Finite Lifetime Fragment Model 2 for Synchronic Band Formation in Dust Tails of Comets

ICARUS

134, 24–34 (1998) IS985935

ARTICLE NO.

Finite Lifetime Fragment Model 2 for Synchronic Band Formation in Dust Tails of Comets Kimihiko Nishioka Olympus Optical Company Ltd., 1-43-2, Hatagaya, Shibuya-ku, Tokyo 151, Japan E-mail: [email protected] Received September 22, 1995; revised March 20, 1998

2. REVIEW OF SEVERAL MODELS FOR SYNCHRONIC BANDS

Some big comets showed type II tails with many narrow striae called ‘‘synchronic bands,’’ the formation mechanism of which is still unknown. A dynamic model for the formation mechanism of synchronic bands, which is based on the following process, is proposed. The complex particles of the aggregates of the unit particles are ejected from the nucleus of the comet and disintegrate repeatedly into individual unit particles at various disintegration speeds. Then, these unit particles break up and their fragments are observed as synchronic bands. These fragments continue to disintegrate or sublimate into smaller pieces and finally they become too small to be seen at a certain normalized lifetime. The structures calculated with this theory fit well the observed shape and orientation of the synchronic bands of Comet West and Comet Seki–Lines. This dynamic model suggests that the radii of the complex particles and the radii of the unit particles are of less than visible wavelength.  1998 Academic Press Key Words: comets; comets, dynamics; dust; materials, physical properties; organic chemistry.

There is no theory that can explain the structures of the SYBs perfectly. However, we review some of the theories presented so far and point out the difficulties with them (Nishioka et al. 1992). 2.1. Fragmentation Model It is generally believed that the particles of the type II tail of a comet move around the Sun along their Keplerian orbits, receiving both solar gravitational force and solar radiation pressure. Hereafter, the ratio of solar radiation pressure to solar gravitational force is denoted by b as in conventional use. The fragmentation model (Sekanina 1976) (hereafter SFM) explains the formation of a SYB by using a time dependent b value of the particles. The outline of this theory is as follows. At time te , parent particles of bp are assumed to be ejected from the nucleus. These parent particles break up into fragments of various sizes at time tf under the influence of the solar radiation. Because of the size distribution of these fragments, they have various values of bf . At the time of observation after tf , these fragments lie on one line. We regard these aligned particles as a SYB. The global structures of the SYBs of Comet Mrkos (C/1957P1) and Comet West (C/1975V1) were explained over several days by this theory (Sekanina and Farrell 1980, 1982). In other words, this theory explains the form of a SYB by choosing appropriate values for the five parameters: te , bp , tf , bfmax , and bfmin , where bfmax and bfmin denote maximum and minimum values of bf , respectively. However, there are two weak points in this model as follows:

1. INTRODUCTION

Many narrow striae can be recognized in the type II tails of some big comets. These striae always appear in the regions of the type II tails far from the nuclei of the comets. These striae are called ‘‘synchronic bands’’ (abbreviated SYBs hereafter). Such SYBs were rarely observed except in a few large comets like Comet Mrkos (C/1957P1), Comet Seki–Lines (C/1962C1), and Comet West (C/ 1975V1). Figure 1 shows the SYBs of Comet West (C/ 1975VI). The formation mechanism of the SYBs is still unknown, although some studies have been carried out (Notni 1964, Sekanina 1976, Lamy and Kouchmy 1979, Uemura 1980, Sekanina and Farrel 1980, 1982, Notni and Tha¨nert 1988, Nishioka and Watanabe 1990, Watanabe and Nishioka 1991). We propose a dynamical theory for the formation mechanism of the SYBs.

1. Uniformity of the b value for the parent particles of one SYB is required. Sekanina and Farrell proposed the chain particle model to explain this situation. However, it is unnatural. 24

0019-1035/98 $25.00 Copyright  1998 by Academic Press All rights of reproduction in any form reserved.

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SYNCHRONIC BAND FORMATION IN TAILS OF COMETS

FIG. 1. Comet West (C/1975V1) on March 6.810, 1976. Many SYBs can be seen in the dust tails. The characters indicate the names of SYBs. Observer, the author; Lens, f.1. 55 mm Fno. 2.8; exposure, 8 min; emulsion; Kodak 103aE, without filter.

2. It is not clear why all the parent particles break up only at time tf . 2.2. Finite Lifetime Fragment Model Equation (1) defines tf as the integrated exposure of the fragment to solar radiation between fragmentation and observation:

tf 5

E

tobs

tf

1/hr (bf)j2 dt.

