Determination of meteoroid orbits from ground-based meteorobservations using numerical integration of equations of motion

Vasily Dmitriev1, Valery Lupovka1, Maria Gritsevich2,3 and Jürgen Oberst4,5

1Moscow State University of Geodesy and Cartography (MIIGAiK), Extraterrestrial Laboratory; 2Finnish Geodetic Institute, Department of Geodesy and Geodynamics & Finnish Fireball Network; 3Russian Academy of Science, Dorodnitsyn Computing Center, Department of Computational Physics; 4German Aerospace Center (DLR),

Institute of Planetary Research; 5Technische Universität Berlin, Institute for Geodesy and Geoinformation Science, Department of Planetary Geodesy

Method Description

We use strict transformations of coordinate and velocity vectors according to IAU International Earth Rotation and Reference Systems Service (IERS) [5] and backward numerical integration [6] of equations of motion. A similar approach was applied in [7] for the Chelyabinsk asteroid orbit reconstruction using the “mercury6” software [8]. In [9] authors have compared Ceplecha meteoroid orbit determination method with the results of numerical integration, which showed good agreement of both approaches. Specifically, the following transformations are in use. Transformation of velocity vector from topocentric to geocentric coordinate system:

where (Vn, Ve, Vu)T and (Vx, Vy, Vz )T are the topocentric and geocentric velocity vectors; R2, R3 and Q2 are appropriate rotation and mirror matrices, accordingly; B and L – geodetic latitude and longitude of the beginning point of the atmospheric trajectory. Diurnal aberration were taken into account as:

where ω is Earth rotation velocity, ae- equatorial Earth radius. Transformation of the beginning point coordinates and velocity vectors from the Earth-fixed geocentric coordinate system ITRF2000 to Geocentric Celestial Reference System (GCRS) realization ICRF2 (J2000) is conducted accordingly to IERS Conventions [5]:

, , R = PNПS,

where P is precession matrix, N – nutation matrix, П – polar motion matrix, and S – apparent Greenwich Sidereal Time matrix. Contributions of polar motion and high order nutation are negligible in comparison to observation errors, so these effects can be skipped in this case. The JPL ephemeris DE421 [10] is used for transformation of meteoroid position and velocity vectors from the geocentric to the heliocentric coordinate system. As a result, required initial conditions for numerical integration - meteoroid position and velocity vectors are obtained in the celestial geocentric coordinate system ICRF2 (J2000).Backward integration of equations of perturbed meteoroid motion:

was performed by an implicit single-sequence numerical method [6]. The equations of perturbed meteoroid motion include central body (Sun) attraction, perturbations from Earth gravity field, Moon, other planets, and atmospheric drag. For obtaining undistorted heliocentric orbit backward integration was performed until the meteoroid intersected with the Hill sphere (i.e. about 4 days before the actual event of a meteor).

It is known, that orbits of meteoroids colliding with the Earth are exposed to significant perturbations before encounter, primarily under the influence of gravity and atmospheric drag at the end of trajectory. Standard methods of preatmospheric meteoroid orbit computation [1], traditionally based on a set of static corrections applied to the observed velocity vector (see

e.g. [2, 3, 4]). In particular, the popular concept of so-called “zenith attraction” is used to correct the direction of the meteoroid trajectory and its apparent velocity in the Earth's gravity field. In this work, we carry out explicit trajectory integrations of the meteoroids, and we wish to investigate the magnitude of errors involved by choosing the mentioned simplifications.

References[1] Ceplecha Z. Geometric, dynamic, orbital and photometric data on meteoroids from photographic fireball networks. Astronomical Institutes of Czechoslovakia, Bulletin (ISSN 0004-6248), vol. 38, 1987, pp. 222-234.[2] Andreev G. The Influence of the Meteor Position on the Zenith Attraction. Proceedings of the International Meteor Conference, Violau, Germany, September 6-9, 1990, Eds.: Heinlein D.; Koschny D., International Meteor Organization, 1990, pp 25-27.[3] Langbroek M. A spreadsheet that calculates meteor orbits // WGN, Journal of the International Meteor Organization, vol. 32 (4), 2004, pp. 109-110.[4] Zoladek P. PyFN –multipurpose meteor software. Proceedings of the International Meteor Conference, Sibiu, Romania, 15-18 September, 2011 Eds.: Gyssens, M.; and Roggemans, P. International Meteor Organization, ISBN 2978-2-87355-023-3, 2011, pp 53- 55. [5] IERS Conventions (2010). IERS Technical Note No. 36, 2010, 160 p.[6] Plakhov Y.V., Mytsenko A.V., Shelpov V.A. Method for the numerical integration of equations of perturbed satellite motion in problems of space geodesy. Geodeziia i

Aerofotos'emka (ISSN 0536-101X), No. 4, 1989, pp. 61-67. In Russian.. [7] Zuluaga J., Ferrin I., Geens S. The orbit of the Chelyabinsk event impactor as reconstructed from amateur and public footage. arXiv:1303.1796. Submitted on March 7, 2013.[8] Chambers J. E. A hybrid symplectic integrator that permits close encounters between massive bodies. Monthly Notices of the Royal Astronomical Society, vol. 304,(4), 1999, pp. 793-799. [9] Clark D. L., P. A. Wiegert, A numerical comparison with the Ceplecha analytical meteoroid orbit determination method, Meteoritics & Planetary Science, vol. 46, no. 8, pp. 1217–1225,. 2011.[10] Folkner W., Williams J., Boggs D. The Planetary and Lunar Ephemeris DE 421. IPN Progress Report 42-178, August 15, 2009, 34 p.[11] Borovička J., Tóth J., Igaz A., Spurný P., Kalenda P., Haloda J., Svoreň J., Kornoš L., Silber E., Brown P., Husárik M.The Košice meteorite fall: Atmospheric trajectory, fragmentation, and orbit. Meteoritics & Planetary Science, vol. 48 (10), 2013, pp. 1757-1779.

