Meteor orbits and impactsDavid Garfinkle and Alberto G. Rojo Citation: American Journal of Physics 82, 1130 (2014); doi: 10.1119/1.4896575 View online: http://dx.doi.org/10.1119/1.4896575 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/82/12?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Introducing the Moon's Orbital Eccentricity Phys. Teach. 52, 460 (2014); 10.1119/1.4897579 Gravitational orbits in one dimension Am. J. Phys. 74, 1115 (2006); 10.1119/1.2348890 Catastrophic bolide impacts on the Earth: Some estimates Am. J. Phys. 74, 789 (2006); 10.1119/1.2205880 Of Orbits, Conics, and Grammar Phys. Teach. 43, 84 (2005); 10.1119/1.1855742 Dynamics of meteor impacts Chaos 14, S4 (2004); 10.1063/1.1821711

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Meteor orbits and impacts

David Garfinklea) and Alberto G. Rojob)

Department of Physics, Oakland University, Rochester, Michigan 48309

(Received 18 June 2013; accepted 16 September 2014)

We study the motion of meteors and calculate the speed at which they hit the Earth. The calculation

is done in a rotating system where the Earth and the Sun are at rest. Our treatment is a pedagogical

account of some well established celestial mechanics results, suited for presentation in an

intermediate mechanics course. The impact speed of meteors in several regularly occurring meteor

showers agree very well with our calculation. VC 2014 American Association of Physics Teachers.

[http://dx.doi.org/10.1119/1.4896575]

“Like a comet pulled from orbit as it passes a sun.”Stephen L. Schwartz, Wicked.1

I. INTRODUCTION

Meteors impact the Earth with velocities that range from16 km/s to 72 km/s. Why? In this paper, we present an an-swer to this question using a self contained explanation thatcan be of pedagogical use in intermediate mechanicscourses. Finding the impact velocity seems at first a compli-cated problem, since the meteor is subject to the gravitationalinfluence of both the Earth and the Sun. However, since theEarth is (to a very good approximation) in a circular orbit,there is a constant of motion for the meteor: the energy inthe rotating frame. The presence of this constant of motionmakes the calculation of impact speeds appropriate in an in-termediate level mechanics course. We compare our calcula-tions to the impact speeds of meteors in several regularlyoccurring meteor showers and demonstrate excellent agree-ment with the observed impact speeds. Our result can also beused to calculate the escape velocity from the solar system, aproblem that was treated using a different method in thisjournal.2–4

II. METEOR IMPACT

Consider the motion of a meteor of mass m in a frame thatrotates at an angular velocity x, the rotational velocity of theEarth around the Sun given by

x ¼ffiffiffiffiffiffiffiffiffiffiMSG

r3O

s; (1)

where MS is the Sun’s mass and rO is the radius of theEarth’s orbit around the Sun. In the inertial frame inwhich the Earth is moving, the meteor’s motion takesplace in the presence of a time-dependent potential, andthe energy of the meteor (kinetic plus potential) is not aconstant of motion. In the rotating frame, the Earthand the Sun (neglecting the rotation of the Earth aroundits axis) are at rest, and the energy is a constant ofmotion. The effect of the rotation manifests itselfthrough the addition of the centrifugal potential given byV¼�mx2(x2þ y2)/2, where we assume that the angularvelocity of the Earth’s orbit is along the z-axis, and thevariables x, y, and z refer to the meteor’s coordinates inthe rotating frame.

If we call E the energy per unit mass of the meteor in therotating frame, we have

E ¼ v2r

2� GME

r2

� GMS

r1

� x2

2x2 þ y2� �

; (2)

where r1 is the meteor-Sun distance, r2 the meteor-Earth dis-tance, and ME the mass of the Earth. Since we are in a framethat rotates with the Earth, ~vr is the relative velocity of themeteor with respect to the Earth. When the meteor is veryfar from the Earth, one can neglect the gravitational influ-ence of the Earth and the meteor’s motion is therefore wellapproximated by the elliptical orbit that it would have if onlythe Sun acted on it. As the meteor approaches the Earth, theinfluence of the gravitational potential of the Earth becomesnoticeable, and the orbit will deviate from the ellipse, but Ewill remain constant.

