Dynamics of Meteor Outbursts and Satellite Mitigation Strategies
Glenn E. Peterson
Chapter 1: Assessing the Threat (cont).
1.2 Orbit Dynamics
Because the nature of meteor outbursts is governed by how close the comet is to the Earth at meteor stream encounter and how close the Earth is to the center of the stream, it is imperative to understand the basic dynamics of the system. This section examines the parent comet's orbit, the sublimated particles' orbit, and the time it takes for the newly created particles to disperse throughout the stream into an even distribution.
This book assumes a basic knowledge of orbital mechanics and dynamics, but to refamiliarize the reader, an orbit is defined by these six parameters:
- Semimajor axis a
- Eccentricity e
- Inclination i
- Longitude of ascending node
(also called the right ascension of ascending node when referring to Earth-centered spacecraft) - Argument of periapse
- True anomaly f
This constitutes the classical Keplerian set. The semimajor axis and eccentricity define the size and shape of the orbital ellipse; the inclination, node, and argument of periapse set the orientation of the orbit in space (Fig. 1.3). The inclination is the angle the orbit makes with either the ecliptic plane (Sun-centered orbits) or the equatorial plane (Earth-centered). The ascending node location is the angle between the vector pointing from the central body to the first point of Ares (x-axis) and the location where the orbit crosses the ecliptic or equatorial plane moving from south to north (z-axis is perpendicular to this plane). The argument of periapse is the angle from the ascending node to the periapse location. Finally, the in-plane location of the satellite or comet is given by the true anomaly. For near-circular orbits, the eccentricity becomes small, resulting in the periapse location becoming ill-defined. In this case, the argument of periapse and the true anomaly are replaced by the more well-defined argument of latitude u (= + f). Another way of indicating the in-plane location is through a fictitious angle, the mean anomaly M, which is linearly related to the mean motion (i.e., the semimajor axis) of the central body through Kepler's Equation. These orbital elements will be discussed repeatedly in this text. For a more in-depth treatment of these definitions and the vast field of orbital mechanics, the interested reader is referred to additional sources in astrodynamics (Bate, Mueller, and White, 1971, and Vallado, 1997).
|
Fig. 1.3. Basic orbit element angles. |
1.2.1 Parent Orbit
Because meteoroids are created by a parent comet, it is often useful as a first approximation to model the movement of the meteoroids as following the orbit of the parent body. But in modeling the motion of an arbitrary comet, the Sun is not the only gravitating body that needs to be considered. Other large bodies in the solar system can significantly influence periodic comets as well. Jupiter and Saturn, most notably, can create resonant types of effects that alter the behavior of either the comet or the generated dust stream. One theory holds that the rare storms associated with certain meteor streams that occur when their parent comets are far from perihelion are caused by a resonance within the dust cloud dictated by Jovian and Saturnian perturbations, which are far-comet types of outbursts (Jenniskens, 1995). The planetary-disturbing function must therefore include all significant bodies (i.e., Jupiter and Saturn, certainly) and possibly ones whose importance may not be immediately apparent. While the Earth, for example, is not an important gravitating body for virtually any comet, if a close approach to the Earth occurs, it can be quite significant in the resulting trajectory of that comet. Care must then be taken when excluding the smaller bodies. Note also that problems can develop during very close approaches to various third bodies. Some comets, such as Shoemaker-Levy 9, are more appropriately treated as in the influence of Jupiter as the dominant central body for at least a portion of their orbits. Such switching between central bodies (in this case, Jupiter and the Sun) creates special problems in the numerical integration, of which analysts must be aware (Vallado, 1997).
Simple relativistic effects also are typically included in the reduction of observational data. Because the best available ephemerides for the planets and the Sun include general relativity, small errors can result in the reduction of the comet's orbit if the same effect is not included in the comet's force modeling when the reduction occurs. Owing to the small sizes of the relativistic perturbations, only the Sun's additional curvature on the fabric of space time is necessary (Shahid-Saless and Yeomans, 1994).
