SIGNATURES OF RECENT ASTEROID DISRUPTIONS IN THE FORMATION AND EVOLUTION OF SOLAR SYSTEM DUST BANDS

In the IRAS observing strategy a lune is defined as a bin of ~30° ecliptic longitude. Each lune contains up to ~50 usable thermal emission profiles at solar elongation () angles varying from about = 85°–95°. As the spacecraft incremented the solar elongation angle of the observations (the observing angle measured from the Sun–Earth line), the effective distances to the dust bands changed (Figure 1). The reason for the difference is that most of the dust band signal is coming from near 2 AU due to the dynamical dispersion of the dust band structure interior to this region (e.g., Kehoe et al. 2007). Observations taken at solar elongation angles greater than 90° result in the spacecraft viewing dust band material that is closer than the dust band material observed by the spacecraft when the observations had a solar elongation angle less than 90° (as shown in Figure 1). Observing the dust bands at different distances results in a parallax effect: dust band pairs that are closer to the observer (higher elongation angles) will appear to show a greater latitudinal separation than dust that is farther away (smaller elongation angles). Figure 2 shows an example of the variation in the separation of the 10° Veritas dust band pair peaks for observations taken at solar elongation = 84°, 90°, and 95°, in a single lune.

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The 10° Veritas band pair components (seen at an ecliptic latitude of about ±10°) were used as markers of the data for the coadding process. For each observation, we did systematic fits to determine the precise latitude of the peak flux of the northern and southern components of the 10° band. When the latitudes were plotted against the solar elongation angle, we found that the location of the latitude of the peak varied approximately linearly with the solar elongation angle of the observation over the ecliptic longitude range of a single lune. For each lune, we determined the slope and y-intercept of the linear variation then normalized the separation of the peaks in that lune to their = 90° values. An example of a normalized scan is shown at the bottom of Figure 2.

Because the dust band midplane is inclined to the ecliptic (e.g., Dermott et al. 1999), the north and south components of the dust bands will be equally spaced above and below the ecliptic only at the nodes of these two planes (Figure 3). For observations taken away from the nodes, the bands will have a center of symmetry that is shifted away from the ecliptic, either to northern or southern latitudes depending on the ecliptic longitude of the observation. This effect can be clearly seen if we look at observations taken at different ecliptic longitudes around the sky. Figure 4 shows examples for observations taken at a range of ecliptic longitudes (all at = 90°, representing the approximate center of each lune) in the direction of the Earth's motion. The seasonal shift of the center of symmetry of the 10° band pair due to its inclination to the ecliptic is clearly evident. The resulting variation of this shift is sinusoidal around the sky (e.g., Dermott et al. 1999). Because the inclination of the dust band midplane to the ecliptic is well understood, the magnitude of the shift of the band center with ecliptic longitude can be characterized and used to normalize the observations of each lune so that the dust band midplane is centered on an ecliptic latitude of 0°. This removes the forced inclination of the dust band midplane with respect to the ecliptic (mostly due to the presence of Jupiter) and projects the dust band midplane into the ecliptic plane. After we normalize the data in each lune to a solar elongation angle of 90°, we then normalize these coadded lunes to the ecliptic plane (an ecliptic latitude of 0°). This allows virtually the entire IRAS ZOHF data set to be coadded (with the exception of a few regions contaminated by the galactic center) to yield a high signal-to-noise ratio in the data (Jayaraman 1995; Grogan et al. 1997).

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We used the coadding procedure described in Section 2 to combine virtually the entire IRAS 25 μm ZOHF data set into a single thermal emission profile. This profile (Figure 5) reveals the existence of a new dust band pair at an ecliptic latitude of approximately ±17° (shown marked by vertical lines). Based on the latitude of this new band pair, we believe it to be a confirmation of the M/N pair originally seen by Sykes in the IRAS data (1988) and also by Reach in the COBE data (1997).

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As an additional check that this band is, in fact, a real dynamic structure and not a relic of the Fourier-filtering process (which separates the fine structure dust bands from the broad background cloud), we did a test to determine the plane of symmetry of the new structure relative to the existing band pairs. We fitted Gaussians to both the northern and southern Veritas 10° band components to determine their precise locations and used these to determine the plane of symmetry. We repeated the process for the northern and southern components of the 17° band to determine if both band pairs have the same plane of symmetry, as would be expected if the new dust band is a real dynamical structure. The results of the Gaussian fitting to the partial band and the 10° band show that the partial band does, in fact, have a common plane of symmetry to the other dust band pair to within 001, which would be surprising if the band was simply a relic of the Fourier-filtering process. A common plane of symmetry is expected for all band pairs because, in the asteroid belt where the dust band structure is located, the plane of symmetry is largely determined by the forced inclination imposed by Jupiter (e.g., Dermott et al. 2001). The inclination of the dust band midplane with respect to the ecliptic was found to be 116 ± 009 for the central and 10° inclination band pairs by Grogan et al. (2001), consistent with the values we determined for the partial band. These values are close to that of Jupiter's orbit which has an inclination of 131 for the same epoch.

