Gravitational focusing and shielding of meteoroid streams

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1 Mon. Not. R. Astron. Soc. 375, (007) doi: /j x Gravitational focusing and shielding of meteoroid streams J. Jones 1 and L. M. G. Poole 1 Physics and Astronomy Department, University of Western Ontario, London, Ontario N6A 3K7, Canada Department of Physics and Electronics, Rhodes University, Grahamstown 6140, South Africa Accepted 006 November 7. Received 006 November 1; in original form 006 August 30 ABSTRACT The gravitational attraction of planets can cause significant perturbation of the trajectories of meteoroids. The resulting deflection can result in significant enhancement of the flux of meteoric particles in the neighbourhood of the planet. We give an analytical method for calculating the relative flux of stream meteoroids in the vicinity of a planet. We include the effect of shielding of certain regions that are not accessible to the meteoroids that have impacted the planet. We compare our results with those of Divine et al. and although we confirm the accuracy of their trajectory equations, the fluxes predicted by their calculations are not consistent with integrated fluxes over a planet predicted by the classic work of Öpik. Our method yields predictions for the integrated flux enhancement factor that are identical to Öpik s. We present the results of these calculations and find that in all cases, tail-like regions of enhanced meteoroid flux appear downstream of the planet, with very large enhancements possible in the case of the giant planets for all probable stream velocities. Key words: meteors, meteoroids. 1 INTRODUCTION The trajectories of meteoroids moving in the vicinity of a planet are deflected by its gravitational attraction that causes the trajectories to both converge and diverge depending on the geometry. Some regions are shielded from the stream meteoroids whose trajectories intersect the planet. This results in a shadow region downstream of the planet. The gravitational field does not focus the stream to a point, since the degree of deflection of the trajectory decreases with increasing impact parameter. In optical terms, the gravitational attraction results in severe spherical aberration. It is easy to see on qualitative grounds that the degree of focusing is speed-dependent and that the effect increases for slower meteoroids. Gravitational perturbations by a planet also affect the rate at which meteors impact on its own surface. For quantitative discussions, we define meteor flux at a given point in space as the rate at which particles pass through unit area (normal to the trajectory) in the vicinity of the point. Calculations of the degree of enhancement of meteor fluxes on a planet under the influence of its gravitational field have traditionally been dealt with using Öpik s (1951) formulation. Referring to Fig. 1, the capture cross-section of the planet in the absence of gravity is just its cross-sectional area A NG. When gravity is present, the capture crosssection is increased to A G. The factor, H F, by which the total number of meteoroids impacting the planet in a given time is increased is just the ratio of A G to A NG. Using the principles of conservation of jimeteor@uwo.ca (JJ); g.poole@ru.ac.za (LMGP) both mechanical energy and angular momentum, it is readily shown (see Appendix A) that ( H F = A G /A NG = 1 + V ) esc, (1) Vi where V esc is the escape speed at the planetary surface and V i is the initial speed of the meteoroid when it is far from the planet. Our results as presented in the following sections are likewise based on the above conservation laws. In practice, the trajectories of meteoroids are not perfectly collimated, but nevertheless can be thought of as a collection of perfectly collimated streamlets so that equation (1) applies just as well to meteoroids emanating from diffuse radiants as it does to those from point radiants. Apart from the slight approximation that the planet is spherical, Öpik s calculation is rigorous, but equation (1) is of limited usefulness since it refers specifically to enhancement of the integrated flux over a planet s surface whereas we require the flux at a specified point in space (the local flux). Öpik s simple formulation provides a benchmark for assessing the validity of more complicated derivations that yield expressions for the local flux in the vicinity of the planet, since integrated fluxes are easily computed if the local fluxes are known (see Section 3). Both the magnitude of the flux enhancement and its spatial extent are the quantities of crucial importance. Operators of commercial satellites need to know the locations of potentially hazardous regions of space. Satellites in Earth orbit are subject to impacts from both stream and sporadic meteoroids. Since meteor streams are highly collimated and even sporadic meteors originate predominantly from Downloaded from by guest on January 019 C 007 The Authors. Journal compilation C 007 RAS

