- Email: [email protected]
A kinetic model for dust coagulation
Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 1}9
A kinetic model for dust coagulation Cesare Cecchi-Pestellini *, Luigi Barletti, Aldo Belleni-Morante, Santi Aiello Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA Dipartimento di Astronomia e Scienza dello Spazio, Universita% di Firenze, Largo E. Fermi 5, I-50125, Firenze, Italy Dipartimento di Matematica U. Dini, Universita% di Firenze, Viale Morgagni 67/A, I-50134, Firenze, Italy Dipartimento di Ingegneria Civile, Universita% degli Studi di Firenze, Via S. Marta 3, I-50139, Firenze, Italy Received 10 March 2000; accepted 21 May 2000
Abstract A Boltzmann-like model is developed for particle transport in presence of coagulation, in which evolution equations for the number densities of small and large particles are derived. Unlike the standard Boltzmann equation, number densities have a dependence on the particle mass. 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Interstellar dust composed of small submicron-sized solid particles pervades interstellar space in the Milky Way and other galaxies. Despite the fact that its mass comprises a minimal fraction of the total galactic mass, interstellar dust has turned out to be an ubiquitous active factor in the galactic evolution. Dust grains decisively a!ect the thermal balance of di!use clouds, shield cloud interiors from ultraviolet di!use stellar radiation, and provide a site for heterogeneous catalysis. The existence and the properties of interstellar dust particles are inferred mainly from their interaction with radiation, in particular from the selective (frequency-dependent) extinction of stellar light. Interstellar extinction has been measured in a large interval of wavelengths from the far
* Correspondence address: Dipartimento di Astronomia e Scienza dello Spazio, Universita` di Firenze, Largo E.Fermi 5, I-50125, Firenze, Italy. Tel.: #39-055-432858; fax: #39-055-4222475. E-mail address: [email protected] (C. Cecchi-Pestellini). 0022-4073/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 0 ) 0 0 1 0 6 - 0
2
C. Cecchi-Pestellini et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 1}9
ultraviolet (+100 nm) to the infrared. The main features of the interstellar extinction law are more or less similar everywhere in the Galaxy: a mainly linear growth from the far-infrared side to a maximum value at 217.5 nm, followed by a "nal rise towards the far ultraviolet. The interstellar extinction curve has been interpreted [1] as the result of the mix of three distinct populations of solid particles, i.e. relatively large grains, responsible for the infrared and the visual extinction, small size carbonaceous grains revealing as the carriers of the 217.5 nm bump, and small size silicates producing the ultraviolet rise. Dust particles are in permanent metamorphosis. Indeed, since they are injected into the interstellar space from circumstellar formation sites, dust particles experience a variety of local conditions which may signi"cantly modify their optical and morphological properties. Far ultraviolet observations along a number of lines of sight towards "eld stars and OB associations of di!erent ages have shown di!erent dust responses to radiation, depending on the local physical conditions along the extinction paths [2,3]. As a general result, the ultraviolet portion of the extinction curve appears to be depressed in dense environments, what is considered evidence of growth of dust particles. There are two mechanisms which can lead to grain growth in dense regions: accretion of gas-phase atoms and molecules onto dust grains forming icy mantles, and coagulation of colliding grains. Evidences of the presence of molecular mantles are provided by infrared absorption features observed along the line of sight towards young stellar object embedded in dense clouds. Since, as shown by Draine [4], accretion cannot explain the signi"cant dust growth inferred by changes in the extinction curve, coagulation is considered as a more viable mechanism for an e$cient modi"cation of the dust grain mass spectrum in dense regions [5]. The physics underlying coagulation has been applied in the "elds of friction, lubri"cation, surface deformation, material sciences. Drawing on these studies, in recent years an increasing number of computational studies dedicated to the collisional dynamics of grains has been performed in the astrophysical context, both in static and dynamical media [6,7]. Chokshi et al. [5] and Dominik and Tielens [8,9] studied the microphysics of the coagulation process in the collision of two smooth, elastic, spherical grains. They found that sticking will occur when the relative collision velocity is less than a critical velocity, which depends on size, elastic properties and surface energy of the dust material. The main result of their study is an e$cient swept up by large grains of the lighter components, which thus are removed from the grain mass distribution. The results obtained by Chokshi et al. [5] are, in some sense, the starting point of the study presented in the following sections. In fact, we shall consider a Boltzmann-like model for the evolution of the number densities N and N of large and small dust particles, respectively. We * 1 note that N and N depend also on the masses of the particles under consideration. Thus, for * 1 instance, N (r, v, , t) dr dv d is the number of small particles that, at time t, are in the volume 1 element dr around the location r, have velocities in the range between v and v#dv, and are characterized by masses between and #d. In the same way, N (r, m, t) dr dm determines the * number of large particles with masses between m and m#dm contained, at time t, in a volume element dr around r. The dependence of N and N on the masses m and is a peculiarity of our model; in the * 1 standard Boltzmann equation, the number density does not depend on masses because particles have an assigned and constant mass.