While coadding the entire IRAS data set serves to clearly reveal and confirm the existence of the 17° band, we can also view the band in smaller coadded bins of ~30° longitude around the sky (called lunes) to examine the longitudinal structure variations. Examining the band in these individual coadded lunes shows how the magnitude of the band varies around the sky. These individual coadded lunes, shown in Figures 6 and 7, are for observations taken in the leading and trailing directions of the Earth's motion, respectively. The 17° dust band is not present at all longitudes and the intensity of both the northern and southern components of the band varies. This behavior of the bands gave the first hint that the band at 17° may be a young structure that is still forming and as such, its nodes are not fully differentially precessed around the sky. To understand this reasoning we consider the formation process of a dust band.

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When an asteroid breaks up, the orbits of the particles released in the disruption begin to disperse in semimajor axis, resulting in a differential precession of their nodes. This results in the dust orbits slowly forming a torus and a dust band pair. At the initial stage of this process, the particles ejected in the disruption will have slightly different velocities and thus slightly different semimajor axes. Usually the velocities of the particles will be on the order of the escape velocity, which for a 10 km diameter asteroid is about 5 m s⁻¹ (and for a 100 km asteroid is about 50 m s⁻¹). These ejection velocities are small compared to the orbital velocity of 15–20 km s⁻¹, yet result in the particles spreading into a ring of material along the orbit of the parent asteroid on a timescale of hundreds to thousands of years. Not only are the particles initially spread slightly in semimajor axis from the initial disruption but they are also spiraling inwards under the effect of P–R drag (Wyatt & Whipple 1950), which causes their semimajor axes to decay over time at a rate roughly inversely proportional to particle size.

When the material has just spread around the orbit into a ring, very early in the evolution, no dust band is evident, just a ring of dust along the orbit of the source body. As the nodes begin to precess, the dust bands will start to appear as just a small amount of material at two different ecliptic longitudes 180° apart in the sky, one at a northern latitude and one at a southern latitude. Slowly, as the nodes precess and disperse around the ecliptic, the northern and southern band will spread in longitude and eventually create a full dust band pair when the nodes are completely differentially precessed. At the intermediate stages of formation there exists at some longitudes a dust band pair, at some only a northern band, and at some only a southern band. Fully dispersed nodes and a fully formed dust torus are the situation we see for the Veritas, Karin, and Beagle dust bands, which each have full band pairs (e.g., Nesvorný et al. 2003, 2008; Espy 2010). However, the new 17° dust band appears to be in an intermediate stage of formation, where the nodes are not completely dispersed and the dust bands do not extend all the way around the sky as both a northern and southern band pair.

We can further investigate the structure of the 17° band by examining how the band intensity varies with longitude. By fitting Gaussians to the northern and southern 17° band profiles in each lune of the leading and trailing coadded observations, we can determine the variation of the intensity of the band emission with longitude. The leading and trailing observations each cover only a portion of the sky. Thus in order to see the band variation around the whole sky we combine these observations by projecting the viewing directions onto the sky. In order to do this, we apply a transformation from the longitude of the Earth when the observations were taken to the longitude of the the structure being observed. If the dust band structure is considered to be at 2 AU, we can use simple geometric triangulation to determine the longitude variation between the location of the spacecraft and the location of the structures it was observing (in both the leading and trailing directions). The assumption that most of the flux is coming from 2 AU is not crucial to the result, because if the bands were located in the range of 1.8–2.5 AU, the longitudinal location of the structure relative to the observations would only shift by a few degrees in either direction. The resulting intensity variation of the band is shown in Figure 8. The bands appear to be bright over a range of longitudes, but show a reduced brightness at other longitudes, and the northern and southern bands show this pattern 180° out of phase. We overlay a sine curve to guide the eye to the band intensity variation pattern (but it is not a fit to the data). This pattern of intensity variation is precisely what would be expected by the dynamics of a forming dust band: the northern and southern bands begin to form 180° out of phase and spread from these points. Given some assumptions about the gaps in the data, the northern band appears to begin at about 130° longitude and the southern band appears to begin at about 310° ecliptic longitude. Based on the way in which the dust band forms, the northern band should be about 90° ahead of the ascending node and the southern band should be about 90° behind it. For the locations of the band observed in the coadded data, 90° behind the start of the northern band would put the ascending node of the source at approximately 40°. Thus, the variation of the dust band components reveals a pattern that suggests this new dust band is a partial band structure that is still forming and came from a source body with a node of about 40°.