In addition to the third-body perturbations and the central-body relativistic effect, comet orbits are also best simulated when a nonconservative force based upon Whipple's ejection model is used (Yeomans and Chodas, 1989). Whipple's model assumes that particle ejection occurs in a rocket-like manner as a result of solar heating of a rotating icy ball. If the nucleus of the comet were not spinning, the heating would be symmetrical about the radial axis (comet-Sun vector). But with a rotating snowball, a thermal lag angle develops between the point of maximum ejection and the subsolar point. Generally speaking, the ejection force will thus have transverse and normal components and not just the radial aspect. However, the normal component is small and periodic in nature and thus tends not to influence the long-term behavior of the given comet. Taken together, the equations of motion for a given comet are best modeled by
 |
(1.1) |
where µsun is the Sun's gravitational parameter, 
and 
are relativistic parameters describing the curvature of space time (both are equal to unity in general relativity), c is the speed of light, N is summed over all the required planetary bodies (note again that very close approaches may require a different third-body formulation), r and v are the comet's heliocentric position and velocity vectors, and the index j is summed over all the necessary planetary bodies. The third term of Eq. (1.1) depicts Whipple's icy snowball model, where R and T are unit vectors in the radial and transverse directions in the plane of the comet's orbit, and g is a function describing the aqueous physical composition of the comet:
 | (1.2) |
and
 |
(1.3) |
where DT denotes a comet-specific asymmetry in the cometary ejection about perihelion passage; m, n, k, r0, and 
are parameters dictated by the comet's physical nature (i.e., icy snowball) and therefore are assumed invariant between individual comets. A1 and A2 are comet-specific values determined from observations for the radial and transverse components of the acceleration (the normal component is again typically insignificant). Because they are empirically determined from measurements, A1 and A2 tend to absorb some of the other nonconservative forces, such as solar radiation pressure, that would otherwise act in the radial and transverse directions. This is the traditional approach to generating cometary orbits. The reader should be aware that when tabulated values of A1 and A2 appear, the above force model (Eq. [1.1]) has been used. Adding in solar radiation pressure or other forces on the comet as well as the empirical ejection forces could therefore generate a slightly erroneous orbit.
Table 1.1 shows values typical for Whipple's ejection model. The first two columns are parameters dictated by the assumption of the icy-snowball model and are assumed applicable for all comets. The third column contains parameters specific to each comet (A1 and A2 are in AU/day2). The values shown are for comet Tempel-Tuttle, the parent body of the Leonids. The orbit elements for Tempel-Tuttle are additionally shown in Table 1.2. The orbit elements of some other significant comets (at least as far as meteor showers are concerned) are given in Appendix A. The orbit elements presented here were obtained from the Solar System Exploration Division of the NASA Jet Propulsion Laboratory.
Table 1.1. Comet Tempel-Tuttle Model Parameters
| m 2.15 | r0 2.808 | A1 5.4 (±2.7) e-10 |
| n 5.093 | 
0.111262 |
A2 9.129 (±0.015) e-11 |
| k 4.6142 | DT 0.0 |
For near-comet types of meteor outbursts, the intensity of encounter is driven by two factors—how close typically the comet's (or streamlet's) orbit is to the Earth's orbit at closest approach (a positive value indicates that the Earth passes inside the comet's orbit) and how much time has elapsed between the Earth's passage and the comet's passage of that point (a positive value indicates that the Earth is behind the comet). Although the stream particles will have a distinct distribution—it has been shown through modeling (Brown and Jones, 1993) that it may resemble a twisted braid-like structure—the exact distribution is not determinable before the encounter occurs. Therefore the only way to predict whether an encounter may be intense is through studying the dynamics of the encounter.
Table 1.2. Comet Tempel-Tuttle Keplerian Elements for Epoch 1998 Mar 8.0
| Perihelion distance (AU) | 0.9765849 |
| Eccentricity | 0.9055036 |
| Inclination (deg) | 162.48614 |
| Longitude of ascending node (deg) | 235.25883 |
| Argument of perihelion (deg) | 172.49731 |
| Time of perihelion passage | 1998 Feb 28.09666 |
By comparing the ephemeris found from Eq. (1.1) with that of the Earth's orbit, the point of closest approach can be calculated for any encounter. Table 1.3 shows the data for the Leonid storms that were computed using the above model. The zenith hourly rate (ZHR) is a measure of the observational intensity of a meteor stream's activity (discussed in further detail in Sec. 1.4). Historical values of the ZHR are taken from recent sources (Jenniskens, 1994 and Yeomans et al., 1996). It should be noted that the values given here are for the peak returns for that particular return of Tempel-Tuttle. They do not necessarily correspond to the years where the Earth passes closest behind the comet. For example, the Earth passed just 129 days behind the comet in 1899, but that storm had a ZHR estimated in the 1100 range only. Two years later, the 1901 storm was quite a bit more active than that, with a peak ZHR of approximately 7000. In 1903, 4 years after the comet's passage, the Leonids again had a ZHR over 1000. It is important to note that just because the Earth is close to the comet temporally does not mean that a storm will take place, nor does it mean this encounter will be the most active for that particular comet passage. A more telling measure of activity than closeness in time is the distance of closest approach. But even that comes with a caveat. The 1932-33 encounter was spatially closer to the center of the stream than the 1899-1901-1903 events, but was disappointing in its behavior. These discrepancies are representative of the complexity and uncertainty of evaluating meteoric events.