We are interested in the impact velocity vI when themeteor is at a distance r2 ’ RE; r1 ’ rO; x2 þ y2 ¼ r2

O. Here,RE is the radius of the Earth. Expressing E in terms of thesequantities, we obtain

E ¼ v2I

2� GME

RE� GMS

rO� x2

2r2

O ¼v2

I

2� GME

RE� 3

2

GMS

rO;

(3)

where we have used Eq. (1) for x. For a point in the orbit farenough from the Earth, we can neglect its gravitational influ-ence. If the velocity and position of the meteor are ~vf and~r f ¼ ðxf ; yf ; zfÞ, we get

E ¼ v2f

2� GMS

rf

� x2

2x2

f þ y2f

� �(4)

from Eq. (2). Now, we can express~vf in terms of the corre-sponding inertial frame velocity ~va and position ~ra

¼ ðxa; ya; zaÞ at the distant point as

v2f ¼ ð~va � ~x �~raÞ2

¼ v2a � 2~va � ð~x �~raÞ þ x2ðx2

a þ y2aÞ: (5)

Substituting Eq. (5) in Eq. (4), and usingrf ¼ ra; x2

a þ y2a ¼ x2

f þ y2f , and~va � ð~x �~raÞ ¼ ~x � ð~ra �~vaÞ

we obtain

E ¼ v2a

2� GMS

ra� ~x � ~ra �~vað Þ ¼ e� x‘ cos h; (6)

where e is the energy per unit mass of the meteor’s orbit, ‘its angular momentum per unit mass, and h the inclination ofthe orbit—the angle between the axes of rotation of theEarth’s and the meteor’s orbits around the Sun.

1130 Am. J. Phys. 82 (12), December 2014 http://aapt.org/ajp VC 2014 American Association of Physics Teachers 1130

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We make two notes here. First, that the individual (x, y)components of the position vector in the two frames are dif-ferent, but the magnitudes are the same. And second, in thecelestial mechanics literature, the symbol i is used for the in-clination; we find h to be in better accordance with the inter-mediate mechanics nomenclature. The quantity e� x‘ cos hin Eq. (6) appears in the celestial mechanics literature as theJacobi constant6 or Tisserand parameter.7 The orbit of a ce-lestial object of negligible mass can change during its pas-sage close to a planet in a circular orbit. During that passagethe energy e, the angular momentum ‘, and the inclination h(measured in the inertial frame) will change to e0; ‘0 and h0

such that e� x‘ cos h ¼ e0 � x‘0 cos h0. Equating the expres-sions for E from Eqs. (3), Eq. (5) we obtain

v2I ¼ 2

GME

REþ 3

GMS

rOþ 2 e� x‘ cos hð Þ: (7)

To compare our expression with observational data, we needto express e and ‘ for the meteor in terms of the orbit’s geo-metric parameters. The relevant expression is known in thecelestial mechanics literature as the vis viva equation.5 Wepresent a simple derivation for completeness.

For an elliptical orbit, the motion takes place between twoextremal radii rmin and rmax, the turning points of the orbit.Equating the energies at these two points and calling va andvb, the meteor velocities at r¼ rmin and r¼ rmax, respec-tively, we have

e ¼ �GMS

rmin

þ v2a

2¼ �GMS

rmax

þ v2b

2: (8)

Also, since~r �~v vanishes at the turning points, the magnitudeof the angular momentum is simply

‘ ¼ varmin ¼ vbrmax: (9)

Using Eqs. (8) and (9), with a little algebra, we obtain

e ¼ � GMS

rmin þ rmax

(10)

and

‘ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2GMSrminrmax

rmin þ rmax

r: (11)

One interesting aspect of the above equations is that theenergy depends on the semi-major axis a¼ (rminþ rmax)/2 ofthe orbit but not on the orbit’s eccentricity �, defined asrmin¼ a(1 � �) and rmax¼ a(1þ �).

Combining Eqs. (10) and (11) with Eq. (7) and using Eq.(1) for x, we obtain

vI ¼�

2GME

REþ GMS

rO

�3�

�2rO

rmin þ rmax

þ 2 cos h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2rminrmax

rO rmin þ rmaxð Þ

s �1=2

; (12)

or, expressed in terms of the semi-major axis A in astronomi-cal units (A¼ a/rO),

vI¼2GME

REþGMS

rO3� 1

Aþ2cosh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 1��2ð Þ

p� �� ( )1=2

:

(13)

It is interesting that the term in square brackets in our Eq.