(1)

Here, r (bf) is the heliocentric distance in units of AU of the fragment of bf , and tobs is the time of observation of the SYB. By using tf values obtained by applying SFM to the observations, we found that tf , the normalized lifetime

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KIMIHIKO NISHIOKA

FIG. 2. Schematic of the fragmentation process of the first particle model. The complex particle is composed of the unit particles. The unit particle consists of the refractory particles and the absorbent particles as defined in Section 3.1. A circle indicates the refractory, and a triangle means the absorbant. The complex particles disintegrate several times into the individual unit particles. At time tf the unit particles break up and soon a SYB made of the fragments of the unit particles is formed. The fragments of the refractory particles and the absorbent particle continue to disintegrate and at last they become too small to be seen.

of the fragment that forms a SYB defined by Eq. (1), lies between 25 and 70 days for the SYBs of three comets: Comet Mrkos (C/1957P1), Comet Seki–Lines (C/1962C1), and Comet West (C/1975V1). Then, the Finite Lifetime Fragment Model (Nishioka and Watanabe 1990) (hereafter FLM) postulates that the parent particles may break up at any time by the solar radiation while the fragments must have the finite normalized lifetime: tf . The fragments created after the fragmentation of the parent particles are thought to break up continuously by the solar radiation. Consequently they become too small to be observed in the visible wavelength. The idea of continuous fragmentation is first proposed by Uemura (pers. commun., 1980). To sum up, this theory explains the form of a SYB by choosing appropriate values for the five parameters, te , bp , tf , bfmax , and bfmin , and solve problem 2 of SFM. However, the first weak point of SFM still remains unsolved even in this model. 3. FINITE LIFETIME FRAGMENT MODEL 2

3.1. Dynamic Model To solve the weak point of FLM, we propose the following Finite Lifetime Fragment Model 2 (hereafter abbreviated FLM2). This dynamic model applies, for example, to the process as in Fig. 2, which shows the formation process of a SYB. Hypothetical complex particles which consist of the aggregates of the unit particles combined by ‘‘the first glue’’ are ejected from the nucleus of the comet and after breaking up many times in the order of the sketches from left to right, fragments of unit particles form a SYB. The first glue is made of volatile matter. A unit particle is assumed to be constructed from the component particles of various materials combined by ‘‘the second glue.’’ The

second glue is also made of volatile matter. Since the solar radiation energy sublimates the first glue gradually, the complex particle disintegrates several times at various speeds as time passes by and smaller complex particles are created. Until time tf the complex particle is separated into individual unit particles. We assume that the radius of the unit particle is Sc . Next, the component particles of each unit particle break up since the second glue is sublimated by the solar radiation, and these fragments of component particles form a SYB. The component particles repeat disintegration creating smaller fragments and finally they become too small to be observed in the visible wavelength. The triggers for the disintegrations of the complex particles and unit particles will be given precisely in Section 4.2. In FLM2 it is assumed that complex particles ejected from the nucleus of the comet experience both solar gravity and solar radiation pressure and break up many times while orbiting around the Sun following Newtonian dynamics. The method of numerical calculations is essentially equal to that of Bessel (1836) and Bredichin (1886) except that the b value varies with time. The dynamic model of FLM2 is as follows. At time te , complex particles of bp0 of which radii are S0 are released from the nucleus of the comet with zero initial velocity. Even if we assume the velocity of the particles ejected from the nucleus to be 1 km/s, the calculated width of SYB is almost the same as that observed. The initial radius of the complex particle S0 has the upper limit S0u . In other words, only those particles of which radii are less than S0u are released from the nucleus. S0u is needed to limit the calculated width of the SYB (see Table I). The complex particles disintegrate several times and decrease their radii S(t) as S(t) 5 S0 exph2E(t)/cj,

(2)

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SYNCHRONIC BAND FORMATION IN TAILS OF COMETS

TABLE I Correlations between Parameters of FLM2 and the Characteristics of SYB Parameters of FLM2 Characteristics of SYB Center coordinate X, Y Inclination Length Width Maximize S0u /S0L

te

h

c

bfmax

bfmin

X

X

1

X

X

X

X

•

•

tf

SC

X

• 1 X X

1 •

S0U

X

Note. 3, correlation is strong; 1, correlation is medium; •, correlation is weak.

where

and by E(t) 5

E

t

te

1/hr (S )j2 dt9.