2 96 J. Jones and L. M. G. Poole b max R P Figure 1. The meteoroids approach from the left in a collimated stream. R P is the radius of the planet and b max is the max value of the impact parameter of particles colliding with the planet. If gravity were not acting the capture cross-sectional area of the planet would be A NG = R P, with gravity present, it is effectively A G = b max. a few identifiable (albeit rather diffuse) sources, gravitational focusing may appreciably exacerbate the meteoroid impact hazard. Divine, Grün & Staubach (1993) and Staubach, Grün & Jehn (1997) have published analytic expressions for the focusing enhancement and shielding factors based on work by Divine (199). Unfortunately, Divine s original work, which involved a great deal of algebra, never appeared in the open literature and was applied only to a few representative cases for sporadic meteors. In this paper, we compare the predictions of the Divine method with those based on a different analytical approach and present results for a selection of applicable cases. THE METHOD We need to avoid complications arising from the different masses and sizes of the planets that are likely to be considered, so we have adopted a normalized set of units with the unit of speed equal to that of a body in a circular orbit just above the surface of the planet (V 0 ) and the unit of distance being the radius of the planet. In this system of units, a meteoroid trajectory is characterized by the parameter F given by F = v i r, () where v i is the normalized value of V i (i.e. V i /V 0 ), and r is the normalized distance of the observer from the planet. The focusing geometry is depicted in Fig.. If r is the vector from the planet centre to an observer, the initial and final directions of the meteoroids at the observer are defined by the angles ξ and φ, respectively, as shown in the diagram. From Fig. 3 it is readily seen that there are two trajectories by which a particle initially travelling in a given direction can arrive at a specified point P in space, i.e. two b db r dt r sin( ) Figure. Focusing geometry. Meteoroids are initially travelling horizontally from the left. ξ and φ are, respectively, the angles between r and the initial and final meteoroid velocities. The gravitational field of the planet causes the thickness d t of the initially parallel sheath-like bundle of trajectories to increase, while its cross-section decreases. Figure 3. Particles initially travelling in a given direction in the gravitational field of the planet can reach the point P while moving by either of two trajectories T S (shorter) and T L (longer). The corresponding values of φ are φ S and φ L. The path T L, because it undergoes the greater deflection, must make the closer approach to the planet. values of φ for a given ξ. We will refer to the shorter, more direct path as T S and to the longer path as T L. To estimate the intensification factor due to focusing for a particular trajectory, we treat the problem as one of fluid flow whereupon we note that the flux enhancement factor is just the ratio by which the cross-sectional area of the flow tube has decreased as it arrives at the specified location in space as shown in Fig.. It is convenient to consider that the meteoroids flow in a thin sheath, in which case the initial cross-sectional area of the sheath far from the planet is just πbdb, where b is the impact parameter given by b = r sin(φ) F + F. (3) The cross-sectional area A R of the sheath at some target location specified by (r, ξ) isgivenby A R = πr sin(ξ) sin(φ)dr (4) assuming constant ξ, leading to the following expression for the focusing factor: η F (ξ, F) = b db r sin(ξ) sin(φ) dr = 1 sin(ξ) F + db F dr. (5) The factor db/dr is expressible as an analytic function of ξ and F (Appendix B), leading to η F (ξ, F) = sin(φ) F sin(ξ) F sin(ξ) sin(φ) { [1 + cos(ξ)][cos(ξ) B] ± 1 } cos (ξ). (6) F + FB The quantity B(ξ, F) is defined in Appendix B, and the upper and lower signs, where they appear, refer to the trajectories T S and T L, respectively. Results obtained with equation (6) have been confirmed using form (5) with db/dr evaluated numerically. To make use of equations (5) or (6), we need to know both ξ and φ. It is a straightforward, if lengthy, matter to find an analytic expression for ξ(φ, F), but in practice it will be ξ and F that are known and we require analytic expressions for φ(ξ, F). Since we use a Monte Carlo approach involving large numbers of particles to estimate the flux distribution in the vicinity of the planet, speed of calculation is of the essence. Fortunately, Divine (199) has succeeded in finding a set of expressions for the trajectory solutions Downloaded from by guest on January 019 C 007 The Authors. Journal compilation C 007 RAS, MNRAS 375,

