Evolution of Cometary Dust Particles to the Orbit of the Earth: Particle Size, Shape, and Mutual Collisions

In our model, we ejected hypothetical dust particles from the orbits of selected "actual" comets. We considered nine different values of β, the ratio of the solar radiative acceleration to the solar gravitational acceleration: β = 0.57, 0.285, 0.114, 0.057, 0.0114, 0.0057, 0.00285, 0.00114, and 0.00057. When spherical particles with a density of 0.8 g cm⁻³ are assumed, these β values are equivalent to diameters of 2.5, 5, 12.5, 25, 125, 250, 500, 1250, and 2500 μm (Burns et al. 1979). Details of the source selection and dust ejection will be explained in the subsections below. All dust particles were ejected at the same epoch, a Julian date of 2457054.5 (A.D. 2015 February 1, 0:00) for convenience in our simulation. Planetary ephemerides for each epoch were calculated with an N-body simulation using the initial data in Chambers (1999). During the numerical calculation of the orbital evolution of dust particles, we accounted for the Sun and eight planets as massive objects that exert gravitational accelerations and ignored the gravitational effects of other objects. In addition, radiative acceleration, including P–R drag due to solar radiation, was considered, while such effects due to other sources were ignored because of their weakness. Solar radiation was treated as constant over time. The effect of drag from the solar wind is assumed to be equal to 30% of the P–R drag (Burns et al. 1979). We employed a numerical integrator applying the RADAU15 algorithm (Everhart 1985), originally coded by Chambers (1999, MERCURY 6.2), and modified by Jeong (2014) to take into account the effects of radiation. Dust particles were excluded from the calculation when their heliocentric distance became larger than 80 au or smaller than 0.05 au. The initial integration time step was 3.6525 days, and the variable time step was chosen to achieve an accuracy of at least ∣ΔE/E∣ ≤ 10⁻¹² in energy during a single step. We stopped integrations after 2 million years, when most dust particles were excluded from the numerical integration by the above conditions.

According to previous research (Tancredi et al. 2006; Snodgrass et al. 2011), even though JFCs have not been completely surveyed, it is suggested that the distribution of known JFCs is not observationally biased, and therefore the SFDs of all known JFCs and those of JFCs with small perihelion distances are statistically the same (Fernández et al. 2013), implying that known JFCs can statistically represent the whole population. Therefore, supposing that JFCs are the main source of the IDP cloud complex, we began our simulation from existing cometary nuclei. Furthermore, we assumed that number of comets has been constant over a few million years, as we explain in Section 2.4. This approach of source selection is basically similar to but slightly different from the idea in Wiegert et al. (2009). The difference in source selection between Wiegert et al. (2009) and this work is threefold. First, they assumed that the whole asteroid population in their model ejects a small number of dust particles as a result of mutual collisions, whereas we considered that active asteroids (rather than the entire population) eject dust particles. Second, we considered small dust particles with a large cross section to mass ratio (i.e., β > 0.057), which were not considered in Wiegert et al. (2009). Lastly, we included three times as many comets as Wiegert et al. (2009) did. We chose the source comets from the JPL Horizons comet list as of 2015 January 23 (http://ssd.jpl.nasa.gov/dat/ELEMENTS.COMET).

Among the 3321 comets in the list, we chose periodic comets with eccentricity e < 1 and excluded fragments to avoid duplication from dust particles from the same comets. In total, we chose 1049 comets for consideration. We classified these comets into JFCs, Encke-type comets (ETCs), Chiron-type comets (CTCs), and Halley-type comets (HTCs) following the criteria in Levison (1996). HTCs were further divided into two classes: one with semimajor axis a larger than that of Jupiter, a_J (HTC-1), and the other with a ≤ a_J (HTC-2). ETCs were further categorized into two classes: one case having a Tisserand parameter with respect to Jupiter of T_J < 3.01 (ETC-1), the other with T_J ≥ 3.01 (ETC-2). In our simulation, dust particles from ETC-1 experienced close encounters with Jupiter and had similar evolutionary tracks to dust particles from JFCs. Dust particles from ETC-2 are less influenced by close encounters with Jupiter. Note that this criterion is different from the working definition of active asteroids (AAs), that is, T_J ≥ 3.08 (Jewitt et al. 2015), because our concern is not the orbital similarity with asteroids but the dynamical interaction with Jupiter. The orbit of 2P/Encke is largely different from those of AAs (occasionally referred to as main-belt comets) in that 2P/Encke has a large eccentricity and a short perihelion distance. However, dust particles from 2P/Encke are less susceptible to a close encounter with Jupiter than those from JFCs, similar to AAs. For this reason, both 2P/Encke and AAs are categorized into the same group (ETC-2) in our definition.

