I N T E R S T E L L A R G R A I N SIZE S P E C T R U M AND C I R C U M S T E L L A R

G R A I N - G R A I N C O L L I S I O N S

J. D O R S C H N E R

Universitiits-Sternwarte Jena, G.D.R.

(Received 10 June, 1981)

Abstract. In this paper, grain-grain collisions, which were recently suggested by Biermann and Harwit (1980) to occur in cool circumstellar envelopes and to be responsible for the interstellar grain size spectrum, are investigated. On the basis of the author's fragmentation theory, it is shown that in the result of such collisions size distributions of the type n(a) oc a -p arise. In toe steady-state case the exponent p ranges from 3.4 to 3.7. This result matches well with grain size spectra derived from the interstellar extinction curve.

1. Introduction

Up to now, the exact shape of the interstellar grain size spectrum is unknown. Its determination represents a long-standing unsolved problem in astrophysical dust research. In spite of numerous efforts made in the last 50 years, it has so far proved impossible to derive the size-distribution function of the interstellar dust grains in a way independent of the assumptions of a special dust model. In recent years, grain size distributions described by power laws of the type n(a) oc a -v , have become favoured in the discussion on the physical properties of interstellar dust. It is a well-known fact, which has been proved by direct measurements of space probes, that micrometeoroides in interplanetary space follow distributions of this type (cf. McDonnell, 1978).

Considering permanent production of small particles in the Solar System, the present author (Dorschner, 1967) proposed that solar nebulae and evolved planetary systems in the Galaxy could account for considerable amounts of silicate dust in the interstellar space. The dust grains were suggested to originate from collisions and other disintegra- tion processes of macroscopic condensates in this circumstellar environment. In this connection, dust production by crushing of colliding objects was thoroughly studied (Dorschner, 1968, 1970, 1971).

The foundations of mathematical description of particles created by crushing colli- sions were laid in Piotrowski's (1954) pioneering work on the fragmentation of aster- oids. Independently of each other, Dorschner (1968), Dohnanyi (1969), and Hellyer (1970) tackled this problem anew and derived improved fragmentation equations. These authors also showed that fragment distributions of the type n(a) oc a - v are special solutions of the non-linear integro-differential equation describing fragmenta- tion processes in a closed system of colliding particles. A survey of all solutions discussed up to that date was given by Dorschner (1974).

Numerical calculations by Mathis et al. (1977) formed a new impact to the discussion of power law distributions in connection with interstellar dust. These authors showed that, if a dust model consisting of silicate and graphite grains is accepted, from the

Astrophysics and Space Science 81 (1982) 323-328. 0004-640X/82/0812~)323500.90. Copyright �9 1982 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

324 J. DORSCHNER

interstellar extinction curve grain size spectra of this very type can be derived in a rather direct way. The exponents p range from 3.3 to 3.6. The particle radii a must belong to the interval 0.005 ~ a ~< 0.25 #m. Mathis (1979) also demonstrated that infinite silicate cylinders following the radii distribution n(a) oc a-2.5, can explain the observed pro- perties of the interstellar polarization.

In a recent paper, Biermann and Harwit (1980) suggested that grain-grain collisions in the envelopes of red giants could be responsible for the occurrence of a general power-law spectrum n(a) = Ka-3.5 in the interstellar space. Without being aware of it, these authors revived the discussion of the significance of fragmentation processes for an explanation of the interstellar grain size spectrum, which commenced more than a decade ago.

It is the purpose of this paper to treat grain-grain collisions taking place in a nearly closed system, based on the author's fragmentation theory, in a more quantitative manner and in greater detail than the above-mentioned authors did in their letter.

Term A:

Term B: Term C:

A, B, and C are the efficiencies respectively.

2. Basic Relations of the Mathematical Description of Fragmentation

In an approximately closed system of colliding bodies or particles, the resulting size distribution of fragments is described by the integro-differential equation

Ot A ~a jol a'3N(a ',t) d a ' -

f: ,;: . . . . . . .