3 that does not involve much computational overhead and which we have verified by substitution in an independent inverse calculation to recover ξ(φ, F). Not only does this give us confidence in Divine s trajectory solutions, it also assures us that we are using a common geometry. Appendix C1 shows how Divine s trajectory solutions enable us to find sin (φ)(ξ, F), and Appendix C presents Divine s own expression for the focusing factor expressed in our normalized units, η FD. Some of the particles are on trajectories that intersect with the planet so that certain regions of space are shielded from them. We take this into account by assigning a shielding factor, η S as follows. The closest distance of approach to the centre of the planet (periapsis), q, isgivenby [ ] q = r 1 + F(F + ) sin (φ) 1 /F. (7) If q > 1 there is no shielding and η S = 1. If q 1, the particle is stopped as soon as it reaches the surface of the planet so that η S = 0, unless it reaches the target point P before periapsis. The position of a meteoroid relative to periapsis determines the sign of the radial component of its orbital velocity v r, given in Appendix C (equation C), which leads to the condition that η S = 0ifq < 1 and, for trajectory T S, cos(ξ) > 1/(F + 1), but otherwise η S = 1. Clearly, we also have η S = 0 for any point inside the planet. The combined effect of focusing and shielding is given by a composite weighting factor η F η S. In general, we need to specify a different η F and η S for each of the trajectories T S and T L,so that an overall weighting factor at the observer could be written as η = η FS η SS + η FL η SL. 3 COMPARISON OF FOCUSING ENHANCEMENT FACTORS Fig. 4 shows how the focusing enhancement factors at a point in space (the local enhancement factors) predicted by the present work and those generated by Divine s method compare for a perfectly collimated beam of meteoroids with identical speeds. The methods agree well for high values of F (with the best agreement being for very small values of ξ), but the discrepancies at low values of F show that at least one of the methods is in error. Figure 4. Comparison of focusing factors calculated using Divine s (199) method and the present work for ξ = 10. D and J& P refer to the methods of Divine and the present work, respectively, while the subscripts S and L refer to the shorter and longer trajectories, respectively. Both enhancement factors have the same asymptotic behaviour at large values of F. Gravitational focusing of meteoroid streams 97 ξ r θ θ max Figure 5. Focusing geometry for impacts on the planet surface. The meteor stream is initially perfectly collimated and approaches from the left as indicated by the arrows. The angle θ = ξ. We need a test to discriminate between the two. The Öpik method mentioned in Section 1 provides in an easily understood fashion the focusing factor, H F, for the integrated flux over the surface of the planet and Öpik s formula (equation 1) for H F can be written as ( H F = 1 + ) (8) vi in our normalized units. As discussed in Section 1, we can use the Öpik expression for the integrated flux enhancement factor to compare the two methods. We call this the Öpik test. For this we need to integrate the local flux enhancement factors over the surface of the planet. Fig. 5 shows the geometry for impacts with the planet by particles initially in a perfectly collimated stream. The impacts occur over slightly more than the forward hemisphere because of the gravitational deflection of the trajectories. With the angle θ defined as π ξ (see Fig. 5), we have 0 <θ<θ max, where θ max is obtained by stipulating that it corresponds to the periapsis distance, enabling us to find the eccentricity e of that grazing trajectory. It can then be shown that cos(θ max ) = 1/e = 1/(F + 1). (9) The enhancement factor, H F, for the integrated flux of impacting particles is given by H F = θmax 0 η FS (π θ, F) cos(φ) sin(θ) dθ (10) since ξ = π θ. We need to consider only the shorter trajectories here because in this case the meteoroids on the longer trajectories would have to pass through the planet to reach the impact point. The results are shown in Fig. 6. The present method gives excellent agreement with Öpik s, whereas Divine s formula yields appreciably lower integrated flux enhancement factors for v i < 4.0, which corresponds to about 3 km s 1 for the Earth. At v i = 1 and below, a region which becomes important for the giant planets, the discrepancy exceeds 50 per cent. Similar results were found when comparing our local values of the focusing enhancement factor with those obtained using the Divine method (Fig. 4; see also Figs 11 and 1). For normalized speeds below 10 ( 80 km s 1 for the Earth), Divine s procedure yields integrated flux enhancement factors that differ from Öpik s by more than 9 per cent. We do not know why Divine s formulation fails the Öpik test and regard the question as beyond the scope of this paper. We will therefore adopt the present method as a basis for the remaining discussions. Downloaded from by guest on January 019 C 007 The Authors. Journal compilation C 007 RAS, MNRAS 375,