(3)

r (S ) denotes the heliocentric distance of the complex particle at time t9 when its radius is S(t9), and E(t) is proportional to the solar radiation energy which poured on between time te and time t per unit area. 1/c is the speed of the disintegration of the complex particle and c is the time required for the complex particle to decrease its radius 1/e at 1 AU. Equation (2) is an approximation for an expected step function S (t) between times te and tf of the complex particle. In this disintegration process, it is assumed that at any time between time te and tf all debris of the initial complex particle have the same radius where DS/Dt 5 2S/[hr (S )j2c].

(4)

In this process the numbers of the broken complex particles N increase approximately as N 5 (S0 /S )3.

(5)

Equations (2) and (3) mean that disintegrations of the complex particle to the smaller complex particles are proportional to the solar radiation energy that acts on the complex particle per unit area and the disintegration of this mechanism occurs in the case where the thickness of the first glue combining the unit particles is independent of S. It is assumed that the complex particles disintegrate at various speed within a certain range so that cmin % c % cmax . b of the complex particle: bp is a function of S and is given by bp 5 h 3 F(S ),

(6)

where F(S ) is given by F(S ) 5 0.48 3 1021.3hlog(S/0.24)j

2

for S % 0.6 em,

(7)

F(S ) 5 0.16 3 S 21.2

for S . 0.6 em.

(8)

Here, F(S ) indicates the ratio of the radiation pressure force to the gravitational force on the silicate particles (Mukai 1989), h is a proportional constant for fitting F(S ) to the b value of the real matter other than pure spherical silicate, and log means a common logarithm. According to Eqs. (7) and (8), when S0 ^ 0.24 em, bp first increases with disintegration and reaches its maximum value: bpm 5 0.48 3 h at S 5 0.24 em. After reaching its maximum bp begins to decrease. The complex particles continue to disintegrate, and at time tf , when S reaches its critical value of Sc , namely, when the complex particles are separated into the individual unit particles, they break up into fragments of various bf values. bf is the b of the fragment. After tf , the fragments move along the orbit breaking up many times by the solar radiation energy. FLM2 approximates the time variations of the b value of the process of Fig. 2 to the function illustrated in Fig. 3a. Thin line of Fig. 3a illustrates the time variations of the b of a real particle and thick line shows the approximations of them according to FLM2. We accordingly dynamically approximate bf to be constant, though bf of real fragments change with time. The validity of this approximation is represented in Fig. 3b. Therefore, Eqs. (6), (7), and (8) are applied only to the complex particles and not to the fragments of the unit particles. The fragments of the unit particles created from the complex particles of same S0 and c lie on the straight line at a certain time after tf . The complex particles with different S0 and/or c make different lines. We regard the concurrences of these lines of various S0 and c values as a SYB. It is postulated that the maximum and minimum bf values of the fragments forming one concurrence are constants for one SYB. The fragments continue to break up until they become too small to be seen at the visible wavelength. We assume that only those fragments of the unit particles that satisfy

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KIMIHIKO NISHIOKA

3.2. The Applications of FLM2 to the Observations of SYBs

FIG. 3. (a) Relationship between b values of real particle and corresponding b parameters of FLM2. The vertical axis denotes the b values of particles. The horizontal axis denotes time. The thin line indicates the time variation of b of the real particle schematically. The thick line shows the corresponding time variation of b of FLM2. The approximate time variations of b of the real particle are represented by the parameters of FLM2. (b) The time variations of the b values of the particles of FLM2 and the validity of the dynamical approximation of bf . The vertical axis denotes the b values of particles. The horizontal axis denotes time. The dashed lines indicate the time variation of b of the particle in FLM2 that forms the upper end of line ‘‘a’’ of SYB IV of Comet West in Fig. 4a. The thick lines indicate the time variation of b of the particle in FLM2 that forms the upper end of line ‘‘c’’ of SYB IV of Comet West in Fig. 4a. The thin line shows realistic time variation of b of the particle corresponding to the thick lines. Though FLM2 approximates bf to be constant, calculated position of the particle corresponding to thin line coincides with the calculated position of the particle corresponding to thick line at tobs (X 5 0.1388, Y 5 0.4528).

