Dynamics of Meteor Outbursts and Satellite Mitigation Strategies
Glenn E. Peterson
Chapter 1: Assessing the Threat (cont).
1.4 Particle Flux
With the dynamical behavior of the particles determined at Earth encounter, the next issue to ponder is just how many particles have the potential of impacting an orbiting vehicle and causing damage. The temporal variation in the meteor event as viewed by the Earth is examined in terms of the activity curve as well as the distribution of the particle flux expected at encounter. This leads to a computation of the probability of impact for meteoroids likely to consist of a threat to a satellite.
1.4.1 Meteoroid Activity Curve
The Zenith Hourly Rate (ZHR) is the typical means by which the intensity of a meteor event is characterized and describes the number of meteors seen by the average observer in 1 hour under ideal sky conditions with the radiant in the zenith direction. It is most often computed from naked-eye Earth observations that have been normalized to account for atmospheric conditions and observer bias. Measurements of the meteor shower activity can also be derived from radar and infrared instruments. However, this is not done as often as naked-eye observations because of cost, availability, and the difficulty such instruments have in distinguishing background sporadic meteors and those associated with a given shower. So even with today's technology, many observations of meteor shower activity are performed by amateurs and professionals alike using visual techniques.
The observer biases inherent to naked-eye techniques are affected by the quality of the watcher's eyesight, attention span, style of observations, and judgment of meteor brightness, speed, and direction. This makes the counting of meteors highly individualistic. It is only through repetitive comparisons between observers that an individual's personal corrective factor can be determined and the observer's own biases removed from the raw data. The basic equation governing the normalization procedure (Jenniskens, 1994) is
 | (1.24) |
where N is the raw observational count in the time interval 
t (it is typically assumed that t must be greater than 0.4 hour with a level of observational downtime less than 20%), cp is the observational bias factor (standard observer = 1.0), Lm is the sky limiting magnitude (Lm = 6.5 is ideal), hr is the radiant altitude of the particular stream (90 deg = zenith), 
is an exponent greater than or equal to 1, but is usually taken as unity for visual observations, and r is the population index determined from magnitude observations. There is a slight variation in r as a function of the solar longitude (Brown and Rendtel, 1996). In addition, there is an argument for an observer dependence as well (Jenniskens, 1994). However, for the first approximation purposes a satellite operator would be interested in, it can be assumed that r is constant.
Because the Earth is moving through the meteor stream for any given event, there will be a time-varying component to the measured ZHR. This activity curve defines the level of ZHR activity as a function of time for a given meteor shower (Jenniskens, 1994) as:
 | (1.25) |
where ZHRmax is the maximum zenith hourly rate for the shower under consideration, B is the slope of a straight line in a log-normal plot experimentally fitted from observational data, 
is the solar longitude, and max is the solar longitude at the peak value of the ZHR. Appendix B gives the activity curve data for a variety of meteor showers.
For most meteor showers, there is an exponential growth to a sharp peak and then a fairly symmetrical shrinkage as the Earth moves away from the cometary debris. However, some of the larger meteor showers (Boo, PAu, Per, Leo, Gem, Urs) are more accurately described as the sum of two exponentials. This is largely due to the greater densities of these showers and the subsequent better observations made of them. A more definitive structure can then be determined. In an opposite sense, some showers are so sparse that modeling an exponential fit to them is problematic at best. In these cases, it is best to assume a constant ZHR over the shower's duration (aHy, dVe, tCe, Vir, CAu, eEr, dEr). For a more detailed treatment of the reduction of real observations to the ZHR activity curve, see additional sources (Jenniskens, 1994; Brown and Rendtel, 1996; and Koschack and Rendtel, 1990a and 1990b). Figure 1.12 shows a representative activity curve for the typical annual components of the Leonid shower as given by the values of Appendix B.
|
Fig. 1.12. Typical activity curve for Leonid annual shower. |
The length of time that the Earth spends in the stream can be approximated by integrating the time-varying component of the activity curve (Eq. [1.25]) over the entire cycle of the solar longitude:
 | (1.26) |
where t is in the same units as B (i.e., per degree, per day; Appendix B gives the B values per degree). The index i ranges over the ascending and descending branches. In the case of showers such as the Leonids, where the activity curve is more accurately described by two curves (a peak and a background), the index will range over four values.
Some care must additionally be used in defining the stream boundaries when using this expression. For the Leonids of Fig. 1.12, this expression would give an approximate length of the shower as 30 days. However, much of the activity leading up to the maximum is very sparse in nature and almost lost in the background-sporadic meteors unless the observer knows what to look for. Several of the more well-defined meteor showers show a similar characteristic. This is most likely due to the Poynting-Robertson drag pulling smaller particles in toward the Sun, making the side of the stream facing the Sun more diffuse than the side facing away. In this case it is more appropriate to assume the activity curve is symmetrical with the more well-defined branch taking the place of the ill-defined value of B. For the Leonids (with two components to the activity curve), Eq. (1.26) becomes:
 | (1.27) |
where p refers to the B value for the peak, b for the background component, and the minus sign indicates the descending branch. Using the Leonid values found in
Appendix B, the time spent in the stream becomes 7.5 days. Assuming the Earth travels in a straight line over the course of the shower at 30 km/sec, the Earth will travel a distance in the Leonid stream tube of 19.4 million km. From the geometry of the encounter, the corresponding tube width can be found to be approximately 3 million km (the Earth goes almost directly down the tube).
