DYNAMICAL MODEL FOR THE ZODIACAL CLOUD AND SPORADIC METEORS

We only consider JFCs in this paper, because previous works showed that they are the main source of the ZC particles and helion/antihelion meteors (e.g., Jones et al. 2001; W09; N10). The asteroid meteoroids have low impact speeds and are not detected by meteor radars. The meteoroids released from long-period comets contribute to apex meteors and are not modeled here (see Nesvorný et al. 2011 for a discussion of apex meteors).

The orbital distribution of JFCs was taken from Levison & Duncan (1997, hereafter LD97), who followed the orbital evolution of bodies originating in the Kuiper belt as they are scattered by planets, and evolve in small fractions into the inner solar system. For each critical perihelion distance, q*, we selected bodies from LD97's simulations when they reached q < q* for the first time. Particles were released from these source orbits.⁸ We used 10 values of q* equally spaced between 0.25 AU and 2.5 AU, particles with D = 10, 30, 100, 300, 1000, and 3000 μm, which should cover the interesting range of sizes, and particle density ρ = 2 g cm⁻³. Our tests show that this size resolution is adequate because the orbital dynamics of, say, a D = 150 μm particle is similar to that of a D = 100 μm particle. The results for a continuous range of sizes were obtained by interpolation. Varying particle density has only a small effect on their orbital dynamics. Ten thousand particles were released for each q* and D for the total of 0.6 million of the initial orbits.

Upon their release from the parent object, particles will feel the effects of radiation pressure. These effects can be best described by replacing the mass of the Sun, m_☉, by m_☉(1 − β), with β given by

$$\begin{equation} \beta = 5.7\times 10^{-5}\ {Q_{\rm pr} \over \rho s}\, \end{equation} \tag{ 1 }$$

where the particle's radius s = D/2 and ρ are in cgs units. Pressure coefficient Q_pr can be determined using the Mie theory (Burns et al. 1979). We set Q_pr = 1, which corresponds to the geometrical optics limit, where s is much larger than the incident-light wavelength. Note that all particles considered here are large enough to stay on bound heliocentric orbits after their release (see, e.g., Nesvorný et al. 2011).

In Section 3, we combine the results obtained with different q* and D to mimic the continuous distributions dN(q) and dN(D). This is different from N10 where the source particle distributions were parameterized by the "fading time," particle production rate was assumed to be q-independent, and dN(D) was approximated by single size. Here we consider dN(q) and dN(D) that can be approximated by simple power laws. For dN(q), we thus have dN(q)∝q^γdq, where γ is a free parameter. For dN(D), we have dN(D) = N₀D^−αdD, where N₀ is a normalization constant and α is the usual slope index (at source). Alternatively, we use the two-slope SFD with ${\it dN}(D) \propto D^{-\alpha _1} {\it dD}$ for D < D* and ${\it dN}(D) \propto D^{-\alpha _2} {\it dD}$ for D > D*, where α₁, α₂, and D* are free parameters.

Parameter γ can be inferred from the number of JFCs found at each q, and their disruption probability as a function of q. If the former can be approximated by q^ξ, where ξ ∼ 0.5 (LD97; Di Sisto et al. 2009), and the latter is proportional to q^−ζ, where ζ ≃ 0.5–1 (Di Sisto et al. 2009), it would be expected that γ = ξ − ζ ∼ −0.5–0. We use γ = 0 as our starting value, and test the sensitivity of results for γ < 0 and γ > 0. As for dN(D), we set α ∼ 4 for the whole size range, as motivated by meteor radar observations (e.g., Galligan & Baggaley 2004), or D* ∼ 100 μm, α₁ < 3 and α₂ > 4, as motivated by space impact experiments (e.g., Love & Brownlee 1993). Given the various uncertainties of these measurements (see, e.g., Mathews et al. 2001), we also test D* < 100 μm and D* > 100 μm.

The particle orbits were numerically integrated with the swift_rmvs3 code (Levison & Duncan 1994), which is an efficient implementation of the Wisdom–Holman map (Wisdom & Holman 1991) and which, in addition, can deal with close encounters between particles and planets. The radiation pressure and P-R drag forces were inserted into the Keplerian and kick parts of the integrator, respectively. The change to the Keplerian part was trivially done by substituting m_☉ by m_☉(1 − β). We assumed that the solar-wind drag force has the same functional form as the P-R term and contributes by 30% to the total drag intensity.

