Collision of comets with meteoroids

ICARUS 27, 323--329 (1976)

Collision of C o m e t s w i t h M e t e o r o i d s OSCAR T. MATSUURA Department of Astronomy, InstiSuto AstronSmico e Geofisico Universidade de Sito Paulo, Sdo Paulo, Brazil R e c e i v e d J u l y 10, 1975; revised A u g u s t 7, 1975 S t a t i s t i c a l a n a l y s i s of t h e q u a n t i t y o f d u s t in t h e c o m e t a r y a t m o s p h e r e in relation t o t h e d i r e c t i o n o f m o t i o n o f t h e c o m e t a b o u t t h e Sun suggests a n excess of d u s t for t h e r e t r o g r a d e comets. This excess is a n a l y z e d in t h e light o f H a r w i t ' s t h e o r y o f t h e cloud o f " b o u l d e r s " a n d o f (~pik's i m p a c t t h e o r y . A c o m p a r i s o n is also m a d e b e t w e e n t h e s e excesses a n d o t h e r c o m e t a r y p h e n o m e n a such as s p l i t t i n g s a n d outbursts.

I. INTRODUCTION There exist in the literature determinations of the mass of solid material in the comas of comets (see, for example, Whitney, 1955; Van~sek, 1958, Houziaux, 1959; Sekanina, 1962; Rozhkovskii, 1966). Basically, all the methods of determination are equivalent since they all use photometric data and assume the scattering of solar light by solid grains through single scattering. Each author assumes a distribution function of sizes of grains and a type of material of the grains in order to determine their mass density and photometric properties. According to Van~sek's method (1958), the uncertainties in the logarithm of the mass of solid material in the coma, log M=(g), is of order unity. He computed the value of M= for six comets with strong emission in the continuum. New calculations of the dust mass were done by Van~sek (1965) for comets observed between 1954 and 1964. Van~sek (1972) then added data regarding two new comets. Sekanina (1962) also calculated masses of coma dust. The above-mentioned papers include practically everything t h a t exists on the mass of coma dust. In the majority of cases in which the heliocentric distance r of the comet is mentioned, its value is about 1AU; furthermore, 8 < logMg(g) < 12. Various physical processes can be directly associated with the production of Copyright C) 1976 by A c a d e m i c Prcas, Inc. All fights of reproduction in a n y form reserved. Printed in Great Britain

coma dust. One is the simple vaporization of the nuclear surface, consisting of water or hydrates in mixture with meteoric material, under the effects of solar radiation (Delsemme and Miller, 1971). Also, processes of an explosive nature may be a source of the solid material of the coma, such as (a) exothermic reactions of free radicals liberated by the nucleus (Donn and Urey, 1956; Donn, 1963), (b) phase transition of the uncrystallized ice to ice with cubic crystalline structure (Patashnick et a/., 1974), (c) effects of tidal forcesdue to Jupiter or to the Sun, and (d) collisions with meteoroids (Harwit, 1967). In this work we make an analysis of the masses of dust in the coma, with its principal basis in alternative (d) above. II. E x c e s s oF MAss M s Assuming t h a t the meteoroids move about the Sun in the direct sense near the ecliptic plane, some asymmetrical properties of cometary comas in relation to their direction of motion might be expected due to collisions with meteoroids. With this in mind, the most probable values of M s described in the previous section are presented in Fig. 1, as a function of the inclination angle of the comet orbit, i. The scattering of points in this figure clearly shows an excess of mass M s for comets with an inclination angle i > 90 °. 323

324

OSCAR T. MATSUURA 12

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FIo. 1. Values of log Mg(g) as a function of the inclination angle of the comet orbit. Crosses refer to Van~sek (1965, 1972), open circles to Sekanina (I962). " P / " indicates short-period comets.

Such excess is of order 103, larger than the uncertainties in the calculations. Unfortunately, the data of M s utilized do not indicate whether they were taken before or after perihelion passage. To give a greater statistical consistency in the treatment of the few available data, when values of log, Ms(g ) could be determined for the same comet more than once, we used those values t h a t were the most probable. I t is important to emphasize t h a t not one of the processes cited above, except collision, is capable of explaining the asymmetry observed. According to Harwit (1967) the meteoroids of intermediate size (boulders) with diameters of the order of tens of meters are produced in a stationary state by means of collision (of the meteoroids) with asteroids. These meteoroids originally have orbits of small inclination and eccentricity. When they slowly approach the orbit of the E a r t h under the action of the Poynting-Robertson effect, their orbits undergo a scattering t h a t raises the value of eccentricity and of inclination. This limits their number colliding with the Earth to only a fraction of their total, the rest creating a cloud of meteoroids with orbits having parameters randomly distributed. Since the time scale, according to Harwit, for the action of the