E

tobs

tf

1/hr (bf)j2 dt % tf

(9)

can survive and form a SYB at the time of observation: tobs , where r (bf) is the heliocentric distance of the fragment of bf (in AU) at time t. FLM2 assumes that the normalized lifetime of the fragments, tf , is a constant value which is independent on bf . In FLM2, we mentioned that a SYB is constructed from the fragments of the unit particles that compose the complex particles of which initial radii are less than upper limit value, S0u ; on the other hand, such complex particles of which radii are less than lower limit value, S0L , do not satisfy Eq. (9), so that at tobs we cannot observe at all the fragments created from the complex particles of which radii %S0L . S0L depends not only on tf but also on c.

FLM2 has eight independent parameters: te , h, c, bfmax , bfmin , tf , S0u , and Sc . On the other hand, there are the following five observational characteristics of a SYB of one observation and one condition to be explained: X,Y coordinates of the central position of a SYB, the inclination of a SYB to the X,Y coordinate, the length, the width, and the maximization of the value of S0u /S0L under the condition; fragments created from the complex particles of S0 % S0L do not satisfy Eq. (9). The last requirement arises as follows. It is desired for the theory that the brightness of a calculated SYB shall be sufficiently bright to be observed. Since the brightness of a SYB increases as the range of S0 increases, if we fix S0L , maximizing S0u /S0L is desired for the better visibility of a SYB. Therefore, the number of the parameters which can be selected arbitrarily is 8 2 6 5 2, and for example we can choose the values of te and c freely so long as they are physically reasonable. Namely, after we select the value of te and c for one observation of a SYB, the remaining six parameters are nearly uniquely determined by FLM2. Table I shows how five characteristics of the SYB and one condition are subjected to the eight parameters of FLM2. The center position of a SYB is mainly controlled by te , h, c, bfmax , bfmin . The inclination to the axis of the coordinates of a SYB is principally determined by tf . The width of a SYB depends on c, S0u , and Sc . Thus it is not allowed from the above discussion that we may choose the values of the arbitrary two parameters of FLM2. In addition, if we explain two observations of a certain SYB of several days’ interval, te value will be determined uniquely. It may be worth pointing out that in the case of SFM or FLM, we may arbitrarily choose one of the values of te , bp , and the combination of bfmax and bfmin to explain one observation of the SYB. Table II lists the parameters of FLM2 used for the calculations of the SYBs observed in the tails of two comets. The parameters of the Table II are determined as follows. First, we superpose the calculated synchrones on the photograph of the dust tail of the comet. A synchrone is a line on which particles of various b values ejected from the nucleus at the same moment align at the time of observation. Examples of the synchrones are presented in Saito et al. (1981). Next, we select the te value, the ejection time of the synchrone which intersects a certain point of the SYB between the center of the SYB and the lower end of the SYB that is closer to the nucleus of the comet. Although c covers a certain range for one SYB, for the present a ‘‘center value of c’’ is taken as a parameter. The center value of c should be selected as one to a few months, because the width of a SYB calculated by FLM2 becomes wider than that of the observed one if the center value of

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SYNCHRONIC BAND FORMATION IN TAILS OF COMETS

TABLE II Parameters of Finite Lifetime Fragment Model 2 Comet name: SYB name: Perihelion distance q (AU) Time of obs. (UT) tobs (days) te (days)a ha c (days)a cmax (days) cmin (days) tf (days)a bfmaxa bfmina Sc (em)a S0u (em)a S0L (em) S0L /Sc S0u /S0L bpm bpC zf [L 5 26.45] zf [L 5 13.225] Line name S0 (em) a bp0 tf (days) S0 b bp0 tf S0 c bp0 tf S0 d bp0 tf S0 e bp0 tf

West C/1975V1 VI 0.196626 Mar. 4.83 1976 8.6087 0.23 1.45 100 140 70 61 6 2.5 1.3 0.65 0.125 0.35 0.2 1.6 1.75 0.696 0.547 11.1 6 0.4 3 1027 20.2 6 0.9 3 1024 0.2 0.683 2.33 0.24 0.696 3.38 0.28 0.687 4.53 0.337 0.6525 6.58 0.35 0.642 7.18

West C/1975V1 VI

West C/1975V1 G2

Seki–Lines C/1962C1 2

Mar. 6.81 1976 10.589

Mar. 5.845 1976 9.6234 3.0 3.5 50 70 35 38 6 1.5 2.5 0.98 0.125 0.424 0.2 1.6 2.12 1.68 1.321 2.65 6 0.13 3 1027 9.16 6 0.47 3 1024

0.031397 Apr. 10.172 1962 8.509 0.6667 1.1 100 140 70 66 6 5.5 1.5 0.01 0.1 0.6 0.3 3.0 2.0 0.528 0.343 47.3 6 2.0 3 1027 29.6 6 2.0 3 1024