C. Cecchi-Pestellini et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 1}9
3
2. The mathematical model We consider a Boltzmann-like model for a system of two species of interacting particles, characterized by the number densities N (r, m, t) of large particles and N (r, v, , t) of small * 1 particles. In what follows, we assume that (a1) N is non-zero only if m'm , where m is the smallest mass of large particles; * (a2) N does not depend on v because large particles are (approximately) at rest with respect to * some suitable reference system; (a3) N is non-zero only if ( , where is the largest mass of small particles; 1
(a4) only collisions between a small particle and a large one are considered: if the speed v"v of the small particle, of mass , is lower than a critical speed v , then the particle will stick to the large one, of mass m, whose mass will become m# after the collision; on the contrary, if v'v , the small particle will be de#ected, undergoing an elastic or inelastic scattering collision; (a5) during the time interval between two consecutive collisions, there are no forces acting on small and large particles. Assumptions (a1)}(a5) lead to the following balance equation for the density of small particles: N (r, v, , t)"![(v ) r )#v( (v)# (v))N I (r, t)]N (r, v, , t) * 1 t 1
#N I (r, t) *
R
[v (vPv)N (r,v,, t)] dv, 1
(1)
where r3R, v3R, 3(0, ], and
N I (r, t)" *
(v)"
R
K
N (r, m, t) dm, *
(2)
(vPv) dv"
R
(v ) v) dv.
(3)
Note that N I (r, t) is the spatial density of large particles at time t, independently from their mass. * In Eq. (1), the "rst term on the right-hand side represents the free streaming of small particles; the second is a sink term describing the loss of small particles due to coagulation (with cross-section (v)) and to out-scattering (with total cross-section (v) given by Eq. (3)). Finally, the third term is a source-like channel due to in-scattering (small particles with v'v collide with large particles and their velocity changes from v to v). Note that the coagulation cross-section (v) is zero if v'v because of the assumption a4, whereas, for simplicity, (v) may be assumed to be constant if 0(v(v .
4
C. Cecchi-Pestellini et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 1}9
The balance equation for large particles reads N (r, m, t)"!N (r, m, t) * t *
dv
#
dv
R
I
R
I
[v (v) N (r, v, , t)] d 1
[v (v) N (r, v, , t)N (r, m!, t)] d, 1 *
(4)
where r3R, 3[m ,#R), and N (r, m!, t)"0 if m!(m . (5) * There is no free-streaming term on the r.h.s. of Eq. (4) because of the assumption a2. Large particle free streaming would only imply some formal complication. The "rst term on the r.h.s. is a sink term due to coagulation that changes the mass m of large particles into m# (out-growing), whereas the second term changes m! into m (in-growing). We remark that Eqs. (1) and (4), together with some initial conditions for N and N , lead to an * 1 abstract semilinear evolution problem in a suitable ¸-like Banach space. By using the theory of semigroups [10] it can be shown that such an evolution problem has an unique positive (mild) solution.