-- B N(a', t (a' + a")2N(a ", t)F(a, a, a , vc, Z) da da + a i

+ C N(a, t) (a + a')2N(a ', t) da' = 0. (1) *1 r o a

In this equation N(a, t) is the time-dependent distribution function of the particle radii a, which describes the state of the system with respect to fragmentation at a given time t. The function F(a, a', a"; vc, X) is the comminution law describing the fragment distribu- tion of a single crushing collision of a particle of the radius a' with another particle of the radius a". This function depends on the relative velocity of the colliding particles, v~, and on the physical properties of the particle material described symbolically by the argument X. The meaning of the three integral terms (which will hereafter represent by efficiencies of the corresponding processes, A, B, and C) in Equation (1) is as follows:

rate of change of N due to erosion of particles by cratering which only reduces the radii, but does not crush the particles; rate of change of N due to new particles created by collisions of larger ones; rate of change of N due to the loss of particles suffering from destructive collisions.

of cratering, particle creation and particle loss,

INTERSTELLAR GRAIN SIZE SPECTRUM 325

The parameter t/is the ratio of the size of the larger of the two colliding particles of a collisional event to the size of the largest fragment created in the result of this collision: i.e.,

max (a', a") t / - (2)

arnax

The parameter r o denotes the critical value of the ,size ratio r of the two colliding particles. The ratio r is defined by

min (a', a") r - max (a', a")' (3)

If r < ro, then only cratering occurs. If, however, r > r 0 then fragmentation takes place, which creates fragments distributed according to the comminution law defined by the function F.

The fragment distribution function F has the general shape

F = ~ Fijkaia'Ja "k. (4) i , j ,k

This function gets a very simple shape, if we assume that a' and a" are of comparable magnitude and that F fulfils the reasonable requirement of mass conservation during the collision. It is clear that F must be symmetric with respect to a' and a". Mass conservation implies that

~-'max(a"a")Fa3da = 2(a '3 + a"3). (5)

The factor 2 takes into account the fact that during the collision the portion (1 - 2) of the particles' material is lost by evaporation. From impact experiments and from the study of natural terrestrial rock fragmentation it follows that F oc a-~, where 7 is a material parameter. Thus, we obtain for the fragmentation function

F = 2(4 - T)tl4-~2a-~'a '~- 1, (6)

where 7, q, and 2 are characteristic parameters depending on material properties and on the relative velocity of the colliding particles (cf., Figure 1).

According to the survey of special solutions of the fragmentation equation (Dor- schner, 1974), a grain-size spectrum with the exponent p = 3.5, as it was suggested by Biermann and Harwit (1980), points to the steady-state ease. In this case the evolution of the system of colliding particles is neglected by putting ON/& = 0. This very case was first studied by Dohnanyi (1969).

In the steady-state case, the problem of solving Equation (1) reduces to finding zeros of the following transcendental equation

1 ( 1 1 ) ~7-2p - - + 2(4 -- y)2 + r~ -1 - 0. (7) 1 -- p ~ p - y 2p - - 7 ~ 3

326 J. DORSCHNER

Fig. 1.

8.0

7.0

6.0

5.0

z,.O

3.0

2.0

1.0

0.0

i i i i i i

~tkk

0.08

0.07

0.06

0.05 l

0.0/-, ~o

0.03

0.02

0.01

lb lg 2b 25 3b V c

Dependence of the collisional parameters ~ (solid line) and ro (dashed line) on the relative velocity vc (in km s- 1) of the colliding particles.

For details of the derivation, cf. Dorschner (1974): Equation (7) clearly shows the distinguished role that the case p = 3.5 plays. We get the estimate

-> 4. (8) 7 + 3 if ~ 2

3. Numerical Results for the Steady-state Case and their Discussion

In this section numerical solutions of Equation (7) for important combinations of the parameters 7, ~, and 2 will be presented. The extremely weak dependence ofp on r o has

been neglected. In order to evaluate the exponent 7, the results published by Hartmann (1969) are

used. He investigated fragment distributions of terrestrial, lunar and interplanetary rock fragmentation and found that y varies between the limits 2.8 and 4.6 for different kinds of natural and experimental rock comminution. For the evaluation of the parameters t/and r o we adopted semi-empirical relations compiled by Dohnanyi (1972, 1978) from experimental high-velocity impact results. The parameters i/ and r o are connected with Dohnanyi's parameters A and F by the relations

r I = A 1/3 and r o -- (50F) -1/3. (9)

The factor 50 within the brackets is typical of basaltic rocks. F and A are related to the impact velocity vc in the following way

F = 5v~ and A = 0.5v~. (10)

INTERSTELLAR GRAIN SIZE SPECTRUM 327

In Figure 1 the dependence of the parameters t /and r o on vc is represented. Solutions of Equation (7) for different values of tb, e parameters 7, t/, and 2 are

represented in Figure 2. This representation shows the following interesting properties of the solutions:

(i) All curves p ( j belonging to a fixed value of r/cross each other at the point 7 = 4 and reach there the singular value p = 3.5 according to the estimate (8).