The code tracks the orbital evolution of a particle that revolves around the Sun and is subject to the gravitational perturbations of seven planets (Venus to Neptune; the mass of Mercury was added to the Sun) until the particle impacts a planet, is ejected from the solar system, evolves to within 0.05 AU from the Sun, or the integration time reaches 5 Myr. We removed particles that evolved to R < 0.05 AU, because the orbital period for R < 0.05 AU was not properly resolved by our one-day integration time step.⁹ The particle orbits were recorded at 1000 yr time intervals to be used for further analysis.

The solar system meteoroids can be destroyed by collisions with other particles and by solar heating that can lead to sublimation and vaporization of minerals. Here we explain how we parameterize these processes in our model.

The JFC particles will rapidly lose their volatile ices. We do not model the volatile loss here. The remaining grains will be primarily composed from amorphous silicates and will survive down to very small heliocentric distances. Following Moro-Martín & Malhotra (2002), Kessler-Silacci et al. (2007), and others, we adopt a simple criterion for the silicate grain destruction. We assume that they are thermally destroyed (sublimate, vaporize) when the grain temperature reaches T ⩾ 1500 K.

Using the optical constants of amorphous pyroxene of approximately cosmic composition (Henning & Mutschke 1997), we find that a dark D ≳ 100 μm grain at R has the equilibrium temperature within 10 K of a black body, $T \simeq 280/\sqrt{R}$ K. According to our simple destruction criterion, T ⩾ 1500 K, the silicate grains should thus be removed when reaching R ≲ 0.035 AU. On the other hand, the smallest particles considered in this work (D = 10 μm) have T = 1500 K at R ≃ 0.05 AU. Thus, we opted for a simple criterion where particles of all sizes were instantly destroyed and were not considered for statistics, when they reached R ⩽ 0.05 AU. Note that, by design, this limit is the same as the one imposed by the integration time step (Section 2.2).

The collisional lifetime of meteoroids, τ_coll, was taken from G85. It was assumed to be a function of particle mass, m, and orbital parameters, mainly a and e. For example, for a circular orbit at 1 AU, particles with D = 100 μm and 1 mm have τ*_coll = 1.5 × 10⁵ yr and 7.3 × 10³ yr, respectively, where τ*_coll denotes the collisional lifetime from G85. Also, τ*_coll increases with a. To cope with the uncertainty of the G85's model, we introduced a free parameter, S, so that τ_coll = Sτ*_coll. Values S > 1 increase τ_coll relative to τ*_coll, as expected, for example, if particles were stronger than assumed in G85, or if the measured impact fluxes were lower (e.g., Dikarev et al. 2005; Drolshagen et al. 2008). See Nesvorný et al. (2011) for a fuller description.

Collisional disruption of particles was taken into account during processing the output from the numerical integration described in Section 2.2. To account for the stochastic nature of breakups, we determined the breakup probability p_coll = 1 − exp (− h/τ_coll), where h = 1000 yr is the output interval, and τ_coll was computed individually for each particle's orbit. The code then generated a random number 0 ⩽ x ⩽ 1 and eliminated the particle if x < p_coll.

We caution that our procedure does not take into account the small debris fragments that are generated by disruptions of larger particles. Instead, all fragments are removed from the system. This is an important approximation whose validity needs to be tested in the future.

Meteoroids were assumed to be isothermal, rapidly rotating spheres. The absorption was assumed to occur into an effective cross section πs², and emission out of 4πs². The infrared flux density (per wavelength interval dλ) per unit surface area at distance r from a thermally radiating particle with radius s is

$$\begin{equation} F_\lambda = \epsilon (\lambda,s) B(\lambda,T) {s^2 \over r^2} \, \end{equation} \tag{ 2 }$$

where is the emissivity and B(λ, T) is the energy flux at (λ, λ + dλ) per surface area from a black body at temperature T:

$$\begin{equation} B(\lambda,T) = {2 \pi h c^2 \over \lambda ^5} \left[ {e}^{hc/\lambda k T} - 1 \right]^{-1}. \end{equation} \tag{ 3 }$$

In this equation, h = 6.6262 × 10⁻³⁴ J s is the Planck constant, c = 2.99792458 × 10⁸ m s⁻¹ is the speed of light, and k = 1.3807 × 10⁻²³ J K⁻¹ is the Boltzmann constant.

Since our model does not include detailed emissivity properties of dust grains at different wavelengths, we set the emissivity at 25 μm to be 1 and fit for the emissivities at 12 and 60 μm. We found that the relative emissivities at 12 and 60 μm that match the data best are 0.70–0.75 and 0.95–1, respectively. Such a variability of MIR emissivity values at different wavelengths is expected for small silicate particles with some carbon content. T(R) was set to be $280/\sqrt{R}$ K, as expected for dark D ≳ 10 μm particles. See Nesvorný et al. (2006) for a more precise treatment of (λ, s) and T(R) for dust grains composed of different materials.