Poynting-Robertson effect is of the order of 1 0 7 y r , while the time of collision in the cloud of boulders is only 104yr, the relative population of meteoroids with orbits of low eccentricity and inclination is large. Under these conditions, the frequency v¢~¢ with which the excess of dust mass in the coma would be observed during the passage of a comet in the vicinity of the Sun should be expressed as a function of the inclination angle of the cometary orbit by Vexc(i) = NmZNVr©l(i)Sin i see -l, (1) where N m is the number density of the meteoroids t h a t collide with the comet and 2:N is the cross section for collisions of the cometary nucleus. In the present case we can take the cross section to be simply the geometric cross section Z N = 1rRN 2,

(2)

where R N is the radius of the cometary nucleus. Vr, i(i) is the relative velocity of impact and is given, in practice, by V . I = V®, (3) where V~ is the velocity of approach of the comet to the meteoroid when they are very far apart. The relative velocity of impact is given by 0pik (1963) by V~.I ---- Vc21r{3- 2[a(1 - e2)]lncos i - a-I}, (4)

325

COLLISION OF COMETS W I T H METEOROIDS

distance for comets. The fact is t h a t the f r e q u e n c y calculated for excess of mass M s is p r o n o u n c e d for r e t r o g r a d e comets e v e r y time the radius of the orbit o f the meteoroid approaches the perihelion distance of the comet. T h e histogram presents a depression near i = 110 °. The statistics regarding the distribution of i b y P o r t e r (1963) also claims this absence of points, possibly suggesting catastrophic collisions o f nuclei with meteoroids.

where Vci, is the velocity o f an object in circular orbit and in direct m o t i o n a t a heliocentric distance r = 1AU; e and a are the eccentricity and semimajor axis of the comet orbit. F o r a comet in a parabolic orbit, (4) is r e w r i t t e n as V~¢l = V~l,[3 -- 2(2q) 1;2 cos i],

(5)

where q is the perihelion distance of the comet. Since we are c o m p u t i n g the f r e q u e n c y with which the mass excess M s is observed, the factor sin i is i n t r o d u c e d in order to a c c o u n t for the relative area of the sky observed at angle i. This distribution function in the inclination of the orbit of comets is shown t o hold for all comets (excluding the case o f short-period comets) (Porter, 1963). I n the sample in question only three comets are periodic. Figure 2 presents a curve with normalized a m p l i t u d e corresponding to (1) for q = 1 AU. I t also gives a normalized histogram of observed f r e q u e n c y of mass excess in each interval of 10 ° angle of inclination. F o r the p r e p a r a t i o n of the histogram, it was necessary to define a mass excess as corresponding to the condition logM~(9) > 10. This value was stipulated on the basis of a m e a n value of logMg(9) of direct comets, and allows a margin of error corresponding to an error b a r o f w i d t h 2. I n Fig. 2 the curve of e x p e c t e d f r e q u e n c y of t h e excess of mass M s is valid for t h e case in which the meteoroid is in a circular orbit 1 A U from the Sun. N a t u r a l l y , t h e r e exists an interval of values of perihelion

I I I . PHYSICS OF THE IMPACT AND DUMBER D E N S I T Y OF M E T E O R O I D S

The flux o f solar r a d i a n t e n e r g y for the heliocentric distance r = 1 AU is of order 106 erg em -2 sec-l. Using the probable value of n u m b e r density o f meteoroids o f intermediate size, having a radius of 100m and a mass density of 5 g c m -3, the flux o f kinetic energy becomes comparable with the flux of solar e n e r g y above, assuming a velocity of i m p a c t of order 5 0 k m s e c -I. This comparison is i m p o r t a n t if the collision of comets with meteoroids competes with solar radiation when producing coma dust. T o analyze the i m p a c t we should define the n a t u r e of the nucleus. This is supposedly composed of h y d r a t e s in a solid state (Delsemme, 1965), with physical properties v e r y similar to those o f ice. T h e plastic limit, lp, of commercially available ice was d e t e r m i n e d to be 107 d y n cm-2. The m e t h o d of d e t e r m i n a t i o n is the same as t h a t described b y B u d d h u e (1942).

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Fxo. 2. Normalized expected frequency of observations of mass excess for a perihelion distance q--'-- 1AU compared to the normalized observed frequency.

326

OSC~m T. MATSUURA

The physics of impacts at high velocities is a very complex problem and the estimates of erosion are not in agreement (Gault, 1973). The semiquantitative study developed by 0pik (1957) allows us to obtain estimates with uncertainty less than a factor of 2, on the basis of conservation principles. This model of impact will be used here. The critical velocity, Vcr, for erosion is given by

Vcr =

([p/~N) I/2,

(6)

where 8 N is the nucleus density (8 N = 2.4g em-3). For the value of the plastic limit determined above, this critical velocity is of order 2 x 103cmsec -~. The eroded mass of the nucleus, M*, is given by

M* =Cz Mm(Vre~/V¢,),

(7)

where M m is the mass of the meteoroid and C~ a constant in the interval 2 < C~ < 5. Since only about 3% of the kinetic energy of the projectile is used for the fragmentation, the larger part is transformed into kinetic energy of the fragments. We can estimate the velocity of ejection through the conservation of energy ~,j = ( v o r v ~ , , / c , ) '/~.