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To be certain that this variation pattern is a function of the new dust band's structure and not due to a background variation, we also determined the intensity variation of the northern and southern components of the 10° band peaks using the same method. By fitting Gaussians to the northern and southern components of the 10° dust bands in the leading and trailing coadded bins (Figures 6 and 7), the band magnitude variation of the 10° Veritas band around the sky can be compared to the magnitude variation of the 17° band. If the pattern of variation seen in the 17° was due to a background asymmetry, the 10° bands would also be expected to show a similar variation pattern. Figure 9 shows the longitudinal intensity variation of the northern and southern components of the 10° band compared to the 17° band. The variation of the 10° and 17° bands shows variation in a different pattern: the two bands show different nodes as well different regions of northern or southern band component intensity domination. This implies that the longitudinal intensity variation of the 17° band is not merely background cloud fluctuation and therefore provides further evidence that the 17° band is a young and still-forming partial band.

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Based on the way in which the dust bands form, the particle orbits retain the proper orbital inclination of the parent body producing the band. This process creates a dust band above and below the midplane at the latitude of the proper inclination of the parent body and allows the dust bands to be attributed to possible source bodies in the main belt. Using the inclinations of the asteroid families, we can link the latitude of the new band with asteroid families at that proper inclination to determine possible sources (e.g., Dermott et al. 1984; Nesvorný et al. 2003). At 17° inclination, the possible sources are 1400 Tirela at a proper inclination of 169, and the Emilkowalski cluster at a proper inclination of 172. Because of the intermediate nature of the structure of this new band, we think it should result from a recent disruption and, through backward integration of the orbits, the Emilkowalski cluster was found to be 2.2 ± 0.3 × 10⁵ years old (Nesvorný & Vokrouhlický 2006) and is at a semimajor axis of 2.6 AU, where the timescale for nodal dispersion and thus band formation is on the order of 10⁶ years. Thus, we would expect a dust band created from the Emilkowalski disruption to be a partial band, much as we see in the coadded IRAS data. The average node of the Emilkowalski cluster is ~41° degrees (Nesvorný & Vokrouhlický 2006) and consistent with the node needed to fit the data. The other source at 17°, the Tirela family, has not been dated because of complications of the backward integrations from chaos due to overlapping mean-motion resonances (Nesvorný et al. 2003). As Emilkowalski is the most likely candidate at present, we built a full dynamical model of the dust band that would be associated with the Emilkowalski cluster and compared it to the coadded IRAS data. Because the three previously known dust bands result from older disruptions, only the inclination remains to determine the source body. This new band however, allows us more information on the source, because we can model the precession of the nodes of the dust orbits to constrain both the age and the node of the source body.

4.1.1. Radiation Pressure

Radiation pressure is the component of the radiation force that points radially away from the Sun. It stems from a transfer of momentum due to the impact of solar photons with the dust particles. The strength of the force of radiation pressure varies inversely proportional to the square of the particles' distance from the Sun, just as gravity does. Radiation pressure therefore cancels out some of the Sun's gravitation pull. The motion of a dust particle is the same as the gravitational motion of a particle around an object of mass (1-β), where β is the ratio of the radiation force to the gravitational force (Burns et al. 1979):

$$\begin{eqnarray}&&\beta (D)=\displaystyle \frac{{F}_{\mathrm{rad}}}{{F}_{\mathrm{grav}}}.\end{eqnarray} \tag{ 1 }$$

