Dynamics of Meteor Outbursts and Satellite Mitigation Strategies
Glenn E. Peterson
Chapter 1: Assessing the Threat (cont).
1.3 Radiant Point
With the stream dynamics now defined, the encounter at Earth crossing must be examined. One of the important characteristics for an Earth encounter is the radiant point of the resulting meteor event. The radiant point is the apparent location in the sky that the meteors come from when viewed by an observer on the Earth. Assuming the meteoroids generated by a comet closely follow the comet's orbit, the radiant point is simply computed from the relative velocity components between the meteor particles and the Earth at the point of closest approach between the Earth's orbit and the comet's orbit (Fig. 1.6).
|
Fig. 1.6. Relative velocity between the Earth and the comet at closest approach. |
Note that the relative velocity must be taken from the Earth to the comet.
 | (1.14) |
The relative velocity at that point is then rotated into the geocentric equatorial plane with the right ascension and declination subsequently withdrawn from the geocentric relative velocity components as (Eq. [1.15]):
 |  | (1.15) |
where the right ascension must be tested against the x- and y-components of the relative velocity to make sure the quadrant is correct and where the magnitude of the relative velocity is simply the approach velocity of the meteors with respect to the Earth (hereafter referred to as V∞). For the Leonids, the relative velocity vector as computed from the comet orbit in Sec. 1.2 at Earth crossing in km/sec is shown in Eq. (1.16).
 | (1.16) |
The results of the radiant point computation using the generated ephemeris for Tempel-Tuttle are given in Table 1.4, along with the same computation for several different currently available ephemerides. The newer ephemerides all give very close results, which instill confidence in the methodology employed here. A formal error study shows that the errors on the radiant points can be conservatively estimated at less than 0.1 deg.
The best geographical viewing locations are tied to the time of closest approach (from Sec. 1.2) and the radiant vector. Table 1.5 gives these values for the Leonids over the 1998–2000 time span.
| Right Ascension (deg) | Declination (deg) | |
|---|---|---|
| Peterson, 1998 | 153.45 | 21.87 |
| Yeomans, 1997 | 153.45 | 21.88 |
| Yeomans, 1996 | 153.34 | 21.76 |
| Muraoka, 1997 | 153.43 | 21.88 |
| MPC catalog (Marsden), 1997 | 153.42 | 21.88 |
| Yeomans, 1981 | 152.66 | 22.00 |
| LTEP (Carusi), 1985 | 152.69 | 22.15 |
| Cook, 1973 | 152.30 | 22.20 |
| Date (±2 h) | Best Visibility (deg E) |
|---|---|
| 1998 Nov 17: 20.64h UT | 130 ±30 |
| 1999 Nov 18: 03.36h UT | 36 ±30 |
| 2000 Nov 17: 10h UT | 313 ±30 |
1.3.1 Motion of the Earth through the Meteor Stream
Equation (1.15), however, computes the ideal radiant point at the time of the expected peak activity (i.e., closest approach). It does not take into account the motion of the Earth in its orbit through the entire stream. To do so requires a computation of the right ascension and declination during the complete time interval in question. Although the comet's orbit is fixed at the encounter time (i.e., all particles are assumed to be traveling parallel to the comet's orbit), the Earth is traveling through the debris ring. The relative velocity between the Earth and the particles will therefore change as a function of time, as described in Eq. (1.17):
 | (1.17) |
The Leonids are typically active for several days around the time of peak activity. To make sure the time span was not too short, however, a longer 2-week interval centered on the expected time of peak activity for the Leonids of November 17 for each year was chosen. The resultant radiant point motion is presented in Fig. 1.7. Curve-fitting results over the 2-week interval give the following equations for the mean motion of the right ascension and declination:
 |
(1.18) |
 |
where day refers to the date in November desired (i.e., November 14, 0 hour, would compute as 14.0, while hour 12 of November 17 would be 17.5).