(8)

The velocity of escape, V**¢,for a typical comet is about 103 cmsec -~. For the condition ptj > V,,c to be satisfied, it is necessary thatVrel > 1.75 x 103cmsee-~,forCl ---3.5. Now, for i = 100 °, q = 1, a typical value of V**l is 5.55 × 106cmsec -1. Using the value Vr, l ---7 × 10ecmsec -~, t h a t is, near the maximum value, we can determine a lower limit for the mass M m of the meteoroid, with the help of (7), namely, M m > 7.9 x 1014g, so t h a t the erosion will just destroy the whole nucleus. Under these conditions the radius of the meteoroid will be Rm > 3.35 × 10*cm. I f the quantity of the eroded mass M~ can be set equal to the quantity of the coma dust of comets with mass excess, then the mass of the meteoroid should be 8.2 × 107g; its radius should consequently be 1.57 × 10'cm. Inserting the values used up to now in (8), we obtain /sej = 6.3 ×

104cmsee -l, which is compatible with the observations of expansion of halos in comets. This velocity also affirms a time scale of the order of 105 see (about 2 days) for the observation of the mass excess. The analysis of impact could be taken further, supposing t h a t the crater has the form of a hemisphere. For the loss of 10~2g of material, this hemisphere should have a radius of 5.7 × 103cm. I f the velocity of sound in the ice [about 3 x 10Scmsec -I (Koshkin and Shirkevich, 1968)] is used to provide a time scale for erosion, this time should then be 1.9 × 10-2sec. The mean rate of production of solid material would then imply t h a t the density of mass in the neighborhood of the nucleus would be unreasonably high. Setting the density equal to the value of the nuclear density, the radius of the crater should be 104cm, with a depth of 1.2 × 103 cm. Certainly, the crater should be wider than it is deep. More recent studies on hypervelocity impacts are presented by Dohnanyi (1972). The eroded mass can be calculated by assuming the comet nucleus as a basalt target and using Dohnanyi's results. The results are in approximate agreement with ~)pik's formula given in (7) for the energy region of interest. Also, the calculation performed above for the lower limit of M m required to have a catastrophic collision is in good agreement with Dohnanyi's (1972) work. However, the acquisition of experimental parameters for icy targets is still desirable. In the interval of inclination angles 90°< i < 140 °, in which the mass excess Mg is most important, the number of times in which there is an excess is 10 for the 14 comets analyzed. The ratio 10/14 = 0.71 can be set equal to the frequency, vexc, given in (1) for a typical time of passage of a comet in the neighborhood of the Sun (107see ~ 4 months). We then require a number density of meteoroids with radii of 1.57 × 102cm equal to 4 × 10-27cm-3. On the basis of the number density of grains scattering zodiacal light in the proximity of the Earth (10-13cm-3), the number density of these meteoroids can be taken to be of order 10-3z cm -3 (Whipple, 1967). The required density of meteoroids

C O L L I S I O N OF COMETS W I T ~ M E T E O R O I D S

is 4 × 104 times larger. However, support for this is found in the ideas of Harwit (1967). IV. DISCUSSION

A. Vaporization I f we assume t h a t the number density of grains decays radially roughly following a p-2 law, where p is the radial distance to the center of the nucleus of the comet, t h a t the light scattering is single scattering, and t h a t there exists only a population of scattering grains already produced in their final form at the surface of the nucleus, then the total number density of the grains, Zr,, will be proportional to

RN2RJ Pt,,

(9)

where ~t~ is the terminal velocity of the grains. I f we assume t h a t the photometric measurements were calculated for the masses enclosing the whole coma of radius R v and if we suppose t h a t the terminal velocity of grains is unique for all of them, then Ng will be only proportional to RN2. In this case, the extreme limits for R s (Rahe eta/., n.d.) permit us to vary N s from a minimum value up to a maximum value 2500 times larger. However, nothing can justify the existence of greater radii RN for retrograde comets. Also, we must take into account the fact t h a t if the radius RN is increased, the critical diameter for a grain able to be dragged by the gas flowing from the nucleus is in turn less, which then makes R~ less, consequently augmenting the number density of grains and the optical depths. Under these conditions the calculated mass M S (assuming single scattering) would be underestimated and would show smaller excesses of mass M s. During the vaporization due to solar radiation of pieces of volatile nuclear material, nonvolatile and small grains about 0.1/zm in size are produced. I t can be shown t h a t these nonvolatile grains are photometrically more efficient t h a n the larger volatile grains and t h a t the photometric data in the continuum can be treated exclusively by the processes of scattering by grains smaller t h a n 1 ~m.