4 98 J. Jones and L. M. G. Poole Figure 6. Comparison of H F versus v i obtained by various methods. Solid line equation (1) (Öpik s expression); Filled circles equation (10) with equation (6) (present work); dashed line equation (10) with equation (C6) (Divine s formulation). 4 THE FLUX DENSITY DISTRIBUTION IN THE VICINITY OF A PLANET In this section, we apply our method to estimate how gravitational focusing and shielding modifies the flux distribution in the planetary environment. We adopt a Monte Carlo approach involving a large number of test particles ( 10 4 ) with a specified scatter in the initial velocities of the approaching meteoroids, V i. The mean flux enhancement factor, η F j,atthejth point on a grid in the vicinity of the planet is given by η F j = 1 N [η SS η FS (ξ i, j, F i, j ) + η SL η FL (ξ i, j, F i, j )], (11) N i where N is the number of test particles, and ξ i,j is the angle between V i and the vector R j from the centre of the planet to the jth point in the sampling grid. F i,j is calculated using equation () with the appropriate values for v i and r. For present purposes, the abovementioned scatter in V i for meteors associated with a particular radiant region will be characterized by a scatter in v i in combination with w rad, the rms width of the angular dependence of radiant density about the mean radiant, assumed Gaussian. Fig. 7 illustrates the regions of enhancement and shielding found by applying equation (11) to the representative case of a diffuse radiant with scatter, w rad = 1 and with normalized initial speed v i = 4( 3 km s 1 for the Earth). Figs 8 1 show the variation of the enhancement factor along the central axis of Fig. 7 for various values of v i and radiant scatter w rad. Figs 10 onwards incorporate an rms spread of 10 per cent in v i, although this makes little difference to the form of the plots (compare Fig. 8 with Fig. 10). Fig. 11 represents flux enhancements that might be expected in the vicinity of Jupiter. Fig. 1 is a larger scale plot of the region of Fig. 11 near the surface of the planet. The degree of the gravitational focusing enhancement in the vicinity of a given planet depends on the degree of collimation of the stream particles, the appropriate V 0 (as defined in Section ) as well as relative speed of the approaching meteoroids which, in turn, depends on the orbit of the stream. To illustrate the effect, we have calculated the quantities shown in Table 1 for a meteoroid stream approaching each of the main planets with a relative speed equal to the heliocentric orbital speed, V orb, of the planet. The actual relative speed can be anything up to.4 V orb, corresponding to a particle in a parabolic heliocentric orbit involved in a head-on collision with the Earth. This, together with the fact that the radiant scatter may be less than 1, means that the degree of focusing can be much greater than that shown in Table 1. It is evident that while the focusing effect for the inner planets is not insignificant, for the outer giant planets it can be much larger. It should be noted, however, that these very large enhancements are confined to the region close to the central axis, the half-width of the tail at z = z max being less than 0.1 of a planetary radius. 5 CONCLUSIONS The results presented here show that significant meteor flux enhancements may be expected in the vicinity of planets that intercept well-collimated meteor streams, and can become very large for v i < 1. Although this regime is of little relevance to observers near the Earth or the other terrestrial planets, it becomes all-important for the outer giants and indicates a possible serious hazard to any space vehicle orbiting or visiting satellites near the surface of one of these planets. In the case of Jupiter, the two inner Galilean satellites and four others lie within the region z < 10, where large enhancements are possible. The enhancement factor due to gravitational focusing decreases rapidly with increasing radiant width (compare Figs 8 and 9), and in the case of Earth, approaches unity for the more diffuse streams or sporadic sources (w rad 0 ). Satellites in the Solar system are generally too small to cause a significant focusing effect on the parent planet. Interception by the Moon of the stream illustrated in Fig. 7 could, for example, result in an increase of no more than per cent in the flux incident on the Earth. Downloaded from by guest on January 019 Figure 7. The relative flux density in the vicinity of the planet for a radiant width (w rad )of1 and a fixed reduced speed (v i ) of 4. The light circle denotes the outline of the planet. Meteors are travelling from left to right, and the radiant and planet centres both lie on the central horizontal axis. The horizontal and vertical markers are separated by an interval of one planet diameter. C 007 The Authors. Journal compilation C 007 RAS, MNRAS 375,