For large spherical particles ($D\geqslant 20\;\mu {\rm{m}}$) in the solar system, β can be approximated (Gustafson 1994):

$$\begin{eqnarray}&&\beta (D)\approx \displaystyle \frac{1150}{\rho D},\end{eqnarray} \tag{ 2 }$$

where ρ is the density of the particle in kg m⁻³, and D is the particle diameter in μm. Immediately after the disruption of a parent body (which itself was too large to be affected by radiation forces), the dust particles created in the disruption will be perturbed by radiation pressure. The result is an instantaneous alteration of the orbit of the dust particle calculated by Burns et al. (1979) to be

$$\begin{eqnarray}&&{a}^{\prime }=\displaystyle \frac{a(1-\beta )}{[1-2\beta (1+e\mathrm{cos}f)/(1-{e}^{2})]}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{e}^{\prime }=\displaystyle \frac{\sqrt{{e}^{2}+2\beta e\mathrm{cos}f+{\beta }^{2}}}{(1-\beta )}\end{eqnarray} \tag{ 4 }$$

where a' and e' are the new semimajor axis and eccentricity of the particle's new, altered orbit, a and e represent the values of the parent body's semimajor axis and eccentricity, and f is the true anomaly which denotes the position in the orbit at the time of disruption. The orbits of the largest particles, (those with the smallest β) experience the least perturbation from radiation pressure effects and remain on orbits similar to the orbit of the parent body. The smaller particles, however, experience a significant orbital perturbation from radiation pressure. The smallest fragments will end up on hyperbolic orbits (for initially circular orbits, those for which β > 0.5; Gustafson 1994) and thus be removed from the solar system on the order of an orbital period. These are the so called β-meteoroids. For asteroidal type densities (2–3 g cm⁻³; Hilton 2002), particles with diameters less than a few microns are the population lost from the system due to radiation pressure, which is referred to as the "blowout threshold."

4.1.2. P–R Drag

P–R drag is the component of the radiation force that acts tangential to the particle's orbit. The effect of P–R drag is a decrease in both the semimajor axis and eccentricity of the dust particle's orbit. P–R drag does not change the plane of the particle's orbit, though, and thus results in no variation of the inclination or the node of the orbit. The semimajor axis decay caused by the effect of P–R drag is the main transport mechanism of the particles created by asteroid disruptions into the inner solar system. Larger particles decay in more slowly than smaller particles and a useful scaling is that the rate of the semimajor axis decay is roughly inversely proportional to the particle size (see Equation (2)). Thus a 200 μm particle will require 10 times as long as a 20 μm particle to decay into the Sun. The equations for the rate of change of the semimajor axis and eccentricity of the particle orbit were derived by Wyatt & Whipple (1950):

$$\begin{eqnarray}&&{\dot{a}}_{\mathrm{PR}}=-\displaystyle \frac{\alpha }{a}\displaystyle \frac{2+3{e}^{2}}{{(1-{e}^{2})}^{3/2}}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\dot{e}}_{\mathrm{PR}}=-\displaystyle \frac{\alpha }{{a}^{2}}\displaystyle \frac{2.5e}{{(1-{e}^{2})}^{1/2}}\end{eqnarray} \tag{ 6 }$$

where a and e represent the initial semimajor axis and eccentricity of the particle and $\alpha =6.24\times {10}^{-4}\beta {{\rm{AU}}}^{2}{{\rm{yr}}}^{-1}$.

4.1.3. Solar Wind

To account for the orbital changes that result from the ejection velocities initially imparted to the dust particles during the asteroid disruption, we use Gauss' equations. These equations allow the changes in velocity to be converted into changes to the dust particle orbits. The zeroth order form of Gauss' equations (Zappala & Cellino 1996) are

$$\begin{eqnarray}&&\displaystyle \frac{\delta a}{a}=\displaystyle \frac{2}{{na}}{V}_{T}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\delta e=\displaystyle \frac{1}{{na}}[2(\mathrm{cos}f){V}_{T}+(\mathrm{sin}f){V}_{R}]\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\delta I=\displaystyle \frac{1}{{na}}\mathrm{cos}(\omega +f){V}_{W}\end{eqnarray} \tag{ 9 }$$

where V_T, V_R, V_W are the tangential, radial, and vertical components of the ejection velocity, respectively, and n is the mean motion.

We calculate the evolution of the dust particle orbits under the effects of secular planetary perturbations from the rate of change of the orbital elements of a particle with time. These are found from the Lagrange perturbation equations and using the disturbing function and eigenfrequencies from Murray & Dermott (1999). They are given in their final form by