1.2.2 Particle Ejection
Once the basic orbit of the meteor stream is established based upon the behavior of the parent comet, the ejection of the future meteoric particles must commence. Although near-comet outbursts can be modeled based primarily upon the orbit of the parent, general meteor stream behavior might not follow the parent closely. Streamlets generated during past perihelion passages will follow the parent orbit closely but will deviate slightly over time. Far-comet outbursts such as the Lyrids of 1992 can behave even more differently, and some streams are not associated with any known parent at the current time (Quadrantids). The particle ejection behavior is thus significant because that provides a fundamental, random addition to the initial velocities of the particles when computing the subsequent evolution of the meteor stream.
Table 1.3. Tempel-Tuttle Earth Encounter Data from Past Leonid Storms
| Year | Distance (AU) | Time (days) | ZHR |
|---|---|---|---|
| 1799 | -0.0034 | -116 | >5000 |
| 1833 | -0.0012 | +307 | >5000 |
| 1866 | -0.0050 | +298 | ~17,000 |
| 1901 | -0.0168 | +859 | ~7,000 |
| 1933 | -0.0118 | +126 | <1000 |
| 1966 | -0.0035 | +561 | 15,000–150,000 |
| 1998–2000 | -0.0085 | +257 | ~10,000 (expected) |
Consider a particle that has broken away from the parent body. Such a particle is far enough away from the cometary nucleus that the specific sublimation mechanism that created the particle as being separate from the comet body can be ignored. In this case the future meteoroid will have momentum imparted to it from the more highly energetic gas that is sublimating off the comet due to the solar heating over the hemisphere that is facing the Sun. When this sublimating force is larger than the local gravity acting to pull the particle back to the surface, the particle will escape and become part of the meteor stream (Fig. 1.4).
|
Fig. 1.4. Forces acting on a particle in the near-comet environment. |
Assuming the two forces act purely radially, the velocity computed at infinity (i.e., the delta to be added to the initial velocities) (Whipple, 1951) is:
 |
(1.4) |
where Fsun is the solar constant (315 cal/sec at 1 AU), µc (km3/sec2) and Rc (km) are the gravitational parameter and radius of the parent comet, r is the solar distance in AU, sp (cm) and 
p (g/cm3) are the radius and density of the particle, H is the mean heat of sublimation (~450 cal/g), 1/neff represents the fraction of the solar heating that goes into sublimation (1/neff can be assumed as unity for most comets; however, it can be much smaller: comet Encke's fraction is assumed to be ~0.1), and vgas is the average speed of the gas carrying the particle away (Whipple, 1950):
 |
(1.5) |
where the constant in the numerator is found as a function of the temperature and the gas constituents typical of comets. The magnitude of the velocity to be added in random directions to the particles' initial conditions for a typical comet is:
 |
(1.6) |
Once again, in this expression Rc is in kilometers, sp is in centimeters, and r is in AU. Note that there is a factor of 4 difference in the first term of this expression and one noted elsewhere (Whipple, 1951). The earlier model assumed that the meteor grains embedded in the icy comet were rocky in nature, resulting in a particle density of 4 g/cm3. More recent studies (McDonnell et al., 1991) show that the comet particles vary in composition from ice fragments to fluffy silicates. The particle densities thus range from 1 to 2.5 g/cm3. Because the lower density gives a higher ejection velocity (and hence greater perturbation), the 1 g/cm3 value is assumed in Eq. (1.6).
This perturbation on the initial velocity acts radially away from the comet nucleus. However, comets will be spinning as they move through their orbit. There will therefore be an additional transverse perturbation to the initial velocities of the particle cloud. This creates the braid-like structure in the resulting stream that was mentioned earlier. But current determinations of actual comet spin rates and directions are highly uncertain and are currently best left to the realm of theoretical modeling.
1.2.3 Ring Creation
Once the particles have escaped from the comet's influence, the small deviations in the velocities from that of the parent body will cause the particles to continue to move away from the basic orbit. The stream will disperse but remain fairly centered on the parent comet's orbit as long as the parent comet is in a fairly short period (several Jovian cycles or less). For these parents, the orbits of the child particles do not have time to be significantly dispersed or altered before a fresh batch of children is birthed by the parent. For long-period comets, substantial deviations can occur, resulting in long-period streams that may be quite different in their behavior than the parent comet.
