Radar Detectability Studies of Slow and Small Zodiacal Dust Cloud Particles. III. The Role of Sodium and the Head Echo Size on the Probability of Detection

The mass-loss rate in CABMOD is calculated using the Langmuir treatment of evaporation kinetics (Vondrak et al. 2008), which assumes that the rate of evaporation into a vacuum is equal to the rate of evaporation needed to balance the rate of uptake of a species i in a closed system. The rate of mass release of species i with molecular weight ${\mu }_{i}$, from a particle of area S and a temperature T, is given by the Hertz–Knudsen expression

$$\begin{eqnarray}&&\displaystyle \frac{{{dm}}_{i}}{{dt}}{| }_{A}=f(T){\gamma }_{i}{p}_{i}S\sqrt{\displaystyle \frac{{\mu }_{i}}{2\pi {k}_{{\rm{B}}}T}},\end{eqnarray} \tag{ 2 }$$

where the subscript A refers to thermal ablation; k_B is the Boltzmann constant, p_i is gas-liquid equilibrium vapour pressure, and ${\gamma }_{i}$ is the apparent evaporation coefficient, which is equal to the probability that the molecule is retained on the surface, or within the particle, after collision. Safarian & Engh (2013) concluded that ${\gamma }_{i}$ is unity for pure metals but may be lower than 1 in silicate melts (Alexander et al. 2002) because diffusion from the bulk into the surface film may become rate-limiting. Due to the lack of experimental data, the apparent evaporation coefficient for each metal is assumed to be the same for all the compounds of the bulk, such that ${\gamma }_{i}=\gamma$ (Vondrak et al. 2008). The range of temperature where the solid and the liquid are in equilibrium is treated by applying a phase transition factor f(T) (Gómez Martín et al. 2017), which is represented by a sigmoid temperature dependence weighting that varies between 0 and 1:

$$\begin{eqnarray}&&f(T)=\displaystyle \frac{1}{1+\exp \left(\tfrac{-(T-{T}_{c})}{\tau }\right)},\end{eqnarray} \tag{ 3 }$$

where τ is a constant that characterizes the width of the sigmoid profile and T_c is the temperature such that $f({T}_{c})=0.5$. The values T_c = 1800 K and τ = 14 K are prescribed for the olivine Fa₅₀ solid–liquid equilibrium temperature range, meaning essentially that $f(T\lt 1700\,{\rm{K}})=0$ and $f(T\gt 1800\,{\rm{K}})=1$ (Vondrak et al. 2008).

The previous version of CABMOD (hereafter referred as CABMOD v1, Vondrak et al. 2008) simulated poorly the width of the Na and Fe peaks, and the magnitude of the Ca ablation relative to Na (see Figure 14 of Gómez Martín et al. 2017), which is an important consideration since it is the instantaneous properties of single meteors that play a primary role for radar detection (Janches et al. 2014). In a first attempt to improve the agreement between CABMOD and MASI, the model was upgraded with the latest version of the MAGMA code (Fegley & Cameron 1987), which includes revised data for Na and K alumino-silicates (Holland & Powell 2011). Revision of the thermodynamics of Na resulted in lower equilibrium vapour pressures and thus in a lower evaporation rate, which fits better the high-temperature tail of the Na pulses measured by MASI. A temporary solution for the Na profile has been found by increasing the width of τ (which defines the width of the temperature dependence) in the transition between the liquid and solid phase. Due to its low ionization potential and high volatility, ablated Na is a major contributor to radar detectability, in particular for the slow-velocity particles.

The effect that the new version of CABMOD (hereafter referred to as CABMOD v2, Gómez Martín et al. 2017) introduces to the calculation of ${N}_{{\rm{e}}}(m,V,\alpha )$ is shown in Figure 1. The top panel of this figure shows the altitude profile of ${N}_{{\rm{e}}}(m,V,\alpha )$ for a particle with m = 10 μg, α = 45° entering the atmosphere at two velocities: 14 km s⁻¹ (blue lines) and 40 km s⁻¹ (black lines). The bottom panel shows the results for the case of a particle with m = 1 μg. The solid lines represent the electron density calculations using the original version of CABMOD presented in J14 and J15 while the dotted lines utilize the new one. The dashed lines represent the results derived from further improvements discussed in Section 2.2. The figures show that the main differences between CABMOD v1 and v2 is a wider altitude range over which electrons resulting from the ablation of Na are produced. Also, the peak production occurs at lower altitudes. The magnitude of the peak is somewhat lower but not significantly so. For the case of smaller masses, electrons resulting from the ablation of the main constituents is somewhat larger than for the case of the CABMOD v1, which likely results from the overlap with the wings of the broader CABMOD v2 alkali peaks, since no changes have been made in CABMOD related to Fe, Mg, or Si. Overall, the differences introduced by using the CABMOD v2 will not produce significantly different results in the overall detectability of meteors than those presented in J14 and J15.