$$\begin{eqnarray}&&\displaystyle \frac{d{\rm{\Omega }}}{{dt}}=\displaystyle \frac{1}{\sqrt{1-{e}^{2}}\mathrm{sin}I}\left({BI}+\displaystyle \sum _{j=1}^{N}{B}_{j}{I}_{j}\mathrm{cos}({\rm{\Omega }}-{{\rm{\Omega }}}_{j})\right),\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}\displaystyle \frac{{dI}}{{dt}} & = & \displaystyle \frac{-\mathrm{tan}(\displaystyle \frac{1}{2}I)}{\sqrt{1-{e}^{2}}}e\displaystyle \sum _{j=1}^{N}{A}_{j}{e}_{j}\mathrm{sin}(\varpi -{\varpi }_{j})\\ & & +\displaystyle \frac{1}{\sqrt{1-{e}^{2}}\mathrm{sin}I}I\displaystyle \sum _{j=1}^{N}{B}_{j}{I}_{j}\mathrm{sin}({\rm{\Omega }}-{{\rm{\Omega }}}_{j}),\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{de}}{{dt}}=-\sqrt{1-{e}^{2}}\displaystyle \sum _{j=1}^{N}{A}_{j}{e}_{j}\mathrm{sin}(\varpi -{\varpi }_{j}),\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}\displaystyle \frac{d\varpi }{{dt}} & = & \displaystyle \frac{\sqrt{1-{e}^{2}}}{e}\left({Ae}+\displaystyle \sum _{j=1}^{N}{A}_{j}{e}_{j}\mathrm{cos}(\varpi -{\varpi }_{j})\right)\\ & & +\displaystyle \frac{\mathrm{tan}(\displaystyle \frac{1}{2}I)}{\sqrt{1-{e}^{2}}}\left({BI}+\displaystyle \sum _{j=1}^{N}{B}_{j}{I}_{j}\mathrm{cos}(\varpi -{\varpi }_{j})\right),\end{eqnarray} \tag{ 13 }$$

where the orbits of the planets, denoted by ${{\rm{\Omega }}}_{j},{\varpi }_{j},{I}_{j},{e}_{j}$, are calculated using the eigenfrequencies of the system. These equations are used to describe evolution with constant a, but the semimajor axes of the asteroidal dust particle orbits are decaying with time under the effects of P–R and solar wind drag. Combining Equations (5) and (6) with Equations (10)–(13) and expanding the Laplace coefficients as a series, we can describe the evolution of the dust particles under both radiative and gravitational forces simultaneously. We evaluate these equations over a finite time step using a fourth-order Runge–Kutta method (Press et al. 1998). The size of the time step is left as a variable function of particle size due to the different rates of decay of the dust orbits under P–R drag, and we chose the time-step to obtain a specific step in semimajor axis, ${\rm{\Delta }}a$. We empirically checked the required value of ${\rm{\Delta }}a$ and chose it to be ${\rm{\Delta }}a={10}^{-4}$, AU which yields a time-step of ~10 years for 100 μm diameter particles and ~1000 years for 1 cm diameter particles.

For each model we use the current orbital elements of the Emilkowalski cluster (from Nesvorný & Vokrouhlický 2006) and evolve them backwards in time over the age of the family to determine the orbital elements at the epoch of family formation. At this epoch we release the dust particle, using a size range from 10 μm to 5 cm (a size distribution will be applied later). A velocity is imparted to each dust particle (to account for the initial impact) as it is released, and the orbits are altered to reflect this using Equations (7)–(9). The ejection velocity of the particles is allowed to be just slighter greater than the escape velocity of the parent body (~10 m s⁻¹) as the body is statistically most likely to be disrupted by a second body that has just enough energy to do so. The effect of radiation pressure is then switched on, yielding another instantaneous orbit alteration (Equations (3) and (4)) dependent on the particle size. These orbital elements provide the starting point for the dynamical evolution outlined above. The dust orbits are tracked into the inner edge of the dust bands near 2 AU. Interior to this semimajor axis, the effect of the secular and mean-motion resonances disperses the dust orbits and diffuses the band structure into the background cloud (e.g., Espy 2010). The specific semimajor axis location at which each dust particle is removed depends on its size and is characterized by the results of previous numerical simulations (Kehoe et al. 2007). Smaller particles retain their band structure farther into the inner solar system due to their faster P–R drag based orbital decay which results in quicker passage through the resonances and thus reduced perturbation to their orbits. We record the orbital elements of the dust particles at each time and use them to produce the thermal emission models.

