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(meteorobs) On limiting magnitude (long)
Hi folks,
In last week's discussion, Marko Langbroek wrote:
"There's nothing wrong with the correction for limiting magnitudes (sorry
Norman, but this is the way it is)."
This type of statement strikes a discord with me, because in science,
nothing is as simple as "this is the way it is," but is instead "this is the
way we think it is right now." In science all "truths" are provisional, and
every theory, hypothesis, and correction factor is subject to revision,
modification, or outright discard. All of these are models of what we hope
is a fair representation of reality, but is not the reality itself.
I wanted to discuss the limiting magnitude correction factor in particular,
and point out some of the uncertainties involved with it. To understand the
uncertainties, you have to understand the correction factor's derivation.
The IMO handbook (at least the old edition) does not explain this, so I will
do so here. To those of you who know this stuff already, I apologize, but I
wanted to carry everyone along.
The first assumption is that a particular meteor population (sporadics or a
particular shower) undergoes an exponential increase in numbers as the
magnitude gets fainter. The base for this exponential function is called
the population index, or "r" value. For example, if r = 3, and we know that
we have 5 1st magnitude meteors in a chunk of atmosphere over a certain
amount of time, then by this factor there are 5(3^1) = 15 2nd magnitude
meteors, 5(3^2) = 45 3rd magnitude meteors, and so on, down to the fainter
magnitudes, such that in this population there would be 405 5th magnitude
meteors.
From our own observing experiences it should be obvious that an observer is
not going to notice all of these faint magnitude meteors, and this is where
perception factors come in. Observers will see only x% of the 1st
magnitude, y% of the 2nd magnitude, and z% of the 3rd magnitude meteors,
with x, y, and z becoming successively smaller as the magnitude gets
fainter. If we assign the letters a, b, and c to these percentiles (i.e. a
= x%), then to find the total number of 1st magnitude meteors seen by our
observer we use:
T1 = aN1 or T1 = aN1(r^0)
where N1 is our known number of 1st magnitude meteors.
To find the total number of meteors seen by our observer for all magnitude
classes, the totals for each magnitude class are added together to give:
T = aN1(r^0) + bN1(r^1) + cN1(r^2) + ...
Now for the next assumption. What happens if our observer's sky brightens
by a factor of 1 magnitude, that is, his limiting magnitude goes up by 1?It
is assumed that the perception factors for each magnitude class all shift up
by one class, such that if the observer previously saw x% of the 1st
magnitude meteors, he will now only see y% of these meteors. The new
equation for the 1st magnitude is bN1(r^0), and the a corrrection factor now
shifts to the 0 magnitude meteors, giving:
Tl = aN1(r^-1) + bN1(r^0) + cN1(r^1) + ...
where Tl = the new total of meteors seen under our brighter skies.
Note that I have left the perception factors in there old place, but have
shifted the magnitude class to which each applies.
In order to develope an expression to correct the number of meteors seen in
our brighter sky up to the total number that would have been seen in our
first, darker sky, we can look at the ratio of T to Tl:
T/Tl = (aN1(r^0) + bN1(r^1) + cN1(r^2) + ...) /
(aN1(r^-1) + bN1(r^0) + cN1(r^1) + ...)
If we factor r from each term in the numerator, and then simplify, this
entire expression reduces to:
T/Tl = r or T = Tl * r^1
This becomes the basis for the IMO correction factor. According to this
expression, each increase in sky brightness by a factor of 1 magnitude gives
a corresponding drop in the number of meteors seen by a factor of r^1. If
6.5 is adopted as the standard limiting magnitude to which all others are
corrected, our correction factor becomes:
T = Tl * r^(6.5-lm)
where lm = limiting magnitude.
This brings up one of the problems which Norman pointed out. If your
limiting magnitude is 6.5 or less, than the correction factor is 1.0 or
greater. If you have extremely dark skies, as he does, then the correction
becomes less than 1 and reduces the total number of meteors seen to match
the standard conditions. Marko corrected for this effect by assuming that
Norman had less than standard perception, multiplying the total by 2.5 to
bring the number back up again.
