One of my favorite meteor observing trips occurred in 1989, when I joined a crew from Meteor Group Hawaii to view the Eta Aquarid meteor shower from the slopes of Mauna Loa on the big island of Hawaii. The alien volcanic landscape, the complete lack of city lights or other signs of civilization, and a beautifully bright, clear sky dome overhead made for some very memorable observing sessions. Due to the late radiant rise time of this often difficult to observe shower, we began each observing session long before any Eta Aquarids appeared, recording sporadic and minor shower activity while waiting for these swift cousins of the Orinids. Later during each night, generally in the last half hour before radiant rise, we would be treated to at least one very long, "earth grazing" Eta Aquarid, covering up to 90 degrees of arc in about 3 seconds -- enough time to follow these often trained beauties with your head as they tried to zip out of your field of view and over your shoulder. While the Eta Aquarids never reached spectacular rates in their race with the morning twilight during that trip, they frequently produced a very pretty opening "salvo."
A close inspection of my data from that trip also revealed another interesting aspect to those rare pre-radiant-rise Aquarids: dividing the meteor's angular distance by the meteor's duration showed a higher average angular speed than any others recorded during that night, especially if the grazer passed near the zenith in its path. Those meteors appearing at lower altitudes (closer to the horizon), and especially those meteors appearing near the radiant point were often dramatically slower. Obviously, the term "swift" to describe the Eta Aquarids was more of a general observation than a set speed, with a whole range of angular speeds possible depending upon the meteor's location with respect to the radiant point and the observer's zenith. In this article we explore this observation in detail, along with the equations governing meteor angular speeds.
Without going through the full derivation, the complete solution to the meteor angular speed equation is given by:
V_theta = (180/pi) * [(V_m * sin(theta)) / d], [Eq. 1]
d = sqrt(r^2 * cos^2(z) +2*r*h +h^2) - r*cos(z),
d = sqrt(r^2 * sin^2(alt) +2*r*h +h^2) - r*sin(alt),
d = distance between meteor and observer (km),
r = Earth radius (mean = 6369 km),
h = meteor height above ground (usually between 70-120 km),
z = meteor zenith angle (deg),
alt = meteor angular altitude (deg),
theta = meteor angular distance from radiant (deg),
V_m = meteor linear speed (km/sec),
V_theta = meteor angular speed (deg /sec).
If we substitute the geometric first order approximation in for the meteor distance (d = h / cos(z)), we get the more familiar forms of this equation:
V_theta = (180/pi) * [(V_m * sin(theta) * cos(z)) / h], [Eq. 2]
V_theta = (180/pi) * [(V_m * sin(theta) * sin(alt)) / h].
These equations show the meteor angular speed to be a function of four independent variables, such that:
V_theta is (in the first order approximation) --
* directly proportional to the meteor's linear speed (V_m),
* directly proportional to the sine of the meteor's radiant distance (theta),
* directly proportional to the meteor's angular altitude (alt),
* inversely proportional to the meteor's height (h).
Since the meteor's linear speed, V_m, will be essentially constant for a given meteor shower, and the meteor's height, h, will also stay within a relatively narrow band of values, the angular speeds for meteors from a particular given shower will primarily be functions of the meteor angular distance from the radiant and the meteor angular altitude. This is shown graphically in the next section.
As a function of four independent variables, the angular speed equation is rather difficult to depict graphically; however, by using multiple plots, each one depicting a constant meteor speed (V_m) and meteor height (h), and graphing several level curves on each plot for the angular speed (V_theta) as a function of the radiant angular distance (theta) and meteor angular altitude (alt) only, a reasonably good idea as to how this function behaves can be obtained. Figures 1 to 5 show the meteor angular speed, V_theta, plotted on the y-axis as a function of meteor-radiant angular distance, theta, on the x-axis, for four constant meteor angular altitudes (each shown as a separate curve): alt=0 deg (dash-dot), alt=30 deg (dash), alt=60 deg (dot), and alt=90 deg (solid). These curves are shown for five different meteor linear speeds, V_m, and an associated average meteor height, h, for that speed. These are:
* Figure 1: V_m = 12 km/sec, h = 85 km,
* Figure 2: V_m = 27 km/sec, h = 90 km,
* Figure 3: V_m = 42 km/sec, h = 95 km,
* Figure 4: V_m = 57 km/sec, h = 100 km,
* Figure 5: V_m = 72 km/sec, h = 105 km.