In order to compare the results of our dynamical evolution code to the coadded IRAS observations, we need a mechanism by which these orbital distributions can be visualized. We use the visualization tool SIMUL (Dermott et al. 1988; Grogan et al. 1997) for this purpose. In short, the SIMUL suite of codes allow the creation of a three-dimensional model of the cross-sectional area of the distribution of dust that would result from a given orbital distribution. We then visualize the 3D model as line of sight thermal emission profiles which can be compared directly with observational data. The SIMUL algorithm is based on the idea that a cloud can be represented by a large number of individual dust particle orbits. The total cross-sectional area of the cloud is divided equally among the orbits. The particles are distributed along the orbits according to Kepler's second law. The 3D space is divided into O(10⁷) ordered cells, which are filled with cross-sectional area representative of the orbits that pass through the cell. In this way, the model generates a large three-dimensional array which describes the spatial distribution of the cross-sectional area of dust particle material. The model is then converted to line of sight thermal emission profiles that match the observing geometry of IRAS (or any telescope) by calculating the Sun–Earth distance and ecliptic longitude of Earth at the observing time and setting up appropriate coordinate systems. These profiles can be compared directly to the IRAS scans in order to constrain the properties of the dust particles composing the bands.

We implement a size distribution in the models and use the comparison with the coadded data to constrain the parameters of the distribution. The size-frequency distribution of particles resulting from a catastrophic collision can be described by a cumulative power-law of the form

$$\begin{eqnarray}&&N(\gt D)=\displaystyle \frac{1}{3(q-1)}{\left(\displaystyle \frac{{D}_{0}}{D}\right)}^{3(q-1)}\end{eqnarray} \tag{ 14 }$$

where N(D) is the number of particles with diameter greater than D, q is the size-frequency index, D is the particle diameter, and D₀ is a constant as

$$\begin{eqnarray}&&{D}_{0}=\sqrt[3]{3(2-q)}{D}_{e}{\left(\displaystyle \frac{{D}_{0}}{{D}_{\mathrm{max}}}\right)}^{2-q}\end{eqnarray} \tag{ 15 }$$

where D_max is the diameter of the largest fragment of the disruption and D_e is the equivalent diameter of the entire distribution. A distribution in a closed box in collisional equilibrium (Dohnanyi 1969) is described by a power law with q = 11/6 and, in this case, the cross-sectional area of the distribution is dominated by particles at the small end of the size range. The value of q = 5/3 represents a turnover value where distributions described by a larger q will have area dominated by the smallest particles of the distribution and a distribution described by a smaller q will have area dominated by the largest particles present in the distribution.

We implement the size distribution of the particles in the models through the choice of particle sizes making up each of the bins that come together to produce a final dust torus model. For each size distribution that is to be modeled, the particle sizes are divided into 20 bins. These bins are chosen using a logarithmic distribution of sizes such that there is an equal amount of cross-sectional area in each bin. As the size distribution parameter, q, changes between models, so does the distribution of particle sizes in the bins. For example, a size distribution dominated by small particles (higher q) has more bins of small particles and fewer large particles than would those for a model with a size distribution described by a smaller value of q. This method also allows the cross-sectional area present in the model to be constrained separately from the size distribution since changing the area in this regime would increase (or decrease) the amount of area in the bins together but would not affect the relative contributions of each bin. These bins are used to create 20 individual SIMUL models, each with the size, node and semimajor axis ranges describing the particles in that bin. Each SIMUL model is then converted into a thermal emission torus, described by the emission of the average size particle in that bin. The 20 thermal emission tori are then added together to produce a full model of the partial dust band resulting from the contributions of all particle sizes. In this way, we can compare observations to models to constrain the size distribution of the dust particles by matching the variation of the band intensity with ecliptic longitude.

The semimajor axis location of particles (due to P–R drag) in this early stage of dust band formation is a function of only particle size, assuming collisions are not yet dominating the evolution (Section 5). Since the precession rate of the node is a function of semimajor axis, the node location of the particle's orbit is also a function of particle size. This results in an interesting aspect of the early stages of dust band formation, whereas the particles are distributed around the sky (in both node and semimajor axis) continuously with particle size. The resulting spiral structure persists only for a short time in the early stages of dust band formation (between 10⁵ and 10⁶ years) until collisions begin to dominate. This is shown schematically in Figure 11 in which, as function of particle size, the location of the node and semimajor axis of the particle are shown for a series of time steps. Once the particles are released from the parent body, the smallest particles (~100 μm shown in purple) begin to evolve away from the largest particles (~5 cm shown in green), which remain near the source. As the semimajor axes of the smaller particles decay, the nodal precession rates slow due to their larger distance from Jupiter and its perturbing effects. Since these smaller particles are precessing at a slower rate, they lag behind the larger particles (and the source) which results in a differential dispersion of the nodes around the sky, sorted by semimajor axis and particle size. Older, fully formed dust bands, in contrast, represent a situation where particles of all sizes exist at all semimajor axes and with randomized nodes since they have had enough time to reach this state through inter-particle collisions and orbital decay.

