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Migration of small bodies and dust in the Solar System

S. I. IpatovDTM

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Initial data. Migration of small bodiesThe orbital evolution of ~30,000 Jupiter-crossing objects (JCOs) with initial period

P<20 yr was considered under the gravitational influence of 7 planets (Venus-Neptune).

First series (n1): Orbital elements were the same as those of 20 real comets (with numbers 7, 9, 10, 11, 14, 16, 17, 19, 22, 26 30, 44, 47, 51, 57, 61, 65, 71, 73, 75) with 5<P<9 yr, but different values of the mean anomaly were considered.

Second series (n2): Orbital elements were the same as those of 10 real comets (with numbers 77, 81, 82, 88, 90, 94, 96, 97, 110, 113) with 5<P<15 yr, but different values of the mean anomaly were considered.

In each of other series initial orbits were close to the orbit of one comet (2P, 9P, 10P, 22P, 28P, 39P, or 44P) or to test long-period or Halley-type comets. For 2P we considered also Mercury.

Bodies initially located at the 3:1 and 5:2 resonances with Jupiter. Methods of integration. We used the SWIFT package by Levison and Duncan (Icarus, 1994,

v. 108, 18-36). Evolution of N orbits was calculated using the Bulirsh-Stoer method (BULSTO) with the error per integration step less than =10-9, or =10-8, or some value between these two values. Also =10-12 and =10-13 were used. We also used a symplectic method with an integration step 3≤ds≤10, or ds=30 days (RMVS3).

The considered time interval in one run usually was equal to the largest dynamical lifetime of bodies in the run (until all bodies reached 2000 AU or collided with the Sun). Sometimes it reached several hundreds Myr.

Earth, Moon, and Planets, v. 92, 89-98 (2003); Advances in Space Research, v. 33, 1524-1533 (2004). Annals of the New York Acad. of Sci., v. 1017, 46-65 (2004); Proc. IAU Symp. 236 “Near-Earth Objects, Our Celestial Neighbors: Opportunity and Risk” in press (2007)

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Motion of JCOs in NEO orbits for a long time• The motion of former JCOs inside Jupiter’s orbit for a long time was obtained both

for integration with the use of BULSTO and with a symplectic method.• A few JCOs got orbits located inside Jupiter’s orbit and moved in

such orbits for millions or even hundreds of millions of years. The probability of a collision of such object with a terrestrial planet can be greater than the total probability of thousands of other JCOs.

• For one object (from 10P runs with BULSTO) its probability of collisions with Earth and Venus was 0.3 and 0.7, respectively.

• For another object (from 2P runs) during its lifetime (352 Myr) its probability of collisions with Earth, Venus and Mars was 0.172, 0.224, and 0.065, respectively. For 12,000 other objects with BULSTO such probability was 0.2, 0.18, and 0.04, respectively.

• For series 2P, at ds=3 days (i.e., for a symplectic method), a lifetime of one object was 400 Myr, and it moved on Inner-Earth, Aten, Apollo, and Amor orbits during 2.5, 2.2, 44.9, and 80.8 Myr, respectively. At t=6.5 Myr this object got an orbit with e=0.03 and a=1.3 AU, and then until 370 Myr the eccentricity was less than 0.4 and often was even less than 0.2. The probability of a collision of this object with the Earth was about 1, and it was greater than that for all other 99 objects in that run by two orders of magnitude, i.e. greater by four orders of magnitude than the mean probability for one 2P object.

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10P (a, d), 2P (b), resonance 3:1 ©. BULSTO (a-c), RMVS (d)

•

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Distribution of all migrating objects in runs with BULSTO(time in Myr during which a was in interval with a width of 0.005 AU (a-b) or

0.1 AU (c-d))

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Calculations of the characteristic time elapsed up to

the encounter of two objects to radius of sphere rs • T2=6.28·kp·Ts ·R ·kv/(rs·kfi) - planar model, • T3=T2·Δi·R/rs - spatial motel; • R is the distance of encounter from the Sun, kfi is the sum of angles

(in radians) with apices in the Sun, within which the distance between orbits is less than rs, Ts is the synodic period of revolution, kp=P2/P1 where P2>P1, Pi is a period of revolution of the i-th body around the Sun. In order to take into account that velocity at distance R from the Sun differs from the mean velocity, we used coefficient kv=sqrt{2a/R-1}.

• In contrast to the approach used by Opik (1951) and Arnold (1965), T depends on orbital orientations and on a synodic period.