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Estimating the rate of electron production from a meteoroid requires the ionization efficiencies ${\beta }_{\mathrm{ip}}$ of the individual elements once they ablate and collide with air molecules (O₂ or N₂) at hyperthermal velocities. Unfortunately, there is relatively little laboratory data available. In our previous study reported in J14 and J15 we used the ionization cross section of K atoms in collision with O₂ and N₂ over a large range of collision energies (Cuderman 1972) as a reference point. We used these results because this was the only metallic atomic species for which experimental data with both collision partners over a wide collision energy existed. The most challenging aspect of these beam-gas experiments is determining the absolute cross section for ionization. Although Cuderman (1972) appears to have carried out a careful study, we have now concluded that the cross sections were underestimated, based on very recent results reported by Thomas et al. (2016). The authors measured ${\beta }_{\mathrm{ip}}$ for Fe in N₂ and air by reproducing the experimental arrangement and results reported by Friichtenicht et al. (1967). Figure 2 illustrates ${\beta }_{\mathrm{ip}}$(Fe) as a function of impact velocity, where the red line represents the results derived by Friichtenicht et al. (1967) and later utilized by Jones (1997), the black line is the recent measurements by Thomas et al. (2016), and the blue and green lines are the values obtained for K and Fe, respectively, using the measurements of Cuderman (1972) and reported in J14. At a representative velocity of 20 km s⁻¹ (vertical dash line in Figure 2), ${\beta }_{\mathrm{ip}}$(Fe in air) = 0.03. In contrast, using ${\beta }_{\mathrm{ip}}$(K in air) as the reference point from Cuderman (1972) indicates that ${\beta }_{\mathrm{ip}}$(Fe in air) is only 0.0015 (Janches et al. 2014), roughly a factor of 20 smaller. We therefore now use the new measurements of ${\beta }_{\mathrm{ip}}$(Fe) from Thomas et al. (2016) as the reference point for the present study, and estimate ${\beta }_{\mathrm{ip}}$ for the other metals using the following procedure (also described in J14).

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The ionization probability for the first collision after ablation, ${\beta }_{0}$, is expressed as a function of collision V by the analytic expression of Jones (1997)

$$\begin{eqnarray}&&{\beta }_{0}=\displaystyle \frac{c{(V-{V}_{0})}^{2}{V}^{0.8}}{1+c{(V-{V}_{0})}^{2}{V}^{0.8}},\end{eqnarray} \tag{ 4 }$$

where V₀ is the threshold velocity, given by

$$\begin{eqnarray}&&{V}_{0}=\sqrt{\displaystyle \frac{2({M}_{{\rm{e}}}+{M}_{a})e\psi }{{M}_{{\rm{e}}}{M}_{a}}},\end{eqnarray} \tag{ 5 }$$

where M_e and ψ are the mass and ionization potential of the atom, respectively, e is the electronic charge, M_a is the molecular mass of O₂ or N_2, and c is a fitted parameter. For Fe in air, c = 1.97 × 10⁻⁵ (km s⁻¹)^−2.8, and V₀ = 8.91 km s⁻¹. ${\beta }_{0}$ is then increased to allow for ionization through subsequent collisions of the metal atom as it loses momentum (Jones 1997):

$$\begin{eqnarray}&&{\beta }_{\mathrm{ip}}(V)={\beta }_{0}(V)+2{\int }_{{V}_{0}}^{V}\displaystyle \frac{{\beta }_{0}(V^{\prime} )}{V^{\prime} }{dV}^{\prime} .\end{eqnarray} \tag{ 6 }$$

These new results, in principle, worsen the agreement between ZoDy predictions and HPLA observations because the larger ${\beta }_{\mathrm{ip}}$ values will produce stronger radar returns signals. However, as discussed in J14 (see discussion for Figure 9 in Janches et al. 2014), the detectability of the slow and small particles, which are the particles in question, will strongly depend on the ability to detect electrons produced from the ablation of alkali atoms (i.e., K and Na), as they are the elements that will ablate more easily off the meteoroid's body according to differential ablation (Vondrak et al. 2008; Janches et al. 2009).