3. The properties of model Eqs. (1) and (4) can be considerably simpli"ed by using the integrated densities N I given by * Eq. (2) and N I de"ned by 1 I N (r, v, , t) d. (6) N I (r, v, t)" 1 1 In fact, after integration with respect to , Eq. (1) becomes
N I (r, v, t)"![(v ) r )#v( (v)# (v))N I (r, t)]N I (r, v, t) * 1 t 1
#N I (r, t) *
R
[v (vPv)N I (r, v, t)] dv. 1
(7)
In order to integrate Eq. (4) with respect to m, it is convenient to distinguish the two regions m 4m(m # and m # 4m. If m 4m4m # , Eq. (4) reads as follows:
I [v (v) N (r, v,, t)] d N (r, m, t)"!N (r, m, t) dv 1 * t * R K\K [v (v) N (r, v,, t)]N (r, m!, t) d (8) # dv 1 * R because of relation (5). On the other hand, if m5m # then m!5m # ! 5m ,
∀3[0, ]. As a consequence, the equation for N has just the form (4). If we now integrate
*
C. Cecchi-Pestellini et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 1}9
5
Eq. (8) with respect to m, between m and m # , and Eq. (4) between m # and #R,
after some simple manipulation we obtain
N I (r, t)"!N I (r, t) * t *
R
[v (v)N I (r, v, t)] dv 1
dv v (v) R
#
dv v (v) R
#
K >I
K >
K >I
dm
K\K
dm
I
[N (r, v, , t)N (r, m!, t)] d 1 *
[N (r, v, , t)N (r, m!, t)] d. 1 *
(9)
Since
K >I
>
J "
K I
dm
K\K
K >I
[N (r, v, u, t)N (r, m!, t)] d 1 *
[N (r, v, u, t)N (r, m!, t)] dm 1 * K >IY K >I \IY d [N (r, v, u, t) N (r, m, t)] dm " 1 * K "
d
I
and J "
dm
I
[N (r, v, , t)N (r, m!, t)] d 1 *
K >I I > d [N (r, v, , t) N (r, m, t)] dm " 1 * K >I \IY we obtain that
J #J "N I (r, v, t)N I (r, t). 1 * Substitution of Eq. (10) into the right-hand side of Eq. (9) gives N I (r, t)"0, t *
(10)
(11)
which is simply the statement of the conservation of the number of large particles at r. Eq. (11) implies that N I (r, t)"N I (r,0)"N I (r) ∀t50, (12) * * * where N I (r) is the initial density of large particles (integrated with respect to m) and it can be * thought as a given function of r.
6
C. Cecchi-Pestellini et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 1}9
By substituting Eq. (12) into Eq. (7), we get an equation for N I (r, v, t), 1 N I (r, v, t)"![(v ) r )#v( (v)# (v))N I (r)]N I (r, v, t) * 1 t 1 #N I (r) *
R
v (vPv)N I (r, v, t) dv. 1
(13)
Note that Eq. (13) has the form of the classic linear Boltzmann equation, which is used to study the di!usion and transport of particles (e.g. small particles) in a given host medium (e.g. the background of large particles). We remark that Eq. (13), together with some initial condition for N I , leads to an abstract linear 1 evolution problem in the Banach space ¸(R;R). By using the theory of linear semigroups [10] it can be shown that, under suitable, technical assumptions, such a problem has an unique positive (strict) solution, such that the total number of small particles
n (t)" 1
dv
R
R
N I (r, v, t) dr 1
(14)
decreases exponentially as tP#R. In fact, the integration of both sides of Eq. (13) with respect to v gives t
R
N I (r, v, t) dv"! 1
v ) r N I (r, v, t) dv!N I (r) 1 *
R
R
v (v)N I (r, v, t) dv, 1
(15)
where we used Eq. (3). After integration of Eq. (15) with respect to r, we obtain
d n (t)"! dt 1
R
dr N I (r) *
v (v)N I (r, v, t) dv , 1
R
(16)
because
dr
R
R
v ) r N I (r, v, t) dv" 1
dv v )
R
R
r N I (r, v, t) dr "0 1
by Gauss theorem and taking into account that N I (r, v, t)P0 as rP#R. Therefore, Eq. (16) 1 shows that n (t) has a negative time derivative at any t50. 1 Furthermore, we have
R
v (v) N I (r, v, t) dv+v 1
R
N I (r, v, t) dv, 1
where v is some suitable representative velocity. Hence,
R
dr N I (r) *
R
v (v)N I (r, v, t) dv +v N I n (t), 1 * 1
C. Cecchi-Pestellini et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 1}9
7
where N I is some representative value of N I (r) in the region under consideration. Thus, we * * "nally obtain d n (t)+!v N I n (t) * 1 dt 1
(17)
and consequently n (t)+n (0) exp (!v N I t), 1 1 *
t50.