327

I f 10~2g of solid material of the coma is in a sphere of radius R, ~ 109 cm, then the grains, for example, of iron, having a diameter of 5 x 10 -5 cm will have a number density of order 4 . 5 x 10-4cm -3, which assures optical depths of orderl0 -3. In these conditions the hypothesis of single scattering is valid. However, we have cases in which the optical depth measured along the line of sight at 600kin from the nucleus is of order unity (Dossin, 1962). In this case the total number of grains can be grossly estimated to be about a thousand times greater. Thus, the mass excess Mg should be still larger.

B. Fracture of the Nucleu.s In a sample of 13 cases of splitting of nuclei presented by Stefanik (1965), 5 cases correspond to comets with retrograde orbits. I f the periodic comets are excluded from this sample, the occurrence of retrograde comets in the sample is 5/10. I f we also exclude those cases in which the comet has a very close passage to Jupiter or the Sun, the ratio above becomes 3/8. This ratio is less than t h a t referring to the occurrence of mass excess for retrograde comets and does not show a preference for retrograde comets. Therefore, the apparent concentration of the occurrence of fractures of the nucleus near the ecliptic plane (Harwit, 1967) does not argue in favor of the idea of collisions with meteoroids. One explanation for these phenomena was proposed on the basis of the possible occurrence of thermal shocks in the nuclear crust (Whipple and Stefanik, 1965). This, however, does not explain the larger occurrences of fractures in the proximity of the ecliptic plane. On the other hand, it is necessary to have more data in order to make a better evaluation of the statistics, and also, new models of the fracture process are needed. It is also important to note t h a t the physical processes of impact apparently exclude the possibility of explaining the fractures by collisions. I f the typical velocity of separation between the fractured parts is of order 15msec -~ (Pittich, 1972), the corresponding kinetic energy

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OSC~LR T. MATSUURA

is of order 102Serg. I f VrcI = 5.5 x 106cm sec -~, the mass of the meteoroid must be of order 7.4 × 101~g. Following (7), the mass eroded by such a meteoroid will be M* = 7.1 x 10'~g. This mass will constit u t e fragments and not one monolithic piece unless the nucleus possesses inhomogeneities in its mechanical properties. C. Explosive Phenomena

Explosive phenomena correspond to sudden variations of magnitude of the comets (outbursts). Pittich (1971) made a statistical analysis of these phenomena and verified t h a t their occurrence is not pronouncedly concentrated near the ecliptic plane. In another work (Pittich, 1969), he presented a list of 41 comets, observed between 1950 and 1967, which were probably discovered shortly after a sudden variation in brightness. These 41 comets correspond to about 8% of the total number observed in this same period. Eliminating from this sample those t h a t are periodic yields a ratio of 18/36 occurrences of retrograde comets in the sample of comets with explosive phenomena. This number demonstrates t h a t there does not exist a preference for the occurrence of these explosive phenomena in relation to the sense of direction of the comet's motion. Thus, a collision of the nucleus with meteoroids cannot be the dominant mechanism for these outbursts. In fact, there exist explanations for these outbursts based on chemical reactions in the gas phase (Whitney, 1955; Donn and Urey, 1956) as well as phase transitions from amorphous ice to a cubic crystalline structure (Patashnick et aL, 1974). Such transitions must occur when the temperature of the ice surface is of order 150-160°K ; these temperatures are attained b y the surface of the nucleus, in vaporization equilibrium, at heliocentric distances in the interval 3.7 to 6.9 AU (Delsemme, 1965). These transitions explain the outbursts t h a t occur in t h a t region of interplanetary space. Consequently, the dust coma observed at r ~ 1 AU does not have this process at its origin. The reactions of free radicals m a y be a good explanation for a large part of these explosive phenomena.

V. CONCLUSIONS Collisions of nuclei with meteoroids remain a rather controversial subject. The purpose of this work, however, consists in showing t h a t these collisions are a plausible explanation for the mass excess observed in the comas of retrograde comets. Larger numbers of determinations of masses M s and a better understanding of the physics of collisions are required for a more profound analysis. ACKNOWLEDGMENTS I t is a great pleasure to thank Eng. S. A. de Souza, of the Institute of Technological Research ofS~o Paulo, for his support in the determination of the plastic limit of ice. Special thanks go to Dr. C. M. Burgoyne for interesting discussions and orientation during the preparation of the doctoral thesis of the author, partially related in this work. The figures were prepared by Mr. Afonso R. Neto. REFERENCES BUDDnU~, J'. D. (1942). The compressive strength of meteorites. Pop. Astron. 50,

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COLLISION OF COMETS W I T H M E T E O R O I D S

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