To compare our results with IRAS observations illustrated in Figure 2,¹⁰ we developed a code that models thermal emission from distributions of orbitally evolving particles and produces infrared fluxes that a space-borne telescope would detect depending on its location, pointing direction and wavelength. See Nesvorný et al. (2006) for a detailed description of the code.

In brief, we define the brightness integral along the line of sight of an infrared telescope (defined by fixed longitude l and latitude b of the pointing direction) as

$$\begin{equation} \int _{a,e,i}\! da de di\! \int _{0}^{\infty } \!dr\ r^2\! \int _D \!dD\, F_\lambda (D,r) N(D;a,e,i) K(R,L,B) \, \end{equation} \tag{ 4 }$$

where r is the distance from the telescope, F_λ(D, r) is the infrared flux (evaluated at the effective wavelength of the telescope's system) per unit surface area at distance r from a thermally radiating particle with diameter D. K(R, L, B) defines the spatial density of particles in the Sun-centered coordinates as a function of R, ecliptic longitude, L, and latitude, B. N(D, a, e, i) is the number of particles having effective diameter D and orbits with a, e, and i.

We evaluate the integral in Equation (4) by numerical renormalization (see Nesvorný et al. 2006). F_λ(D, r) is calculated as described in Section 2.4. N(D, a, e, i) is obtained from our numerical simulations (Section 2.2). K(R, L, B) uses analytic expressions for the spatial distribution of particles with fixed a, e, and i, and randomized orbital longitudes (Kessler 1981).

We assume that the telescope is located at (x_t = r_tcos ϕ_t, y_t = r_tsin ϕ_t, z_t = 0) in the Sun-centered reference frame with r_t = 1 AU. Its viewing direction is defined by a unit vector with components (x_v, y_v, z_v). In Equation (4), the pointing vector can be also conveniently defined by longitude l and latitude b of the pointing direction, where x_v = cos bcos l, y_v = cos bsin l, and z_v = sin b. We fix the solar elongation l_☉ = 90°, so that l = ϕ_t + 90°, and calculate the thermal flux of various particle populations as a function of b and wavelength. The model brightness profiles at 12, 25, and 60 μm are then compared with the mean IRAS profiles shown in Figure 2.

We used the Öpik theory (Öpik 1951) to estimate the expected terrestrial accretion rate of JFC particles in our model. Wetherill (1967), and later Greenberg (1982), improved the theory by extending it more rigorously to the case of two eccentric orbits. Here we used a computer code that employs Greenberg's formalism (Bottke et al. 1994).¹¹

We modified the code to compute the radiants of the impacting particles. In doing so we properly accounted for all impact configurations and weighted the results by the probability with which each individual configuration occurs, including focusing. The radiants were expressed in the coordinate system, where longitude l was measured from the Earth's apex in counterclockwise direction along the Earth's orbit, and latitude b was measured relative to the Earth's orbital plane. Note that our definition of longitude differs from the one more commonly used for radar meteors, where the longitude is measured from the helion direction. The radiants were calculated before the effects of gravitational focusing were applied.

The meteor radars use different detection methods (i.e., trail versus echo)¹² and have different sensitivities. Their detection efficiency is mainly a function of the particle's mass and speed, but it also depends on a number of other parameters discussed, for example, in Janches et al. (2008). Following W09, we opt for a simple parameterization of the radar sensitivity function, where the detection is represented by the ionization function

$$\begin{equation} I(m,v) = {m \over 10^{-4}\, {\rm g}} \left({v \over 30\,{\rm km\,s^{-1}}} \right)^{3.5} . \end{equation} \tag{ 5 }$$

All meteors with I(m, v) ⩾ I* are assumed to be detected in our model, while all meteors with I(m, v) < I* are not detected. The ionization cutoff I* is taken to be different for different radars. For example, I* ∼ 1 for the Canadian Meteor Orbit Radar (CMOR; Campbell-Brown 2008) and I* ∼ 0.01–0.001 for Advanced Meteor Orbit Radar (AMOR; Galligan & Baggaley 2004, 2005). For reference, a JFC particle with v = 30 km s⁻¹ and m = 10⁻⁶ g, corresponding to D ≃ 100 μm with ρ = 2 g cm⁻³, has I(m, v) = 0.01, i.e., a value intermediate between the two thresholds. These meteoroids would thus be detected by AMOR, according to our definition, but not by CMOR. The particle size detection threshold is shown, as a function of v, in Figure 3.

