NonPower Law Behavior in Fragmentation Cascades
Abstract
Collisions resulting in fragmentation are important in shaping the mass spectrum of minor bodies in the asteroid belt, the Kuiper belt, and debris disks. Models of fragmentation cascades typically find that in steadystate, the solution for the particle mass distribution is a power law in the mass. However, previous studies have typically assumed that the mass of the largest fragment produced in a collision with just enough energy to shatter the target and disperse half its mass to infinity is directly proportional to the target mass. We show that if this assumption is not satisfied, then the power law solution for the steadystate particle mass distribution is modified by a multiplicative factor, which is a slowly varying function of the mass. We derive analytic solutions for this correction factor and confirm our results numerically. We find that this correction factor proves important when extrapolating over many orders of magnitude in mass, such as when inferring the number of large objects in a system based on infrared observations. In the course of our work, we have also discovered an unrelated type of nonpower law behavior: waves can persist in the mass distribution of objects even in the absence of upper or lower cutoffs to the mass distribution or breaks in the strength law.
1 Introduction.
Collisional evolution of manybody astrophysical systems in which the relative velocity between colliding objects is large compared to the escape speed and coagulation is unimportant is dominated by fragmentation. Examples of such systems include the asteroid belt (Durda & Dermott, 1997; Bottke et al., 2005), the Kuiper Belt (Davis & Farinella, 1997; Pan & Sari, 2005), and debris disks around young stars (Kennedy & Wyatt, 2011; Kenyon & Bromley, 2010). As the large bodies are slowly ground down, a collisional cascade is launched, in which mass flows unidirectionally to smaller objects until at some scale it is flushed out of the system by some removal process (e.g. by PoyntingRobertson drag, radiation pressure, or gas drag).
In real astrophysical systems there is normally a large dynamic range between the scale at which mass is injected into the cascade and the scale at which mass is removed from the system. The mass can be defined as the mass for which the collisional destruction timescale is comparable to the age of the system. In highly evolved systems, such as the Kuiper Belt, this scale corresponds to the characteristic mass of the largest bodies (Pan & Sari, 2005; Fraser, 2009). The collision time for the smallest bodies is usually much shorter than for the largest ones, so a steadystate can be set up for . In such a steadystate, the mass distribution of particles evolves on the collision timescale of bodies at the injection mass scale and can be considered static on shorter timescales.
Dohnanyi (1969) was the first to construct a model of a fragmentation cascade that aimed at explaining the mass distribution of objects in the asteroid belt. He assumed that the internal strength of colliding objects is independent of mass with the implication that , which we define as the mass of the smallest projectile capable of dispersing one half the mass of a target of mass to infinity, has its mass linearly proportional to . Dohnanyi also assumed that , the mass of the largest fragment produced in a collision between a target of mass and a projectile of mass , scales linearly with and is independent of . Under these assumptions, he showed that a fragmentation cascade allows a steadystate power law solution
(1) 
where is the number of objects in the size distribution with mass between and .
Later, Tanaka et al. (1996) generalized Dohnanyi’s result by assuming a selfsimilar model of fragmentation, which again entails , but now , where is an arbitrary function. They confirmed the power law form of the mass spectrum and showed that the value of is determined by the mass dependence of the collision rate and that reduces to if the collision rate is proportional to (geometrical crosssection with massindependent relative velocities).
In reality, however, fragmentation does not have to be selfsimilar because the minimum energy necessary for disruption is not always linearly proportional to the target mass. For instance, if an object’s internal strength is dominated by gravity, then , the energy per unit mass required to shatter an object and disperse half of its mass to infinity, scales as , which is the case for objects larger than km (Benz & Asphaug, 1999; Holsapple, 1993; Benz et al, 1994). O’Brien & Greenberg (2003) considered a model in which scales as a power law in , but took linearly proportional to . Under these assumptions, O’Brien & Greenberg (2003) found that a steadystate power law solution for still exists, but that differs from unless is constant, even when the collision rate scales as .
The steadystate power law solutions of Dohnanyi (1969), Tanaka et al. (1996), and O’Brien & Greenberg (2003) have since been confirmed numerous times by simulations, which have also shed light on nonpower law effects present in astrophysical fragmentation cascades. These effects typically manifest themselves as waves superimposed on top of the steadystate power law solution and are caused either by noncollisional mass sinks, e.g. due to the removal of small particles ( m for ) by radiation pressure (Thébault & Augereau, 2007; Campo Bagatin et al., 1994; Durda & Dermott, 1997); a change in the power law index of induced by a transition from the strengthdominated to the gravitydominated regime (O’Brien & Greenberg, 2003, 2005); or a transition from a primordial to a collisionallyevolved size distribution (Fraser, 2009; Pan & Sari, 2005; Kenyon & Bromley, 2004).