0.2 1.649 4.5 0.24 1.68 5.25 0.28 1.658 6.0 0.337 1.574 7.15 0.424 1.399 9.3

0.3 0.513 1.822 0.337 0.495 2.122 0.424 0.44 3.097 0.55 0.358 5.722 0.6 0.329 7.627

Same as left

0.215 1.72 1.63

0.215 a9 0.692 2.73

Same as left

Note. bpm is maximum value of b of the complex particle in the process of disintegration. bpc is b of the complex particle at tf . c is the center value of c. cmax is the upper limit value of c. cmin is the lower limit value of c. a, b, c, d, e; names of the lines calculated by FLM2. The concurrence of these lines form a SYB. As line ‘‘a’’ cannot be seen on Mar 6.81, 1976, yet, ‘‘a9’’ of the third column from the left instead of ‘‘a’’ means the name of the line which is formed from the smallest complex particle of which fragments can be seen at tobs . bp0 is b of the complex particle at time of ejection. The blank parameters of the third column from the left are same as those of second column. Inside of the parentheses indicates the unit. a Eight independent parameters of FLM2. SYB G2 from our findings for comet West perhaps corresponds to SYB XIV in the analysis by Sekanina and Farrell (1980).

c is lower than this range. If the center value of c exceeds that range, the S0u value decreases. This leads to the fall of the brightness of the SYB since the quantity of the fragments created from the complex particle decrease. The remaining six parameters are determined almost uniquely if we choose them so that the calculation may fit to the observation and the value of S0u /S0L may be maximized on the condition that fragments of the unit particles created

from the complex particles of S0 % S0L do not satisfy Eq. (9). After we determined the values of the eight independent parameters, the allowance of c value is investigated on the condition that the other seven parameters are fixed. Figures 4a–4d show the comparison of the observed structures of SYBs with those obtained by the theoretical calculations based on FLM2. The parameters used for the calculations of Fig. 4 are listed in Table II. Even if it was

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KIMIHIKO NISHIOKA

FIG. 4. (a) Comparison of finite lifetime fragment model 2 with observation. Dashed and dotted lines show bright and faint parts of the observed shape of SYB VI of Comet West, respectively. The concurrence of the solid lines shows the shape calculated by FLM2. The origin is the position of the Sun. The X axis is parallel to the Sun–perihelion line. The parameters of FLM2 used for the calculations are listed in the second column from the left in the Table II. (b) Comparison of finite lifetime fragment model 2 with observation. Dashed and dotted lines show bright and faint parts of the shape of SYB VI of Comet West as in (a) observed another day. The parameters of FLM2 used for the calculations are listed in the third column from the left on Table II. Other definitions are same as in (a). (c) Comparison of finite lifetime fragment model 2 with observation. Dashed and dotted lines show bright and faint parts of the observed shape of SYB G2 of Comet West, respectively. The parameters of FLM2 used for the calculations are listed in the fourth column from the left on Table II. Other definitions are same as in (a). (d) Comparison of finite lifetime fragment model 2 with observation. Dashed and dotted lines show bright and faint parts of the observed shape of SYB 2 of Comet Seki–Lines, respectively. The parameters of FLM2 used for the calculations are listed in the fifth column from the left on Table II. The photograph of the SYB 2 of Comet Seki–Lines is presented in Fig. 2 of Nishioka and Watanabe (1990). Other definitions are as in (a). (e) Comparison of finite lifetime fragment model 2 with observation. Dashed and dotted lines show bright and faint parts of the observed shape of SYB VI of Comet West, respectively. Solid lines show the theoretical shape calculated by FLM2 using lower limit c value of 70 days. Other independent parameters are same as the second column from the left of the Table II. Lines ‘‘a’’ and ‘‘b’’ have already disappeared. Solid lines with open circles at both ends show the theoretical shape calculated by FLM2 using upper limit c value of 140 days. Other independent parameters are same as in the second column from the left on Table II. Lines ‘‘c’’, ‘‘d’’, and ‘‘e’’ have not been created yet. Other definitions are same as in (a). (f) Comparison of finite lifetime fragment model 2 with observation. Dashed and dotted lines show bright and faint parts of the observed shape of SYB 2 of Comet Seki–Lines, respectively. Solid lines show the theoretical shape calculated by FLM2 using lower limit c value of 70 days. Other independent parameters are same as in the fifth column from the left on Table II. Lines ‘‘a’’, ‘‘b’’, and ‘‘c’’ have already disappeared. Solid lines with open circles at both ends show the theoretical shape calculated by FLM2 using upper limit c value of 140 days. Other independent parameters are same as in the fifth column from the left on Table II. Lines ‘‘c’’, ‘‘d’’, and ‘‘e’’ have not been created yet. Other definitions are same as in (a).