The most notable observation from these figures is that the faster is the meteor linear speed for a particular shower, the wider is the range of angular speeds over which the meteors from that shower can vary. That is, very slow showers will display only a narrow range of possible angular speeds, while very swift showers will display a wide range of possible angular speeds -- depending upon the individual meteor's angular distance from the radiant and meteor angular altitude. Since it is very common for observers to divide meteors into various angular speed classes, we adopt here a typical grouping of angular speeds into five ranges, as follows:
Now we look at the five different shower meteor speeds shown in Figs. 1-5 to see what possible speed classes emerge:
* Fig. 1: V_m = 12 km/sec; V_theta = 0-8 deg/sec; speeds = 1
* Fig. 2: V_m = 27 km/sec, V_theta = 0-17 deg/sec; speeds = 1, 2
* Fig. 3: V_m = 42 km/sec; V_theta = 0-25 deg/sec; speeds = 1, 2, 3
* Fig. 4: V_m = 57 km/sec, V_theta = 0-33 deg/sec; speeds = 1, 2, 3, 4
* Fig. 5: V_m = 72 km/sec; V_theta = 0-39 deg/sec; speeds = 1, 2, 3, 4, 5
Note: the extreme uppermost angular speeds for each linear speed will generally not be seen, since this represents the radiant at the horizon and the meteor exactly at the zenith. Thus, the speed classes listed are rounded down slightly to the more commonly encountered speeds.
These numbers show how very ambiguous the common speed classifications of shower meteors can become in common usage. It is typical to describe a shower as being "slow" or "swift," which seems to imply that all of the meteors from a particular shower will appear to move across the sky at a common speed. However, Figures 1-5 show that this is patently not the case, and that the descriptive speed for a shower only indicates the maximum possible angular speed -- with a full range of meteor angular speeds actually present for a particular shower linear speed. This is illustrated more clearly in the following section.
While the above graphs show how the angular speed equation works mathematically, it does not show how real meteors behave in the field very well. In order to illustrate how actual meteors behave with regard to angular speed verses angular distance from the radiant, we employ a meteor shower computer simulation written by the author for the Maple mathematics package. using this aid, we can trace out the path of a simulated meteor on the celestial sphere while at the same time tracking its angular speed.
Figure 6 shows eight computer simulated 0 magnitude Geminid meteors (V_m = 34.4 km/sec) coming from a radiant at 45 degrees of altitude. The tiny circle shows the radiant location, and the long curved line near the bottom of the figure shows the horizon. The zenith is near the top center of the projection, which is gnomic (sphere projected on to a flat plane tangent to the sphere at the radiant point). The width of the projection is 120 degrees, about the full field of view for a single observer.
Figure 7 shows the resulting plots of meteor angular speed vs. distance from the radiant for the meteors depicted in Figure 6. The eight meteors are evenly distributed at distances of 5, 10, 15, 20, 25, 30, 35, and 40 degrees from the radiant point, with a "near" and "far" pair placed in each directional quadrant. A few pertinent observations can be made based upon this plot: (1) meteors appearing closer to the radiant will move slower than meteors appearing farther away from the radiant; (2) meteors generally accelerate in angular speed as they move away from the radiant; (3) meteors which are seen moving down towards the horizon are significantly slower than those moving up towards the zenith; (4) meteors which are seen moving down towards the horizon have significantly shorter path lengths than those moving up towards the zenith -- in addition to being generally dimmer (magnitude extinction). Note also that the meteors displayed show a full range of angular speeds, depending upon each meteor's circumstance, and with an upper limit set by the shower's linear speed, as was implied from Figures 1-5.