(13) appears in €Opik’s book on interplanetary encounters8

as an “imaginary encounter or collision velocity (U), (incase the encounter takes place), when there is no significantgravitational interaction between the encountering par-ticles, one in an arbitrary orbit, the other on a ‘standard’ cir-cular orbit, both ruled exclusively by the gravitational fieldof the Sun.” We can write our impact velocity in terms of U

and another ideal velocity ve ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2GME=RE

p, the escape ve-

locity of the particle from the gravitational field of theEarth; the result is

vI ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2

e þ U2

q: (14)

Noting that G¼ 6.67� 10�11 N m2/kg2, MS¼ 1.99� 1030 kg,ME¼ 5.97� 1024 kg, rO¼ 1.50� 1011m, and RE¼ 6.38� 106 m, it then follows thatffiffiffiffiffiffiffiffiffiffiffiffiffi

2GME

RE

r¼ 11:2 km=s (15)

and ffiffiffiffiffiffiffiffiffiffiGMS

rO

s¼ 29:7 km=s: (16)

Therefore, Eq. (12) can be written as

vI ¼ 29:7 km=s 3:14� 1

Aþ 2 cosh

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 1� �2ð Þ

p� �� 1=2

:

(17)

Note that Eq. (12) simplifies if the meteor comes from suffi-ciently far out in the solar system so that rmax � rO. In thiscase, we find

vI ¼ 29:7 km=s 3:14� 2 cos h

ffiffiffiffiffiffiffiffiffiffi2rmin

rO

s8<:

9=;

1=2

: (18)

Thus, for meteors described by Eq. (18), it follows that themaximum impact velocity occurs for rmin¼ rO and h¼pand is

vImax ¼ 72:6 km=s; (19)

while the minimum impact velocity occurs for rmin¼ rO andh¼ 0 and is given by

vImin ¼ 16:6 km=s: (20)

This last result immediately gives the solution of a seem-ingly unrelated problem: the escape velocity of the solarsystem. A spaceship that starts from the surface of theEarth and barely escapes the solar system follows a trajec-tory that is the time reverse of the trajectory of one of our

1131 Am. J. Phys., Vol. 82, No. 12, December 2014 D. Garfinkle and A. G. Rojo 1131

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meteors, in the limit as rmax !1. The speed of the space-ship on launch is thus given by Eq. (18). The minimumsuch launch speed—the escape velocity of the solar sys-tem—is therefore given by Eq. (20). The solar systemescape velocity was treated by Hendel,2 who divided themotion into an initial time period where the influence of theSun can be neglected and a later time period where theinfluence of the Earth can be neglected. The result of Ref. 2agrees with our Eq. (20).

The analysis of the maximum and minimum impact veloc-ity in the more general case of meteors described by Eq. (13)is a little more complicated and goes as follows. First con-sider the minimum impact velocity. It follows from Eq. (14)that vI� ve with equality if U¼ 0. But it is clear intuitivelythat we can make U¼ 0, and thus vI¼ ve, simply by choosingthe orbit of the meteor to be the same as that of the Earth.This intuition is verified by using Eq. (13) and choosingA¼ 1, �¼ 0, and h¼ 0. We thus find that the minimum valueof the impact velocity is

vImin ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi2GME

RE

r¼ 11:2 km=s; (21)

which is equal to the escape velocity from the Earth. To findthe maximum impact velocity, first note that this requiresh¼p in Eq. (12). Then, differentiating the square of Eq. (12)with respect to rmax we find

@

@rmax

v2I

� �¼ 2GMS

rminþ rmaxð Þ21þ rmin

rO

� �3=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirminþ rmax

2rmax

r" #:

(22)However, the right hand side of Eq. (22) is positive. Thismeans that we can always increase the impact velocity byincreasing rmax. In physical terms, the maximum impact ve-locity occurs for meteors that come from far away. But theseare precisely the meteors that are well described by Eq. (18).Therefore, even in the general case of meteors described byEq. (12), the maximum impact velocity is still given by Eq.(19) (this, of course, for meteors that are bound to the Sun inelliptical orbits).