(18)
4. Discussion In this work we present a general formalism for describing particle transport in presence of coagulation. We have considered a Boltzmann-like model in which the number density depends on the particle mass. As many of time-dependent equations in mathematical physics, Eq. (1) is non-linear in nature due to the dependence on the integral of the unknown distribution function N . Integral}partial di!erential equations are usually solved by integrating forward in time from 1 given initial conditions using a predictor}corrector algorithm or a global relaxation method. The convergence behavior of these methods is di$cult to predict in advance and usually depends on the nature of the initial conditions. We will not address the di$cult subject of numerical simulation techniques. However, it is possible to estimate the timescale of removal of small grain component from the dust size distribution, using the simple exponential damping (18) derived in the previous section. Astronomical observations indicate that the visible extinction per unit length (which is mainly produced by large grains) through normal interstellar matter is A "2 mag kpc\ [11], 4 where the subscript V indicates the visible wavelength "555 nm and 1 pc corresponds to 3.018;10 cm. A is related to the usual optical depth by the relation A "1.086; . Since 4 4 4 4 "Q an ¸, where Q is the e$ciency factor for extinction, ¸ is the photon geometrical 4 path, a the grain radius, and n the numerical density of dust, then Q an " 6.0;10\ cm\. In the previous relation the quantity an is the average along the photon path. Mathis et al. [12] have proposed a simple power-law for dust-size distribution dn " n a\ da, a (a(a , (19) & K + where n is the gas density (mainly hydrogen), and the estimated lower and upper cut-o! to the size & are a "5;10\ m and a "2.5;10\ m, respectively. Choosing a "5;10\ m as smallK + * er size for large grains [13] we obtain
?+
6;10\ a\ da" an " , (20) Q ?* which gives +3.9;10\ m /H if Q "2, value appropriate for large grains at visible * wavelengths [14]. *
8
C. Cecchi-Pestellini et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 1}9
We can now estimate the numerical density of large particles
?+ N I * " a\ da"2.8;10\ * n ?* & and the coagulation cross-section 3;10\ "1.1;10\ cm, " N I /n * & where is the sticking coe$cient for coagulation. Finally, Eq. (18) reads as
(21)
(22)
n (t)+n (0) exp (!3;10\ v n t), t50. (23) 1 1 & If v (v then +1. From Ref. [5] we select v +5;10 cm/s, the characteristic velocity for a large particle with a mean size derived from Eq. (22). Substituting in Eq. (23), we "nd that small grains are swept up by the larger component on a time-scale t "2;10/n yr, which is about an order of & magnitude larger than the accretion time [15]. Consequently, this suggests that coagulation will occur when dust grains have already been accreted an ice mantle. It is evident that coagulation will be important only in regions where the gas density is in excess of 10 cm\ (t (2 Myr). As a result, the ultraviolet portion of the interstellar extinction curve in dense cloud cores will be quite di!erent from that in the di!use interstellar medium. In conclusion, dust coagulation can play a signi"cant role in the modi"cation of the ultraviolet extinction curve. There are observational evidence that the extinction properties are changing along di!erent line of sight [16] and inside the same cloud [17]. Coagulation could drive the process operating on dust size distribution. Since, as expected, the coagulation time-scale is decreasing with increasing density, the observed modi"cations could be explained by a systematic increase of dust size from boundary to center of the cloud. The solution to the Eq. (1), which gives the local action of coagulation, could support this scenario.
Acknowledgements The present work was supported by Italian MURST and by CNR-GIFCO. The research of C.C-P. was supported by the Division of Astronomy of the US National Science Foundation.
References [1] Hong SS, Greenberg JM. Astron Astrophys 1980;88:194. [2] Aiello S, Barsella B, Chlewicki G, Greenberg JM, Patriarchi P, Perinotto M. Astron Astrophys Suppl Ser 1988;73:195. [3] Fitzpatrick EL, Massa D. Astrophys J Suppl Ser 1990;72:163. [4] Draine BT, Protostars and Planets II. University of Arizona Press, 1985. p. 621. [5] Chokshi A, Tielens AGGM, Hollenbach D. Astrophys J 1993;407:806. [6] Ossenkopf V. Astron Astrophys 1993;280:617. [7] Weidenschilling SJ, Ruzmaikina TV. Astrophys J 1994;430:713. [8] Dominik C, Tielens AGGM. Phil Mag A 1995;72:783.
C. Cecchi-Pestellini et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 1}9 [9] [10] [11] [12] [13] [14] [15] [16] [17]
Dominik C, Tielens AGGM. Phil Mag A 1996;73:1279. Belleni-Morante A, McBride AC. Applied nonlinear semigroups. New York: Wiley, 1998. Duley WW, Williams DA. Interstellar Chemistry. London: Academic Press, 1984. Mathis JS, Rumpl W, Nordsiek KH. Astrophys J 1977;217:425. Cecchi-Pestellini C, Williams DA. Mon Not R Astron Soc 1998;296:414. Bohren CF, Hu!man DR. Absorption and Scattering of Light by Small Particles. New York: Wiley, 1983. Whittet DCB. Dust in the Galactic Environment. London: Institute of Physics Publishing, 1992. Cardelli JA, Clayton GC, Mathis JS. Astrophys J 1989;329:L33. Vrba FJ, Rydgren AE. Astrophys J Suppl Ser 1984;283:123.