In this work, we describe a new source of nonpower law behavior in fragmentation cascades. We consider a model similar to the one used by O’Brien & Greenberg (2003), but with a more general form for . More specifically, previous researchers (Petit & Farinella, 1993; O’Brien & Greenberg, 2005; Williams & Wetherill, 1994; Kobayashi & Tanaka, 2010; de Elía & Brunini, 2007) have typically assumed that , where is the kinetic energy of the colliding particles in the center of mass frame, and is a function that varies from author to author. We instead consider the more general dependence , which is motivated in §2. We find that unless , there is no steadystate power law solution. Instead, is described by a power law with the same power law index as found by O’Brien & Greenberg (2003), but multiplied by a slowly varying function of mass (i.e. ). The nonpower law effects caused by the slowly varying function show up as a smooth deviation from powerlaw behavior, which is quite different from the wavelike features described earlier. This deviation is significant when extrapolating over many orders of magnitude in mass, as demonstrated in §6.3.
In the course of our investigation, we have also discovered that it is possible for waves to appear and persist in a collisional cascade, even if the particle size distribution does not contain an upper or lower mass cutoff, and the strength is described by a pure power law with no breaks. This is a completely independent type of nonpower law behavior from the one caused by not proportional to . In astrophysical systems, such waves may be triggered by stochastic collision events between large planetesimals (Kenyon & Bromley, 2005; Wyatt & Dent, 2002; Wyatt, 2008). However, the main focus of our paper is on the nonpower law behavior that results when is not proportional to , and we defer a detailed discussion of these waves for the future.
The paper is organized as follows. In §2, we introduce the general equations describing fragmentation and discuss specific assumptions relevant to our model. In §3 we demonstrate that pure power law solutions for a fragmentation cascade are indeed possible if , consistent with Kobayashi & Tanaka (2010). We show in §4 that if the assumption is not satisfied, then solutions are given by the product of a power law with the same index as in the case, and a slowly varying function of mass. We then find analytic solutions for this slowly varying function for monodisperse (all fragments having the same mass) and power law fragment mass distributions. We find that in the monodisperse case, the solutions for the steadystate distribution are not unique and can support waves. We confirm our analytical results numerically in §5 and discuss their validity and possible applications in §6.
2 Basic setup
The number density distribution of particles in mass space, , obeys the continuity equation (Tanaka et al., 1996):
(2) 
where the mass flux, is defined to be the amount of mass which flows past a point in mass space per unit time. We will consider the evolution of the mass spectrum for in which case we can adopt the steadystate assumption. Under this assumption, is constant, so there is no accumulation of particles at any scale.
In order to write down the explicit form of , we need to make some definitions. Disruption of a target in a collision produces a spectrum of fragments characterized by the function , where is the number of fragments in the mass interval coming only from the target in a collision between bodies with mass and . Mass conservation then requires that
(3) 
Kobayashi & Tanaka (2010) have shown that erosive collisions provide the dominant contribution to the mass flux, and in order to take them into account, it is useful to split into two components
(4) 
Here, is the contribution to the mass flux from the continuous distribution of fragments ejected from the target and is normalized such that
(5) 
where is the total amount of mass ejected from the target and dispersed to infinity. In catastrophic collisions with , . However, in collisions with , the core of the target is left almost intact (Fujiwara et al., 1977), leaving behind a remnant of mass . This yields
(6) 
When , then by definition of , . A collision is commonly referred to as catastrophic when and as erosive when . We also clarify that is the largest fragment in the continuous distribution of ejecta, , to which does not belong.
We next define as the mass fraction of debris with , which comes from the target only in a collision between a target of mass and a projectile of mass :
(7) 
This is consistent with Tanaka et al. (1996) and Kobayashi & Tanaka (2010) and allows for projectiles that are larger than targets. We next split into two parts, just as we did with :
(8) 
Using the definitions (5), (6), and (7), we have
(9) 
and
(10) 
We can now write down the mass flux in the form
(11) 
where is the collision rate between bodies with mass and . The utility of splitting into two components (Eq. 8) will become apparent shortly in §2.2.