5 Gravitational focusing of meteoroid streams 99 Figure 8. The relative flux density along the central axis of Fig. 7 for the same set of parameters (v i = 4, w rad = 1 ). z is the normalized distance from the centre of the planet. Also shown are the separate contributions to the flux density due to particles following the shorter ( ) and longer ( ) trajectories. Figure 9. As for Fig. 8, but w rad = 5. Figure 10. As for Fig. 8, but with a 10 per cent scatter in v i. This paper has dealt with relative changes in meteor fluxes in the vicinity of planets. An appraisal of absolute fluxes to be expected in these regions requires knowledge of the distribution and motion of meteoric material near the planetary orbits. Our method is readily adapted to handle this additional information through the introduction of another weighting factor (qv; Section ) which specifies the incident flux as a function of planetocentric direction. ACKNOWLEDGMENTS An early version of this project was funded in part by the NASA Space Environment and Effects Program. The authors wish to thank Figure 11. As for Fig. 10, but for v i = 0.3, characteristic of Jupiter. Also shown for comparison is the prediction of Divine s method (broken line). Figure 1. Detail of Fig. 11, near the surface of the planet. Table 1. Representative values of the maximum focusing enhancement factor, η max, for meteor streams with a spread of 1 in radiant directions and with a relative speed equal to the heliocentric orbital speed of the planet (V orb ). The column headed z max gives the approximate position of maximum enhancement. Planet V 0 (km s 1 ) V orb /V 0 η max z max Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Heather McNamara for helpful discussions and especially for providing a copy of the Divine (199) interoffice memo. One of us (JJ) wishes to thank Almighty God for the joy of exploring this small corner of His universe. REFERENCES Divine N., 199, JPL-Interoffice Memorandum, Divine N., Grün E., Staubach P., 1993, Proc. First European Conf. Space Debris, Modeling the Meteoroid Distributions in Interplanetary Space and Near-Earth, ESA SD-01. ESA Publications Division, Noordwijk, p. 45 Öpik E. J., 1951, Proc. R. Ir. Acad., 54A, 165 Staubach P., Grün E., Jehn R., 1997, Adv. Space Res., 19, 301 Downloaded from by guest on January 019 C 007 The Authors. Journal compilation C 007 RAS, MNRAS 375,

6 930 J. Jones and L. M. G. Poole APPENDIX A: ÖPIK S GLOBAL FOCUSING ENHANCEMENT FACTOR For a particle that just grazes the planet, the conservation of mechanical energy yields the speed, v p at its closest approach to the planet as V p = Vi + Vesc. (A1) Conservation of angular momentum requires (see Fig. 1) b max = V p R p /V i. (A) The global enhancement factor, H F = A G /A NG, is therefore given by H F = bmax/ R p = 1 + V / esc V i. (A3) It is important to note that H F is the enhancement factor of the total flux of meteoroids averaged over the entire surface of the planet rather than at any specific location. H F is global rather than local. APPENDIX B: LOCAL FOCUSING ENHANCEMENT FACTOR η F The derivative factor in equation (5) is db dr = sin(φ)(f + 1) F(F + ) + F(F + ) d[sin(φ)] df with sin (φ) given by equation (C5), from which we also obtain d[sin(φ)] df (B1) = 1 sin(φ) { [1 + cos(ξ)][cos(ξ) B] ± 1 } cos (ξ). (F + ) F(F + )B (B) B is obtained from equation (C3), and the upper and lower signs refer to the shorter and longer trajectories, respectively. Substitution of these equations into equation (5) yields η F explicitly in terms of ξ and F (equation 6). APPENDIX C: RELEVANT ASPECTS OF DIVINE S METHOD In this appendix, we have expressed Divine s equations in our normalized units. C1 Calculation of φ(ξ, F) based on Divine s trajectory solutions The speed, v(f), of the particles at the observer is given by F + v = v i F. (C1) After much algebra, Divine obtained the following expression for the magnitude of the radial component of the particle velocity at the observer, v r (ξ, F), [ ] 1 + cos(ξ) v r = v i B, (C) where we have followed Divine in defining the quantity B as [ 1 cos(ξ) B = 1 cos(ξ) + 4 ] (C3) 4 F and the upper and lower signs in equation (C) refer to the shorter and longer trajectories, respectively. (Note that the above definition of B differs from Divine s by a factor of V i.) The transverse component of the velocity, v t, at the observer is given by v t = v vr. The value of sin (φ) needed in equations (5) (7) is given by sin(φ) = v t v. (C4) (C5) C Divine s expression for the local focusing enhancement factor The focusing enhancement factor, η FD, using Divine s method, corresponding to our η F,is { [ η FD = 1 ± 1 cos(ξ) + F ] } 1. (C6) 4B The upper and lower signs are now reversed compared to equation (C), but again refer to the shorter and longer trajectories, respectively. This paper has been typeset from a TEX/LATEX file prepared by the author. Downloaded from by guest on January 019 C 007 The Authors. Journal compilation C 007 RAS, MNRAS 375,