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WGN, the Journal of the IMO XX:X (200X)

1

General Relativistic Precession in Meteoroid Orbits
A. Sekhar1
,2 1

There are only very few past works related to the application of general relativity in the orbital evolution of small solar system bodies. Previous calculations have shown that mean motion resonances due to Jupiter and Saturn can enable meteoroids to stay resonant for the order of few thousand years. Now we study the general relativistic precession for such long term evolution of meteoroid particles and its subsequent effects in nodal displacements. Our calculations show that although the Newtonian model works very well for almost all practical purposes for the well known showers during present epochs, there could be some exceptional combinations of orbital elements for which the general relativistic precession in argument of pericentre and its influence on nodal distances could become significantly pronounced and decisive for predicting any earth-meteor intersection. In this work we present some calculations proving these aspects in the context of well known and active meteor showers namely Geminids, Orionids and Leonids. A similar analysis is done on low perihelion distance and low semi-ma jor axis meteoroid streams taken from the list of established showers at the IAU-Meteor Data Center. Submitted on 2013 May 17; Revised 2013 Oct 3;

1

Intro duction

One of the greatest triumphs of general relativity (GR) was the prediction (Einstein 1915) and subsequent confirmation of the precession of perihelion of Mercury. Ever since this important discovery, only very few works (Fox, Williams & Hughes 1982, Sitarski 1992, ShahidSaless & Yeomans 1994, Venturini & Gallardo 2009) have undertaken applications of general relativity in the long term orbital evolution of small solar system bodies. Previous calculations (Rendtel 2007, Sato & Watanabe 2007, Sekhar & Asher 2013a, 2013b) have shown that the resonant structures (due to both Jupiter & Saturn) in meteoroid streams can retain their compact structures for the order of few thousand years. During that time frame, changes in a & e are quite small for the purpose of study presented here. There are various previous works (Yeomans 1981, Asher & Clube 1993, Jenniskens et al. 1998, Asher, Bailey & Emel'yanenko 1999, McNaught & Asher 1999, Brown 2001, Ryabova 2003, Lyytinen & van Flandern 2004, Rudawska, Jopek & Dybczynki 2005, Watanabe, Sato & Kasuga 2005, Wiegert & Brown 2005, Vaubaillon, Lamy & Jorda 2006, Jenniskens et al. 2007, Maslov 2007, Rendtel 2007, Sato & Watanabe 2007, Christou, Vaubaillon & Withers 2008, So ja et al. 2011, Sekhar & Asher 2013a, 2013b) which focus on dust trails evolving for hundreds to many thousands of years. It would be worthwhile to look at the effects of general relativistic precession for such long term evolution of meteoroid orbits and check whether such effects are important in the long term prediction of meteor outbursts or storms. Relativistic effects would get more pronounced when a body moves with high velocities. Hence low perihelion distance (q ) would lead (due to Kepler's second law) to greater precession per revolution. Since this precession occurs during every perihelion passage, a larger number of revolutions means this effect accumulates very
1 Armagh Observatory, College Hill, Armagh BT61 9DG, United Kingdom 2

efficiently over a long period of time. In short, a body with small q and small a will have maximum contribution due to relativistic precession. Another important effect when a body comes very close to a massive rotating body is the Lense-Thirring effect which is manifested due to the dragging of space-time by a rotating body (Iorio 2005). It is not included in our calculations in this work mainly because it is typically four orders of magnitude smaller (Iorio 2005) than the effect discussed here.

2

Drift in argument of pericentre due to GR and its subsequent effect on no dal distances

Change in the argument of pericentre ( ) of an orbit is given by (page 197, Weinberg 1972): = 6 GM a(1 - e2 ) (1)

where a and e are the semi-ma jor axis and eccentricity of the orbit respectively. Equation 1 gives the result in radians/revolution. The same expression can be applied to any cometary/meteoroid orbit in the solar system (Fox, Williams & Hughes 1982, Shahid-Saless & Yeomans 1994). ra = rd = a(1 - e2 ) (1 + e cos ) a(1 - e2 ) (1 - e cos ) (2) (3)