In orbital element space, we constructed 17 step bins for semimajor axis a (0.0–0.5, 0.5–1.0, 1.0–1.5, 1.5–2.0, 2.0–2.5, 2.5–3.0, 3.0–3.5, 3.5–4.0, 4.0–4.5, 4.5–5.0, 5.0–7.0, 7.0–9.0, 9.0–14.0, 14.0–19.0, 19.0–24.0, 24.0–30.0, 30.0–40.0 in au), 5 bins for eccentricity e (0.0–0.2, 0.2–0.4, 0.4–0.6, 0.6–0.8, 0.8–1.0), and 6 bins for inclination i (00–300, 300–600, 600–900, 900–1200, 1200–1500, 1500–1800), and counted the number of cometary orbits within each bin three-dimensionally. With this choice of bins, each box includes 0 to 63 cometary orbits. In each box, we chose sample comets as representative of the whole population of the given orbital class and performed the numerical calculation with their orbital elements. To eliminate sampling bias and reduce computational load, we chose multiple samples in a box if the number of orbits in a box is larger than 5.5% of the total number of JFCs, ETC-1s, ETC-2s, and HTC-2s. With this assumption, we implied that dust ejected from other comets in the box should experience orbital evolution similar to that of representative comets. Table 1 shows the list of selected comets included in the calculation and their weighting factor w_i, which is the number of comets whose dust particles are assumed to have similar evolution to the particles from the listed comets. Later in Section 2.4, when we made clones of cometary dust particles, the number of clones was determined to be proportional to w_i. The orbital elements of selected comets, 64 JFCs, 45 HTC-1s, 2 HTC-2s, 3 ETC-1s, 6 ETC-2s, and 6 CTCs are listed in Table 1 along with their w_i values.

As mentioned above, we assumed that our sample of JFCs, HTC-2s, ETC-1s, and ETC-2s is free from discovery bias and used the data in making the IDP cloud complex model in Section 2.4. CTCs and HTC-1s were not included in the model but are referred to for comparison. The orbital distribution of these comets is shown in Figure 1. As a comparison (see Tables 2 and 3), we also performed a numerical simulation of asteroidal dust particles. We tested a simple situation in which one dust particle per β value was ejected from the largest 1933 asteroids (>15 km in diameter), considering that these large asteroids are completely detected.

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We ejected dust particles from the orbits of the above actual cometary nuclei by using the following methods. At first, the orbits of comets at the epoch t₀ = JD 2457054.5 were numerically calculated from the orbital elements of the JPL Horizons comet lists. Then, five orbital elements of the comets were fixed, except for anomalies. All dust particles were ejected simultaneously at the epoch on the orbits of comets with randomized true anomalies. We ejected 100 dust particles per given size per cometary orbit for particles with β ≥ 0.00285 and 50 for particles with β ≤ 0.00114.

When we distributed the true anomalies, we made the number density of ejected dust particles becomes a function of heliocentric distance which reflects the observed change in the dust ejection efficiency (Ishiguro et al. 2007; Mazzotta Epifani et al. 2009; Hanayama et al. 2012). Where N(r_h) denotes the initial number density of ejected particles on the given cometary orbit in cm⁻³, and r_h denotes the heliocentric distance in au, we distributed the particles as

$$\begin{eqnarray}&&N({r}_{{\rm{h}}})\propto {r}_{{\rm{h}}}^{-3.0},\end{eqnarray} \tag{ 1 }$$

The directions of ejection are randomly distributed over the sunlit hemisphere following an uniform probability distribution, and the speeds of ejection are given as a function of β and r_h (Ishiguro et al. 2007; Hanayama et al. 2012), hence the ejection speed v_ej(r_h) are given by

$$\begin{eqnarray}&&{v}_{{ej}}({r}_{{\rm{h}}})=200[{\rm{m}}\,{{\rm{s}}}^{-1}]{\beta }^{0.5}{r}_{{\rm{h}}}^{-0.5}.\end{eqnarray} \tag{ 2 }$$

Although we assumed a relatively realistic situation in ejecting dust particles compared to the cases with zero ejection velocity, numerically, the added ejection velocities function as a kind of cloning process because they are small compared to the orbital velocities. Likewise, even though we ejected dust particles simultaneously at different positions on the cometary orbits instead of at the exact position of nuclei at the epoch, this approach is dynamically the same as changing the anomaly without changing other orbital elements, and therefore this method works as if it were a cloning process.