2.1 Some simplifications
When gravitational focusing is unimportant, the collision rate is given by
(12) 
where is an averaged collision velocity, which can be a function of and . To limit the number of parameters in our study, we will assume
(13) 
so the collision rate becomes
(14) 
but our results are easily extended to the forms of considered by Tanaka et al. (1996), who varied the power law dependence of on and .
It is natural to expect that disruption of targets with mass is dominated by collisions with projectiles having masses near or below the breaking threshold . This is because the crosssection for catastrophic collisions (defined by (§2)) is dominated by the smallest particles as long as (Dohnanyi, 1969), and the mass flux from erosive collisions drops off for (Kobayashi & Tanaka, 2010). We now assume
(15) 
which is typically valid for astrophysical fragmentation cascades. Then, for collisions that are responsible for the majority of the mass flux. This allows us to rewrite (11) in the following form:
(16) 
The normalization of is unimportant since the only thing that matters for a steadystate solution is that is constant.
2.2 Fragmentation Model
Experimental data on collisional breakup (Gault & Wedekind, 1969; Fujiwara et al., 1977)
and numerical simulations of highvelocity collisions
(Benz et al, 1994; Benz
& Asphaug, 1999) suggest that the
mass spectrum of fragments ejected from the target in a single collision
can be reasonably well fit within a broad range of masses by a
powerlaw with a cutoff at
(17) 
Equation (17) is difficult to analyze for arbitrary
dependencies of and . Thus, we make the
simplification, motivated in §2.3 that
has the form
(18) 
In the limit ,
(19) 
which together with the assumptions (15) and the
definition
(20) 
implies
(21) 
It now helps to define the variables
(22)  
(23)  
(24) 
Then, with the form of given by Eq. (18), becomes
(25) 
The normalization of is such that , for , which follows from Eq. (5) and Eq. (9).
The prefactor in Eq. (25) can be written as
(26) 
If we now make the assumption that
(27) 
we can absorb the prefactor in Eq. (25) and write
(28) 
One can consider more general prescriptions for than Eq. (27), but the latter suffices to illustrate the nonpower law behavior. It also follows from Eq. (27) that has the form
(29) 
so that finally we arrive at
(30) 
2.3 Nonlinear scaling of .
We now address the natural question of whether one should expect a nonlinear scaling of in practice? Experimental (Fujiwara et al., 1977) and numerical (Benz & Asphaug, 1999) studies of collisional fragmentation suggest that the mass of the largest fragment formed in a highspeed catastrophic collision decreases with increasing collision energy. In particular, we focus on the experiments of Fujiwara et al. (1977), who fired polycarbonate projectiles of a constant mass and kinetic energy ( g km ) into basalt targets with masses in the range . They fit their data in the catastrophic regime with the following relation:
(31) 
The vertical bar denotes the fact that the experiments of Fujiwara et al. (1977) were performed at constant projectile mass and velocity. Since for , what Fujiwara et al. (1977) have shown is that . However, we now make the following extension to their results. We assume Eq. (31) to be valid as a function of as well, so we write
(32) 
In the highly catastrophic fragmentation regime (), we expect all of the fragments to be a part of the continuous fragment distribution, so that there is no remnant mass remaining (). This means we can simplify Eq. (32) to the form
(33) 
Using in Eq. (19) and making the usual assumption (13) then yields
(34) 
Thus, unless , is not proportional to . Moreover, we see that if varies as a power law, then varies as a power law as well. One assumption that we have made in deriving Eq. (34) is that Eq. (33) is valid for , even though Fujiwara et al. (1977) obtained the power law relationship (Eq. (31)) by fitting primarily to data in the regime . Nevertheless, our analysis has shown that a power law dependence for is plausible, and we discuss the matter further in §6.2.
We now deduce what we would expect for the power law exponent of in the range of target masses considered by Fujiwara et al. (1977). Experiments and simulations (Benz & Asphaug, 1999; Holsapple, 1993; Housen et al., 1991; Benz et al, 1994) show that is welldescribed over a large range of masses by the expression
(35) 
where the exponent is different for the strengthdominated and gravitydominated regimes. If is constant as we have been assuming, then it follows from Eq. (21) that
(36) 
and consequently
(37) 
From simulations of impacts into basalt with km s, Benz & Asphaug (1999) found that in the strengthdominated regime. Using , the value measured by Fujiwara et al. (1977) yields . This is a small deviation from (i.e. ), but enough to cause noticeable effects for real systems as we demonstrate in §6.3.