In order to determine ${\beta }_{\mathrm{ip}}$ for Na and other meteoric metals, we first estimate the maximum interaction distance between a metal atom and a collision partner (O₂ or N₂) as the curve-crossing (or harpoon) distance, R_c given by⁷

$$\begin{eqnarray}&&{R}_{c}=\displaystyle \frac{e}{4\pi {\epsilon }_{0}(\psi -\gamma )},\end{eqnarray} \tag{ 7 }$$

where γ is the vertical electron affinity of O₂ and N₂, which is close to zero and thus R_c is inversely proportional to the ionization potential of the metal atom ψ. The ionization cross section should then scale as R_c² (i.e., a target of radius R_c), and hence to the inverse square of the ionization potential (${\psi }^{-2}$). The premise of the harpoon mechanism is that the electron transfer occurs at a distance where the energy involved (ionization potential of the metal atom minus the electron affinity of the O₂ or N₂) is balanced by the resulting Coulomb attraction of the metal ion and the O₂-or N₂-anion (Smith 1980).

The probability of ionization, ${\beta }_{\mathrm{ip}}$, in Equation (4) is (to the first order) proportional to c, since the second term in the denominator is less than 1 for $V\lt 45$ km s⁻¹, and less than 3 even at the highest V of 72 km s⁻¹. ${\beta }_{\mathrm{ip}}$ and thus c are proportional to the cross section for ionizing collisions. Hence, c is proportional to ${\psi }^{-2}$ and thus knowing c(Fe), the c parameter can be estimated for other metals as

$$\begin{eqnarray}&&c(X)=c(\mathrm{Fe})\times {\left(\displaystyle \frac{\psi (\mathrm{Fe})}{\psi (X)}\right)}^{2}.\end{eqnarray} \tag{ 8 }$$

Using Equation (8), the c parameter for Na, K, Mg, Si, and O can then be estimated by multiplying the value of 1.97 × 10⁻⁵ (km s⁻¹)^−2.8 for Fe (in air) by factors of 3.30, 2.36, 1.06, 0.94, and 0.34, respectively. V₀ in Equation (5) is estimated for each element from its respective ψ (Equation (5)). The values of c and V₀ for calculating ${\beta }_{\mathrm{ip}}(V)$ using Equations (4) and (6) are listed in Table 1, and the resulting curves of ${\beta }_{\mathrm{ip}}$ versus V₀ in air are illustrated in Figure 3. To check on this procedure we compare our results with existing experimental data for Na + O₂ and N₂ collisions. Bydin & Bukteev (1960) showed that at collision velocities below 50 km s⁻¹, the ionization cross section for O₂ is more than 70 times larger than for N₂, so that air collisions with N₂ will produce very few electrons. Moutinho et al. (1971) measured an absolute cross section for electron production in O₂ of 0.4 Ų. This may be compared with a capture cross section of 15.4 Ų computed from long-range capture theory, using the experimental polarizabilities and ionization potentials for Na and O₂ to calculate the C₆ parameter (=3.3 × 10⁻⁷⁷ J m⁶) for the London intermolecular potential (Smith 1980). This implies ${\beta }_{0}$(Na in O₂) = 0.4/15.4 = 0.026. In air this will be reduced by a factor of four (since ${\beta }_{0}$(Na in N₂) is so much smaller). At a low collision velocity, ${\beta }_{\mathrm{ip}}(\mathrm{Na})=1.27{\beta }_{0}$(Na) from Equation (6), so ${\beta }_{\mathrm{ip}}$(Na in air) = 0.0082. As shown in Figure 3, this experimental point from Moutinho et al. (1971) is in very good agreement with the value of ${\beta }_{\mathrm{ip}}$(Na in air) based on the measurement of ${\beta }_{\mathrm{ip}}$(Fe in air; Figure 2 and Thomas et al. 2016). In addition, a preliminary comparison of this methodology with recent measurements of ${\beta }_{\mathrm{ip}}$ for Al suggests that the calculation matches the measurements within a factor of five (DeLuca & Sternovsky 2016, personal communication).

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A comparison of the new proposed ${\beta }_{\mathrm{ip}}$(Na) with those used in Vondrak et al. (2008) and Janches et al. (2014) is shown in Figure 4. The top panel of this figure compares the revised values of ${\beta }_{\mathrm{ip}}$(Na in air) recalculated using ${\beta }_{\mathrm{ip}}$(Fe in air) measured by Thomas et al. (2016) as a reference point (black line) with those originally utilized by Vondrak et al. (2008; red line) and the revised values reported in J14 using K atoms in collision with O₂ and N₂ over a large range of collision energies (Cuderman 1972) as a reference point (blue line). The bottom panel of Figure 4 shows the ratio between the different results. It can be seen from this figure that the revision reported in J14 significantly decreased ${\beta }_{\mathrm{ip}}$(Na in air) over the entire velocity range, while the results from the approach proposed here mostly affect the detectability of slower particles (V < 30 km s⁻¹).