The initial orbital elements of ejected dust particles were calculated after the ejection velocities were added to the orbital velocities, taking account of the change of orbital elements due to radiative acceleration.

We assumed that all of the above cometary nuclei eject dust particles at the same rate when we calculate the average over single revolutions. We also assumed that all nuclei eject dust particles with same shape and SFD once we fixed these parameters. Under these assumptions, the ratio between initial dust masses JFCs:HTC-2:ETC-1:ETC-2 is 486:6:19:11.

We constructed an IDP distribution covering different ages of particles for given β values using the following methods. First, we recorded the positions and velocities of dust particles every 100 yr and cloned 1000 × w_i particles from each individual dust particle. We set the number of the clones to be proportional to the parameter w_i. During the cloning process, we kept three orbital elements (i.e., the semimajor axes, eccentricities, and inclinations) constant and randomized their arguments of perihelion, longitudes of ascending node, and mean anomalies for each clone. Consequently, we made snapshots composed of clones with single β values and from the same class of cometary population every 100 yr. Assuming a steady state (i.e., that dust ejections have occurred constantly over 2 million years), we regarded dust particle records after a given time of integration from the current epoch as records from possible ejections at the same amount of time before the current epoch. This assumption is similar to that made in previous works such as Wiegert et al. (2009). Finally, the snapshots were added together, constructing an IDP distribution covering the full dynamical evolution from sources to sinks.

Using the above information from our numerical calculation, we validated the initial conditions of dust ejection via following methods. Whenever the initial conditions about dust particle shape, mass density, and initial SFD were given, the resulting IDP cloud complex models were derived. We constructed a three-dimensional grid in Cartesian coordinates. Grids of size 0.1 au × 0.1 au × 0.1 au were constructed throughout interplanetary space, and the number of dust particles of a given β value within a single grid was counted. The results for different β values were weighted by the given initial conditions (see Section 3.2). As a next step, we derived the cumulative number, cross section, and mass distribution of IDPs in the grid box, assuming a piecewise exponential SFD within every single grid box. Then, after summing the numbers from different source populations, the resulting SFDs around the Earth's orbit were normalized at β = 0.00285 and compared with the observational SFD in Grün et al. (1985), which was obtained through the Pegasus and HEOS-2 missions and widely applied as the reference SFD model. We examined the ratio between our normalized SFD and the observed one between β = 0.00057 and β = 0.114 and we treated the given initial conditions as valid if the ratio in SFD is between 0.5 and 2 for every β value. For such cases, we derived a goodness of fit as the averaged absolute value of the logarithm of the ratios.

Finally, the total IDP cloud complex model was scaled to match the observed zodiacal light brightness in the antisolar direction (i.e., gegenschein) (Leinert et al. 1998; Ishiguro et al. 2013). As we mentioned in Section 2.1, we limited the largest dust size to β = 0.00057. This cutoff is expected to have negligible effects on the total cross section but may exert a considerable influence on the total mass according to the initial conditions. Therefore, we tabulated the total mass supply in two cases, namely up to β ≥ 0.00114 and β ≥ 0.00057, along with validated initial conditions.

In this work, the sinks of dust particles were determined by close encounters with planets for most cases. We will explain the role of close encounters, which concurs with previous research. Dust particles ejected from comets fall toward the Sun because of P–R drag if the Sun's gravity and radiation are the only two factors affecting their orbits. P–R drag continuously reduces a and e of the dust particles. The effects of resonances with planets (i.e., the mean motion and secular resonances) do not themselves change sinks of dust particles significantly because the resonances change e rather than a. Note that the lifetime of dust particles may be changed by resonances, which would increase it temporarily by trapping them. In contrast, close encounters with planets, mainly Jupiter, can completely change the orbits of dust particles and usually prevent their falling toward the Sun. As a consequence of close encounters, a may decrease or increase, and any e value may occur from a circular orbit to a hyperbolic orbit. Dust particles ejected from the JFCs are prone to close encounters with Jupiter because the initial orbits of their source comets intersect or are close to the Jovian orbit and their velocities relative to Jupiter are low.