Queen's University of Belfast, BT7 1NN , United Kingdom

University Road,

Belfast

Email: asw@arm.ac.uk , asekhar01@qub.ac.uk

Equation 2 gives the expression for the heliocentric distance of ascending node (ra ) for Orionids. Equation 3 gives the expression for the heliocentric distance of descending node (rd ) for Leonids and Geminids. These two quantities are critical for any meteor shower prediction calculations because the heliocentric distances of ascending or descending node should be close to Earth's orbit in order to produce any meteor activity. Hence significant changes in these parameters can directly decide the outcome of shower prediction models. The relationship between the change in nodal distances (r) with respect to the change in argument of


2

WGN, the Journal of the IMO XX:X (200X)

Table 1 ­ and r due to general relativistic effects for different parent b odies and meteoroid streams in 1000 years.

Body/Meteoroid Stream Icarus Phaethon Geminids Halley Orionids Tempel-Tuttle Leonids

q (AU) 0.187 0.140 0.141 0.575 0.578 0.977 0.984

a (AU) 1.078 1.271 1.372 17.871 18.000 10.337 10.300

e 0 0 0 0 0 0 0 .8 .8 .8 .9 .9 .9 .9 2 9 9 6 6 0 0 7 0 0 8 8 6 4

(Degrees) 31.348 322.148 324.420 112.279 81.500 172.499 172.400

(в10

-2

Degrees) 2.8 2.7 2.3 0.013 0.012 0.017 0.017

(в10

-4

r AU) 8.3 8.0 7.7 0.054 0.018 0.0020 0.0019

Table 2 ­ and r for different low q ( 0.15 AU) and low a ( 1.5 AU) meteoroid streams (taken from the list of established meteor showers in IAU-MDC) due to general relativistic precession in 1000 years.

IAU Code 004 164 390 165 152 171 GEM NZC THA SZC NOC ARI

Meteoroid Stream Geminids Northern June Aquilids November Aurigids Southern June Aquilids North. Daytime Cetids Daytime Arietids

q (AU) 0.141 0.114 0.116 0.110 0.108 0.085

a (AU) 1.372 1.348 1.130 1.150 0.967 1.376

(Degrees) 324.420 329.500 330.070 152.000 25.600 25.900

(в10

-2

Degrees) 2.3 3.1 4.0 4.1 5.4 4.0

(в10

-3

r AU) 0.8 1.3 1.4 1.6 1.9 2.0

pericentre (d ) could be computed by differentiating in this list are not confirmed, it is still worthwhile to calculate and compare the GR precession and the subequation 2 and 3. sequent nodal displacement in these streams. We find ae(1 - e2 ) sin d that the Northern Daytime Cetids have the highest (4) dra = rate of GR precession in ( 5.4 в 10-2 degrees in (1 + e cos )2 1 kyr). Low values of q and a make this stream apt for -ae(1 - e2 ) sin d efficient accumulation of the GR effect over many revdrd = (5) olutions. However the maximum r ( 2 в 10-3 AU (1 - e cos )2 in 1 kyr) is exhibited by Daytime Arietids due to low q The values of and r given in Table 1 and 2 and low a compounded by the favourable value in . are calculated using the equations 1, 4 and 5. The orbital elements a,e and of 1P/Halley (JD 2456400.5), 55P/Tempel-Tuttle (JD 2450880.5), 3200 Phaethon (JD 3 Values of argument of p ericentre for 2456400.5) and 1566 Icarus (JD 2456400.5) are taken for maximum change in no dal distances epochs (mentioned in brackets) from IAU-Minor Planet Center. Orbital parameters for various meteoroid streams Equations 4 and 5 show that r at any instant would are substituted from IAU-Meteor Data Center. It can depend on for a constant value of . Numerical solutions were done (see figures 1,2 and be clearly seen that in Geminids is about 100 times that of Leonids and Orionids for an orbital evolution 3) to compute the limiting values of . Figure 1 and of 1000 years. Subsequently our calculations show that 3 shows that r in Geminids and Leonids has extreme r due to in Geminids can be around 1000 times of values when 16 and 343 . Figure 2 indicates that that in Leonids. Overall the substantial effect of GR in r in Orionids has peak values when 171 and low q showers compared to other showers can be under- 188 . Understanding the maximum change in nodal stood from this analysis. Calculations for Icarus were distances is crucial in meteor forecast models. done because it is a well known low q body and has the Please note Y-axis scale for figure 1 representing highest precession rate due to GR among small solar Geminids. It clearly shows how substantial the error (of system bodies. Hence it is a good example to compare the order of 10-3 AU) in nodal distances (if GR effects with other parent bodies. Although the orbital elements are not included) could be, when the particles have of meteoroid streams are slightly different from those of 343 during its past or future. In such cases, general the corresponding parent bodies, the changes in and relativistic precession could actually play as decisive a r are practically small in terms of order of magnitude factor in the intersection or miss of a concentrated dust (as shown in Table 1). trail with Earth (diameter 10-4 AU) when long term Table 2 shows the list of established meteor showers predictions (of the order of kyr) are involved. During which have low q ( 0.15 AU) and low a ( 1.5 AU). All present times, Geminids have 324 (IAU-Meteor the orbital elements are taken from IAU-Meteor Data Data Center) which is not too far from producing an Center. Although the parent bodies of most showers error of the order of 10-3 AU when GR effects are ig-