We next motivate our form for in §2.2 by assuming the more general form
(38) 
and showing that it reduces to Eq. (18) if is given by Eq. (33) (with the assumption ). Using Eq. (19) and Eq. (34) in Eq. (33), we can write
(39) 
Substituting this expression into Eq. (38) and using the definition of from Eq. (23), yields
(40) 
Making the definition , we arrive at Eq. (18).
3 Power Law Solutions
We now look for a steadystate power law solution for the mass distribution. Plugging Eq. (1) into Eq. (16), using , and using the form of given in Eq. (30) yields
(41) 
We now demonstrate how the power law solutions previously derived in the literature follow from this equation and elucidate under what conditions they fail.
3.1 Dohnanyi (1969) and Tanaka et al. (1996) case
Dohnanyi (1969) and Tanaka et al. (1996) assumed a scalefree model of fragmentation with , which in Dohnanyi’s case was stated simply as with constant, and with constant. From Eq. (30), we see that this is equivalent to assuming and is constant. Changing the variables of integration to and in Eq. (41), and using Eq. (8) we have
(42) 
Taking results in being constant in agreement with Dohnanyi (1969) and Tanaka et al. (1996), and we discuss the conditions under which the integrals in Eq. (42) converge in Appendix A. We will subsequently call a fragmentation model with and a “Dohnanyi model”.
3.2 O’Brien & Greenberg (2003) and Kobayashi & Tanaka (2010) case
O’Brien & Greenberg (2003) and Kobayashi & Tanaka (2010) went one step further and considered a power law dependence of the strength as given by Eq. (35). If in Eq. (35) (i.e. in Eq. (36)), then this reduces to the Dohnanyi model. At the same time, O’Brien & Greenberg (2003) and Kobayashi & Tanaka (2010) still took , so in their case and . Changing variables again to and in Eq. (41) and using Eq. (8), we find
(43) 
The mass flux is independent of mass if
(44) 
which was derived by O’Brien & Greenberg (2003) and Kobayashi & Tanaka (2010), and the reader is again referred to Appendix A for the conditions under which the integrals in Eq. (43) converge. The arguments of Pan & Sari (2005) are analogous to the calculations of O’Brien & Greenberg (2003) and Kobayashi & Tanaka (2010), but their qualitative nature has rid them of the need to worry about the scaling of with . We will subsequently call a fragmentation model with and an “OBG model”.
3.3 Failure of the power law solution.
We now demonstrate that the power law solution (1) does not in general make the mass flux completely independent of for any , unless as in §3.1,3.2. To make the calculations tractable, we assume a power law dependence for in the form given by Eq. (36), and for in the form given by Eq. (37).
Using Eq. (41), we make the definitions
(45)  
(46) 
where . Because , in contrast to the Dohnanyi and OBG models, we change variables to and for the remnant flux and to and for the ejecta flux. This yields
(47)  
(48) 
As before, is given by Eq. (44) in order for the dependence outside both of the integrals to vanish. Now, however, appears in the upper limit of integration in the integral over in the expression for , and unless (i.e. ), is not independent of . This is one of the key conclusions of this work, and in the rest of the paper we will investigate the nonpower law behavior of fragmentation cascades in detail.
To keep things simple, we will assume that the mass flux from remnants is negligible so that . This is consistent with Fig. 3 of Kobayashi & Tanaka (2010), which shows that when is constant.
4 NonPower Law Behavior
From Eq. (48) we see that the mass flux corresponding to the power law solution of the OBG model depends only weakly (logarithmically) on . This motivates us to look for a solution of Eq. (16) in the form
(49) 
where is a slowly varying function of :
(50) 
This property can be verified a posteriori, after the explicit form of is obtained.
With given by Eq. (49), Eq. (41) becomes
(51) 
As discussed in Appendix A, the value of typically drops off below and above , which means that is peaked at , where is a constant. In the case when erosion is neglected , since collisions with projectiles of mass contribute no mass flux, but if erosion is included, then (Fig. 6 of Kobayashi & Tanaka (2010)). Together with the condition that is a slowly varying function of (Eq. (50)), this allows us to expand in a Taylor series about :
(52) 
With this approximation, the inner integral of Eq. (51) becomes
(53) 
From Eq. (50), , so as long as the peak in at is sharp enough that over the width of the peak, then the second term is negligible in comparison with the first. Thus, up to terms of order , Eq. (51) becomes
(54) 
Changing the inner variable of integration from to , using the value of from Eq. (49), and using the definition of from Eq. (36), we have
(55) 
Finally, defining
(56) 
and introducing the auxiliary function
(57) 
we arrive at
(58) 
with given by Eq. (22). Equation (58) is the master equation for the twostep determination of :

First, given the explicit form of one needs to solve this integral equation under the assumption const to determine the behavior of the auxiliary function .