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To study the orbital properties of different meteor sources, these sources need to be isolated. This is typically done by selecting meteors with specific radiants. To test how the radiant cutoff affects the results, we select the helion meteors with −90° < l < −45° and −30° < b < 30°, and antihelion meteors with 45° < l < 90° and −30° < b < 30°. Since our code computes the same impact speed and orbit distributions for the helion and antihelion sources, we combine the results from the two radiant cutoffs together. Note, therefore, that our method cannot capture the suspected asymmetry between the helion and antihelion sources (see, e.g., W09, and the references therein).

We start by discussing the results relevant to AMOR, because AMOR is capable of detecting particles with D ∼ 100 μm (Figure 3), and can thus provide constraints on the particle sizes that are thought to be dominant in the ZC. Figure 4 shows the distributions of impact speeds and orbits of JFC meteoroids for D* = 100 μm, γ = 0, S = 1, and several values of the ionization cutoff. With I* = 0, corresponding to no cutoff on mass or impact speed, the impact speed distribution, dN(v), is strongly peaked toward the Earth's escape speed (v_esc = 11.2 km s⁻¹). When I* = 0.003 cutoff is applied, as roughly expected for the AMOR detections, dN(v) has a maximum at v ≃ 25 km s⁻¹. This illustrates the crucial importance of the ionization cutoff for the interpretation of meteor radar observations.

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Given the strong effect of the ionization cutoff it is difficult to imagine how the radar observations can be correctly "debiased," based solely on the measurements, corrections, and considerations of the Earth-impact probability of different orbits (e.g., Taylor & McBride 1997; Galligan & Baggaley 2004; Campbell-Brown 2008), to obtain the real distribution of meteoroids at 1 AU. As shown in Figure 4, the real distribution can be very different from the observed one; although meteors at low speeds may dominate the real distribution, only a tiny fraction are detected. This highlights the importance of dynamical modeling.

In a similar fashion, the observed eccentricity distribution, dN(e), is strongly biased toward large values by the ionization cutoff, while the underlying distribution has more small and moderate values (Figure 4(d)). This explains, at least in part, why N10 were unable to obtain meteor-like dN(e), because they did not model the meteor detection in detail. The model semimajor axis and inclination distributions, dN(a) and dN(i), are also strongly affected by the ionization cutoff. With I* = 0.003, both distributions become significantly broader than those computed for I* = 0 (Figures 4(b) and (c)).

The distribution of impact speeds obtained in our model has a peak value of v = 20 km s⁻¹ for I* = 0.001 and v = 30 km s⁻¹ for I* = 0.01, in good agreement with the AMOR measurements of helion/antihelion meteors that show a peak at v = 20–25 km s⁻¹. The spread of model dN(v) (Figure 4(a)), however, is slightly narrower than the one indicated by observations (Figure 1(a)). We will discuss this difference later in this section and show that it can be related to the initial SFD of particles produced by JFCs.

Figure 5 illustrates the effect of the radiant cutoff for I* = 0.003 and the case described above. The radiant cutoff, as defined in Section 2.6, moves dN(v) to slightly larger values (Figure 5(a)) and leads to narrower distributions of a, e, and i. More aggressive radiant selection criteria, such as the ones used in Campbell-Brown (2008) to define the helion/antihelion sources, would produce a slightly larger effect. On the other hand, Galligan & Baggaley (2005) adopted a very broad radiant cutoff (−120° < l < −30° and 20° < l < 120°, respectively, for our definition of l, and no condition on b). According to our tests, these broad selection criteria give results that are similar to those with no radiant cutoff.

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The effect of the radial distribution of initial orbits is illustrated in Figure 6. As expected, γ < 0 produces dN(v) that peaks at larger values, and dN(e) that is more skewed toward e = 1. This is because more particles are produced with small q values in this case, and these particles tend to have larger v and e values when they impact. The effects of γ > 0 are opposite to those of γ < 0. Interestingly, dN(a) and dN(i) obtained with I* = 0.003 are not very sensitive to changes of γ.

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Together with Figure 4, these results show that it can be difficult to constrain the value of γ from the fits to the AMOR measurements alone, because the effects of γ are similar to those produced by slight changes in the detection efficiency, and can be confused with them. A detailed knowledge of the instrument sensitivity, that goes beyond the simple concept of the ionization cutoff described in Section 2.6, will be required for a more constrained modeling (see, e.g., Fentzke & Janches 2008; Fentzke et al. 2009).