III. METEOR SHOWERS AND COMET ORBITS

To apply Eqs. (17) or (18) to predict the impact velocityof a meteor, one would need to know the orbital parametersof the meteor. However, most meteors are sufficiently smallthat they are not noticed (much less recorded and tracked)before they hit the Earth. Nonetheless, many regularly occur-ring meteor showers are associated with comets and sharethe orbital parameters of the comet they are associated with.It is therefore possible to use the formulas of the previoussection to predict the impact speed of the meteors in theseshowers. The reason for the association is that the meteorsare debris that were once part of the comet. It is for this rea-son that the showers are regularly occurring: small relativevelocities between the meteors and the associated cometspreads the meteors out over the orbit of the comet. Themeteor shower occurs when the Earth crosses the orbit of thecomet. In Table 1 and in Fig. 1, we show a comparison, withvery good agreement, between our calculation and the mostfrequent meteor showers.

ACKNOWLEDGMENTS

The authors would like to thank Fred Adams and DiegoJanches for helpful discussions. This work was partiallysupported by NSF Grant Nos. PHY-0855532 and PHY-1205202 to Oakland University.

a)Electronic mail: garfinkl@oakland.edub)Electronic mail: rojo@oakland.edu1From the song For Good, in Wicked: A new musical, music & lyricsby Stephen Schwartz (Hal Leonard Corporation, Wisconsin, 2003),

p. 105.2A. Z. Hendel, “Solar escape,” Am. J. Phys. 51, 746–748 (1983).

Table I. Comparison of observed impact velocities vI, obs for different meteor showers with our theoretical velocities vI computed from Eq. (12).9

Shower Comet of origin h rmin/rO rmax/rO vI (km/s) vI, obs (km/s)

Quadrantid 2003 EH1 70.98 1.19 5.06 41.06 41.36

Lyrid C/1861 G1 Thatcher 79.78 0.92 111.0 48.24 47.6

Eta Aquarid 1P/Halley 162.38 0.856 35.1 67.2 65.0

Delta Aquarid 96P Macholz (suspected)10 59.98 0.12 5.9 45.2 42.3

Perseid 109P/Swift-Tuttle 113.458 0.959 51.22 60.82 59.49

October Draconid 21P/Giacobini-Zinner 31.88 1.038 6.014 22.9 20.4

Orionid 1P/Halley 162.38 0.856 35.1 67.2 66.5

Andromedid 3D/Biela 13.28 0.88 6.19 19.7 17.2

Leonid 55P/Tempel-Tuttle 162.58 0.976 19.7 70.56 70.66

Geminid 3200 Phaethon 22.248 0.14 2.4 35.14 34.58

Fig. 1. A comparison of the data of Table 1 with those of Eq. (18), using the

meteor showers for which rmax> 6rO.

1132 Am. J. Phys., Vol. 82, No. 12, December 2014 D. Garfinkle and A. G. Rojo 1132

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3A. Z. Hendel and M. J. Longo, “Comparing solutions for the solar escape

problem,” Am. J. Phys. 56, 82–85 (1988).4V. J. Menon and D. C. Agrawal, “Solar escape revisited,” Am. J. Phys. 54,

752–753 (1986).5See, for example, C. D. Murray and S. F. Dermott, Solar System Dynamics(Cambridge U.P., Cambridge, 2006), p. 31.

6See Ref. 5, pp. 68–74.7D. Merritt, Dynamics and Evolution of Galactic Nuclei (Princeton U.P.,

Princeton, New Jersey, 2013), p. 477.

8Ernst Julius €Opik, Interplanetary Encounters: Close-range GravitationalInteractions (Elsevier, Amsterdam & NY, 1976) p. 23.

9The data for the parent body of the meteor shower, as well as the impact veloc-

ity, was taken from the Meteor Data Center <http://www.astro.amu.edu.pl>,

and the information for the inclination h, perihelion rmin, and aphelion rmax

was taken from <http://ssd.jpl.nasa.gov>.10Our calculation of vI assumes that the parent body is (the sus-

pected) 96P Macholz. Our calculation is consistent with the

suspicion.

1133 Am. J. Phys., Vol. 82, No. 12, December 2014 D. Garfinkle and A. G. Rojo 1133

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