possible from the dynamical point of view to choose the sets of values other than those of the Table II, they were not physically acceptable if they were largely apart from the values listed in Table II. As indicated by Fig. 4, the calculations based on FLM2 coincide very well with observations of the SYBs for two comets with different perihelion distances. Furthermore, in the case of Comet West,

FLM2 could explain the observations of SYB VI of the interval of 2 days by the same parameters of FLM2. Figures 4e and 4f represent the calculated results using the upper limit value or the lower limit value of c. The other seven independent parameters are the same as in Table II. We can find very good agreement between observations and calculations in them. Moreover, though figures

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SYNCHRONIC BAND FORMATION IN TAILS OF COMETS

TABLE III Photographic Data on SYBs Used in This Study SYB’s Name

Date

tobs

Observer

r

Reference

Seki–Lines 2

Apr 10.172 1962

8.509

A. McClure

0.430

West VI West G2 West VI

Mar 4.831 1976 Mar 5.845 1976 Mar 6.810 1976

8.61 9.623 10.589

Y. Kato S. Sugimoto K. Nishioka

0.367 0.395 0.422

McClure, A 1962 Nishioka and Watanabe 1990 Nishioka et al. 1992 Nishioka et al. 1992 Nishioka et al. 1992

Note. Date is time of observation (UT). tobs is time of observation measured from the date of perihelion passage. r is heliocentric distance of the comet at tobs in units of AU.

are not presented, also the calculations of SYB VI on Mar. 6.81 and SYB G2 of Comet West fit very well to the observations. If c value largely exceeds cmax , no lines that form a SYB are created yet. If c value is largely less than cmin , all the lines that formed a SYB already disappeared (refer the definitions of cmax and cmin to the caption of Table II). Notwithstanding other seven parameters are fixed, two times the range of c value is allowed. Thus, the allowance of the value of c is very wide. Namely, the range of the speed of the disintegrations of the complex particles is very wide. As the real particles may disintegrate at various rates, the wide range of c value explains this situation well. Therefore, we can reach the conclusion that a SYB is a sum of the concurrence of the lines formed by fragmented complex particles with various c values. Photographic data on SYBs used in this study can be found in Table III. 4. DISCUSSION

4.1. The Physical Meaning of FLM2 We will study the physical interpretations of the values of Table II. In SFM and FLM, the problem of the uniformity of bp is not solved. On the contrary, in FLM2, the uniformity of bp is not explicitly required; however, the radii of the complex particles are restricted to values lower than S0u . This is the greatest merit of FLM2 and we solved the first problem of the models proposed so far. The reason why the value of S0 can have the range is as follows. In the fragmentation process, bp of the complex particles of which S0 is somewhat larger than 0.24 em increases at first and after reaching its maximum, bp begins to decrease. Therefore the time average of bp looks substantially independent on S0 of any disintegrating speed 1/c. As shown in the b-particle radius graphs of Figs. 7a and 7b of Burns et al. (1979), bp value reaches its maximum when the radii of the refractory (quartz, basalt, etc.) and/or the absorbant (iron, graphite, etc.) are of less than visible wavelength. Thus, FLM2 strongly limits the radii of the complex particles. The second problem of SFM is explained by FLM2

