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Capture Resonance of the Asteroid 1685 Toro by the Earth by Venus, Earth, Mars, Jupiter, and Saturn Are Included in the Orbit Inte- Abstract

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Capture Resonance of the Asteroid 1685 Toro by the Earth by Venus, Earth, Mars, Jupiter, and Saturn Are Included in the Orbit Inte- Abstract The present orbital elements of Toro show that its period is almost exactly 1.6 years. Further, the relative posi- tions of Toro and the Earth exhibit a certain symmetry, including close ap- Reports proaches. This capture resonance, if we may call it so, has been investigated in more detail by integrating the per- turbed or-bit of Toro for 100 years backward and forward, that is, for a total of 200 years. The perturbations Capture Resonance of the Asteroid 1685 Toro by the Earth by Venus, Earth, Mars, Jupiter, and Saturn are included in the orbit inte- Abstract. The asteroid 1685 Toro has its perihelion inside the Earth's orbit gration according to Cowell's method. and a period which is 8/5 of that of the Earth. A calculation of Toro's orbit The integration step size is 5 days covering 200 years shows that this asteroid at present is captured in resonance per step. If there are close encounters with the Earth. The capture is due to the gravitational interaction at close en- between Toro and Venus, Earth, or counters between the bodies. Mars, the step size can be divided to be 2.5, 1.25, and so on, days per step .1 In the solar system there are many because, as found by Cohen and Hub- so that the accuracy is retained. resonance phenomena due to different bard (2), these two planets are in a The orbits of Toro and the terrestrial mechanisms. The motions of two plan- 3/2 resonance such that the distance planets are at present oriented as shown Downloaded from ets or satellites can couple together so- between them is always larger than 18 in Fig. 1 (projection on the ecliptic that the ratio of their orbital periods astronomical units (A.U.). This effect plane). The possible positions of the has the value of p/q where p and q implies that Neptune and Pluto may Earth and Venus when Toro is at peri- are small integers (1). This kind of have coexisted since the beginning of helion are marked. The orbit of Toro coupling also occurs between the spin the solar system. crosses the orbits of both Mars and the of one body and the orbital motion of This result leads to the question Earth; the orbit of the Earth is crossed http://science.sciencemag.org/ another. The most remarkable exam- whether stray bodies could also be at the longitudes that the Earth will have ple is the orbital coupling between stored in the inner part of the interplan- during the months of January and Au- Neptune and Pluto. Due to its large etary space by similar effects, namely, gust. eccentricity, the orbit of Pluto inter- by resonances with the Earth or Venus, In a coordinate system rotating with sects the orbit of Neptune. In spite of or both. In this report we discuss the the Earth, the orbit of Toro is as shown this the two bodies can never collide asteroid 1685 Toro. in Fig. 2. This particular loop is pro- on September 17, 2018 Fig. 1 (left). Projection of the orbits of the terrestrial planets and Toro onto the ecliptic plane. The positions of Venus and Earth when Toro passes its perihelion are marked. The data used are from the perihelion epoch of December 1967 and four following perihelions. Fig. 2 (right). Projection of Toro on the ecliptic plane in a coordinate system rotating with the Earth. The fact that Toro's period oscillates around 1.6 years is shown by superimposing an oscillation on the rotation of the coordinate system. This makes the Earth oscillate along a 160 arc with a period of 144 years. The arrow indicates the position of the Earth with respect to the orbital pattern of Toro in the year 2020. 906 SCIENCE, VOL. 176 1.602- position of the Earth has moved to the 70 midpoint, which is equidistant from the Phase at perihelion 1.601- points of the January and August en- 60 counters. In this region, the accelerat- w ing force cancels the decelerating force 50 uC Z:1.600- .2 and a zero slope appears. As soon as the midpoint is passed the relative (A 40 w 1.599- distances at the January encounters w - 0. - are smaller than those at the August 30 encounters, and as the accelerations 1.598- dominate the decelerations of Toro, an upward slope in the plot of the 20 period 1- 1.597 fi~~~~ results. 