Fig. 1.7. Leonid radiant point motion. |
1.3.2 Apparent Radiant Including the Motion of the Satellite
In describing the mean motion of the radiant over the course of a shower, it was assumed that the relative velocity that dictated the radiant point location was determined by the Earth and the comet (Eq. [1.14]). However, vehicles orbiting the Earth are also going to be traveling rather quickly (~7.5 km/sec for low Earth orbit [LEO]). This means that the radiant point, as far as a spacecraft is concerned, will experience a high-frequency, once-per-orbit variation about the mean values just presented. The complete relative velocity is more properly given by:
 | (1.19) |
where vs is the spacecraft geocentric velocity and can be written in terms of the orbital elements for near-circular orbits as:
 | (1.20) |
where vs is the circular velocity of the satellite, a is the semimajor axis, and µ is the gravitational parameter for the Earth:
 | (1.21) |
and where 
, i, and u are the right ascension of ascending node, inclination, and argument of latitude (Fig. 1.8).
Fig. 1.8. Orbit geometry with meteor radiant vector. |
The radiant point including the satellite's motion (where vrel is the relative velocity between the Earth and the comet at closest approach, Eq. [1.14]) is:
 | (1.22) |
 |
These expressions are complex functions of several variables:
- The approach velocity of the particles V∞
- The spacecraft velocity (i.e., the semimajor axis)
- The in-plane location u
- The orientation of the orbit plane as determined by the node and inclination, and i
To simplify the problem somewhat, it is appropriate to examine the maximum envelope of values for the deviations from the mean values. Assume an orbit perpendicular to the oncoming stream (i = 90 deg – 
Lmean, = 
Lmean + 90 deg). In this case, relative velocity between the particles and the spacecraft will be at its largest angular extent, subsequently giving the widest range of the radiant point locations possible (Fig. 1.9). In essence, an envelope of maximum angular values is created for any given altitude. Using the small-angle approximation, the limiting values of the maximum angular deviation are:
 | (1.23) |
|
Note that the plus/minus signs are independent of each other. Consider an orbit that is edge-on with respect to the meteors (Fig. 1.9, in which the viewer is looking along the meteor approach vector). In this case, the deviation can occur either in the right ascension or declination, but not both. If the orbit is perpendicular to the approach vector, then the maximum deviation can happen in both components. The two components can therefore be independent in their behavior but are not necessarily so. Care must therefore be taken in using Eq. (1.23) to find the possible deviations. This simply determines an approximate envelope; the actual behavior will depend upon the orbit's specific orientation in space (Eq. [1.22]).
Fig. 1.9. Radiant point deviation geometry. |
Figure 1.10 shows the variations of the maximum deviations for the Leonid meteor shower, assuming that the particular orbit of interest occurs at the time of the storm (Lmean = 153.45 deg, Lmean = 21.87 deg). For LEO, the deviations from the mean can be quite large; even for geosynchronous (GEO) orbits, the values are still significant. However, this represents the maximum values possible for the Leonids. Again, each specific orbit should be examined distinctly. Each meteor event will also produce different radiant point notions. The Leonid deviations shown here can be small when compared with slower moving streams such as the Draconids.
Fig. 1.10. Maximum angular deviation as function of altitude for near-circular orbits for the Leonids. |
Consider two common examples: a near-equatorial (i = 5 deg) GEO satellite and a Sun-synchronous LEO satellite (i = 98.60 deg, = 156.93 deg). The LEO case is special in that for the Leonids, a Sun-synchronous satellite with this node and inclination is edge-on with respect to the radiant. The GEO case is significant because there are many satellites in Earth orbit with these approximate parameters.
Figure 1.11 shows the deviation of the radiant from the mean value for these two cases over the course of an orbital cycle. For comparison purposes, note that the maximum values of the deviations for the same altitude orbit as shown in Fig. 1.10 are 6.4 deg and 8.7 deg in the right ascension and declination for the LEO altitude and 2.7 deg and 3.7 deg for GEO. From Fig. 1.11, the specific case maximums are 1.1 deg and 6.3 deg for the LEO, and 2.6 deg and 1.0 deg for GEO in the right ascension and declination, respectively. For the Sun-synchronous LEO orbit, the deviation in the declination can be almost as large as the maximum possible; since the orbit is edge-on, the right ascension variation is only about a degree from the mean value. For GEO, the declination deviation is less than a degree, while the right ascension can vary noticeably from the mean value. The larger implications of this once-per-revolution apparent motion in the radiant will be examined in Sec. 4.3.
|
Fig. 1.11. Radiant point motion with once-per-rev effect. |
Next: Particle Flux