Beyond the assumptions already made, the potential problem with this
correction factor is that it hinges entirely upon a correct value for r.
Thus, we require the proper determination of a population index for the
stream we are studying. This determination, in turn, hinges upon the proper
derivation of the perception factors a, b, c, ... mentioned previously.These
perception factors are then applied to the number of meteors seen in each
magnitude class to derive a value for r.
The situation is analogous to a dog chasing his tail because the
determination of the different perception factors for each magnitude class
depends upon the study of a meteor population with a "known" value of r. If
a large sample of visual observations is taken, and the number of meteors
seen in each magnitude class is plottted with the numbers seen as the
ordinant and magnitudes as the abscissa, then a bell curve results with peak
numbers seen in the magnitudes from about 2nd to 4th. Beginning at the very
bright end of the magnitude spectrum where it is believed that ALL meteors
of that class would have been seen, you can then use the known r value(s)
to plot the number of meteors actually present for each magnitude class.
The ratio between the total number seen and the total number calculated to
be present will give the perception factor for each magnitude class.
The problem lies in the "known" value of r for even the sporadic meteor
population. The studies indicate that for the sporadics the r value drops
as magnitude decreases from about 3.5 in the -6 to -4 range, to about 2.5
for 5th magnitude. Thus, our total number of meteors in each class
calculated above is not really a simple exponential function, but a complex
one with a decreasing base. This makes are determination of perception
factors even more difficult.
The problem is one which has been studied for some time, and I will only
give a couple of examples:
"With respect to the fraction of meteors observable down to a given limiting
magnitude, Opik (1922) thought that a single observer should see all
second-magnitude meteors in his normal field of view, 90 per cent of third
magnitude, 50 per cent of fourth magnitude, and 8 per cent of fifth
magnitude. On the other hand, Millman believed that his standard team of six
observers should see all zero-magnitude meteors, but only 60 per cent of the
first magnitude, 25 per cent of the second magnitude, 10 per cent of the
third, 2 per cent of the fourth, and less than 0.5 per cent of the
fifth-magnitude objects, visible over the entire sky. Other workers have
evolved still different correction factors for this effect, for example,
Ceplecha (1950) and Kviz (1958)." (McKinley, 1961, pp. 106-107)
As Marko pointed out, perhaps the most extensive study of this type was
conducted by M. Kresakova (1966) of the Scalnate Pleso Observatory,
Czechoslovakia, using 11 years of visual meteor observations (over 48,000
total meteors), which yielded the following perception factors: 6m
(0.00007), 5m (0.008), 4m (0.064), 3m (0.232), 2m (0.343), 1m (0.420), 0m
(0.48), -1m (0.57), -2m (0.73), -3m (0.87), -4m (0.95), -5m (0.98), and -6m
and greater (1.00).
Marko also points out that Dr. Jenniskens has derived his own values for
these perception factors.
While the values obtained have probably improved over the years, they are
still subject to revision as our understanding increases. Thus r values
obtained using these factors should be treated with a grain of skepticism,
because the "r standard" from which they have been derived is not entirely
nailed down yet.
Meteor showers exacerbate this problem because they tend to become
"stratified" over time due to radiation pressure and other effects on the
individual particles. Marko alluded to this as well in applying a different
r value for fainter Leonids over Europe than for those over North America.
Shower streams are not isotropic, and the population index and particle
distribution can change depending upon what part of the stream is examined.
We probably have a better chance at deriving good r values for the older,
stable streams than we do for the younger ones.
The point of this exercise is to say that no correction factor should be
considered "sacred," especially when there are many assumptions and
uncertainties involved. I have wondered recently if a set of correction
factors for limiting magnitude might be derived empirically as opposed to
theoretically by studying the effects of moonlight and light pollution on
sporadic rates and "old reliable" shower rates.
Thank you for listening to my long presentation, and I hoped that I have
helped our understanding a bit.
Take care, everyone,
Jim Richardson
Graceville, FL
Richardson@DigitalExpdot com