• Ipatov, S.I., Evolution times for disks of planetesimals, Soviet Astronomy, v. 32 (65), 560-566 (1988).

• Ipatov, S.I. and Mather, J.C., Comet and asteroid hazard to the terrestrial planets, Advances in Space Research, v. 33, 1524-1533 (2004).

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Comparison of the results obtained by symplectic and BULSTO integrators.

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V V E E M M Method N Pr T Pr T Pr T n1 BULSTO 10-9 1900 2.42 4.23 4.51 7.96 6.15 30. n1 RMVS3 ds10 days 1200 25.4 13.8 40.1 24.0 2.48 35.2 n1 RMVS3 ds10 days 1199 7.88 9.70 4.76 12.6 0.76 16.8 2P BULSTO 10-9 501 141 345 110 397 10.5 430 2P RMVS3 ds=10 days 251 262 307 344 348 28 364 2P BULSTO =10-12 50 >434 >388 >213 >433 >17.5 >450 2P BULSTO =10-13 50 58 373 40 432 2.7 448 n2 BULSTO 10-9

4000 9.9 24.4 11.6 35.6 2.12 56.3

n2 RMVS3 ds10 days 10000 14.7 24.8 14.9 36.1 2.88 56.1

There is no principal difference in the values of Pr=1,000,000P and T, obtained by different methods and with different values of and ds (at ds10 days), or if we delete objects with perihelion distance q<2rSun (exclusive for some runs with Comet 96P in series n2). Such difference is about the difference obtained for different N for the same method and integration step. One additional object can increase the values of P and T by an order of magnitude. In this table the collision probabilities with the Earth for a symplectic method and for 10-12 are not smaller than those for 10-910-8.

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Migration of trans-Neptunian objects to the near-Earth space

• The number of TNOs migrating to the inner regions of the Solar System can be evaluated on the basis of simple formulas and the results of numerical integration. Let NJ=PNpJNNTNO be the number of former TNOs with d>D reaching Jupiter's orbit for the given time span TSS, where NTNO is the number of TNOs with d>D; PN is the fraction of TNOs leaving the trans-Neptunian belt and migrating to Neptune's orbit during TSS; and pJN is the fraction of Neptune-crossing objects which reach Jupiter's orbit for their lifetimes.

• The current number of Jupiter-crossers that originated in the zone with 30<a<50 AU equals NJn=NJtJ/TSS, where tJ is the average time during which the object crosses Jupiter's orbit. According to [1], PN0.1-0.2 at TSS=4 Gyr and pJN=0.34. As mutual gravitational influence of TNOs also takes place, we take PN=0.2.

• Hence, at tJ=0.13 Myr and NTNO=5109 (d>1 km), we have NJn=104. • The number of former TNOs now moving in Earth-crossing orbits equals

NE=NJnT/tJ. For T=0.0065 Myr (it is less than for series n1 and n2) and tJ=0.13 Myr, we have NE=500. It is about 2/3 of the estimated number of Earth-crossers with d>1 km (750). The latter number doesn’t include NEOs with large values of e and i. It is also probable that the number of 1-km TNOs is smaller by a factor of several than 5109, and PN can also be smaller. We do not consider destruction of bodies at these estimates.

• [1] M.J. Duncan, H.F. Levison, and S.M. Budd, Astron. J., 110, 3073-3081 (1995).

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Conclusions on migration of TNOs

• Results of our runs testify in favor of at least one of these conclusions:

• 1) the fraction of 1-km former trans-Neptunian objects (TNOs) among near-Earth objects (NEOs) can exceed several tens of percents,

• 2) the number of TNOs migrating inside solar system could be smaller by a factor of several than it was earlier considered,

• 3) most of 1-km former TNOs that had got NEO orbits disintegrated into mini-comets and dust during a smaller part of their dynamical lifetimes if these lifetimes are not small.

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Delivery of water to the terrestrial planets during the formation of the giant planets

• The total mass of water delivered to the Earth during the formation of the giant planets is Mw=MJPJEki, where MJ is the total mass of planetesimals from the feeding zones of these planets that got Jupiter-crossing orbits during evolution, PJE is the probability P of a collision of a JCO with the Earth during its lifetime, and ki is the portion of water ices in planetesimals.

• For MJ=100m (where m is the mass of the Earth), ki=0.5, and PJE=410-6 (this value is even less than those in series n1 and n2), we have Mw=210-4m. This value is about the mass of the Earth’s oceans.