(ii) The value ;t = 0.5 represents another singular case. Independent of ? and t/, at 2 = 0.5 all solutions have the constant value p = 3.5.

(iii) For small evaporation rates (Z > 0.5) p ( j is a monotonously decreasing function for all values of the parameter t/whereas large evaporation rates (2 < 0.5) result in monotonously increasing functions p( j .

(iv) Large values oft/corresponding to large relative velocities vc (cf. Figure 1) show smaller scatter of the solutions around the limiting case p = 3.5 than small values of t/do.

In circumstellar envelopes the relative grain velocities can be expected to be rather small. If we consider the value v~ = 2 km s- t , which was discussed by Biermann and Harwit (1980), then q can be expected to amount to about 2. In this case, p can reach a

32

3.6

3.5

3.4

3.7

I 3.6

o_ 3.5

3.4

3.7

3.6

3.5

3.4

;] ~10 ~=1.0 ~

0.8"

0.5

0.3 t

,-1.0 ~ r~ ~5 0.8

05

0.3

~-~.o ~ ~o2 001-- ,0.3, ~ ,

3'5. ' ' ' ' 3.0 4.0

y ,

Fig. 2. Steady-state solutions of the fragmentation equation. The exponent p of the resulting power law distribution n(a) oc a - p is represented as a function of the parameters ?, r/, and 2 (for explanation see text).

328 J. DORSCHNER

maximum value of about 3.7 if the evaporation losses are small. Large evaporation losses result in smaller values (minimum at about p = 3.4). That means, even in the idealized steady-state case the exponent p in the grain size spectrum of the circumstellar dust can considerably deviate from the commonly discussed value p = 3.5. The exact value of p depends on the combination of the parameters 7, 0, and 2 realized in the circumstellar envelope.

4. Summary and Conclusions

On the basis of the theory of fragmentation in a nearly closed system of colliding particles, steady-state solutions of the integro-differential equation describing the state of the system were investigated. The solutions are of the type n(a) oc a - p where the exponent p amounts to about 3.5. Its exact value depends on the parameters 7, ~/, and 2, which were defined in the frame of the author's fragmentation theory (Dorschner, 1968, 1970, 1971, 1974).

The results of our numerical calculation show that in circumstellar envelopes, which were suggested by Biermann and Harwit (1980) to be the environment where the interstellar grain size spectrum originates, the exponent p can be expected to range from 3.4 to 3.7. It is a very interesting coincidence that the obtained theoretical result exactly agrees with that obtained by Mathis et al. (1977) from a numerical representation of the interstellar extinction law in terms of a dust model consisting of a mixture of graphite and silicate grains. This coincidence must, however, not be overstressed, because there are many uncertainties in the dust model involved. On the other hand, the statistical treatment of circumstellar grain-grain collisions by the fragmentation theory is only a very rough description of what is going on in circumstellar envelopes. Nevertheless, the coincidence is remarkable.

References

Biermann, P. and Harwit, M.: 1980, Astrophys. J. 241, L105. Dohnanyi, J. S.: 1969, J. Geophys. Res. 74, 2531. Dohnanyi, J. S.: 1972, Icarus 17, 1. Dohnanyi, J. S.: 1978, in J. A. M. McDonnell (ed.), Cosmic Dust, John Wiley, Chichester, New York, Brisbane,

Toronto, pp. i27 IT. Dorschner, J.: 1967, Astron. Nachr. 290, 171. Dorschner, J.: 1968, 'Zur Theorie des interstellaren Staubes unter besone[erer Berficksichtigung der zirkum-

stellaren Staubentstehung', Thesis, University of Jena. Dorschner, J.: 1970, Astron. Nachr. 292, 79. Dorschner, J.: 1971, Astron. Nachr. 293, 65. Dorschner, J.: 1974, Astron. Nachr. 295, 141. Hartmann, W. K.: 1969, Icarus 10, 201. Hellyer, B.: 1970, Monthly Notices Roy. Astron. Soc. 148, 383. Mathis, J. S.: 1979, Astrophys. J. 232, 747. Mathis, J. S., Rumpl, W., and Nordsieck, K. H.: 1977, Astrophys. J. 217, 425. McDonnell, J. A. M.: 1978, in J. A. M. McDonnell (ed.), Cosmic Dust, John Wiley, Chichester, New York,

Brisbane, Toronto, pp. 33711". Piotrowski, S. L.: 1954, Acta Astron. 5, 115.