Smaller dust particles have higher chances of avoiding close encounters than larger particles because they fall toward the Sun faster via P–R drag and therefore spend less time in the region where the Jovian gravity is critical. Figure 2 shows the lifetime of dust particles under radiative acceleration. For comparison, we construct the lifetime in the case where there are no planetary perturbations. Because P–R drift is slow for large particles, the lifetime of particles that are larger than 100 μm is determined by close encounters, not by P–R drift. Therefore, the lifetimes of large dust particles are similar to those of their parent bodies, ∼100,000–300,000 yr (∼100,000 yr as a weighted median, ∼250,000 yr as a weighted average) (Levison & Duncan 1994, 1997). Note that the lifetime with planetary perturbation included the effects of resonances. The resonances may increase the lifetime of dust particles by temporarily trapping them, especially for particles that have evolved into the inner solar system.

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We measured the fraction of dust particles transferred into the inner solar system (a ≤ 1 au) for different populations with different sizes (Table 2). The probability of successful transfer to an orbit with a semimajor axis smaller than 1 au will be selectively higher for dust particles meeting the following two conditions: dust particles ejected from objects with aphelion distances shorter than the semimajor axis of Jupiter and which are free from close encounters with Jupiter, ETC-2s, and asteroids; and dust particles that are small. We checked that the dust particles that are not transferred to the inner solar system within 2 million years of integration time satisfy one of the following conditions: they are trapped in the Trojan region; or β < 0.00114 and they are free from planetary encounter and thus still under P–R decay. The latter case includes dust particles ejected from AAs and asteroids, and we performed 5 million years of integration for these particles to complete the process of dust particle evolution. The proportions of dust particles surviving after 2 million years of integration are summarized in Table 3.

In previous subsection, we explained the different evolutionary tracks between the particles with different β values. From now on, we will connect the dust SFD measured around the Earth's orbit with that measured around cometary comae.

According to measurements from the Rosetta mission, the dust SFD at the coma of 67P/Churyumov–Gerasimenko changed over time, but it appears that there are at least two bending points in the SFD that can thus be approximated by a doubly broken power-law function. The bending points were determined at ∼10⁻⁶ and ∼10⁻⁴ g for most cases except for the region around perihelion (Rotundi et al. 2015; Agarwal et al. 2016; Fulle et al. 2016b; Hilchenbach et al. 2016; Merouane et al. 2016). The power-law exponents α for differential mass distributions dn ∝ m^−αdm varied between ∼1.75 and 2.05 for the smallest masses, ∼0.97 and 1.67 for intermediate masses (∼1.9 right after perihelion), and approximately ∼2.0 for the largest masses. The results from the Stardust mission (Green et al. 2004, 2007), Giotto mission (Fulle et al. 1995), and IR observation (Vaubaillon & Reach 2010) coincided with this point. This distribution is different from that measured around the Earth's orbit by the Pegasus and HEOS-2 missions (Grün et al. 1985), but the difference itself is understandable when we consider the content of the previous subsection.

As explained before, what we actually calculated were orbital evolutionary tracks as a function of initial orbits and β values. In this subsection, we first converted β to particle mass assuming the particle shape and density. Next, we found the expected initial dust SFD that can explain the measured SFD around the Earth's orbit. Finally, we compared the expected initial dust SFD with the measured one and derived assumptions that can match the two SFDs.

Cometary dust particles have been conventionally approximated as having a compact structure and a spherical shape. For compact particles, a low mass density of 0.8 g cm⁻³ was derived from modeling porous icy dust via the Rosetta measurements (Fulle et al. 2016a), but a relatively wide density range of 1.9 ± 1.1 g cm⁻³ was measured during the Rosetta mission by direct comparison between cross section and impact momenta assuming particle shape (Rotundi et al. 2015). We thus tested three different values of particle mass density, namely 0.8, 1.9, and 3.0 g cm⁻³ (Rotundi et al. 2015; Fulle et al. 2016a).