• The larger value of P for Earth we have calculated compared to those argued by Morbidelli et al. 2000 (P (1-3)10-6) and Levison et al. 2001 (P = 410-7 ) is caused by the fact that in our runs we considered a larger number of Jupiter-crossing objects and the main portion of the probability of collisions was caused by a small fraction (0.01-0.001) of bodies, each of which moved at least several hundreds of thousands years in the Earth-crossing orbits with Q<4.7 AU. We also considered that the mass supply from the Uranus-Neptune region is about 100m , in contrast to (20-30)m estimated by other authors, which increase the volatiles delivery. The total mass of water delivered to Venus can be of the same order of magnitude and that delivered to Mars can be less by a factor of 3 or 4 than that for Earth. Ancient Venus and Mars could have large oceans.

• Proc. of the IAU Symposium No. 236 “Near-Earth Objects, Our Celestial Neighbors: Opportunity and Risk”, in press (2007). Also several previous papers.

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Conclusions on migration of small bodies• The mean collision probabilities of Jupiter-crossing objects (from initial

orbits close to the orbit of a comet) with the terrestrial planets can differ by more than two orders of magnitude for different comets.

• For initial orbital elements close to those of some comets (e.g. 2P and 10P), about 0.1% of objects got Earth-crossing orbits with semi-major axes a<2 AU and moved in such orbits for more than a Myr (up to tens or even hundreds of Myrs).

• A relatively small fraction (~0.0001-0.001) of JCOs which transit to orbits with aphelia inside Jupiter’s orbit (Q<4.2 AU) and reside such orbits during more than 1 Myr may contribute significantly in the NEO supply. A probability of a collision of one object with a terrestrial planet can be larger than that of 10,000 other objects.

• Former JCOs can get any Amor, Apollo, Aten and inner-Earth objects’ orbits and move in such orbits for millions years.

• The total mass of water delivered to the Earth during planet formation could exceed the mass of the Earth’s oceans.

• The mass of water delivered to Venus can be of the same order of magnitude and that delivered to Mars can be less by a factor of 3 or 4. Ancient Venus and Mars could have large oceans.

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Formation of Edgeworth-Kuiper belt objects• In most models of the formation and collisional evolution of the Edgeworth-Kuiper belt

(EKB), the process of accumulation of trans-Neptunian objects (TNOs) took place at small (0.001) eccentricities and a massive belt. In our opinion, it is probable that, due to the gravitational influence of the forming giant planets, TNOs, and migrating planetesimals, small eccentricities of TNOs could not exist during all the time needed for the accumulation of TNOs with diameter d>100 km.

• We consider that TNOs with d100 km moving now in not very eccentric orbits could be formed directly by the compression of large rarefied dust condensations (with a>30 AU), but not by the accretion of smaller solid planetesimals.

• Probably, some planetesimals with d100-1000 km in the feeding zone of the giant planets and even some large main-belt asteroids also could be formed directly by the compression of rarefied dust condensations. Some smaller objects (TNOs, planetesimals, asteroids) could be debris of larger objects, and other such objects could be formed directly by compression of condensations.

• As in the case of accumulation of planetesimals, there could be a “run-away” accretion of condensations. It may be possible that, during the time needed for compression of condensations into planetesimals, some largest final condensations could reach such masses that they formed planetesimals with diameter equal to several hundreds kilometers.

• Ipatov S.I., (2001) LPSC, (#1165). http://www.lpi.usra.edu/meetings/lpsc2001/pdf/1165.pdf • Ipatov S.I. (2004) Proc. of the 14th Annual Astroph. Conf. in Maryland “The Search for

Other Worlds”, ed. by S.S. Holt and D. Deming, American Inst. of Physics, 277-280. • Ipatov, S.I. (2004) http://www.astro.umd.edu/~ipatov/ipatov-darwin-conf-p045.doc • Ipatov, S.I. (2005) http://www.lpi.usra.edu/meetings/ppv2005/pdf/8054.pdf

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Formation of scattered disk objects (SDOs)• In 1987 (Ipatov S.I., Earth, Moon, and Planets, v. 39, 101-128, 1987), five years before the

first TNO was discovered in 1992, based on our runs of the formation of the giant planets I supposed that there were two groups of TNOs and, besides TNOs formed beyond 30 AU and moving in low eccentric orbits, there were former planetesimals from the zone of the giant planets in highly eccentric orbits beyond Neptune. A very small fraction of such planetesimals could left in eccentrical orbits beyond Neptune and became so called ''scattered disk objects'' (SDOs). Later on similar model of the formation of SDOs were considered by several authors in more detail. The end of the bombardment of terrestrial planets could be caused mainly by the planetesimals that had got highly eccentric and inclined orbits located mainly beyond Neptune.