9
A kinetic model for dust coagulation Cesare Cecchi-Pestellini *, Luigi Barletti, Aldo Belleni-Morante, Santi Aiello Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA Dipartimento di Astronomia e Scienza dello Spazio, Universita% di Firenze, Largo E. Fermi 5, I-50125, Firenze, Italy Dipartimento di Matematica U. Dini, Universita% di Firenze, Viale Morgagni 67/A, I-50134, Firenze, Italy Dipartimento di Ingegneria Civile, Universita% degli Studi di Firenze, Via S. Marta 3, I-50139, Firenze, Italy Received 10 March 2000; accepted 21 May 2000
Abstract A Boltzmann-like model is developed for particle transport in presence of coagulation, in which evolution equations for the number densities of small and large particles are derived. Unlike the standard Boltzmann equation, number densities have a dependence on the particle mass. 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Interstellar dust composed of small submicron-sized solid particles pervades interstellar space in the Milky Way and other galaxies. Despite the fact that its mass comprises a minimal fraction of the total galactic mass, interstellar dust has turned out to be an ubiquitous active factor in the galactic evolution. Dust grains decisively a!ect the thermal balance of di!use clouds, shield cloud interiors from ultraviolet di!use stellar radiation, and provide a site for heterogeneous catalysis. The existence and the properties of interstellar dust particles are inferred mainly from their interaction with radiation, in particular from the selective (frequency-dependent) extinction of stellar light. Interstellar extinction has been measured in a large interval of wavelengths from the far
* Correspondence address: Dipartimento di Astronomia e Scienza dello Spazio, Universita` di Firenze, Largo E.Fermi 5, I-50125, Firenze, Italy. Tel.: #39-055-432858; fax: #39-055-4222475. E-mail address: [email protected] (C. Cecchi-Pestellini). 0022-4073/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 0 ) 0 0 1 0 6 - 0
2
C. Cecchi-Pestellini et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 1}9
ultraviolet (+100 nm) to the infrared. The main features of the interstellar extinction law are more or less similar everywhere in the Galaxy: a mainly linear growth from the far-infrared side to a maximum value at 217.5 nm, followed by a "nal rise towards the far ultraviolet. The interstellar extinction curve has been interpreted [1] as the result of the mix of three distinct populations of solid particles, i.e. relatively large grains, responsible for the infrared and the visual extinction, small size carbonaceous grains revealing as the carriers of the 217.5 nm bump, and small size silicates producing the ultraviolet rise. Dust particles are in permanent metamorphosis. Indeed, since they are injected into the interstellar space from circumstellar formation sites, dust particles experience a variety of local conditions which may signi"cantly modify their optical and morphological properties. Far ultraviolet observations along a number of lines of sight towards "eld stars and OB associations of di!erent ages have shown di!erent dust responses to radiation, depending on the local physical conditions along the extinction paths [2,3]. As a general result, the ultraviolet portion of the extinction curve appears to be depressed in dense environments, what is considered evidence of growth of dust particles. There are two mechanisms which can lead to grain growth in dense regions: accretion of gas-phase atoms and molecules onto dust grains forming icy mantles, and coagulation of colliding grains. Evidences of the presence of molecular mantles are provided by infrared absorption features observed along the line of sight towards young stellar object embedded in dense clouds. Since, as shown by Draine [4], accretion cannot explain the signi"cant dust growth inferred by changes in the extinction curve, coagulation is considered as a more viable mechanism for an e$cient modi"cation of the dust grain mass spectrum in dense regions [5]. The physics underlying coagulation has been applied in the "elds of friction, lubri"cation, surface deformation, material sciences. Drawing on these studies, in recent years an increasing number of computational studies dedicated to the collisional dynamics of grains has been performed in the astrophysical context, both in static and dynamical media [6,7]. Chokshi et al. [5] and Dominik and Tielens [8,9] studied the microphysics of the coagulation process in the collision of two smooth, elastic, spherical grains. They found that sticking will occur when the relative collision velocity is less than a critical velocity, which depends on size, elastic properties and surface energy of the dust material. The main result of their study is an e$cient swept up by large grains of the lighter components, which thus are removed from the grain mass distribution. The results obtained by Chokshi et al. [5] are, in some sense, the starting point of the study presented in the following sections. In fact, we shall consider a Boltzmann-like model for the evolution of the number densities N and N of large and small dust particles, respectively. We * 1 note that N and N depend also on the masses of the particles under consideration. Thus, for * 1 instance, N (r, v, , t) dr dv d is the number of small particles that, at time t, are in the volume 1 element dr around the location r, have velocities in the range between v and v#dv, and are characterized by masses between and #d. In the same way, N (r, m, t) dr dm determines the * number of large particles with masses between m and m#dm contained, at time t, in a volume element dr around r. The dependence of N and N on the masses m and is a peculiarity of our model; in the * 1 standard Boltzmann equation, the number density does not depend on masses because particles have an assigned and constant mass.