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The structure of these spiral plots also serves to explain why partial dust bands contain additional information that allows us to constrain the size distribution of the material ejected from the asteroid disruption that produced them. Because the particles are effectively sorted azimuthally around the sky as a function of size, the size distribution of the dust will be revealed in the longitudinal distribution of material. Thus, for different values of the size-distribution parameter, q, the thermal emission intensity of the dust bands with longitude will vary. Comparison of the modeled longitudinal variation together with the observed variation of the partial band intensity in the different coadded lunes (e.g., Figure 8) allows this parameter to be constrained. Smaller values of q, which result in a greater proportion of large particles will yield a band with most of its intensity in the large particles nearer the source, whereas larger q values which result from a population dominated by smaller particles will result in a more even distribution of band intensity around the sky. While the pattern of intensity variation allows us to constrain the size distribution, the magnitude of the band intensity allows us to constrain the cross-sectional area of material in the band.

By modeling different parameters describing the material producing the partial dust band at 17° and comparing the resulting thermal emission with the coadded IRAS data, we have been able to constrain the cross-sectional area, the size range of particles, the size distribution, and confirm the source of the band.

6.2.1. Source of the Dust Band

6.2.2. Particle Sizes

Using the method described in Section 4, we created models of the dust band structure resulting from the Emilkowalski disruption. We used particles of diameter ~70 μm–5 cm described by a power-law size distribution ranging from q = 1.6 to q = 1.9. Using the orbital distribution of material described by Figure 11, we produce skymaps of the thermal emission from the resulting dust torus and one example of this is shown in Figure 12. Interestingly, the most intense thermal emission structure, described by a size-distribution parameter of q = 1.8 (which is steep enough that the cross-sectional area is dominated by small particles) is coming from the largest particles (>1 mm) in the distribution. This is surprising because the percentage of cross-sectional area coming from the larger (>1 mm) diameter particles, described by the inverse power-law size distributions used here, is small. Additionally, these larger particles are at a greater semimajor axis, and thus the flux contributed from these larger particles is reduced even further. Due to the combination of these two effects, we would not expect a significant contribution to the thermal emission structure from the largest particles. However, for dust bands at the age of the Emilkowalski cluster, most of the larger particles are still located near the source since they have not yet had much time to decay in semimajor axis (under P–R and solar wind drag) or disperse under differential nodal precession. And, even though the smaller particles dominate both the number density and the cross-sectional area of the material, their nodal distributions are much more dispersed. For the current age of the Emilkowalski disruption (Figure 11) the smaller particles, 100–200 μm in size (shown in purple) are distributed in node fully around the sky, while the larger particles are clustered near the source. So, although the total cross-sectional area from particles $\geqslant 1$ mm and the total flux that they produce are both reduced compared to their smaller counterparts, these larger particles are bunched closely together in the longitudes of their nodes, resulting in an effectively increased "flux-density" compared to the smaller particles, resulting in the mm to cm sized particles dominating the observed thermal emission structure.

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The best fit of the models to the observations is described by a size distribution with q > 1.7 which corresponds to a cumulative inverse power-law index larger than 2.1. This is much steeper (e.g., more small particles) than that found for the older, central and 10° dust bands, whose particles have undergone further processing from inter-particle collisions and dynamical and radiative removal of the smaller particles. These bands are described best by a q = 1.4 (Grogan et al. 2001; Espy 2010), which corresponds to a cumulative inverse power-law index of 1.2. It is also interesting to compare the size distribution found for the young, partial band to that found for the surface of asteroid Itokawa. Tsuchiyama et al. (2011) find a cumulative inverse power-law index that they describe as shallower than 2.8 and "probably around 2.0" for particles down to a few 10's of microns. This value for the regolith on the surface of the 350 m diameter asteroid Itokawa compares well to our findings for the dust generated in the catastrophic disruption of the 8 km diameter asteroid parent to the Emilkowalski cluster. This may imply that the dust released in the catastrophic disruption of an asteroid in this size range is dominated by the release of its surface regolith particles. For the extremely recent disruption of P/2010 A2, the tail of newly ejected dust is in the size range of mm–cm and is described by a cumulative inverse power-law index of 2.3–2.5 (Jewitt et al. 2010; Snodgrass et al. 2010; Hainaut et al. 2010). This is slightly steeper slope which indicates more contribution from particles in the smaller end of the size distribution, but particles below 1 mm are notably absent. For further comparison, the size distribution found for the lunar regolith ranges from a cumulative inverse power-law of 3.1–3.3 for particles 20–500 μm (Heiken et al. 1991).

