[meteorite-list] Earth Trojan asteroids

[Sterling K. Webb]

Hi, Rob, Doug, List


    Why am I always digging myself out of holes that I apparently made with my
own mouth?  Well, better than extracting one's own feet from that same mouth, I
guess.

    There are many definitions of "phase angle."  Leaving out the ones that
apply to periodic and wave functions, I find:

1. The angle between the incident sunlight and the viewing direction when
looking at an illuminated surface. Low phase angles result in relatively few
shadows being cast by the surface relief.

2. The angle between the Sun, an object, and an observer. 0 degrees phase means
the Sun is behind the observer.

3. The angle between a line from the Sun to the center of a body and a line from
the spacecraft to the center of the same body.

4. For a solar system object besides the earth and sun, the angle between the
earth and the sun (or the earth's elongation from the sun) as seen from that
third object. The phase angle is given in ephemerides on IAU Circulars and Minor
Planet Circulars is denoted by either of the lower-case Greek letters beta or
phi.

5. The phase angle states the phase of a celestial body, i.e. the illuminated
fraction of it as it can be seen from the Earth.

    Of course, you might note that I NEVER said "phase angle."  I merely used
the word "phase" (simplified version of definition #5) in its most direct sense,
as in, "What is the phase of the Moon tonight?"  And "2/3rds" means that 2/3rds
of a sphere at the Trojan point would be illuminated and 1/3rd would be dark) as
viewed from the Earth.

    And, considering that the shape of an asteroid is often not only NOT a
sphere, but usually mathematically impossible to characterize, we fall
invariably back on terms like "potato," "lumpy potato," or "it looks like a
butternut squash to me."

    And since the curve of this varying reflectance would also depend on its
rotational period in all axes, any compositional differences on its surface, and
the number of shiny new UFO's parked on it for scheduled maintenance that day, I
did not feel that it was a strictly predictable quantity.

    And while an asteroid would dim by a factor of 2.512 to the .64th power from
opposition to the Trojan position, since we don't have the foggiest notion what
its albedo would be (except that we casually assume it to be like most NEA's or
in their range), it is a result of very high precision and only slight accuracy.

    In ancient days long gone, I used to set up little problems for the
mid-1980's computers that ran MicroSoft Basic Vers. 4.21, in which the answer to
2 + 2 would come out 3.9999997547563954.....  Highly precise; not very
accurate.  The two are not the same.

    I had a lot of fun writing an "arithmetic" program in BASIC which converted
numerical imputs to $tring functions, then digit by digit performing the same
grade school arithmetic every child learns, constructing the "answer" as $tring
functions again, with callable subroutines for each (addition, subtraction,
etc., even roots) arithmetical operation.

    In that era of "big" 16-bit processors and horrendous floating point errors,
people would stare at an old Kapro CPM machine churning out solutions (like the
square root of 2) to 998 decimal points of accuracy like it was the Second
Coming.  They get all excited and yell, "What algorithm are you using?"

    "Arithmetic," I'd answer.

    "No, I know it's doing arithmetic, but what's the algorithm?"

    "No," I reply, "you don't get it. The algorithm is ARITHMETIC."  That's all
I'd say.  They'd figure it out sooner or later, but it was always fun to watch
the struggle to absorb the obvious.

    However, you are dead right about Jupiter's Trojans.  I screwed the pooch on
that one.  Too late at night, wrong column of figures, bad eyes,
just-plain-dumb, all factor in...  Oh, don't forget old data, too

    JUPITER TROJANS EAST a  = 4.90 to 5.37 AU e < 0.30 and i < 40° Lagrangian
point L4 of Jupiter:
    NUMBER KNOWN AS OF MAY 20, 2004: 525  ESTIMATED TOTAL: 1039

    JUPITER TROJANS WEST a  = 4.96 to 5.36 AU e < 0.28 and i < 44° Lagrangian
point L5 of Jupiter
    NUMBER KNOWN AS OF MAY 20, 2004:  352   ESTIMATED TOTAL: 628

    And I can't even figure where I got the 149 number from, now. Pleasant that
we're over last year's estimated total already.

    Congratulations on SEVEN Trojans!

    But there is something that bothers me...

    Always a big fan of the Iliad, even BEFORE it was a movie with Brad Pitt, I
can't recall the names of 1783 Trojan characters in the Iliad!

    In fact, I count only: Hector, Priam, Hecuba, Paris (also known as
“Alexander”), Helen (of course, she's a Trojan now...), Aeneas, Andromache,
Astyanax, Polydamas, Glaucus, Agenor, Dolon, Pandarus, Antenor, Sarpedon,
Chryseis, Briseis, and Chryses.  Not a cast of thousands...

    Where do folks come up another 1765 Trojan names?  Third soldier from the
left in the last row in the Battle in Book Ten?  Just make'em up? Greekify your
children's names?  Just leave it to some gentlemen in Paris at IAU to do it?
Give'em numbers?  Buy one of those "5000 Names for Your Child" books, written in
Greek?  What?!

    Inquiring minds want to know...


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

"Matson, Robert" wrote:

> Hi Sterling, Doug, and any other lurking List members still following the
> earth Trojan thread.  A few comments related to the Earth Trojan magnitude
> calculation.  Sterling wrote:
>
> > Yes, phase would be about 2/3rds if it was spherical, but small bodies
> > rarely are, so that value could be highly variable.
>
> The solar phase angle (the angle, as measured from the asteroid, between the
> sun and earth) is 60 degrees, which results in a noticeable drop in visual
> magnitude compared to how that same asteroid would appear at 1 a.u. and
> at opposition (despite the fact that in the latter case the asteroid is
> twice as far from the sun and thus receives 1/4 the sunlight!).  For
> instance, if an asteroid at opposition and 1 a.u. from earth (2 a.u. from the
> sun) has an apparent magnitude of +18.0, that same asteroid moved to the
> earth-sun L4 or L5 point would dim to magnitude +18.64 for a typical slope
> parameter of G=0.15 -- a drop of a factor of 1.8 in brightness.  What this
> means is that asteroids that wouldn't be missed at opposition could easily
> evade detection at L4/5.
>
> Sterling -- was this a typo? :
>
> > Of course, we all know that Jupiter has Trojans (149 are known -- Jupiter
> > has more of everything!) and even Nepture has one (known).
>
> There are at least 1783 Jupiter Trojans (7 of which were found by me within
> the last year :-).
>
> Cheers,
> Rob
>