Summary and Discussion:

Introduction

Table 1. The calculated pre-impact orbits of meteorites are compared with the results obtained by the other authors obtained by means of traditional approach.

Results

x nT

y e

z u

V V

V V

V V

M

cos sin

cos cos

0

x

y e

z

V B L

V a B L

V

in geoT

in geo

in geo

X X

Y Y

Z Z

Rin geo

Tin geo

in geo

Vx Vx

Vy Vy

Vz Vz

R

),(),(),(),,(3

trrtrrtrrtrSCrrr

GMr atmplanetsMoonnmnmEarth

Sun

Topocentric

radiant

Name Buzzard Coulee Grimsby Neuschwanstein Park Forest Puerto Lápice Sutter's Mill Chelyabinsk Košice

Epoch2008-11-21 00:30:00.0

2009-09-25 01:51:52.545

2000-05-06 11:02:58.41

2003-03-27 05:50:26.4

2007-05-10 17:57:30.0

2012-04-22 14:51:12

2013-02-1503:20:32.0

2010-02-28 22:24:47.0

B 52.996 43.534 47.3039 41.13 39.323 38.803 54.445 48.4667

L -109.848 -80.194 11.5524 -87.9 −2.939 -120.908 64.565 20.705H 86 100.5 84.95 82 24.93 90.2 97.1 68.3

Aapp 167.5 309.4 125 201° 95 272.5 283.2 72.666happ 66.7 55.2 49.75 61 43.4 26 18.3 -60.204

Vapp 18 20.91 20.95 19.5 15 28.6 19.16 15.0

Publishe

d Orbi

t

Source Milley E. et al., 2010

Brown P. , et al., 2011 Oberst J. , et al., 2004 Brown P., et al.

2004Trigo-Rodriguez J.

et al., 2009Jenniskens P. et al.,

2012 Popova et al., 2013 Borovicka J. et al., 2013

a 1.225 2.04 2.40 ± 0.02 2.53 2.09 2.59 1.76 2.71e 0.215 0.518 0.670± 0.002 0.680 0.54 0.824 0.581 0.647i 25.486 28.07 11.418±0.038 3.2 5.6 2.38 4.93 2.0°

Om 238.937° 182.956 16.826648 6.1156 49.61 32.77 326.422 240.072°w 212.019 159.865 241.208±0.068 237.5 ± 1.6 207 77.8 108.3 204.2°

This stud

y

Epoch2008-11-11 00:30:00.0

2009-09-15 01:51:52.545

2000-04-27 11:02:58.41

2003-03-17 05:50:26.400

2007-05-01 17:57:30.0

2012-04-12 14:51:12

2013-02-10 03:20:32.0

2010-02-24 22:24:47.0

a 1.241 2.025 2.403 2.551 2.045 2.371 1.761 2.7225e 0.2247 0.5159 0.6701 0.6826 0.5260 0.814 0.581 0.6486i 25.134 28.0846 11.515 3.040 5.673 2.673 4.992 2.057°

Om 238.953 181.985 16.838 6.156 49.646 32.771 326.454 340.146°w 210.968 159.136 241.205 237.576 207.995 75.304 108.81 204.074°M 337.689 4.222 348.975 18.572 351.051 10.963 17.419 355.1121°

Perturbations Δa, km Δe Δi° ΔΩ° Δω° ΔM°Earth gravity 3779 0.00021 0.00114 -0.000441 -0.01109 0.00675Moon gravity 4086 -0.00016 0.00112 -0.000009 -0.00794 -0.00215

Table 2. Influence of different perturbations on the resulting orbit. Integration was performed 4 days backward.

Perturbing acceleration and associated changes of orbital elements for the Kosice meteoroid before impact with the Earth are presented in the figures below. The orbit is calculated from the initial conditions derived from [11].

ToolsA software tool for determination of the orbit of meteoroids was development. This software has a graphics user-friendly interface and uses SPICE routines and kernels for coordinate transformation and computing ephemeris. In addition, it has a module for visualization of computation results. Screenshots of this software are presented below.

The accuracy of the proposed method does not propagate the observational uncertainties, while obtained results are in good agreement with the traditional orbit-determination method. Unlike the traditional method, the described approach allows to take into account the effects of meteoroid attraction by the Moon and planets, Earth's gravity field, as well as atmospheric drag. Increase in pre-atmospheric velocity and/or lowering beginning height of a meteor result in bigger differences between the proposed and traditional approaches. The proposed technique enables analysis of the meteoroid orbital motion before its collision with the Earth. This only requires further integration (back in time) of the same equations of motion. Portable software with GUI to determine a heliocentric orbit of a meteoroid and analyze it in time before the impact has been developed in accordance with the described here technique.

This work was carried out in MIIGAiK and supported by Russian Science Foundation, project  #14-22-00197

)()90( 321 RRQM