as the particles break up continuously according to the two step disintegration mechanism. Hence, FLM2 solves all the problems of the formation theories of SYB proposed so far. It is also one of the superior points of FLM2 to the previous models that the parameters of FLM2 are supported by the concrete formation process of a SYB and not given by the dynamical assumptions as the models proposed so far. As the ratio of S0L /Sc of Table II is greater than 1.6, the number of the unit particles is at least 1.63, p4 times larger than initial number of the complex particles. The values of tf are several tens of days, which are almost the same as the results of FLM. The values of c are nearly equal to or some what larger than tf . Thus, FLM2 requires the property of the fragile nature of the complex particles and the unit particles likely with FLM. The values of bf in Table II are not more than 2.5. This can be explained, if we assume that the fragments contain the absorbent materials, as the b graph of Fig. 7a of Burns et al. (1979) indicates. As the maximum value of F(s) is 0.48 and the value of h does not exceed 3.5, bp defined by Eq. (6) should be less than 1.68. This can be explained by the radius dependency of b of the solid particles as Fig. 7a of Burns et al. (1979) shows. The models of the particles which satisfy the above conditions will be shown in Section 4.2. 4.2. The Constituent of the SYB The first model of the complex particle is presented in Fig. 2. Several particles made of absorbant and refractory of which radii are less than 0.1 em are combined by the material like the second glue and form a unit particle. Many unit particles are combined by the first glue and construct a complex particle. In the case where the radii of the particle are less than 0.1 em, the b value of the absorbant is greater than that of the refractory as shown in Figs. 7a and 7b of Burns et al. (1979). Therefore, if the density of the fragment is equal to that of solid material, it is possible that the region of a SYB far from the nucleus is made of the absorbant and that the region closer to the nucleus is made either of the refractory or of both materi-

32

KIMIHIKO NISHIOKA

als. Similarly, SYB G2 contains the absorbant because the bf value of the SYB in Table II is larger than that of the refractory. On the contrary, SYB VI might be made of refractory only, because the bf value of SYB VI is comparatively small. Though the constituents of the unit particles of a certain SYB are different from those of other SYBs in the same comet, the comet contains the SYBs of the same constituents as the SYBs of other comets, since the parameters of SYB 2 of Comet Seki–Lines are similar to those of SYB VI of Comet West. Also Sekanina and Farrell (1982) pointed out that there were two kinds of SYBs consisting of different materials in Comet Mrkos (C/ 1957P1). It can be understood as below that bfmax is greater than bpc though the size of a fragment is smaller than the size of a unit particle and the radius of the unit particle is smaller than the radii which gives maximum values of bradius functions. Suppose that a unit particle is made of one absorbant particle and three refractory particles of the size that Fig. 2 illustrates. The radii of those particles are 3

Ï1/4Sc 5 0.63Sc .

(10)

If we put Sc 5 0.1 em, we get 0.063 em as a size of these constituent particles. The b values of the absorbant of 0.063 em, for example, iron and graphite, are 1.6 and 5.0 (Burns et al. 1979, Fig. 7a), respectively. Thus the bfmax values of 1.3–2.5 in Table II can be explained by the absorbant particle. On the other hand, the b value of basalt of 0.063 em as an example of the refractory is about 0.21 (Burns et al. 1979, Fig. 7a). If we assume that the density of the absorbant is equal to that of the basalt, the bpc values of the unit particle are roughly (1.6 1 0.21 3 3)/4 5 0.558 (5.0 1 0.21 3 3)/4 5 1.408, respectively. Thus it is accounted for that bfmax is greater than bpc by this first model. Another explanation is that the fragments of the unit particle receive not only solar radiation but also repulsive force by the solar wind. The smaller the size of the particle is, the greater the repulsive force by the solar wind (Mukai 1989) is. Although the repulsive force by the solar wind is about two or three orders of magnitude smaller than the solar radiation pressure and seems to be not sufficient for the value of bfmax in Table II, cometary plasma may strengthen the repulsive force. One of the candidates for the first glue is CHON found in Comet Halley (Kissel et al. 1986). As the lifetime of the organic material like CHON is still unknown, it is not certain whether it can explain c value or not. The second model for the material of SYB is the complex

particle proposed by Greenberg (1990). One unit particle is composed of some silicate cores doubly covered with organic absorbant inner coat and volatile icy outer coat, and the aggregates of the unit particles form the complex particle. With the sublimation of the volatile coat (i.e., the first glue) the complex particles disintegrate into smaller broken complex particles, and by tf they become such particles of silicate cores covered with organic absorbent. These particles correspond to the ‘‘fragments’’ in the meaning of Section 3.1 and form a SYB. As the coat is absorbant, this particle has large bf value and reflects the visible light efficiently. The organic coat (i.e., the second glue) sublimates gradually by the solar radiation and soon silicate cores emerge. As the size of this silicate is smaller than that of the core of the unit particle of the first model, we cannot observe it probably. As the organic absorbent decomposes in proportion to exp(2L/R0T ) approximately (Lamy and Perrin, 1988), and T varies as T0 / Ïr (S ), Equation (9) should be replaced by Eq. (11). We can observe only those fragments that satisfy Eq. (11),

