goes around slightly faster than the Earth), the position of
the kidney-shaped loop relative to the Earth varies over time. If the
asteroid and the Earth were not in a special arrangement, the Earth
would face potential danger as the asteroid drifted inexorably towards
our planet. However, in their current relationship, the kidney's
direction of drift is reversed every time it approaches the Earth."
    See, I said it was a COMPLICATED horseshoe.
    The webpage which I cited is also cited on the the Lunar and
Planetary Laboratory asteroids page in a positive sense (as in, look at
this; it's truly weird.):
<http://seds.lpl.arizona.edu/nineplanets/nineplanets/asteroids.html>
    Further, NASA's own site <http://impact.arc.nasa.gov/> on the NeoDys
Object List points to the same source: "The motion of this object is
quite interesting. See:
<http://www.asteroid.yorku.ca/~wiegert/>."
    However, this is a broken link, since Wiegert left York and went to
UWO, but it was the same page.
    If there are those who disagree with this very complicated dynamic
argument, I couldn't find them.  Maybe you ran into one, I dunno.  I
doubt I could judge such a dispute if there was one, anyway.  Dueling
ephemerides at twenty paces?
    Holy Moly, Ron!  All I did was use the word "SHARE"!!  They are not
using the word "share" to mean a Trojan or a Earth centered orbit,
either.  It is a helocentric orbit that oscillates back and forth in a
385 year pattern, and that oscillation is dominated by the Earth's
gravity.  Thus, in this very special way, it is "tied" to the Earth.
    OK?


Sterling K. Webb
-----------------------------------------------------------------------------

Ron Baalke wrote:

> >
> > 3753 Cruithne and several other asteroids share the Earth's
> > orbit but are not Trojans, but complicated "horseshoes":
> > <http://www.astro.uwo.ca/~wiegert/3753/3753.html>
> >
>
> 3753 Cruithne does not share Earth's orbit:
>
> http://neo.jpl.nasa.gov/cgi-bin/db_shm?sstr=3753
>
> Cruithne's orbit crosses the orbits of Venus and Mars, and comes near
> the orbit of Mercury.  Earth's orbit is not like that at all. Cruithne
> and Earth do not share an orbit, and they are in separate orbits
> around the Sun.
> Cruithe's orbital period is 0.9966 years, which means it is very close
> to Earth's orbital period.  Cruithne is also not
> a satellite of Earth like some people have claimed. Cruithne
> orbits the Sun, not the Earth.
>
> Ron Baalke
>

Orbital elements:
    "The following elements are for the epoch JD 2450500.5 (Feb 21 1997)
and relative to the equinox J2000.0. Semimajor axis = 0.99778030
astronomical units, eccentricity 0.51478431, inclination i=19.812285
degrees, longitude of the ascending node = 126.373212 degrees, argument
of perihelion = 43.640637 degrees, mean anomaly = 40.048932 degrees, and
epoch of perihelion passage = 1997 Jan 11.50 UT = JD 2450460.00. These
elements are derived from the Asteroid Orbital Elements Database
maintained by Ted Bowell at Lowell Observatory. You can also find them
at Minor Planet Center's Near-Earth Object page."