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(meteorobs) Angular Speed Equations
Hello all,
This thread is a few days old, but out of curiosity I decided to derive the
equations for finding the angular speed and duration of a meteor, given the
following inputs:
* required values:
r = Earth radius (mean = 6378 km)
h = meteor height (80-120 km)
mra = meteor radiant angle (angular distance from shower radiant)
mza = meteor zenith angle (angular distance from zenith)
ms = meteor speed (km/sec)
** angular speed formula
angs = (180 / Pi) * (ms * sin(mra) / d)
angs = angular speed (degrees)
d = distance between observer and meteor
(180 / Pi) = conversion from radians to degrees (if desired)
Exact d:
d = sqrt((r^2 * cos^2(mza)) + (2*r*h) + (h^2)) - r * cos(mza)
approximate d;
d = h / cos(mza) = h * sec(mza)
** Examples
if we substitute the first order approximation for distance into the
angular speed formula, we get;
angs = (ms * sin(mra) * cos(mza)) / h
that is, we get a higher angular speed at a (1) higher meteor speed), (2)
longer meteor angular distance from shower radiant, (3) lower meteor zenith
angle, and (4) lower meteor altitude. The practical upper limit is about
35 deg/sec, for a retrograde, perihelion, parabolic meteor, having the Apex
of the Earth's way as its radiant, and with the Apex on the horizon and the
meteor directly overhead (an "Earth grazer"). This is obviously rather
atypical, with most values being lower.
Here are some crude values to give you an idea as to what can be expected
in the field:
In these the meteor height is fixed at 100 km, and the meteor is fixed at a
meteor zenith angle of 30 degrees (meteor altitude of 60 degrees). We then
look at angular speed as a function of meteor angular distance from the
radiant:
key:
mra(deg), angs (deg/sec), "speed scale"
1. very slow speed (12-24 km/secd)
10 deg, 1.6 deg/sec, 1
30 deg, 4.5 deg/sec, 1
50 deg, 6.8 deg/sec, 2
70 deg, 8.4 deg/sec, 2
2. slow speed (24-36 km/sec)
10 deg, 2.6 deg/sec, 1
30 deg, 7.4 deg/sec, 2
50 deg, 11 deg/sec, 2
70 deg, 14 deg/sec, 3
3. medium speed (36-48 km/sec)
10 deg, 3.6 deg/sec, 1
30 deg, 10 deg/sec, 2
50 deg, 16 deg/sec, 3
70 deg, 20 deg/sec, 4
4. swift speed (48-60 km/sec)
10 deg, 4.7 deg/sec, 1
30 deg, 13 deg/sec, 3
50 deg, 21 deg/sec, 4
70 deg, 25 deg/sec, 5
5. very swift speed (60-72 km/sec)
10 deg, 5.7 deg/sec, 1
30 deg, 16 deg/sec, 3
50 deg, 25 deg/sec, 5
70 deg, 31 deg/sec5
Here I have introduced a linear angular speed scale from 1 to 5, with
increments:
1-6 deg, sec = 1
7-12 deg/sec = 2
13-18 deg/sec = 3
19-24 deg/sec = 4
25 and > deg/sec = 5
While such a scale might help some in shower identification, please note
that the real correlation with meteor linear speeds is actually with the
*RANGE* of angular speeds over which the meteors may vary with increasing
distance from the radiant, rather than restricting the meteors to a
particular set of angular speeds (such as 3-5 or 1-3). That is, all shower
meteors which are very near the radiant will have quite small angular
speeds, regardless of their linear speed. The true variation with linear
speed occurs in that the angular speeds will vary over a much larger range
with increasing distance from the radiant: from 1-2 in the case of the
very slow shower, up to a full 1-5 in the case of the very swift shower.
So why hasn't this been pointed out before? The reason, i think
speculate), is that most visual meteor observers tend to link an idea of
observed angular speed with meteor DURATION. It is a very easy bias to
mentally equate a very short duration with "swift," and a very long
duration with "slow."
** meteor duration equation
ms * dt = d1 * cos(mra1) - d2 * cos(mra2)
dt = delta time = duration in seconds
1 = beginning point of meteor
2 = end point of meteor
using the first order approximation;
ms * dt = h1 * sec(mza1) * cos(mra1) - h2 * sec(mza2) * cos(mra2)
It is a bit more difficult to give examples for this equation which
demonstrate easily what is happening, because their are a few more
variables involved. However, I can illustrate placing an artificial limit
of 10 km on the vertical height range for the meteor, and placing the
beginning point at the zenith.
Only three speeds will be used:
mra (deg), duration (sec)
1. 20 km/sec (height range = 90 to 80 km)
10 deg, 0.5 sec
30 deg, 0.6 sec
50 deg, 0.8 sec
70 deg, 1.5 sec
2. 40 km/sec (height range = 100 to 90 km)
10 deg, 0.3 sec
30 deg, 0.3 sec
50 deg, 0.4 sec
70 deg, 0.7 sec
3. 60 km/sec (height range = 110 to 100 km)
10 deg, 0.2 sec
30 deg, 0.2 sec
50 deg, 0.3 sec
70 deg, 0.5 sec
Note that with durations, all of the numbers tend to move as a group toward
shorter values as the meteor speed is increased, rather than changing the
range of possible values. This is more in keeping with how observers tend
to report "speeds" from a speed scale. In the end, it is probably a
combination of both effects which create an idea of "speed" in the
observers mind.
My own personal preference is for observers to train themselves to mentally
estimate durations directly. When combined with the angular distance
measurement, the two together will give a very easy division to get the
angular speed (done at the kitchen table when recording the data on hard
copy or computer). The duration estimation can be trained for when not
observing, by starting and stopping a stopwatch (wrist-watch with a
stop-watch feature), aiming for different durations randomly. It is
surprising how quickly this skill can be acquired. Out in the field, a
universal time broadcast can be occasionally listened to in order to keep a
1 second duration time span in mind. I did not like continuous listening,
because the monotonous broadcast can easily put you to sleep!
As Malcolm pointed out, a good angular speed measurement can be used to
roughly estimate the shower radiant position, when combined with the path
length. It does have more uses then simple shower association, although
this is the most common use. In the end however, angular velocity, no
matter how estimated is of minor importance in data reduction for non
plotted meteors. It is more important that the observer have a good,
systematic way of associating shower members and sporadic meteors to their
proper class. Well estimated durations have more applicability to
plotting than to "counting," and might more properly be reserved for
plotted meteors. For non-plotted meteors, the method of choice will
suffice, as long as good shower associations are obtained.
Take care,
Jim
James Richardson
Tallahassee, Florida
richardson@digitalexp.com
Operations Manager / Radiometeor Project Coordinator
American Meteor Society (AMS)
http://www.amsmeteors.org
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