Figure 8 shows the exact same eight 0 magnitude meteors, only this time they are simulated Leonid meteors, coming in at V_m = 70.7 km/sec. Although the Leonids will begin at higher atmospheric altitudes, their plotted paths are nearly identical to those of the Geminids depicted in Figure 6. However, the angular speeds of the Leonids displayed in Figure 9 are, on the average, noticeably higher than the Geminids previously shown. The Leonids display a higher maximum speed, and a broader range of speeds under different conditions, while still displaying all of the points made previously about Figure 7 (the Geminids). Note that in both cases, it would be difficult to pin a particular shower down to a narrow range of angular speeds, and that it is actually the breadth of possible angular speeds that is indicative of the linear shower speed, V_m.
Properly measuring and annotating visual meteor angular speeds is one of the few areas of current disagreement amongst the various meteor groups worldwide. The International Meteor Organization (IMO) asks observers to estimate the meteor angular speed directly; the North American Meteor Network (NAMN) utilizes a five class angular speed scale, similar to the one described previously in this article (Table 1), and the American Meteor Society (AMS) continues to ask observers to measure the angular length of each meteor (degrees) and estimate the duration of each meteor (to a fraction of a second), from which the average angular speed can be roughly calculated. The advantage in this last method is that it attempts (although still rather crudely) to remove some of the subjectivity from a single speed estimate by instead asking the observer to make two measurements:
(1) The meteor angular length estimation, usually estimated by holding up a yardstick, rod, or cord to mark the path of the meteor (this device can ruled with 5 degree increment marks, obtained from known distances between bright stars), and
(2) a rough estimate of the meteor's duration in seconds (generally to the nearest 1/5 or 1/10 sec), based upon practice sessions using a digital stopwatch while not observing.
This meteor duration is not an actual measurement, but is instead a mental estimate based upon practice during daylight hours. A rough estimate of the average meteor angular speed is then obtained by dividing the distance by the duration later during initial data reduction on paper. Thus, a meteor covering 5 degrees in about 0.4 sec has a rough angular speed of 13 deg/sec. This last method is a bit more time consuming, however, and many observers (especially beginners) will opt to instead utilize the simpler five-class angular speed scale. This is acceptable, and the associated speed numbers (1-5) can be placed on the AMS visual report form in the velocity column.
The meteor angular speed estimate is primarily used for meteor shower classification verification, using the relationships shown in Equations 1 & 2 and Figures 1-5. Interestingly enough, this article brings up another potential use for an estimated meteor angular altitude (alt) in addition to the estimation of a meteor absolute magnitude, as describe in the previous article in this series. When combined with an estimate of the meteor angular distance from the radiant (theta, taken from a meteor plot), and using the estimated angular speed (V_theta) and an assumption for meteor height (such as the average of h = 95 km) a crude estimate of the meteor linear speed (V_m) can be made -- providing a shower classification validation. On the more accurate side of things, Equations 1 & 2 are also frequently used in single-station photographic / video meteor work, providing single-station meteor height measurements for known meteor shower members (as recently demonstrated by R.B. Minton), or providing rapid meteor shower member classifications in automated computer / intensified video systems (such as those developed by Peter Gural or Sirko Molau). During the heyday of meteor science research in North America, this versatile little equation even permitted the combination of back-scatter radar range information (d in Eq. 1) with single-station photographic data (rotating shutter) for conducting meteor shower studies. From having application to simple backyard meteor observing up to computer simulating meteor showers, the meteor angular speed equation is one of the most useful relationships in this field, in addition to providing a basic understanding of how fast meteors move across the night sky (and that "swift" isn't always so swift).
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