C. Cecchi-Pestellini et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 1}9
3
2. The mathematical model We consider a Boltzmann-like model for a system of two species of interacting particles, characterized by the number densities N (r, m, t) of large particles and N (r, v, , t) of small * 1 particles. In what follows, we assume that (a1) N is non-zero only if m'm , where m is the smallest mass of large particles; * (a2) N does not depend on v because large particles are (approximately) at rest with respect to * some suitable reference system; (a3) N is non-zero only if ( , where is the largest mass of small particles; 1
(a4) only collisions between a small particle and a large one are considered: if the speed v"v of the small particle, of mass , is lower than a critical speed v , then the particle will stick to the large one, of mass m, whose mass will become m# after the collision; on the contrary, if v'v , the small particle will be de#ected, undergoing an elastic or inelastic scattering collision; (a5) during the time interval between two consecutive collisions, there are no forces acting on small and large particles. Assumptions (a1)}(a5) lead to the following balance equation for the density of small particles: N (r, v, , t)"![(v ) r )#v( (v)# (v))N I (r, t)]N (r, v, , t) * 1 t 1
#N I (r, t) *
R
[v (vPv)N (r,v,, t)] dv, 1
(1)
where r3R, v3R, 3(0, ], and
N I (r, t)" *
(v)"
R
K
N (r, m, t) dm, *
(2)
(vPv) dv"
R
(v ) v) dv.
(3)
Note that N I (r, t) is the spatial density of large particles at time t, independently from their mass. * In Eq. (1), the "rst term on the right-hand side represents the free streaming of small particles; the second is a sink term describing the loss of small particles due to coagulation (with cross-section (v)) and to out-scattering (with total cross-section (v) given by Eq. (3)). Finally, the third term is a source-like channel due to in-scattering (small particles with v'v collide with large particles and their velocity changes from v to v). Note that the coagulation cross-section (v) is zero if v'v because of the assumption a4, whereas, for simplicity, (v) may be assumed to be constant if 0(v(v .
4
C. Cecchi-Pestellini et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 1}9
The balance equation for large particles reads N (r, m, t)"!N (r, m, t) * t *
dv
#
dv
R
I
R
I
[v (v) N (r, v, , t)] d 1
[v (v) N (r, v, , t)N (r, m!, t)] d, 1 *
(4)
where r3R, 3[m ,#R), and N (r, m!, t)"0 if m!(m . (5) * There is no free-streaming term on the r.h.s. of Eq. (4) because of the assumption a2. Large particle free streaming would only imply some formal complication. The "rst term on the r.h.s. is a sink term due to coagulation that changes the mass m of large particles into m# (out-growing), whereas the second term changes m! into m (in-growing). We remark that Eqs. (1) and (4), together with some initial conditions for N and N , lead to an * 1 abstract semilinear evolution problem in a suitable ¸-like Banach space. By using the theory of semigroups [10] it can be shown that such an evolution problem has an unique positive (mild) solution.