WGN, the Journal of the IMO XX:X (200X)

3

1.5 1.0

dr

d

vs in Geminids AU)

1.0

dr

d

vs in Leonids

AU)

0.5

0.5 0.0 -0.5

10-3

10-4

0.0

(x

d

dr

dr
90 180 270 36 0

d
-0.5 -1.0 0

-1.0 -1.5 0

(x



(Degrees)

90



(Degrees)

180

270

3 60

Figure 1 ­ Change in heliocentric distance of descending node in Geminids for different values of argument of p ericentre for a constant =2.3 в 10-2 degrees/kyr

Figure 3 ­ Change in heliocentric distance of descending node in Leonids for all p ossible values of argument of p ericentre for a constant =1.7 в 10-4 degrees/kyr

2.0 1.5

dr

a

vs in Orionids

d2 ra =0 d 2

(8)

AU)

1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 0 90

d2 rd =0 (9) d 2 Substituting the expressions in equation 6 and 7 into equations 8 and 9 respectively and further simplification of above expressions yield: e cos2 - cos - 2e = 0 e cos2 + cos - 2e = 0 (10) (11)

dr

a

(x

10-4



(Degrees)

180

270

36 0

Figure 2 ­ Change in heliocentric distance of ascending node in Orionids for various values of argument of p ericentre for a constant =1.2 в 10-4 degrees/kyr

Solving the simple quadratic equations 10 and 11 give two roots each, of which only one corresponds to the real case: = cos
-1

[(1 -

(1 + 8e2 ))/2e]

(12)

nored. All other established meteoroid streams in Table 2 have r 10-3 AU. Previous calculations (Asher, Bailey & Emel'yanenko 1999, McNaught & Asher 1999) have shown that the Leonid meteor storms in the past were caused by dust trails with widths of the order of 10-3 AU. One could find these limiting values in analytically as well, for clarity and rigour. The simple analytical approach (using standard techniques in calculus) is described below.

Equation 12 shows the real root which corresponds to specific values of leading to maximum dra . The extreme values for dra in Orionids occur when 171 and 188 . = cos
-1

[(-1 +

(1 + 8e2 ))/2e]

(13)

Equation 13 gives the real root for specific cases of which can produce extreme values of drd . The maximum values for drd in Leonids and Geminids appear when 16 and 343 . The analytical treatment perfectly matches with the numerical solutions shown in figures 1,2 and 3. Hence it is a parallel verification of ae(1 - e2 )[(1 + e cos ) cos + 2e sin2 ] d2 ra = (6) the whole analysis. d 2 (1 + e cos )3

4
-ae(1 - e )[(1 - e cos ) cos - 2e sin ] d rd = d 2 (1 - e cos )3 (7) Equations 6 and 7 gives the derivative of equations 4 and 5 respectively. In order to find the value of corresponding to the extreme values of ra and rd , equations 6 and 7 can be equated to zero.
2 2 2

Conclusion

In this work we find that, for the well known showers during present epochs, the drifts in and subsequent changes in nodal distances are quite small compared to other errors and effects in meteor shower predictions on a time scale of 1000 yr. Hence a simple Newtonian model for the precise prediction of meteor-earth intersections does apply successfully for most of the cases.