The effects of initial dN(D) of particles, as discussed below, are in many ways similar to those produced by changes of γ and I*. Figure 7 illustrates the effect of D*. Again, dN(v) and dN(e) show more variation than dN(a) and dN(i). While for D* = 30 μm, the velocity peak shifts to v ≃ 30 km s⁻¹, it moves to v ≃ 20 km s⁻¹ for D* = 300 μm. This variation can be linked to the ionization cutoff. For example, with D* = 30 μm, particles tend to be smaller and will be detected with I* = 0.003 only if their speeds are larger.

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The distributions dN(v) shown in Figures 4(a)–7(a) are all slightly narrower than the one indicated by observations (see Figure 1(a)). This difference cannot be resolved by varying γ, S, or I*. Instead, to resolve this problem, we needed to assume that the power index of dN(D) is 3 ≲ α ≲ 4, at least in the size range relevant to AMOR observations. To illustrate this case, Figure 8 shows the distributions for α = α₁ = α₂ = 3.5. While dN(a), dN(e) and dN(i) have not changed much relative to Figure 4, the new distribution dN(v) with I* = 0.01–0.001 becomes broader, thus better mimicking the AMOR measurements.

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This trend can be easily understood. With a sharp SFD break at D* ∼ 100 μm, the particles that produce most meteors with I > 0.01–0.001 are those with D ∼ 100 μm. These particles have similar orbital histories and produce a relatively narrow dN(v). With α₁ = α₂ ∼ 3.5, on the other hand, the size range of particles significantly contributing to AMOR meteors increases, relative to the previous case, producing a larger variability in orbital histories, and thus also a larger spread in v. Figure 9 illustrates these trends.

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By experimenting with different SFDs, we found that the best matches to AMOR observations can be obtained with D* ≲ 50 μm and α₂ = 3.5, or with D* ≳ 200μm and α₁ = 3.5, while the values of γ and S are essentially unconstrained (but see Section 3.2 for a discussion of the collisional model). We opt for D* = 200 μm in Figure 10, which illustrates one of our preferred models, because the original interpretation of spacecraft impact experiments indicates a change of slope at D ≃ 200 μm (e.g., G85; Love & Brownlee 1993). The solutions with D* ≲ 50 μm and α₂ = 3.5, which would better correspond to the reinterpretation of the impact experiments by Mathews et al. (2001), are also plausible. We will discuss this issue in Section 4.

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While our models dN(v), dN(a), and dN(e) in Figure 10 match observations reasonably well, the model dN(i) is slightly narrower than the one measured by AMOR. This indicates that we may be missing sources with larger inclinations. For example, as discussed in Section 2.1, our model for the initial inclinations of JFC particles can be inappropriate if JFCs lose mass gradually by recurrent splitting events (e.g., LD97; Di Sisto et al. 2009; N10). It is also possible, however, that the radiant cutoff of Galligan & Baggaley (2004, 2005) is not sufficiently restrictive to pick up JFC meteoroids only, as hinted on by Figure 1(c), where dN(i) seems to follow different trends for i < 30° and i > 30°. Note that Campbell-Brown (2008), using a more restrictive radiant cutoff, obtained a relatively narrow dN(i) of helion/antihelion meteors.

Figure 11 shows the radiant distributions for our preferred model shown in Figure 10. With I* = 0, the radiants fill the whole sky and show broad concentrations around l = −90°, l = 90°, and b = 0°. With I* = 0.003, however, the radiants become tightly clustered about l = −70°, l = 70°, and b = 0°. This highlights the importance of the ionization cutoff. For a comparison, Galligan & Baggaley (2005) found that the helion and antihelion sources are centered at l = ±70° and their full widths are ≃ 20° in both l and b. The location and spread of our model radiants very closely match these measurements.

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The results discussed in Section 3.1 were obtained with the standard G85 model for the collisional disruption of particles. In G85, the large, millimeter-sized particles have very short physical lifetimes (∼10⁴ yr) and disrupt before they can significantly evolve by P-R drag. Small particles, on the other hand, have long collisional lifetimes and evolve faster by P-R drag. The G85 model therefore implies different orbital histories of small and large particles and, as we found here, produces significantly different distributions of impact speeds and heliocentric orbits for I* = 0.003 and I* = 1. Figure 12 illustrates the case of I* = 1. These results are at odds with observations, because the distributions measured by AMOR and CMOR are not that different.

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To resolve this problem, we needed to assume that τ_coll for D ∼ 1 mm is significantly longer than in G85 (S ≳ 30). Figure 13 shows the results for S = 100. The main improvement with respect to Figure 12 is that dN(a) now peaks at a ∼ 1 AU. This is a direct consequence of longer τ_coll, which allows the large particles to accumulate larger P-R drifts and reach a ∼ 1 AU. On the other hand, models with S ≳ 30 and I* = 0.01–0.001 do not match the AMOR measurements. This shows that the size dependence of the G85 collisional model may be incorrect. Taken together, if S ≳ 30 is needed to match CMOR, while S ∼ 1 is needed to match AMOR, τ_coll(D) should be more constant over the relevant size range, D ∼ 30–1000 μm according to Figure 9, than suggested by G85.