For the outbursts, a similar adjustment may apply, since these events have historically been ill-defined. Appendix C gives the outburst activity curves and, assuming the 1998–2000 Leonid events will be similar to past occurrences (and assuming symmetry in the activity curve), the value of B for the peak will be about 30, while the background component will have a B of approximately 4. This gives the time spent in the region of greater outburst activity to be ~6 hours, with the peak region occupying less than 1 hour.
1.4.2 Distribution
The ZHR activity curve is thus a normalized ground-based observational count of those meteoroids that intercept the Earth's atmosphere and create a visible train large enough for the naked eye to see. So, while the ZHR activity curve gives a relative temporal measure of the meteoroid density, it does not reflect the actual particle flux that a spacecraft will experience away from the protective envelope of the atmosphere. Because the Earth observes a single pass through a meteor stream only once a year, observational confirmation of the flux is very difficult to determine. As a consequence, one of the most important assumptions in meteor stream flux modeling has been to assume that the dust in a meteor stream follows the same general type of distribution as the solar system ambient dust flux.
Although modifications have been proposed (Taylor and McBride, 1997; Wasbauer et al., 1997), the interplanetary dust distribution mass-flux model most widely in use is Grün's model (Grün et al., 1985):
 | (1.28) |
where mp is the mass of the particle. There are three terms in this model: the first line is dominant for larger particles (m > 10–9 g) that pose the greatest threat to spacecraft; the second term is more significant for medium-sized particles (10–14 g < m < 10–9 g); while the third describes the smaller dust motes (m < 10–14 g).
For penetration to occur, only those particles that have a mass greater than 10–7 g need to be considered (see Sec. 4.2.1 for the ballistic limiting equations that assume the worst case of a single wall of aluminum for the fast-moving Leonids); the Grün model can be simplified to:
 | (1.29) |
But while this describes the ambient solar system flux of the dust for threatening-sized particles, it does not describe a particular meteor stream's flux distribution. Since smaller particles resident in the solar system will be "cleaned up" by Poynting-Robertson drag, it is logical to assume that the meteor stream mass index (exponent in Eq.[1.29]) will be different from the solar system dust mass index. The more general form of the instantaneous meteor stream flux at a particular mass value is:
 | (1.30) |
where s is referred to as the mass index for the stream and N0 is a constant particular to that stream. The mass index is related to the population index r that was mentioned in the previous section as (McKinley, 1961):
 | (1.31) |
To find the total number of particles a spacecraft will face, this expression must be integrated from an arbitrary small mass to infinity:
 | (1.32) |
But N0 is not a known value and must be solved for. The only parameter that is available to accomplish this comes from the observations inherent in the ZHR activity curve. The expression relating the ZHR to the spatial number density (Brown and Rendtel, 1996) is:
 | (1.33) |
where the typical units for this expression in the literature is 10–9 km–3. Since the ZHR is normalized about a magnitude of +6.5, N+6.5 is the spatial number density for that magnitude only.
The factor C' adjusts the observed ZHR to a true value accounting for meteors missed by observers. C' is a function of the population index (Koschack and Rendtel, 1990b):
 | (1.34) |
The parameter Asky is the effective collecting area for the meteors in the atmosphere at an assumed height of 100 km that an observer is able to watch. As such, Asky is also a function of the population index (Koschack and Rendtel, 1990a):
 | (1.35) |
The flux at magnitude +6.5 is simply the spatial number density at that magnitude multiplied by the approach velocity:
 | (1.36) |
Once the magnitude +6.5 flux is known, it is a simple matter to scale it for different masses as well as the presence of an outburst:
 | (1.37) |
where Fs is a mass index enhancement factor that describes the greater concentration of smaller particles that would occur in an outburst as compared with the normal shower. In essence, the Poynting-Robertson drag has not had a chance to remove the smaller particles from the system for the near-comet types of outbursts. Fs can be approximated by Table 1.6 (Beech et al., 1997).
| s | 1.5 | 1.75 | 2.0 | 2.25 | 2.5 | 2.75 |
| F | 0.1 | 0.4 | 1.0 | 2.0 | 3.8 | 6.5 |
The magnitude +6.5 particle mass in Eq. (1.37) is found from the mass-luminosity relation:
 | (1.38) |
where Lm is the visual magnitude of the streak (i.e., Lm = +6.5), V∞ is the approach velocity of the meteoroid, and m+6.5 is the mass of the particle. The constants A, B, and C vary between models with three examples given in Table 1.7 (Cooke and Anderson, 1998).