Second, having obtained one must solve the functional equation, Eq. (57), to determine .
We now perform this procedure explicitly for two specific forms of — monodisperse (§4.1) and power law (§4.2).
4.1 Monodisperse Fragment Mass Distribution
We first consider the special case of the monodisperse fragment mass distribution, which puts all fragments at a single mass scale :
(59) 
This singular fragmentation model can be thought of as a very crude qualitative approximation to any fragmentation law that has most of the debris mass concentrated at one scale. It allows us to obtain some interesting analytical results and serves as a simple stepping stone for the more general case considered in §4.2.
The fragmentation law (59) implies
(60) 
where (§2.2). Plugging this into the master equation (Eq. (58)) one obtains
(61) 
where is a new function defined as a function inverse (i.e. ). Upon differentiation with respect to , expression (61) results in
(62) 
where we have used the fact that is constant. This functional equation is valid for arbitrary provided that the fragmentation law is monodisperse.
We now focus on in the form
(63) 
which is valid for given by Eq. (37). Plugging this expression into (62) we find
(64) 
Introducing the new variable
and the new function
(65) 
This has the form of a homogeneous functional equation
(66) 
( and are constants) which has the solution
(67) 
Here is an arbitrary periodic function with period , which can be constant (Polyanin & Manzhirov, 1998). This implies that the solution of Eq. (65) is
(68) 
so that finally
(69) 
Clearly this solution is inapplicable to the OBG case of , because the variable then reduces to a constant. However, for , Eq. (64) already has the form (66), so that its solution is
(70) 
In particular, =const, and consequently =const, is one of the possible solutions, so that simply proportional to with given by (44) is a viable solution for a fragmentation cascade with , in agreement with O’Brien & Greenberg (2003).
However, the existence of periodic solutions brings about the possibility of having waves in the mass distribution of objects while still having constant, even for . The presence of waves at masses in fragmentation cascades having a lower mass cutoff has been previously demonstrated by Campo Bagatin et al. (1994), Durda & Dermott (1997), and Thébault & Augereau (2007); O’Brien & Greenberg (2003) found waves to appear whenever the scaling of specific energy necessary for disruption with object mass changed abruptly (e.g. due to an object’s selfgravity becoming more important than its internal strength); and Fraser (2009), Pan & Sari (2005), and Kenyon & Bromley (2004) have shown waves to be present at the transition from a collisionally evolved to a primordial size distribution (i.e. at ).
The nature of the waves we have found is different from the ones discussed by previous authors, since they exist even when , , and is given by a pure power law without breaks (Eq. (36)). In astrophysical systems, these kinds of waves could be triggered in stochastic collisions of large planetesimals (Kenyon & Bromley, 2005; Wyatt & Dent, 2002; Wyatt, 2008). Most of the mass in such a collision would be in particles of size , and if the density of particles with mass created in the collision is comparable to or exceeds the local disk density of such particles, a wave will be triggered. However, a proper treatment of these waves needs to account for a nonmonodisperse fragment mass distribution, which could damp them, so we leave this subject for future work (Belyaev & Rafikov, in preparation).
4.2 Power Law Fragment Mass Distribution
We now assume that the mass spectrum of fragments produced in a collision is a power law with an index having a cutoff at a maximum fragment mass . This allows us to write
(71) 
where as before (§2.2). This fragmentation model resembles reality, since observational and experimental evidence (Gault & Wedekind, 1969; Fujiwara et al., 1977) as well as numerical simulations (Benz et al, 1994; Benz & Asphaug, 1999) suggest power law behavior of at small fragment masses (§2.2). Various flavors of such a power law model have been adopted in theoretical studies by Dohnanyi (1969), Williams & Wetherill (1994), O’Brien & Greenberg (2003), Kenyon & Bromley (2010), Davis & Farinella (1997), Kobayashi & Tanaka (2010), etc.