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WGN, the Journal of the IMO XX:X (200X)

However it is important to note that precession due to GR is independent of the size of the particle unlike radiation pressure, Poynting-Robertson effect, Yarkovsky effect etc. Furthermore it would accumulate over time and would not get nullified or corrected directly by other effects. It is evident that evolution of small meteoroid particles (with diameters 1 mm) would be dominated by various radiative forces. This would in turn mean that only the large particles accumulate the GR precession effectively over such long time scales. For example in the well known case of Geminids, r is about 1000 times that of present day Leonids because of larger from relativistic precession and initial favouring the near maxima of r. It is found that the low q shower Northern Daytime Cetids has the highest rate of GR precession in ( 5.4 в 10-2 degrees in 1 kyr) out of all the established meteoroid streams so far. The maximum r ( 2 в 10-3 AU in 1 kyr) is seen in Daytime Arietids due to its low q and low a coupled with the favourable value in . Changes in r in this range can be crucial for meteor outburst/storm forecast models. This proves that there could be interesting exceptions (regarding accuracy of Newtonian model) for some particular combinations of q , e & of the meteoroid streams where GR effects have to be taken into account for accurate meteor shower forecasts.

9. Jenniskens P., Lyytinen E., Nissinen M., Yrjola I., Vaubaillon J., 2007, WGN (J. IMO), 35, 125. 10. Lyytinen E., van Flandern T. 2004, WGN (J.IMO), 32, 51. 11. Maslov M. 2007, WGN (J.IMO), 35, 5. 12. McNaught R. H., Asher D. J., 1999, WGN (J. IMO), 27, 85. 13. Rendtel J. 2007, WGN (J.IMO), 35, 41. 14. Rudawska R., Jopek T.J., Dybczynski P.A. 2005, Earth Moon Plan., 97, 295. 15. Ryabova G. O., 2003, MNRAS, 341, 739. 16. Sato M., Watanabe J., 2007, PASJ, 59, L21. 17. Sekhar A., Asher D.J. 2013, Meteorit. Planet. Sci., doi: 10.1111/maps.12117, in press. 18. Sekhar A., Asher D.J. 2013, MNRAS Letters, 433, L8 4 . 19. Sitarski G. 1992, AJ, 104, 1226S. 20. Shahid-Saless B., Yeomans D.K. 1994, AJ, 107, 1885S. 21. So ja R. H., Baggaley W. J., Brown P., Hamilton D. P., 2011, MNRAS, 414, 1059. 22. Vaubaillon J., Lamy, P., Jorda, L., 2006, MNRAS, 370, 1841. 23. Venturini J., Gallardo T. 2009, Proc. IAU Symposium No.263, 106. 24. Watanabe J., Sato M., Kasuga T. 2005, PASJ, 57, L4 5 . 25. Weinberg S. 1972, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, Wiley, New York. 26. Wiegert P., Brown P. 2005, Icarus, 179, 139.

Acknowledgments
Author thanks the anonymous reviewer for the helpful comments. I am grateful to my research supervisor Dr D J Asher for all his suggestions to improve the manuscript. Research at Armagh Observatory is funded by the Department of Culture, Arts and Leisure of Northern Ireland.

References
1. Asher D. J., Clube S. V. M., 1993, QJRAS, 34, 481. 2. Asher D. J., Bailey M. E., Emel'yanenko V. V., 1999, MNRAS, 304, L53. 3. Brown P., 2001, PhD thesis, Univ. Western Ontario. 4. Christou A. A., Vaubaillon J., Withers P., 2008, Earth Moon Plan., 102, 125. 5. Einstein A. 1915, Preussische Akademie der Wissenschaften, Sitzungsberichte, 831. 6. Fox K., Williams I.P., Hughes D.W. 1982, MNRAS, 199, 313. 7. Iorio L. 2005, A&A, 431, 385. 8. Jenniskens, P., Betlem, H., de Lignie, M.,Ter Kuile, C., van Vliet, M. C. A., van `t Leven, J., Koop, M., Morales, E., Rice, T. 1998, MNRAS, 301, 941.

27. Yeomans D. K., 1981, Icarus, 47, 492.