6.2.3. Cross-sectional Area of Dust Band Material

6.2.4. Model Discrepancies

As we produced models of the thermal emission for comparison with observations we encountered one main discrepancy between the models and observations, and that is the amount of inclination dispersion found for the larger particles. Our models show much less dispersion on the larger particles than is seen in the coadded observations. This fact prevented us from being able to put tighter constraints on the size distribution and cross-sectional area of material present. We investigated several different mechanisms for increasing the dispersion and found that the best explanation came from the inclusion of a greater-than-expected ejection velocity at the time of disruption.

Our original model assumed a small disruption ejection velocity, one just greater than the escape velocity of the parent body, as would be expected statistically given that an asteroid is most likely to be broken up by an impactor just big enough to do so. However, even in this scenario the particles would experience a range of ejection velocities that would impart more initial dispersion (following Section 4.2) than we had incorporated into our original models. Laboratory experiments and impact models show that the ejection velocity of particles is likely well above the escape speed (Housen et al. 1979; Housen & Holsapple 2011) in studies that focus mainly on crater formation, an even less energetic event than the catastrophic impacts producing asteroid families and clusters. In order to investigate how much additional orbital dispersion can be gained from higher material ejection velocities, we revisit Gauss' equations with ejection velocities of 8 m s⁻¹ (just over the escape velocity of the parent body), 10 m s⁻¹, 30 m s⁻¹, 50 m s⁻¹, 100 m s⁻¹, and 1 km s⁻¹ (which is expected only for about 1% of the products of the disruption of a solid parent body, e.g., Sykes & Greenberg 1986, and likely would not be expected here). We undertook additional dynamical evolution runs to check how these initial ejection velocities would not only change the initial orbital elements, but how they would effect the final orbital elements for a particle of a given size after those particles dynamically evolve over the age of the Emilkowalski cluster. The results are given in Figure 13, which shows the drastic effect of even a modest (few times the escape velocity) ejection velocity. We find that this effect provides significant inclination dispersion (on the order of a few degrees) and allows a much better match of the model to the observations.

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While the effects of the initial velocity dispersion seem to be the most likely explanation for the observed inclination dispersion, the models still do not give exactly the same same shape of the inclination intensity distribution as is seen in the observations. Thus, likely multiple effects are at work, and this problem should be revisited in the future.

We find that Emilkowalski is the source of the partial dust band, at an ecliptic latitude of 17°, discovered in coadded IRAS data (Espy et al. 2009). The cross-sectional area of dust released in the disruption that produced the band is on the order of 10⁷ km²; this is two orders of magnitude less than that currently observed in the near-ecliptic and 10° bands (Espy 2010), albeit from a much smaller parent body. Scaling arguments imply that the now cometary-dust-dominated zodiacal cloud (Nesvorný et al. 2008) has been dominated by asteroidal dust at the epochs in the past (~8 and 6 Ma) when the larger parent bodies of the near-ecliptic and 10° dust bands broke up. We find that the size distribution of dust released in the catastrophic disruption of the Emilkowalski parent body is described by a q value greater than 1.7 (inverse power-law index greater than 2.1), implying cross-sectional area domination by small particles, which is a much steeper size distribution than the q = 1.4 found for the central and 10° bands (Espy 2010). This implies that the small particles are being removed, as the dust band ages, at a faster rate (due to radiation forces) than they are being replenished (due to inter-particle collisions). The coadded observations show that the structure of the dust band is dominated by large (mm to cm sized) particles due to their closely aligned nodes. This power-law size distribution is, interestingly, in good agreement with the values describing the surface regolith of asteroid Itokawa, which is described by a cumulative inverse power-law index of less than 2.8, and likely closer to 2.0, for a regolith composed mainly of mm to cm sized particles (Tsuchiyama et al. 2011), but is shallower than the distribution determined for the particles released in the extremely recent disruption of P/2010 A2, which is described by a cumulative inverse power-law index of 2.3–2.5 (Jewitt et al. 2010; Snogdrass et al. 2010). The inclination dispersion of the particles released in the disruption that produced the Emilkowalski cluster is more than would be expected for a low ejection velocity disruption. We find that the ejections velocities of dust, at disruption, of a few times the escape velocity provide a better fit of the models to the observations. For the 8 km diameter Emilkowalski parent body, when accounting for all dust particle sizes greater than the radiation pressure blowout threshold, and assuming a q = 1.8, the observed dust would be equivalent to a 3–4 m thick layer of regolith on the surface.