Also shown in Table 1.7 are the masses in grams of the ambient and Leonid (V∞ = 71 km/sec) particles for the ideal limiting magnitude. Nearly an order of magnitude difference in the resulting meteoroid flux can therefore result from the various models. Analysts should note that the flux modeling can be off by a similar order of magnitude as a consequence of these uncertainties.
| A | B | C | Autdor | Ambient | Leonid |
| -6.27 | -1.61 | 9.77 | Jenniskens | 1.4e-3 | 6.6e-6 |
| -8.75 | -2.25 | 11.59 | Brown | 2.4e-3 | 1.2e-5 |
| -8.75 | -2.25 | 12.90 | Kessler | 9.2e-3 | 4.4e-5 |
By combining Eqs. (1.33) through (1.37), the general form for the equation for the cumulative flux of particles having a mass greater than a specified value m for any given meteor event becomes (particles/m2-s):
 | (1.39) |
where km is a constant that is event specific:
 | (1.40) |
Evaluating Eq. (1.39) for the Leonids (assuming a mass index of 2.2 for the outbursts and Brown's value of the m+6.5 particle as the middling value in Table 1.7) gives the final expression for the cumulative flux of particles for the 1998–2000 expected outbursts:
 | (1.41) |
 |
In comparison, the annual Leonid shower has a flux of (mass index of 2.0):
 | (1.42) |
 |
1.4.3 Impact
Damage from a particle impact can come from a variety of phenomena either mechanical (shock waves, cratering, penetration, spall production) or electrical (plasma production leading to electrostatic discharge [ESD], electromagnetic impulse). This study is not designed to address the physics of hypervelocity phenomenology but rather the dynamics of mitigating the threat that they pose. As a consequence, the interest here is to determine the minimum-mass particle that may cause significant levels of danger. This minimum mass provides the basis for determining the flux that has the potential to hurt the vehicle and therefore the probability of a dangerous impact. Assuming the worst case of a single wall of thin 1-mm aluminum and a meteoroid particle impacting perpendicular to the spacecraft surface, the ballistic limiting equation (Sec. 4.2) reduces to a relation between the mass of the particle and its approach velocity:
 | (1.43) |
where V∞ is in km/sec and the mass is in grams. This gives an approximate measure of the minimum-size particle that can be a cause for worry.
The particles that hit a spacecraft of a given cross-sectional area over a meteor event of known duration can be determined from the flux as:
 | (1.44) |
where 
T is the flux (#/m2/s), As/c is the cross-sectional area exposed to the meteors (m2), and the meteor event exists over the time span from t1 to t2 (sec). Figure 1.13 shows this expression for both the Leonid shower and the expected worst-case outbursts of 1998–2000 (Appendix B data were used for the Leonid activity curve with a ZHR of 10,000). Since the ZHR is time-varying, Eq. (1.44) is computed in 1-hour intervals over which the ZHR is assumed constant for the shower, and in 10-minute intervals over the course of the expected storm. The number of impacts is also normalized with respect to the spacecraft area in order to keep the results general.
|
Fig. 1.13. Impact probabilities for critical mass particles. |
The impact probabilities are therefore potentially high during the time of the storm. Note that this is a time profile determined in 10-minute sections. To find the total probability of a hit, the number of impacts under the outburst curve must be totaled. Doing so yields a 0.002 probability. The small time of peak activity clearly dominates the result. A satellite with a 100-m2 area would thus have had roughly a 2-in-10 chance of being hit by a dangerously sized particle during the 1998–2000 Leonids, assuming a peak worst-case ZHR of 10,000. With more than 500 U.S. and international satellites currently in operation, and assuming an average area of 10 m2, approximately five satellites are going to be impacted by a dangerous mass particle during the 1998–2000 event. However, it must be stressed again that the flux estimates could be off by an order of magnitude, either in a positive (i.e., impact chances are lower) or negative (chances become close to unity for a 100 m2 satellite) way.
Figure 1.14 shows the Leonid flux per unit area at the time of peak activity as a function of the particle mass for both the annual shower and the expected 1998–2000 outburst. The smaller particles pose the greatest danger to spacecraft; their presence is orders of magnitude more prevalent than the larger meteoroids. In comparison, launch and mission operations become concerned about the possibility of collision with orbital debris when the probabilities are on the order of 10–6. This level of probability would roughly correspond to a Leonid particle that would have the same kinetic energy as a 0.22 caliber bullet. Clearly, the Leonid outburst will present a danger much larger than typically accepted for mission operations. However, the danger will come from the small-sized particles that cause plasma and perforation damage, which could render the satellite inoperable. Catastrophic destruction of the vehicle is not likely.
|
Fig. 1.14. Impact probabilities at Leonid peak. |