• The total mass of planetesimals in the feeding zone of Uranus and Neptune could exceed 100m. Most of these planetesimals could still move in this zone when Jupiter and Saturn had accreted the bulk of their masses. Our computer runs (published in 1991) showed that the embryos of Uranus and Neptune could increase their semimajor axes from 10 AU to their present values, moving permanently in orbits with small eccentricities, due to gravitational interactions with the planetesimals that migrated from beyond 10 AU to Jupiter, which ejected most of them into hyperbolic orbits. Later on, similar results were obtained by other scientists by numerical integrations using much faster computers. In our old runs the mutual gravitational influence was taken into account by the method of spheres.

• Ipatov S. I. (1987) Earth, Moon, and Planets, 39, 101-128. • Ipatov S.I. (1991) Sov. Astron. Letters, 17, 113-119. • Ipatov S.I. (1991) LPSC, 607-608 ; Ipatov S. I. (1993) Solar Syst. Res., 27, 65-79.

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Migration of embryos of Uranus and Neptune (Sov. Astron. Letters, v. 17, 113-119, 1991; LPSC, 607-608, 1991; Solar System Res. v.

27, 65-79, 1993. Presented at conf. “Planetesimal dynamics" in Sept. 1992 in Santa-Barbara and later). The total mass of a disk equaled to 135Me. Arrows show positions

of planets. Initial semimajor axes of the giant planets: 5.5, 6,5, 8, and 10 AU.

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Migration of embryos of Uranus and NeptuneResults published in 1991-1993. Initial eccentricities of embryos of Uranus and

Neptune (with masses equal to 10 Mearth): 0.75 and 0.82 (left), 0.02 (right). Results obtained by the method of spheres of action are closed to those obtained about ten

years later by E.W. Thommes, M.J. Duncan, and H.F. Levison (Nature, 1999, v. 402, 635-638; Astron. J., 2002, v. 123, 2862-2883) by numerical integrations.

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Uranus and Neptune were not formed if their massive embryos

had not been considered (a figure from Solar System Res., 27, 65-79, 1993) .

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Mixing of planetesimals in the zone of the terrestrial planets

(a figure from Solar System Research, v. 27, 65-79, 1993) .

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Mixing of planetesimals in the zone of the terrestrial planets

Composition of the largest formed planets is close to that of the initial disk

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The choice of the pairs of contacting bodies• The mutual gravitational influence of bodies in disks was taken into account by the action

spheres method, that is two two-body problems were considered. • Initially I used the "probability" algorithm of choosing pairs of encountering objects

during the evolution of discrete systems with binary interactions. The pairs of bodies encountering to the distance equal to the radius rs of action sphere were chosen in the proportion to the probability of their encounter.

• Later on the efficient "deterministic" algorithm was developed, for which for a pair of encountering bodies i and j the moment tij of the first isolated encounter to rs is minimum. In our algorithm we don't store the matrix {tij}. The required memory (exclusive for the memory for parameters of the bodies) is equal to 3N, where N is the number of bodies, but the result of evolution is the same as for the case when we store all this matrix {tij}.

• If the number of bodies in a disk is not small, then for the deterministic algorithm the time needed for the growth of embryos is ten times smaller than that for the "probability" algorithm.

• When the evolution of the disks corresponding to the terrestrial feeding zone was modeled by the deterministic method, it was found that the time to form 80% of the mass of the largest planet (Earth) did not exceed 10 Myr (i.e., was shorter than analytic estimates), and the total evolution times of the disks were of the order of 100 Myr.

• Ipatov, S.I., Computer modelling of the process of solar system formation, Proc. Intern. IMACS Conference on Mathematical modelling and applied mathematics (June 18-23, 1990, Moscow). Ed. by A.A. Samarskii and M.P. Sapagovas. Elsevier. Amsterdam, pp. 245-252 (1992).

• Ipatov, S.I., Methods of choosing the pairs of contacting bodies for investigations of the evolution of discrete systems with binary interactions, Matematicheskoe Modelirovanie (Mathematical Modeling), v. 5, N 1, pp. 35-59 (1993), in Russian.

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Gravitational interaction of two planetesimals• Regions of initial semimajor axis and an initial angle between the directions to bodies from

the Sun vertex were studied for the following types of orbital evolution: motion about triangular libration points on tadpole and horseshoe synodic orbits (types M and N), the case of close approaches of the objects (type A), and chaotic variations of the orbital elements during which close approaches of the bodies are impossible (type C).