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We performed a search in parameter space to see whether we can obtain the impact speed and orbit distributions with S = 100 that would closely resemble those measured by CMOR (Figure 10–12 in Campbell-Brown 2008). We found that the model results with α ∼ 2 work best. With α ∼ 2, at least locally near D ∼ 500 μm, which are the most important sizes for CMOR, the model distributions have the characteristic shapes measured by CMOR (Figure 14). When no radiant cutoff is used, dN(v) has the maximum just below v = 20 km s⁻¹, and dN(e) peaks at e ∼ 0.7. When the radiant cutoff is applied, dN(v) has the maximum just below v = 30 km s⁻¹, and dN(e) peaks at e ∼ 0.8. These trends correspond very well to those in Figures 10 and 12 in Campbell-Brown (2008).

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The model dN(a) with radiant cutoff becomes more tightly clustered at a ∼ 1 AU than in the case without cutoff, in a nice correspondence to the CMOR measurements (Figure 14(b)). Our dN(i) with radiant cutoff is slightly broader than the CMOR distribution (Figure 14(c)), which is logical because the radiant cutoff used by Campbell-Brown (2008) is more restrictive than the one used here. Overall, the agreement between the model and observations is very good.

While the meteoroid SFD can be wavy (e.g., Ceplecha et al. 1998), with α ∼ 3.5 at D ∼ 100 μm (see Section 3.1) and α ∼ 2 at D ∼ 500 μm, these slope determinations can also be artificially imposed on the results by our simple treatment of the radar's detection efficiency.¹³ In addition, it is not clear to us whether it is appropriate to compare our results for the helion/antihelion meteors to Figures 10 and 12 in Campbell-Brown (2008), because their Figure 10 shows the CMOR distributions for all sporadic sources, and the distributions in Figure 12 were weighted to a constant limiting mass, a correction that is not needed for a comparison with our model.

Finally, Figure 15 shows the CMOR radiant distributions for models illustrated in Figures 12 and 13. With longer τ_coll, the radiants are slightly more spread around the helion and antihelion directions. Interestingly, Figure 15 indicates that the JFC meteoroids are capable of producing apex meteors. These apex particles have prograde orbits, low impact speeds, and very low semimajor axes. They impact from the apex direction because their orbital speed near the aphelion at 1 AU is smaller that the Earth's orbital speed. The contribution of JFC meteoroids to apex meteors should be small, however, relative to those produced by the retrograde HTC and/or OCC meteoroids. We verified that only a small fraction (<1%) of the JFC meteoroids can reach retrograde orbits.

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Using the methods described in Sections 2.4 and 2.5, we computed the MIR fluxes for all models considered in the previous section. Here we illustrate these results and compare them with those obtained by IRAS. Before we do so, however, we want to emphasize that the ZC is in all likelihood a mixture of several particle populations, including contribution from asteroids and long-period comets (see, e.g., N10), while here we only model the JFC component. The best fits obtained to IRAS observations in this work are therefore only approximate and could be modified if other components of the ZC were considered.

Figure 16 shows our results for D* = 100 μm, α₁ = 2, α₂ = 5, γ = 0, and S = 1, corresponding to Figures 4 and 5. The model MIR profiles are slightly narrower than the observed ones, but otherwise correspond to IRAS measurements reasonably well. For example, a small, ≲ 10% contribution from a source with a more isotropic distribution of inclinations, such as HTCs and/or OCCs, would easily compensate for the small difference. See N10 for a discussion of additional sources that were not modeled here.

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A different way to bring the model and observations into a closer agreement is to assume that γ < 0. With γ < 0, the distribution of JFC particles is weighted toward low R and is projected to a wider range of b when observed from R = 1 AU. If γ > 0, on the other hand, the distribution is weighted toward large R and is seen closer to the ecliptic. We tested a continuous range of γ values and found that γ ≃ −1.3 provided the best match to the IRAS observations (Figure 17).

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The effects of additional sources and γ ≠ 0 on the MIR profiles are to some degree degenerate in the IRAS model. They would be difficult to separate, based solely on modeling of the IRAS observations, if we included additional sources in the present work. For example, as discussed above, the MIR profiles become broader, and more similar to the IRAS measurements, if γ < 0 and/or if sources with a more isotropic inclination distribution are included (N10). On the other hand, a small asteroid contribution at the ∼5%–10% level (Nesvorný et al. 2006) would produce slightly narrower profiles than those obtained with the JFCs alone (N10). A two-source model with JFCs and asteroids would thus require γ < 0.

