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(meteorobs) Zenith Attraction (was Radiant Rising Time)
Hello all,
I have to chuckle a little here at my friend Mark's expense: I know some
who would consider physics students (like myself) to be "mathematically
depraved" -- complex integration techniques come to mind <grin>. But I
don't consider Mark to be mathematically *deprived* either. I once had an
elderly relative tell me that if it used all those "weird letters and
stuff" and went beyond arithmetic, then it just wasn't "math," in his view.
I am including the below excerpt on zenith attraction, to act as a
supplement to Rob McNaught's comments and program listing. I hope that you
will find it helpful, and not just a case of "Jim blowing his horn."
Important note on nomenclature:
In going from a heliocentric meteor velocity to an observed meteor
velocity, we go through essentially three transformations: (1) heliocentric
to geocentric frame conversion, (2) zenith attraction correction, and (3)
non-rotating Earth to rotating Earth frame conversion (called Diurnal
Aberration). This last is quite minor and is often neglected. Number (1)
and (3) involve vector transformations (pseudo-inertial frame), while
zenith attraction involves the effect of gravitational attraction of the
Earth on the meteoroid.
The most logical nomenclature for the first two conversions would be:
V_heliocentric (Vh) -> V_geocentric (Vg) -> V_atmospheric (or V_observed).
This places the geocentric velocity (Vg) as PRIOR to the zenith attraction
effect being calculated. I have seen both Rob McNaught and Rainer Arlt
using this nomenclature. However, most of the older meteor Astronomy
textbooks (such as Lovell) refer to Vg as the velocity observed AFTER the
zenith attraction effects have been added, and several meteor shower lists
also list Vg in the same fashion. Thus, I was in a quandary when I decided
to write about it myself. I eventually opted to follow the traditional
nomenclature, and use a temporary V_apparent (Va) for the stage after the
heliocentric-geocentric frame conversion, but prior to the addition of
zenith attraction. This makes Vg as the "final" speed (neglecting Diurnal
Aberration). This gives: V_heliocentric (Vh) -> V_apparent (Va) ->
V_geocentric (Vg). This is one of those "roses by another name" that I
hope you don't find too confusing.
excerpt from:
Richardson, J.E., (1999, March). "Colliding with a Moving Earth:
Geocentric Meteor Speed and Radiant Distributions," Meteor Trails, Journal
of the American Meteor Society, No. 3, pp. 7-16.
----------------------------------------
Zenith Attraction, and the Geocentric Meteor Distribution
Note that up to this point we have been careful to refer to our newly
derived meteor speeds only as "apparent" speeds (Va), and have not yet
placed our observer on the Earth itself -- only in orbit at speed Ve. This
is because of a different affect, called Zenith Attraction, which must be
taken into account when we place ourselves on the Earth itself.
Whenever a meteoroid encounters the massive Earth, it will no longer follow
a straight intersection path, but will instead follow a slightly curved
hyperbolic path about the center of the Earth, as the meteoroid "falls"
into the Earth's atmosphere. In addition to the path being slightly
curved, the speed of the entering meteoroid (now a meteor) will also be
increased in accordance with the following relationship:
Vg = sqrt(Va^2 + V_esc^2)
V_esc^2 = 124.9 km^2/sec^2
This will give us our sought after geocentric (Earth centered) meteor speed
(Vg). The number 124.9 comes from V_esc = 11.18 k m/sec, which is the
escape velocity for objects leaving the Earth, as well as an added velocity
to objects falling into the Earth. This is not a direct velocity vector
addition, as we did previously, but instead results from the conversion of
gravitational potential energy for the meteoroid into kinetic energy, given
by 0.5 * mass * V_esc^2. This will be added to the existing kinetic energy
for the meteoroid, 0.5 * mass * Va^2, to give a resulting kinetic energy:
0.5 * mass * Vg^2. Thus, the velocities will follow the relationship:
V_esc^2 + Va^2 = Vg^2. Note that this will have a much more significant
effect upon low Va meteoroids than high Va meteoroids.
The gravitational force will also cause the meteoroid to follow a slightly
curved path, most paths being hyperbolic in shape (Va > 0), but
occasionally parabolic in shape (Va ~0). Meteoroids travelling parallel to
the surface will be the most highly affected, with no path deviation for
meteoroids entering perpendicular to the surface. As a result of this, the
curved path will change the meteor's apparent radiant location as seen by
an observer below, as a function of the radiants zenith angle (90 deg -
radiant altitude). The formula for this correction is given by:
Tan((1/2) * dZa) = ((Vg - Va) / (Vg + Va)) * tan((1/2) * Za0)
Za = Za0 + dZa
Za0 = uncorrected radiant zenith angle
dZa = change to zenith angle
Za = corrected zenith angle
Note that this is a local effect, depending upon the observers latitude and
time of night for a particular radiant. Zenith attraction has its greatest
affect upon slow meteors having heliocentric radiants near the Antapex
point, and has much less effect upon fast meteors originating near the Apex
point. A practical limit for this effect is the addition of about 4-6
km/sec to the speeds of slow, antapex meteors, and a dZa of about 17-20 deg
when these radiants are located at the observer's horizon. Meteor shower
catalogues generally always list the geocentric speeds for shower members
(Vg), taking the zenith attraction speed affect into account. However, for
practical reasons, the radiant locations are given without regard to the
zenith attraction radiant shift effect (dZa), and observers must take this
into account themselves. For most medium speed to fast showers located
significantly above the horizon, this effect can be quite minor and is
often neglected.
---------------------------------------
Take care,
Jim
James Richardson
Tallahassee, Florida
richardson@digitalexp.com
Operations Manager / Radiometeor Project Coordinator
American Meteor Society (AMS)
http://www.serve.com/meteors/
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