• There must be 1.6 mu^{1/2} [not 1.6 mu^{1/3} ] to the right.

• Ipatov S.I. Gravitational interaction of two planetesimals moving in close orbits, Solar System Research, v. 28, 494-512 (1994).

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Mutual gravitational influence of trans-Neptunian objects and planetesimals

• Runs (Preprint KIAM, 1980): Gravitational interaction of three or four TNOs with initially circular orbits and masses close to that of Pluto. For the planar model: Growth of emax to 0.04, 0.1, and 0.2 in 100 Myr, 1 Gyr, and 3 Gyr, respectively (at 50 AU).

• Estimate: For iav=10 deg and N=100, the increase of emax up to 0.2 in 4 Gyr. Sol. Syst. Res. 1995, 29, 9-20.

• Runs of N=50 identical objects (eo=0.0001, io=0.00005 rad). For 1000 km objects (density 2 g/cm^3), growth of eav to 0.04, 0.08, and 0.11 (iav to 0.7, 1.5, and 2 deg) was during 10^6, 10^7, and 10^8 revolutions around the Sun, respectively. For fifty 100 km objects, growth of eav to 0.01 was during 10^8 revolutions, but increase of eav to 0.005 in 10^3 revolutions. LPSC 2007, #1260.

• eav and iav are proportional to sqrt{N encounters}. Estimate: eav reach 0.01 during 3×10^5 revolutions of 100 km objects with a total mass m=Mearth.

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Orbital evolution of asteroids at resonances• The region of initial values of a and e for which orbits become Mars-crossing in 0.1Myr

coincides with the 5:2 Kirkwood gap.• Regions of initial data corresponding to different types of interrelations of between orbital

elements were studied. Limits of variations in e are different for different types.• Ipatov: Sov. Astron. Lett. 1989, 15, 324-328; Icarus 1992, 95, 100-114; Solar System

Research 1992, 26, 520-541.• The probability of a collision of an asteroid (eo=0.15, io=10 deg) during its lifetime with the

Earth is 5×10^{-4} for the 3/1 resonance and 10^{-4} for the 5/2 resonance with Jupiter.

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Delivery of volatiles by dust• Although the role of dust in the accretion mechanism was probably relatively small,

it could have played an important role in the evolution of the terrestrial planets. Specifically, the dust could be more efficient than larger bodies in delivery of organic (prebiogenic) or even biogenic matter to the planets.

• This is explained by the fact that dust particles are not subjected to intense heating (usually higher than 100-150 C) when they enter the atmosphere, because they have a high total surface-to-mass ratio, which allows the excess heat to be radiated effectively. Therefore, they can decelerate in the upper atmosphere for several seconds or tens of seconds (depending on the entry angle) without any change in their internal structure and composition and then softly precipitate onto the surface, delivering in this way the primitive interplanetary or even interstellar matter.

• Dust can be considered as a potential carrier of biogenic material from outer space, without considering the efficiency of the material's transportation between the inner planets. From this point of view, life forms radically different from the terrestrial variants are unlikely to be found, for example, on Mars (if they exist there now or existed before).

• The main sources of interplanetary dust are comets, asteroids, and trans-Neptunian bodies. A considerable amount of dust is produced in the process of the sublimation of the dust-ice material of the nuclei of comets during their orbits to perihelion and in numerous collisions of small bodies and their fragmentation.

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Migration of dust. Initial data and methods of integration

• Migration of 15,000 dust particles under the gravitational influence of all planets (Pluto was considered only for trans-Neptunian particles), radiation pressure, Poynting-Robertson drag, and solar wind drag was integrated using the Bulirsh-Stoer method (BULSTO) with the error per integration step less than =10-8. We used the SWIFT package by Levison and Duncan (Icarus, 1994, v. 108, 18).

• A wide range of particle sizes (from 1 micron to several millimeters) was studied. We considered β (the ratio between the radiation pressure force and the gravitational force) equal to 0.0001, 0.0002, 0.0004, 0.001, 0.002, 0.004, 0.005, 0.01, 0.05, 0.1, 0.2, 0.25, and 0.4. For silicates at density of 2.5 g/cm3, such β values correspond to particle diameters d of about 4700, 2400, 1200, 470, 240, 120, 47, 9.4, 4.7, 2.4, 1.9, and 1.2 microns, respectively. For water ice d is greater by a factor of 2.5 than that for silicate particles.

• In each run we took N250, because for N≥500 the computer time per calculation for one particle was greater by a factor of several than that for N=250.