Despite the conventional treatment of IDPs as compact spherical particles, large fluffy aggregates (larger than hundreds of micrometers in diameter) were recently found around 67P/Churyumov–Gerasimenko via the observations of the Rosetta spacecraft (Fulle et al. 2015; Bentley et al. 2016; Mannel et al. 2016). Although the detailed physical properties such as the structures and/or porosity of the fluffy aggregates are not well investigated, it is likely that these fluffy aggregates have mass densities as low as 0.001 g cm⁻³ (Fulle et al. 2015), and fractal dimensions were estimated to be 1.87 (Fulle et al. 2015; Mannel et al. 2016). Under these conditions, the ratio of cross section to mass of the aggregates is expected to be constant regardless of aggregate mass, or at least to exhibit only insignificant changes, unlike compact particles (Mukai et al. 1992; Skorov et al. 2016). We are not sure about the exact cause of the shallow SFD between 10⁻⁶ and 10⁻⁴ g, but we conjecture that the fluffy aggregates that were discovered by the Rosetta mission (Fulle et al. 2015; Bentley et al. 2016; Mannel et al. 2016) are responsible for this shallow SFD slope. The time of the change in the fluffy aggregate ratio coincides with a change in SFD around perihelion (Della Corte et al. 2015, 2016), and the size of large fragments from fluffy aggregates matches the particle size of the smaller bending point in the SFD (Hilchenbach et al. 2016; Merouane et al. 2016).

As the first trial, we employed the initial SFD of compact spherical dust particles with a single power law. Where dn is the differential number density of dust particles of mass m, the initial SFD is written as dn ∝ m^−αdm. We tested different α values with intervals of 1/12. In this model, we cannot validate any initial parameters that yield less than twice the difference between observations and expectations; therefore, we tabulated the initial parameters that yield results that differ by less than a factor of five from the observation. The derived power index, α, is summarized in Table 4. We also show the results of the fitting in Figure 3 along with other cases. We present the ratio between our best-fit SFD and that of Grün et al. (1985) in Figure 4.

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Second, we employed compact spherical dust particles having a broken power-law initial SFD, namely ${dn}\,\propto \,{m}^{-{\alpha }_{1}}{dm}$ for m ≤ m_c and ${dn}\,\propto \,{m}^{-{\alpha }_{2}}{dm}$ for m ≥ m_c. When fitting the data, we varied m_c between five discrete β values included in the numerical integration. The derived initial parameters are tabulated in Table 5.

Table 5.  Best-fit Parameters and Expected Mass Supply Rate for the SFD Model 2^a

ρ (g cm−3)	mc (g)	α1	α2	dmtotal/dt (t s−1)b	dmtotal/dt (t s−1)c	Goodness of Fitd
0.8	6.5 × 10−9	1.583–1.667	2.167	∼29	∼28	0.16–0.18
0.8	8.2 × 10−7	1.833–2.000	2.167–2.333	30–35	30–34	0.09–0.15
0.8	6.5 × 10−6	2.000	2.333	∼34	∼34	0.18
1.9	1.4 × 10−7	1.583–1.833	2.083–2.250	31–36	30–32	0.06–0.19
1.9	1.2 × 10−6	1.833–1.917	2.083–2.250	34–37	32–34	0.11–0.16
3.0	5.8 × 10−8	1.583–1.833	2.000–2.167	34–41	31–34	0.06–0.16
3.0	4.7 × 10−7	1.750–1.833	2.000–2.167	36–42	33–35	0.11–0.15

Notes.

^a ${dn}\,\propto \,{m}^{-{\alpha }_{1}}{dm}$ for m ≤ m_c, and ${dn}\,\propto \,{m}^{-{\alpha }_{2}}{dm}$ for m ≥ m_c. ^bTotal mass supply rate into the IDP cloud complex for particles with β > 0.00057. ^cTotal mass supply rate into the IDP cloud complex for particles with β > 0.00114. ^dAverage of absolute values of log of ratio between our model and observed model (Grün et al. 1985).

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Third, we tested the case of a doubly broken power-law SFD with compact spherical dust particles. The SFD was assumed in the form ${dn}\,\propto \,{m}^{-{\alpha }_{1}}{dm}$ for m ≤ m_c1, ${dn}\,\propto \,{m}^{-{\alpha }_{2}}{dm}$ for m_c1 ≤ m ≤ m_c2, and ${dn}\,\propto \,{m}^{-{\alpha }_{3}}{dm}$ for m_c2 ≤ m. In this case, we assumed α₁ > α₂, α₃ > α₂, and limited parameters as in the previous case. The results are tabulated in Table 6.