E

tobs

tf

exp(2L/R0T ) dt % zf ,

(11)

where R0 is the gas constant(1.9872 cal mol21 deg21), T is the temperature of the organic coat (degrees), L is the latent heat of sublimation of the organic coat (cal mol21), T0 is the temperature of the organic coat at the heliocentric distance of 1 AU, and the integral is the quantity which is proportional to the total mass sublimated par unit area of the organic coat from tf to tobs ; that is zf is proportional to the normalized lifetime for the organic absorbent. zf instead of tf is the independent parameter for a normalized lifetime in this second model. L is a constant and is not an independent parameter for FLM2. T0 of the absorbant particle of submicron size is about 460 K (Lamy and Perrin 1988). The value of L of the space organics is estimated at 34.5 kcal mol21 by Lamy and Perrin (1988). We take two relatively smaller values of L (26.45, 13.225 kcal mol21) for the calculation. The calculated values of zf are listed in Table II. Other independent parameters are the same as in the first model. The values of zf are very sensitive to the value of L. The smaller the value of L is, the smaller the variations of zf of those SYBs is. Therefore, the material of smaller L value is favorable for various SYBs to have nearly the same normalized lifetime of FLM2, because zf values of smaller L value of Table II are nearly the same. The organic coat might be composed of CHON. If the lifetime of the outer layer of ice is too short, the organic refractory is a probable candidate for the outer coat instead of ice. The second model above has an advantage over the first model as follows. The sizes of the complex particle and the unit particle are about 0.6 and 0.3 em, respectively

SYNCHRONIC BAND FORMATION IN TAILS OF COMETS

(Greenberg 1990), and these unexpectedly coincide with the radii of the particles of FLM2 as listed in Table II. In addition, it is naturally explained that bfmax is greater than bpc , since in this model the components of a SYB are silicate cores covered with organic absorbent, of which b value is rather large. Thus, FLM2 strongly supports the second model as well as the first model. The third model for the materials to form the SYB is the complex particle of the unit particle consists of pure water ice core covered with organic refractory. Both materials are proposed for the candidate of the constituents of the SYB by Watanabe and Nishioka (1991). The bp value of such a particle is about same as that of silicate; it may account for bfmax being greater than bpc if the density of the organic refractory is larger than that of pure water ice. However, the lifetime of the pure water ice of 1 em radius is about 104 s at 1 AU (Patashnick and Rupprecht 1977), and it is much smaller than the value of tf in Table II. Therefore, the core which consists of organic refractory of another kind may be more favorable than that of a pure water ice core. In the above three models, the triggers for the disintegrations are gas pressure of volatile or organic inclusions caused by the solar radiation. Another possible trigger for the disintegrations of the particles is electrostatic breakup (Sekanina and Farrel 1982, Boehnhardt and Fechtig 1987). The smaller the radius of the particle, the easier it disintegrates by the electrostatic force. Sc of FLM2 may be the critical radius for electrostatic breakup. The inhomogeneity of the solar radiation pressure inside the particles may be the third mechanism for the trigger of the disintegrations of the complex particle and the unit particle. In FLM2, the sizes of the complex particles are assumed to be in the submicrometer range, and they are much smaller than the sizes of the particles which are lifted from the nucleus into the coma by the gas pressure. We think particles of sizes greater than 1 em are not fragile and form normal synchrones or disintegrate very rapidly after ejection and reduce their radii to the submicrometer range (i.e., the size of complex particles) within the inner coma. 5. CONCLUSIONS

A new dynamic model for the formation mechanism of the SYB is proposed. This theory solves all the problems of the models for the SYB formation proposed so far. Ups and downs of the time variations of the b of the particles cause the formation of SYBs. FLM2 applies, for example, to the following formation process of SYBs and suggests the properties of the particles that form the SYBs. The complex particles of the unit particle which contains at least the refractory are ejected from the nucleus of the comet and disintegrate continuously into the individual unit particles at various disintegration speeds. The unit

33

particles break up into the fragments and form a SYB. The fragments of the unit particles disintegrate continuously at the normalized lifetime of 10–100 days and finally they become too small to be observed. This model restricts not only the normalized lifetimes of those particles but also the radii of the complex particles and the radii of the unit particles, which are of less than visible wavelength. ACKNOWLEDGMENTS The author expresses my thanks to Dr. Jun-Ichi Watanabe, Dr. Kraus Richter, and Mr. Ryuichi Hasuo for their useful suggestions. We thank also Japanese amateur astronomers Yukihisa Kato, Shirou Sugimoto, and Takaaki Ozeki for supplying us with the photographic materials of the comets.

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