• Ipatov, S.I., Mather, J.C., and Taylor, P.A., Annals of the New York Academy of Sciences, v. 1017, pp. 66-80 (2004). Ipatov, S.I. and Mather, J.C., Advances in Space Research, v. 37, N 1, 126-137 (2006)

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Initial coordinates and velocities• The initial positions and velocities of the asteroidal particles

considered were the same as those of the first numbered main-belt asteroids (JDT 2452500.5), i.e., dust particles are assumed to leave the asteroids with zero relative velocity.

• The initial positions and velocities of the trans-Neptunian particles were the same as those of the first TNOs (JDT 2452600.5). Our initial data for dust particles were different from those in previous papers.

• The initial positions and velocities of cometary particles were the close to those of Comet 2P Encke (a≈2.2 AU, e≈0.85, i≈12 deg), or Comet 10P/Tempel 2 (a≈3.1 AU, e≈0.526, i≈12 deg), or Comet 39P/Oterma (a≈7.25 AU, e≈0.246, i≈2 deg), or test long-period comets (e=0.995, q=a(1-e)=0.9 AU or q=0.1 AU, i was distributed between 0 and 180 deg, particles started at perihelion), or test Halley-type comets (e=0.975, q=0.5 AU).

• We considered Encke particles starting near perihelion (runs denoted as Δto=0), near aphelion (Δto=0.5), and when the comet had orbited for Pa/4 after perihelion passage, where Pa is the period of the comet (such runs are denoted as Δto=0.25).

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Probabilities of collisions of dust particles with the

Earth (during dynamical lifetimes of particles). Maximum is at beta ~0.002-0.01 (i.e., at d~100 micron)

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Distribution of migrating asteroidal dust particles with their semimajor axes. The time in orbits with a<3 AU was greater for smaller beta.

Runs with (right) or without (left) planets.

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Migration of asteroidal dust particles. Resonances n/(n+1) with the terrestrial planets [e.g., 6:7, 5:6,

3:4, and 2:3 resonances with Earth]. The peaks of the distribution of particles over a at resonances with the Earth are better seen for smaller β

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Distribution of dust particles with semimajor axis and eccentricity(a) asteroidal particles at beta=0.01, (b) asteroidal particles at beta=0.4, © trans-

Neptunian particles at beta=0.1

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Distribution of dust particles with distance from the Sun and height above the initial plane of the orbit of the Earth

(a) asteroidal particles at beta=0.01, (b) asteroidal particles at beta=0.4, © trans-Neptunian particles at beta=0.1

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Number density of different migrating dust particles. Observations showed that number density is constant at 3-18 AU from the Sun [Humes

D.H., J. Geophys. Res., 1980, 85, 5841]. Therefore, the obtained plots show that asteroidal dust doesn't dominate at a>3 AU, trans-Neptunian dust does not dominate at 3-7 AU, and a

lot of dust particles at 3-7 AU were produced by comets.

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Our model of calculations of changes of the solar spectrum after the light was scattered by dust particles

• Based on positions and velocities of particles taken from one our run with a fixed β, we studied the variations in solar spectrum after the light was scattered by dust particles and reached the Earth. For each such stored position, we calculated many (~102-104 depending on a run) different positions of a particle and the Earth during the period Pr of revolution of the particle around the Sun. All positions of particles during their dynamical life-times were considered.

• Three different scattering functions were considered:• (1) the scattering function depended on a scattering angle θ in such a way: 1/θ for θ<c, 1+(θ-c)2 for θ>c,

where θ is in radians and c=2π/3 radian. • (2) we added the same dependence on elongation ε (considered eastward from the Sun).• (3) the scattering function didn't depend on these angles at all. • ε is the angle with a vertex in the Earth between directions to the Sun and a particle,• θ is the angle with a vertex in a particle between direction to the Earth and the direction from the Sun to a

particle.• The intensity of light that reaches the Earth was proportional to λ2(R×r)-2, where r is the distance between

a particle and the Earth, R is the distance between the particle and the Sun, and λ is a wavelength of light. • For each considered positions of particles, we calculated velocities of a dust particle relative to the Sun

and the Earth and used these velocities and the scattering function for construction of the solar spectrum received at the Earth after the light was scattered by different particles located at some beam (line of sight) from the Earth. The direction of the beam is characterized by ε and inclination i. Particles in the cone of 2.5 deg around this direction were considered.