The MIR profiles obtained in our model are not overly sensitive to the assumptions on the collisional lifetimes of particles. For example, increasing the collisional lifetime of millimeter-sized particles relative to the G85 model, which may be needed to match CMOR observations (Section 3.2), does not affect the results obtained here, because the ZC's cross section is mainly in D ≲ 200 μm particles (N10).

In addition, the model profiles are also insensitive to the input SFD of the JFC particles. This is because the JFC particles with D ≲ 200 μm have τ_coll that exceeds their P-R drag lifetimes. All these particles therefore have roughly similar orbital histories and produce similar MIR profiles. This explains why we obtain nearly identical results for all D* < 200 μm.¹⁴ The effects of α₁ and α₂ are also minor.

The comparison of our model with the IRAS data is important because it allows us to obtain the absolute calibration of the number of particles in the ZC (or, more precisely, their total cross section). This calibration can then be used to estimate the rate of the terrestrial accretion of interplanetary material, both with and without the ionization cutoff, with the former estimate being relevant to radar observations. Note that it is more difficult to derive the overall terrestrial accretion rate from the meteor radar measurements alone, because different radar instruments have different detection sensitivities, and some, such as the less sensitive CMOR, do not detect the very small and/or slow meteoroids (see discussion in Section 4).

Unless we specify otherwise, all estimates quoted below were obtained for the full size range of particles between D = 5 μm and D = 1 cm (10⁻¹⁰ g to 1 g for ρ = 2 g cm⁻³). These estimates need to be considered with caution because they were obtained with approximate initial SFDs. In reality, the number of particles released by JFCs can be a complicated function of D and should also depend on the circumstances of the splitting/disruption events.

We start by discussing the total cross-section area (σ_ZC) and mass (m_ZC) of particles in the ZC. For the sake of simplicity, we will assume that 30 ≲ D* ≲ 300 μm, α₁ < 3 and α₂ > 4, so that particles with sizes below 30 μm and above 300 μm do not strongly contribute to the cross section or mass, as suggested by the spectral observations of the ZC (e.g., Reach et al. 2003), and various other measurements (see, e.g., Ceplecha et al. 1998, and the references therein).

With these assumptions we find that 1.7 × 10¹¹ < σ_ZC < 3.5 × 10¹¹ km², where the larger values correspond to D* = 300 μm. This estimate is in a good agreement with N10 who found that σ_ZC = (2.0 ± 0.5) × 10¹¹ km². The uncertainty in σ_ZC mainly stems from the uncertainty in D*, with γ producing only a minor effect. For a reference, the models shown in Figures 16 and 17 have σ_ZC = 2.1 × 10¹¹ and 2.0 × 10¹¹ km², respectively (Table 1).

Table 1. A Summary of Different Models

D*	α1	α2	γ	σZC	mZC	$\dot{m}_{\rm ZC}$	$\dot{m}_0$	$\dot{m}_{0.003}$	$\dot{m}_1$
(μm)				(1011 km2)	(1019 g)	(kg s−1)	(tons yr−1)	(tons yr−1)	(tons yr−1)
100	2	5	0	2.1	3.8	4200	15,000	4200	240
100	2.9	4.1	0	2.1	4.6	6200	12,000	4100	860
100	2	5	−1	2.0	3.8	5200	15,000	6100	480
100	2	5	−1.3	2.0	3.9	5800	16,000	7000	590
100	2	5	1	2.3	4.0	4000	14,000	3100	130
30	2	5	0	1.8	1.2	1600	7700	500	27
300	2	5	0	3.4	15	19,000	26,000	19,000	3300
50	2	3.5	0	2.5	12	25,000	18,000	9300	4100
50*	2	3.5	0	2.5	11	13,000	17,000	8500	3300
200	3.5	5.0	0	1.9	2.0	2400	12,000	2900	230
200*	3.5	5.0	0	2.1	3.0	3400	11,000	2,900	220

Notes. See the main text for the definition of parameters shown here. The asterisks denote the cases, where only particles between D = 10 μm and D = 3 mm were considered.

