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Re: (meteorobs) How fast do Meteors go?
This is a followup on the subject of the equation posted to our list yesterday,
describing the theoretical relationship between meteor magnitude and meteoroid
velocity, mass, and incidence angle. This is a pretty technical discussion, so
if you don't enjoy at least ONE of mathematics or physics, HIT DELETE NOW! :)
(As a matter of fact, Marco, I was the opposite of you: I was pretty good at
theoretical math, but was constantly berated by physics teachers for "lacking
physical intuition" and being too "formal" in my thinking. By gosh, I wanted
rigorous definitions of terms, and consistent use of symbology! And for some
reason, physics professors could never be bothered to give me either one. ;>)
Well, the original posted equation relating m to "(Vinf, M, h)" was:
>> log M(m)=2.98-0.44m-3.89log V[inf]-0.67log(sin h)
Marco Langbroek gives the original reference for that formula as:
>Jacchia et al., Smiths. Contr. to Astroph. 10 (1967), p.25
And Robert Gardner references the classic "Meteor Science and Engineering",
for a simpler but well established formula:
>Look on page 121 in McKinley
Finally, Mike Linnolt correctly comments:
>as the radiant approaches the zenith, h->0, sin h ->0, and
>log (sin h) -> infinity... This makes no sense at all.
Mike's comment is right on the money. And in fact, I had just assumed that
'h' was actually meant to designate radiant elevation (that is, *'h'orizon*
angle) rather than radiant zenith angle (which is usually designated Z). If
you assume that, the formula certainly makes (more) sense.
Anyway, I went back to McKinley - and Oepik - last night, and unfortunately, I
was not totally enlightened by either one! Neither text has any information on
the relationship between horizon angle (incidence angle) and luminosity. The
Oepik derivation starts from the assumption that the visual intensity 'j', can
be derived from the formula for kinetic energy, sort of as I expected:
j t = 1/2 beta M (Vinf^2)
Here, 't' is the duration of the meteor in seconds. M is the meteoroid mass.
Vinf we've talked about. And the 'beta' is a "fudge-factor" called "luminous
efficiency": in other words, how MUCH of the kinetic energy of the meteoroid
actually gets turned into visible radiation during its atmospheric entry.
From this formula, Oepik then uses the relationship between "intensity" and
"magnitude" (log j = 9.72 - 0.4 m) to derive the following hefty equation:
log M = 10.02 + log (t Vinf) - log beta - 3 log Vinf - 0.4 m
(NOTE that for Oepik, Vgeo is in cm/sec, and M is in grams! Also note that,
in all of these formulae, 'm' is assumed to be the ABSOLUTE magnitude: the
magnitude if the meteor were seen from 100 km away, with no extinction...)
Anyway, this was at least enough to help me understand that original formula
Marco posted well enough to be able to transcribe it correctly! I write it:
log M(m,Vinf,h) = 2.98 - [0.44 x m] - [3.89 x log Vinf] - [0.67 x log(sin h)]
(Sure enough, 'M' really DOES mean "mass" as Marco indicated, by the way.)
Now when I went off to the Harvard Abstract Data Service to try to read the
Jacchia paper mentioned by Marco in his followup, I found that either it has
not been scanned, or the scanned version isn't accessible to us amateurs!
But I persevered, and found an article by our own Martin Beech that uses the
Jacchia result (Beech, Mon. Not. R. Astro. Soc. (1984) 211, 617-620):
2.25 log M = 55.34 - 8.75 log Vinf - 1.5 log (sin h) - m
And of course, this Beech equation simplifies to:
log M = 24.6 - 3.89 log Vinf - 0.67 log (sin h) - 0.44 m
(Where Vinf is again in cm/sec, as opposed to Marco's km/s!)
Does all of this lead me to some deeper understanding of meteor magnitudes??
Well... MAYBE. :) Clearly by changing units around, it is easy to "fudge" the
Beech and "Langbroek" equations so that they're basically the same... And to
get from Beech to the classic Oepik equation, you can do a neat little trick
and derive the term "log (t Vinf)", which relates M(m) to the length of the
meteor's "burn trajectory", from the incidence angle. (If you assume constant
start and end altitudes, then trajectory length is simply related to sin h!)
Anyway, thanks to Marco "Uncle Albert" Langbroek for his very provocative and
interesting post and followup, and to all the other folks who posted. Geez, I
hope SOMEBODY else got SOMETHING out of all of this - I know I did. :)
Clear skies,
Lew
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