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The shape of the Fraunhofer line produced by our model and at the observations of the zodiacal light spectrum near the solar Mg I λ5184 absorption line by Reynolds,

Madsen and Moseley (ApJ, 2004, 612, 1206) (ast1 and tn1 – for function 1, ast2 and tn2 – for function 2, ast and tn – for function 3; ast – asteroidal dust, tn – trans-

Neptunian dust, com – dust particles started from comet 2P at perihelion)

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Calculation of ‘velocity-elongation’ plots • Based on our plots of the intensity of the scattered light obtained at

the Earth vs. ∆λ (λ is the length of the wave near the solar Mg I λ5184 absorption line and ∆λ=λ-λo, where λo corresponds to the minimum of solar spectrum near this line) we calculated the shift ∆λs of the plot, which is based on our distribution of dust particles, relative to the plot of the solar spectrum. Considering that v/c=∆λs/λ (where v is a characteristic velocity of particles and c is the velocity of light), we calculated the characteristic velocity of particles at different elongations.

• The plots of this velocity vs. elongation (the angle with a vertex in the Earth between directions to the Sun and a dust particle) were compared with those obtained by Reynolds et al. (2004) at observations.

• Velocity-elongation plots for different scattering functions are denoted as c1 and m1 for function 1, as c2 and m2 for function 2, and c and m for function 3.

• For lines marked by ‘c’, we considered a shift of a centroid (the ‘center of mass’ of the region located upper than a plot of intensity vs. ∆λ and restricted by the maximum value of the intensity) for scattered light obtained at the Earth relative to the centroid for the solar light. For lines marked by ‘m’, we considered a shift of the minimum of the plot of intensity of a scattered light from the minimum for the solar spectrum.

• Velocity-elongation plots obtained for different scattering functions and for ‘centroid’ and ‘minimum’ models are close to each other, exclusive for the cases when ε is close to 0.

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Comparison of plots of velocities of Mg I line versus elongation with the observations

• Ipatov, S.I., Kutyrev, A., Madsen, G.J., Mather, J.C., Moseley, S.H., Reynolds, R.J. (e.g., 37th LPSC, #1471, 2006; astro-ph/0608141) compared the plots of velocities of Mg I line (at zero inclination) versus elongation ε (measured eastward from the Sun), with the observational plots obtained by Reynolds et al. (ApJ, 2004, 612, 1206-1213).

• The velocity-elongation plots obtained for different considered scattering functions were close to each other for 30<ε<330 deg, the difference was greater for more close direction to the Sun.

• The difference between different plots for several sources of dust was maximum at ε between 90 and 120 deg. For future observations of velocities of the zodiacal light, it is important to pay particular attention to this interval of ε.

• In our opinion, the main conclusion of the comparison of such curves is that asteroidal dust doesn't dominate in the zodiacal light and a lot of zodiacal dust particles were produced by high eccentricity comets (such as comet 2P Encke).

• This conclusion is also supported by the comparison of a spatial density of different migrating dust particles with the observational result that a spatial density is constant at 3-18 AU from the Sun.

• Significant contribution of cometary dust to the zodiacal dust was considered by several other authors (e.g., Zook, 2001; Grogan et al. 2001).

36

Velocity-elongation plots for asteroidal and trans-Neptunian zodiacal dust. For asteroidal dust, the velocity-elongation curves are below the

observational curve at elongation ε<240 deg. For trans-Neptunian dust, the observational curve is mainly inside the region covered by curves for different β, but at 180<ε<270 deg it is mainly above the trans-Neptunian curves. For particles originated from long-period comets, the

plot was above the observational curve at ε<180 deg and it was below at ε>180 deg .

37

Velocity-elongation curves for particles originated from comets 10P and 39P.

10P curves are below the observational curve, but the difference is smaller for larger particles.

38

Velocity-elongation plots for particles originating from comet 2P at perihelion (per), aphelion (ap), and T/4 (T is the period of revolution) after

perihelion passage (m). The curves for 2P particles originating at perihelion are relatively close to each other for different β, and they are close to the observational

curve.

39

Mean eccentricity of particles at different distances from the Sunat several values of β (see the last number in the legend) for particles originating from comets 2P, 10P, 39P, and trans-Neptunian objects

40

Distribution of dust particles with semimajor axis and eccentricityat several values of β for particles originating from Comet 2P at the

middle of the orbit (2P 0.25t) and at aphelion (2P 0.5t). Right plots are for smaller beta.

41

Mean (for elongation between 30 and 330 deg.) width (FWHM = Full-Width Half-Maximum) of MgI line at several values of β. The values of the width were based on spectra obtained for particles produced by different bodies. The mean observed value was equal to 76.6 km/s. Large 2P particles alone do not satisfy to the observations.