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Mass m_ZC is more poorly constrained then σ_ZC. For 30 ≲ D* ≲ 300μm, α₁ < 3, α₂ > 4 and ρ = 2 g cm⁻³ we estimate that 10¹⁹ < m_ZC < 1.5 × 10²⁰ g, with larger values corresponding to D* = 300 μm. For a more restrictive assumption with D* ≃ 100 μm, we find that 3 × 10¹⁹ < m_ZC < 5 × 10¹⁹ g, where the exact value depends on α₁, α₂, and γ. For a reference, the models shown in Figures 16 and 17 have m_ZC = 3.8 × 10¹⁹ and 3.9 × 10¹⁹ g, which roughly corresponds to a 33 km diameter sphere with ρ = 2 g cm⁻³.

These results compare well with those reported in N10, where it was found that 2.6 × 10¹⁹ < m_ZC < 5.2 × 10¹⁹ g, under the assumption that the continuous SFD can be approximated by a population of the same-size particles with D = 100–200 μm.

N10 estimated that the input mass rate of $\dot{m}_{\rm ZC}=1000$–1500 kg s⁻¹ is needed to keep the ZC in a steady state. Here we obtain larger values, mainly because the population of particles released with low q has shorter lifetimes and needs to be resupplied at a higher rate. If D* ≲ 100 μm, $\dot{m}_{\rm ZC}$ ranges between 3000 and 7000 kg s⁻¹, with the largest values corresponding to γ = −1.3 in the model illustrated in Figure 17. Input rate $\dot{m}_{\rm ZC}$ is also sensitive to D*, roughly in the same proportion as m_ZC. For example, $\dot{m}_{\rm ZC}\sim 1600$ and 19,000 kg s⁻¹ for D* = 30 μm and D* = 300 μm, respectively. These estimates are valid for α₁ < 3 and α₂ > 4. The required input rates can be somewhat smaller or larger if α₁ ∼ 3.5 and/or α₂ ∼ 3.5 (see Table 1).

Finally, we consider the terrestrial accretion rate, $\dot{m}_{I^*}$, where $\dot{m}_{I^*}$ denotes the rate for I > I*. We consider cases with I* = 0, I* = 0.003, and I* = 1, with the latter two roughly corresponding to our expectations for AMOR and CMOR, respectively. Using the IRAS calibration, we find that our standard model with D* ≃ 100 μm, α₁ < 3, α₂ > 4 implies that $\dot{m}_0=({\rm 15{,}000}\pm 3000)$ tons yr⁻¹, $\dot{m}_{0.003}=(5000\pm 2000)$ tons yr⁻¹, and $\dot{m}_1=(500\pm 400)$ tons yr⁻¹, where a large part of the quoted uncertainty comes from the poorly constrained γ. Note that the values for I* = 0.003 and I* = 1 do not include any radiant cutoff.

The uncertainty becomes larger if D* is allowed to vary (Figure 18). For example, with α₁ = 2, α₂ = 5, and γ = 0, we obtain $\dot{m}_0={\rm 26{,}000}$ tons yr⁻¹, $\dot{m}_{0.003}={\rm 19{,}000}$ tons yr⁻¹, and $\dot{m}_1=3300$ tons yr⁻¹ for D* = 300 μm, and $\dot{m}_0=7700$ tons yr⁻¹, $\dot{m}_{0.003}=500$ tons yr⁻¹, and $\dot{m}_1=27$ tons yr⁻¹ for D* = 30μm. Also, our preferred model for the AMOR meteors illustrated in Figure 10 gives $\dot{m}_0= {\rm 12{,}000}$ tons yr⁻¹, $\dot{m}_{0.003}=2900$ tons yr⁻¹, and $\dot{m}_1=230$ tons yr⁻¹. Similarly, the model with D* = 50 μm, α₁ = 2, and α₂ = 3.5 gives $\dot{m}_0={\rm 18{,}000}$ tons yr⁻¹, $\dot{m}_{0.003}=9300$ tons yr⁻¹, and $\dot{m}_1=4100$, or $\dot{m}_0={\rm 17{,}000}$ tons yr⁻¹, $\dot{m}_{0.003}=8500$ tons yr⁻¹, and $\dot{m}_1=3300$, tons yr⁻¹, if the size range of the contributing particles is restricted to 10 < D < 3000 μm (Table 1).

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The above estimates with I* = 0 are a factor of several lower than those found by N10. This difference probably stems from some of the crude approximations used by N10. For example, N10 did not use a continuous SFD of particles and approximated dN(D) by delta functions. Their initial particle orbits had (almost exclusively) q > 1.5 AU, and did not take into account the fact that many JFCs can split and/or disrupt with q < 1.5 AU. Moreover, N10 did not properly include the collisional lifetimes of JFC particles in their model. The results presented here, which include all these components, and which were validated on meteor observations, can be more trusted and should supersede those reported in N10.