Other than these small changes in the initial conditions, the most significant forces that cause the subsequent evolution of the particles to deviate from the parent trajectory are solar radiation pressure and Poynting-Robertson drag. Resonances can also impact ring evolution, but they are beyond the scope of this analysis. Recall that for the comet, the dominant nonconservative force is the rocket-like ejection force governed by Eq. (1.2). The other smaller nonconservative forces that must act on the comet were absorbed into the estimation of the empirically determined parameters A1 and A2. For the particles themselves, no such estimation can be made because the individual particles cannot be tracked. Therefore, in Eq. (1.1) the ejection force must be replaced by explicit expressions for the solar radiation pressure, the Poynting-Robertson drag, and the solar wind drag, as in Eq. (1.7).
 |
(1.7) |
where sw is the ratio of the solar wind drag to Poynting-Robertson drag (~0.3), c is the speed of light, r and v are the position and velocity of the particle, vR is the velocity component in the radial direction, µsun is the gravitational parameter of the Sun, and is the ratio of the solar radiation force to the gravitational force, valid for particles larger than 1 mm in radius (Dohnanyi, 1978):
 |
(1.8) |
where cR is the coefficient of reflectivity for the particle and will vary from 0 (translucent material) to 1 (perfect absorber) and up to 2 (perfect reflector). It should be noted that very small particles do not behave according to this equation, but they do not constitute a threat to spacecraft. The Poynting-Robertson drag is small in comparison to the other forces but significant in that this force, for short-period comets, tends to force the smaller, nonspacecraft-threatening particles to spiral into the Sun and so acts to "clean up" the stream of very small dust.
There are other nonconservative forces that can act on particles in addition to solar radiation pressure, Poynting-Robertson drag, and solar wind drag. Yarkovsky forces can be as large as the Poynting-Robertson effect under certain circumstances (Burns et al., 1979) but only for particles that are too small to be a threat to spacecraft. This force acts as a thrust on the particle along its spin axis because of the differential heating and dissipation that occurs when a rotating body is subjected to thermal radiation. Differential Doppler and Lorentz forces can also influence the behavior of particles but only in a very limited fashion. These smaller forces are typically ignored in stream particle studies. However, the Lorentz force is important for particles still in the coma of the parent (Horanyi and Mendis, 1991).
The dispersive solar radiation force and the small changes in the initial conditions experienced by the particles are the main drivers in determining how fast the larger, more dangerous particles spread out along the stream and thus complete the closure of the meteors into a continuous ring. Because the meteoroids are pushed both in front of and behind the comet by the ejection, the number of orbital revolutions required for the perturbed particles to go through one-half of the parent body's period defines the problem (McIntosh, 1973) as:
 |
(1.9) |
where TP is the period of the comet and dTP the deviation from the comet's period experienced by the perturbed particle (a is the semimajor axis of the orbit). The period and its derivative are given by:
 |
 |
(1.10) |
Assuming that the particle ejection takes place impulsively at perihelion, the differential changes for the two main perturbing forces (ejection and solar radiation pressure) that act to change the semimajor axis are:
 |
 |
(1.11) |
where vc is the speed of the comet at perihelion and e is the eccentricity, producing
 |
(1.12) |
Substituting, the number of revolutions it takes to close the orbit and create a continuous meteor stream is then (where in this expression the semimajor axis is written in AUs, the velocities are in km/sec, and µsun is in km3/sec2):
 |
(1.13) |
Note that the ejection velocity can act in either direction; however, in order to determine the minimum amount of time to complete closure, the maximum value of the combined velocity term should be taken.
Because both the ejection velocity and parameter are functions of the relatively uncertain comet physical properties as well as being highly dependent upon the particle size, the number of revolutions is not an easy matter to express. Figure 1.5 exhibits this property and shows the number of revolutions for several prominent meteor showers as a function of the particle size, assuming the cometary physical properties are constant (p = 1 g/cm3, cR = 1, neff = 1). The time it takes to close a meteor stream into one continuous loop thus varies for different particle sizes. In theory, it may therefore be possible to compute the location of an unknown parent comet not only from the changes in the numbers of meteors that an Earth observer might see, but also from the year-to-year variations in the magnitudes of the observed meteor streaks.
|
Fig. 1.5. Complete stream formation time. |
While particles smaller than 0.01 cm may cause damage to spacecraft, that value was chosen as a limit in the figure not only for presentation purposes but also because for the fastest-moving meteors (i.e., Leonids at 70 km/sec), a particle of mass approximately 1 X 10–5 g will create a magnitude 6.5 streak; slower particles require an even larger radius (see Sec. 1.4). With a particle density of 1 g/cm3, a Leonid +6.5 magnitude-sized particle translates into a radius of 0.013 cm. Therefore, only under rare circumstances will particles smaller than 0.01 cm create noticeably visible meteor streaks in the night sky.
Next: Radiant Point