42

Values of α in n(R)~R^{-α} obtained by comparison of the values of the number density n(R) at distance R from the Sun equal to 0.3 and 1 AU (a), at R=0.8 and R=1.2 AU (b), and at R equal to 1 and 3 AU (c). The values are presented at several values of

β for particles originating from Comets 10P and 39P (10P and 39P), from trans-Neptunian objects (tno), and from long-period comets (lp) at e=0.995, q=0.9 AU and i distributed between 0 and 180 deg. Observational values are 1.3 for (a) and 1.5 for ©.

43

Sources of zodiacal dust particles• There is no principal difference in the ratio of fractions of

asteroidal/cometary/trans-Neptunian dust for our spectroscopic studies at different ε, as our studies showed that the main contribution to the spectrum is from particles on a distance less than 1 AU from the earth and similar vertical shifts of the `velocity-elongation' curves (different for different sources of dust) from the observational curve were obtained for many values of ε. The principal difference in the fractions will be if we consider particles in different parts of the solar system, separated by several AU.

• Besides, our calculations showed that the difference between characteristic velocities of Mg I line for different sizes of particles was usually less than the difference for different sources of particles. It would mean that for our conclusions about the fraction of asteroidal dust we need not know the exact distribution of zodiacal particles in masses.

44

Sources of zodiacal dust particles• Comparison of the WHAM observations (both of velocity and width

of Mg I line) and the observations of the number density with the results based on our models for particles originating from different small bodies testify in favour of a considerable fraction of particles originating from comets, including those originating beyond Jupiter’s orbit, and trans-Neptunian objects, but it did not contradict to 30% of asteroidal dust needed for explanation of formation of dust bands.

• Fractions of asteroidal particles, cometary particles originated inside of Jupiter's orbit, and particles originated beyond Jupiter's orbit can be about 1/3 each, with a possible deviation from 1/3 up to 0.1-0.2.

• The velocity amplitudes of the Mg I line (in plots of the velocity vs. elongation) are greater for greater mean eccentricities and inclinations, but they depend also on distributions of particles over their orbital elements. The mean eccentricities of zodiacal particles located at 1-2 AU from the Sun that better fit the WHAM observations are between 0.2 and 0.5, with a more probable value of about 0.3.

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Publications on dynamics of bodies and dust• Ipatov, S.I. and Mather, J.C., Migration of trans-Neptunian objects to the terrestrial planets,

Earth, Moon, and Planets, v. 92, 89-98 (2003) . • Ipatov, S.I. and Mather, J.C., Comet and asteroid hazard to the terrestrial planets, Advances

in Space Research, Elsevier, v. 33, N 9, 1524-1533 (2004) • Ipatov, S.I. and Mather, J.C., Migration of Jupiter-family comets and resonant asteroids to

near-Earth space, “Astrodynamics, Space Missions, and Chaos”, ed. by E. Belbruno, D. Folta, and P. Gurfil, Annals of the New York Academy of Sciences, v. 1017, pp. 46-65 (2004).

• Ipatov, S.I., Mather, J.C., and Taylor, P.A., Migration of interplanetary dust, “Astrodynamics, Space Missions, and Chaos”, ed. by E. Belbruno, D. Folta, and P. Gurfil, Annals of the New York Academy of Sciences, v. 1017, pp. 66-80 (2004).

• Marov, M. Ya. and Ipatov, S.I., Migration of dust particles and volatiles delivery to the terrestrial planets, Solar System Research (translated from Astronomicheskii Vestnik), v. 39, N 5, 374-380 (2005)

• Ipatov, S.I. and Mather, J.C., Migration of small bodies and dust to near-Earth space, Advances in Space Research, v. 37, N 1, 126-137 (2006)

• Ipatov, S.I. and Mather, J.C., Migration of dust particles to the terrestrial planets, Proceedings of the workshop “Dust in Planetary Systems” (September 26-30, 2005, Kaua'i, Hawaii, USA), ed. by H. Krüger and A. Graps, ESA Publications, SP-643, p. 91-94 (2007).

• Ipatov, S.I. and Mather, J.C., Migration of comets to the terrestrial planets, Proc. IAU Symp. No. 236 “Near-Earth Objects, Our Celestial Neighbors: Opportunity and Risk”, ed. by A. Milani, G.B. Valsecchi and D. Vokrouhlicky, in press (2007).

• See more references and texts of publications on

• http://www.astro.umd.edu/~ipatov; http://www.dtm.ciw.edu/ipatov