Table 6.  Best-fit Parameters and Expected Mass Supply Rate for the SFD Model 3^a

ρ	mc1	mc2	α1	α2	α3	dmtotal/dtb	dmtotal/dtc	Goodness
(g cm−3)	(g)	(g)				(t s−1)	(t s−1)	of Fitd
0.8	6.5 × 10−9	8.2 × 10−7–6.5 × 10−6	1.917–2.250	1.833–2.000	2.167–2.333	31–46	31–45	0.10–0.19
1.9	6.5 × 10−9–8.2 × 10−7	8.2 × 10−7–6.5 × 10−6	1.667–2.167	1.583–1.833	2.083–2.250	32–39	30–36	0.07–0.19
3.0	6.5 × 10−9–8.2 × 10−7	8.2 × 10−7–6.5 × 10−6	1.667–2.250	1.583–1.833	2.000–2.167	34–44	31–38	0.06–0.16

Notes.

^a ${dn}\,\propto \,{m}^{-{\alpha }_{1}}{dm}$ for m ≤ m_c1, ${dn}\,\propto \,{m}^{-{\alpha }_{2}}{dm}$ for m_c1 ≤ m ≤ m_c2, and ${dn}\,\propto \,{m}^{-{\alpha }_{3}}{dm}$ for m_c2 ≤ m. ^bTotal mass supply rate into the IDP cloud complex for particles with β > 0.00057. ^cTotal mass supply rate into the IDP cloud complex for particles with β > 0.00114. ^dAverage of absolute values of log of ratio between our model and observed model (Grün et al. 1985).

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The next model is composed of both compact spherical particles and fluffy aggregates. We assumed that the fluffy aggregates have constant β values regardless of mass (Mukai et al. 1992; Skorov et al. 2016). The SFD of compact spherical particles was assumed to be ${dn}\,\propto \,{m}^{-{\alpha }_{1}}{dm}$ for $m\leqslant {m}_{{c}_{1}}$ and ${dn}\,\propto \,{m}^{-{\alpha }_{3}}{dm}$ for $m\geqslant {m}_{{c}_{1}}$. Then we assumed the total SFD of fluffy aggregates and compact spherical particles between masses of m_c1 and m_c2 to dn ∝ m^−α₂dm. The results are presented in Table 7.

The best-fit initial SFD of the previous paragraph has bending points at smaller mass than the SFD measured at cometary comae, and slopes steeper than measurements in the range 0.000114 < β < 0.0057 (Table 7). As a final model, we applied the SFD with the same functional form as that in the above model but we fixed m_c1 and m_c2 around the observed values. Then, α₁, α₂, and α₃ will be near the observed values, especially when the density of compact particles is higher than 1.9 g cm⁻³. In those cases, the ratio between our estimation and the model of Grün et al. (1985) worsened by a factor of three for some sizes. However, our model using this observed value of input parameters is still not far from the SFD observed in the Earth's orbit as presented in Table 8 and Figures 3 and 4.

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As presented in Figure 4 and Tables 4–8, different dust ejection conditions for dust particle shape and density can be validated with the appropriate initial SFD. It is noteworthy that the dust ejection conditions measured around 67P/Churyumov–Gerasimenko and 81P/Wild 2 concur with dust particle measurements around the Earth's orbit when we assume the existence of fluffy aggregates in the discovered size range. We consider that the various dust environments measured around cometary nuclei are in accordance with our knowledge about the IDP cloud complex and its cometary origin, and that the target comets of space missions (i.e., 1P, 67P, and 81P) are not extraordinary but may represent ordinary dust ejections from comets.

By scaling our model SFD by the observed gegenschein brightness, we calculated the expected mass supply rate to the IDP cloud complex. As presented in Tables 4–8 and Figure 5, the total dust supply rate does not vary greatly according to dust particle shape, density, and the initial SFD when the latter is confined by the observed SFD around the Earth's orbit. The required dust supply rate for particles with β > 0.00057 was 39–45 t s⁻¹ for model 1, 29–41 t s⁻¹ for model 2, 31–49 t s⁻¹ for model 3, 32–41 t s⁻¹ for model 4, and 32–43 t s⁻¹ for model 5. We thus find that the different SFD models in this research do not significantly change the mass supply rate. This value is roughly consistent with but slightly larger than the estimation of Nesvorný et al. (2011) of 1.6–25 t s⁻¹. When we consider that our derived SFD is shallower than that of Nesvorný et al. (2011), we believe that this difference is understandable.