3. The properties of model Eqs. (1) and (4) can be considerably simpli"ed by using the integrated densities N I given by * Eq. (2) and N I de"ned by 1 I N (r, v, , t) d. (6) N I (r, v, t)" 1 1 In fact, after integration with respect to , Eq. (1) becomes
N I (r, v, t)"![(v ) r )#v( (v)# (v))N I (r, t)]N I (r, v, t) * 1 t 1
#N I (r, t) *
R
[v (vPv)N I (r, v, t)] dv. 1
(7)
In order to integrate Eq. (4) with respect to m, it is convenient to distinguish the two regions m 4m(m # and m # 4m. If m 4m4m # , Eq. (4) reads as follows:
I [v (v) N (r, v,, t)] d N (r, m, t)"!N (r, m, t) dv 1 * t * R K\K [v (v) N (r, v,, t)]N (r, m!, t) d (8) # dv 1 * R because of relation (5). On the other hand, if m5m # then m!5m # ! 5m ,
∀3[0, ]. As a consequence, the equation for N has just the form (4). If we now integrate
*
C. Cecchi-Pestellini et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 1}9
5
Eq. (8) with respect to m, between m and m # , and Eq. (4) between m # and #R,
after some simple manipulation we obtain
N I (r, t)"!N I (r, t) * t *
R
[v (v)N I (r, v, t)] dv 1
dv v (v) R
#
dv v (v) R
#
K >I
K >
K >I
dm
K\K
dm
I
[N (r, v, , t)N (r, m!, t)] d 1 *
[N (r, v, , t)N (r, m!, t)] d. 1 *
(9)
Since
K >I
>
J "
K I
dm
K\K
K >I
[N (r, v, u, t)N (r, m!, t)] d 1 *
[N (r, v, u, t)N (r, m!, t)] dm 1 * K >IY K >I \IY d [N (r, v, u, t) N (r, m, t)] dm " 1 * K "
d
I
and J "
dm
I
[N (r, v, , t)N (r, m!, t)] d 1 *
K >I I > d [N (r, v, , t) N (r, m, t)] dm " 1 * K >I \IY we obtain that
J #J "N I (r, v, t)N I (r, t). 1 * Substitution of Eq. (10) into the right-hand side of Eq. (9) gives N I (r, t)"0, t *
(10)
(11)
which is simply the statement of the conservation of the number of large particles at r. Eq. (11) implies that N I (r, t)"N I (r,0)"N I (r) ∀t50, (12) * * * where N I (r) is the initial density of large particles (integrated with respect to m) and it can be * thought as a given function of r.
6
C. Cecchi-Pestellini et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 1}9
By substituting Eq. (12) into Eq. (7), we get an equation for N I (r, v, t), 1 N I (r, v, t)"![(v ) r )#v( (v)# (v))N I (r)]N I (r, v, t) * 1 t 1 #N I (r) *
R
v (vPv)N I (r, v, t) dv. 1
(13)
Note that Eq. (13) has the form of the classic linear Boltzmann equation, which is used to study the di!usion and transport of particles (e.g. small particles) in a given host medium (e.g. the background of large particles). We remark that Eq. (13), together with some initial condition for N I , leads to an abstract linear 1 evolution problem in the Banach space ¸(R;R). By using the theory of linear semigroups [10] it can be shown that, under suitable, technical assumptions, such a problem has an unique positive (strict) solution, such that the total number of small particles
n (t)" 1
dv
R
R
N I (r, v, t) dr 1
(14)
decreases exponentially as tP#R. In fact, the integration of both sides of Eq. (13) with respect to v gives t
R
N I (r, v, t) dv"! 1
v ) r N I (r, v, t) dv!N I (r) 1 *
R
R
v (v)N I (r, v, t) dv, 1
(15)
where we used Eq. (3). After integration of Eq. (15) with respect to r, we obtain
d n (t)"! dt 1
R
dr N I (r) *
v (v)N I (r, v, t) dv , 1
R
(16)
because
dr
R
R
v ) r N I (r, v, t) dv" 1
dv v )
R
R
r N I (r, v, t) dr "0 1
by Gauss theorem and taking into account that N I (r, v, t)P0 as rP#R. Therefore, Eq. (16) 1 shows that n (t) has a negative time derivative at any t50. 1 Furthermore, we have
R
v (v) N I (r, v, t) dv+v 1
R
N I (r, v, t) dv, 1
where v is some suitable representative velocity. Hence,
R
dr N I (r) *
R
v (v)N I (r, v, t) dv +v N I n (t), 1 * 1
C. Cecchi-Pestellini et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 1}9
7
where N I is some representative value of N I (r) in the region under consideration. Thus, we * * "nally obtain d n (t)+!v N I n (t) * 1 dt 1
(17)
and consequently n (t)+n (0) exp (!v N I t), 1 1 *
t50.