2.40 2.42 2.44 2.46 2.48 10- J.D. (10' days) The phase variations in Fig. 4 are 1860 1915 1970 2025 2080 based on the longitudinal distances Time (year) uE l l Ie l l l Il from Toro to the Earth and Venus 2.40 2.42 2.44 2.46 2.48 Fig. 3. Variation of Toro's when Toro is at The data J.D. (106 days) period. The perihelion. 1860 1915 2080 time is in Julian days and years. are reduced so 1970 2025 that the positions of the Time (year) planets always refer to the points around 130 longitude (October) in Fig. 1; that Fig. 4. The difference in longitude be- tween Toro, and the Earth at Venus, Downloaded from duced during 8 years around the year is, n X 720 has been added, where n perihelions 2020. The loop is not exactly closed; in- is 0, 1, 2, 3, cr 4. The equilibrium every eighth year. stead it can be said to oscillate around phase angle would be expected to be its center, the Sun, with an amplitude 36° (for zero inclination), and the One of the objects of studying orbits of about 80 and a period of 144 years. resonance is expected to disappear if like that of Toro in detail is that a res- In Fig. 2 this is best illustrated by the amplitude (relative to equilibrium) onance orbit such as this is one of the keeping the 8-year loop of Toro fixed of the libration exceeds 360 (that is, possible precapture orbits for the Moon http://science.sciencemag.org/ and allowing for oscillations of the if the phase angle passes 00 or 720). (3). Another is that the existence of Earth along a 160 arc. From Fig. 4 it is then concluded that orbits like Toro's may be important At each close encounter (every eighth the phase variafion relative to the for determining the lifetimes of the year) in the vicinity of its ascending Earth is compatible with a capture res- Apollo asteroids (4). node, Toro is ahead of the Earth. Thus, onance during the period studied. In summary, a calculation of the due to the gravitational atttraction by The configuration described would orbit of the asteroid 1685 Toro over the Earth, Toro is retarded, the angular be favorable for a stable resonance, 200 years shows that it is in at least momentum as well as, the period being but two circumstances make it possible a temporary 8/5 resonance with the decreased. This tends to make the sub- that Toro could lose its coupling to the Earth. The amplitude of the libration sequent encounters farther from the Earth. First, Toro's orbit is at a rather around the equilibrium position is on September 17, 2018 Earth. Instead, Toro will approach the large distance from the ecliptic plane at about 80, and the period of libration is Earth in the vicinity of its descending the close encounters. At the August en- about 144 years. Toro is also very close node (in January every eighth year). counter Toro is always at least 2.50 to 13/5 resonance with Venus. At the January encounter Toro is be- ahead of the Earth and 60 above the L. DANIELSSON hind the Earth. It is then accelerated, ecliptic; plane; hence, the perpendicular Department of Plasma Physics, its period is increased, and again it distance is about three times the longi- Royal Inistitute of Technology, tends to move away from the Earth to tudinal distance. Secular variations or S-100 44 Stockholm 70, Sweden approach it in the vicinity of the as- other perturbations might bring Toro to W.-H. IP cending node at the August encounter. the "wrong side" of the Earth there. Department of Applied Physics The variation of the period is shown Second, Toro also has close encoun- and Information Science, in Fig. 3. The upward and downward ters with Venus. Within the time in- University of California, San Diego, slopes of the curve can be explained in terval we have studied, the closest ap- La Jolla 92037 terms of the alternating perturbations proach is about 0.15 A.U. As secular References from the Earth during the January and variations change the node and inclina- and Notes 1. H. Alfven and G. Arrhenius, Astrophy)'s. Space August encounters. Due to the oscil- tion, Venus might either "capture" Toro Sci. 8, 399 (1970). lation of the position of the Earth rela- from the Earth or perturb it out of 2. C. J. Cohen and E. C. Hubbard, Astron. J. tive to the loop of Toro 70, 10 (1965). (Fig. 2), the resonance altogether. 3. Alfven and G. Arrhenius, Science 165, 11 closest distances at the January and Naturally, a time span of 200 years (1969). 4. E. J. Opik, Adv. Astron. Astrophys. 4, 220 August encounters also vary. For ex- is insufficient for deternmining whether (1961); J. Arnold, Astrophys. J. 141, 4 (1964). ample, around the year 1964 the Earth Toro is caught in a permanent reso- 5.
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