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Even though we did not include dust particles larger than β = 0.00057, we expect that this omission will not critically change our conclusion because the mass supply rates for β ≥ 0.00114 and β ≥ 0.00057 are not significantly different from each other, as shown in Tables 4–8.

Collisional probabilities of the objects in the solar system have been investigated since the pioneering works by Opik (1951) and Dohnanyi (1969). The collisional lifetime of IDPs was estimated under the assumption of fixed circular orbits (Dohnanyi 1978; Grün et al. 1985; Steel & Elford 1986). It is noticed that catastrophic collisions among IDPs could be a dominant mechanism for losing IDPs if their sizes are larger than ∼200 μm (Dohnanyi 1978; Grün et al. 1985; Steel & Elford 1986). Despite the simplified models in these references where IDPs revolve on fixed circular orbits, their estimated values have been considered in recent dynamical studies of cometary dust particles with eccentric orbits (Wiegert et al. 2009; Nesvorný et al. 2011; Pokorný et al. 2014). More recently, Soja et al. (2016) derived the collisional lifetime for particles on various fixed eccentric orbits, and the lifetime was comparable with the P–R lifetime for 100 μm particles. On another front, the orbital distribution of helion meteors favors a collisional lifetime longer than the estimate of Grün et al. (1985) (Nesvorný et al. 2011). From the observational aspect, Jenniskens et al. (2016) discovered 7 mm meteorites delivered to the Earth that have orbits with low eccentricity, suggesting that such dust particles migrated via P–R drag before collisional breakup. To sum up this bibliographic background, it is required to calculate the collisional lifetime in a more realistic manner, where the orbits of IDPs evolve under planetary perturbations and P–R drag.

As mentioned in Section 3.1, most dust particles from JFCs are kicked out to orbits with large a values and stay for a long time in the outer solar system. Frequently, the particles eventually fall toward the Sun, are kicked out to orbits with large a values, and spend most of their lifetimes in such orbits. Furthermore, the eccentricity of their orbits does not decrease as in the situation with no planets. When we consider these factors along with the dynamical lifetimes of dust particles that are shorter than the P–R timescale, we can expect a lower probability of mutual collision in the IDP cloud complex. In this paper, we treated mutual collisions quantitatively and discussed whether the effect of impact is significant. We did not consider the impact velocity and the consequence of collision, but calculated the probability of collision regardless of impact velocity. We think this choice can simplify the treatments required during calculation.

We calculated the volume swept by a dust particle during orbital evolution. At each epoch, we also calculated the cross-section density on the positions of the dust particle orbit as a function of particle size. By integrating the product of these two values (i.e., the swept volume and the cross-section density), we derived the cross section swept by the dust particle. By dividing the swept cross section by the cross section of the single particle responsible (i.e., the projectile), we derived the contact probability for given epochs. We defined the contact timescale by dividing the dynamical lifetime by the total contact probability, which is the integral of the contact probability over the entire dynamical lifetime. The contact timescale was calculated as a function of the smallest projectile size considered. The timescales are shown in Figure 2.

As we can see in Figure 2, mutual contact between particles with a size ratio that is larger than 1:100 (a mass ratio of one to million) will not happen in the IDP cloud complex. Because we did not consider the impact speed of these contacts, we are not sure how large a projectile is required to break the target dust particle, but it is expected that a projectile 100 times smaller than the target will not destroy it. In addition, we do not know the projectile size that is required to break the fluffy aggregates. The values in Figure 2 are from the case in which fluffy aggregates have the longest dynamical lifetime and the largest size (3 mm diameter), the case in which particles most easily experience collision. When we consider that fluffy aggregates were fragmented by collisions with impact speeds of ∼cm s⁻¹ in the COSIMA detector on the Rosetta spacecraft (Hilchenbach et al. 2016), we think contact with a 25 μm projectile would break the 3 mm fluffy aggregates at least partially, but we are not sure about the exact consequences of contact. However, overall, we can see that fluffy aggregates will not experience mutual collisions, especially at sizes smaller than a few millimeters in diameter.