(18)
4. Discussion In this work we present a general formalism for describing particle transport in presence of coagulation. We have considered a Boltzmann-like model in which the number density depends on the particle mass. As many of time-dependent equations in mathematical physics, Eq. (1) is non-linear in nature due to the dependence on the integral of the unknown distribution function N . Integral}partial di!erential equations are usually solved by integrating forward in time from 1 given initial conditions using a predictor}corrector algorithm or a global relaxation method. The convergence behavior of these methods is di$cult to predict in advance and usually depends on the nature of the initial conditions. We will not address the di$cult subject of numerical simulation techniques. However, it is possible to estimate the timescale of removal of small grain component from the dust size distribution, using the simple exponential damping (18) derived in the previous section. Astronomical observations indicate that the visible extinction per unit length (which is mainly produced by large grains) through normal interstellar matter is A "2 mag kpc\ [11], 4 where the subscript V indicates the visible wavelength "555 nm and 1 pc corresponds to 3.018;10 cm. A is related to the usual optical depth by the relation A "1.086; . Since 4 4 4 4 "Q an ¸, where Q is the e$ciency factor for extinction, ¸ is the photon geometrical 4 path, a the grain radius, and n the numerical density of dust, then Q an " 6.0;10\ cm\. In the previous relation the quantity an is the average along the photon path. Mathis et al. [12] have proposed a simple power-law for dust-size distribution dn " n a\ da, a (a(a , (19) & K + where n is the gas density (mainly hydrogen), and the estimated lower and upper cut-o! to the size & are a "5;10\ m and a "2.5;10\ m, respectively. Choosing a "5;10\ m as smallK + * er size for large grains [13] we obtain
?+
6;10\ a\ da" an " , (20) Q ?* which gives +3.9;10\ m /H if Q "2, value appropriate for large grains at visible * wavelengths [14]. *
8
C. Cecchi-Pestellini et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 1}9
We can now estimate the numerical density of large particles
?+ N I * " a\ da"2.8;10\ * n ?* & and the coagulation cross-section 3;10\ "1.1;10\ cm, " N I /n * & where is the sticking coe$cient for coagulation. Finally, Eq. (18) reads as
(21)
(22)
n (t)+n (0) exp (!3;10\ v n t), t50. (23) 1 1 & If v (v then +1. From Ref. [5] we select v +5;10 cm/s, the characteristic velocity for a large particle with a mean size derived from Eq. (22). Substituting in Eq. (23), we "nd that small grains are swept up by the larger component on a time-scale t "2;10/n yr, which is about an order of & magnitude larger than the accretion time [15]. Consequently, this suggests that coagulation will occur when dust grains have already been accreted an ice mantle. It is evident that coagulation will be important only in regions where the gas density is in excess of 10 cm\ (t (2 Myr). As a result, the ultraviolet portion of the interstellar extinction curve in dense cloud cores will be quite di!erent from that in the di!use interstellar medium. In conclusion, dust coagulation can play a signi"cant role in the modi"cation of the ultraviolet extinction curve. There are observational evidence that the extinction properties are changing along di!erent line of sight [16] and inside the same cloud [17]. Coagulation could drive the process operating on dust size distribution. Since, as expected, the coagulation time-scale is decreasing with increasing density, the observed modi"cations could be explained by a systematic increase of dust size from boundary to center of the cloud. The solution to the Eq. (1), which gives the local action of coagulation, could support this scenario.
Acknowledgements The present work was supported by Italian MURST and by CNR-GIFCO. The research of C.C-P. was supported by the Division of Astronomy of the US National Science Foundation.
References [1] Hong SS, Greenberg JM. Astron Astrophys 1980;88:194. [2] Aiello S, Barsella B, Chlewicki G, Greenberg JM, Patriarchi P, Perinotto M. Astron Astrophys Suppl Ser 1988;73:195. [3] Fitzpatrick EL, Massa D. Astrophys J Suppl Ser 1990;72:163. [4] Draine BT, Protostars and Planets II. University of Arizona Press, 1985. p. 621. [5] Chokshi A, Tielens AGGM, Hollenbach D. Astrophys J 1993;407:806. [6] Ossenkopf V. Astron Astrophys 1993;280:617. [7] Weidenschilling SJ, Ruzmaikina TV. Astrophys J 1994;430:713. [8] Dominik C, Tielens AGGM. Phil Mag A 1995;72:783.
C. Cecchi-Pestellini et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 70 (2001) 1}9 [9] [10] [11] [12] [13] [14] [15] [16] [17]
Dominik C, Tielens AGGM. Phil Mag A 1996;73:1279. Belleni-Morante A, McBride AC. Applied nonlinear semigroups. New York: Wiley, 1998. Duley WW, Williams DA. Interstellar Chemistry. London: Academic Press, 1984. Mathis JS, Rumpl W, Nordsiek KH. Astrophys J 1977;217:425. Cecchi-Pestellini C, Williams DA. Mon Not R Astron Soc 1998;296:414. Bohren CF, Hu!man DR. Absorption and Scattering of Light by Small Particles. New York: Wiley, 1983. Whittet DCB. Dust in the Galactic Environment. London: Institute of Physics Publishing, 1992. Cardelli JA, Clayton GC, Mathis JS. Astrophys J 1989;329:L33. Vrba FJ, Rydgren AE. Astrophys J Suppl Ser 1984;283:123.
9