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The degree of change that the new ${\beta }_{\mathrm{ip}}$(Na in air) introduces to the calculation of ${N}_{{\rm{e}}}(m,V,\alpha )$ is represented by the dashed lines in Figure 1. From these panels it can be seen that at low velocities, the new ${\beta }_{\mathrm{ip}}$(Na in air) introduces up to a two-order-of-magnitude decrease in the amount of electrons produced, mostly by the ablation of Na, while at higher speeds, the change is about a factor of 10. As expected, the changes are greater for smaller sizes. In particular, as can be seen in the blue lines of the bottom panels, since alkalis are basically the only elements ablating and producing electrons for small and slow particles, the new estimation of ${\beta }_{\mathrm{ip}}$(Na in air) will have a critical effect in the detectability of the incoming flux. This can be seen in Figure 5 where the modeled and observed rates (top panel) and radial (i.e., line of sight) velocity distributions (bottom panel) are displayed. The black line histograms are the observations, the blue lines are the original results presented in J14, which uses CABMOD v1 and the original estimate of β(Na) (Vondrak et al. 2008), and the red line shows the results derived using the new version of CABMOD as well as the new calculation of ${\beta }_{\mathrm{ip}}$(Na in air). The rates and velocity distributions are estimated using ZoDy, constrained by the Planck satellite measurements (i.e., D* = 30 μm). As can be seen from this figure, while our treatment with the new modifications still overpredicts the observed quantities, the agreement is closer by an order of magnitude. More particles with slower radial velocities are filtered out, as seen in the lower panel. Another new effect is that at about 04:00 a.m. (28 hr from midnight of July 15th), the detected rates experience a decrease, in agreement with the observations, but in contradiction with our previous results, which showed an increase in the detected rates.

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As discussed earlier, our detectability model requires a physical assumption of the radar scattering target or HE, which is commonly believed to be "a ball of plasma" traveling at or near the speed of the meteoroid (Mathews et al. 1997; Janches et al. 2000; Close et al. 2002). In our current model of the HE radar cross section (RCS) reported in J14 and J15, we followed the results reported by Mathews et al. (1997) by assuming it to be an ensemble of electrons given by

$$\begin{eqnarray}&&\sigma {(V,\alpha ,m)=4\pi ({N}_{{\rm{e}}}(V,\alpha ,m)\times {r}_{{\rm{e}}}\times \mathrm{MFP}\times F)}^{2},\end{eqnarray} \tag{ 9 }$$

where r_e is the classical electron radius ($2.8179402894\,\times {10}^{-15}$ m) and ${N}_{{\rm{e}}}(V,\alpha ,m)$ is the altitude profile of electron production, provided by CABMOD. This model of the HE assumes that the scattered radio waves are the result of all electrons (single) scattering in-phase, producing a scattered power that is a function of N_e². This is valid while the characteristic size of the ensemble is small compared with the radar wavelength, and so we assume also that, at a given time, the diameter of the cloud of electrons producing the HE is of the order of the atmospheric MFP multiplied by a scale factor F given by⁸

$$\begin{eqnarray}F=\left\{\begin{array}{ll}1 & \mathrm{if}\ \mathrm{MFP}\lt \displaystyle \frac{\lambda }{4}\\ {\left(\displaystyle \frac{\lambda }{4\times \mathrm{MFP}}\right)}^{2} & \mathrm{if}\ \mathrm{MFP}\gt \displaystyle \frac{\lambda }{4},\end{array}\right.\end{eqnarray} \tag{ 10 }$$

where λ is the radar wavelength. The scale factor F accounts for the fact that for meteor plasma larger than a quarter-wavelength, the radar wave cannot completely penetrate the plasma, and so the scattering ability of the interior electrons is reduced. Equation (9) is then utilized with the radar equation and a parameterization of the radar beam pattern to calculate the meteor S/N (Janches et al. 2014, 2015b).

In this section we first examine how well this approximation fares with a recent and more comprehensive model of the meteor HE, and determine what improvements can be made to find reconciliation between radar observations and ZoDy. Specifically, we will compare our results to the model reported by Marshall & Close (2015), who have developed a three-dimensional Finite-difference Time-domain (FDTD) model to investigate the scattering of radar waves from the meteor HE, treating it as a cold, collisional, magnetized plasma. Solving Maxwell's equations and the Langevin equation simultaneously and self-consistently in and around the plasma, the model explores the dependence of the meteor RCS with physical variables such as plasma densities, meteor HE scale sizes, and wave frequencies. The authors found that the computed RCS disagrees with previous analytical theory at certain meteor HE sizes and densities; in some cases the discrepancies could be over an order of magnitude. The authors concluded that for overdense meteors, the meteor head RCS is given by the overdense area of the meteor, defined as the cross-section area of the part of the meteor where the plasma frequency exceeds the wave frequency. For underdense meteors, the model provides a monotonic relationship between the meteor plasma size and peak density and the resulting RCS. These results provide a physical measure of the meteor HE size and density that can be inferred from measured RCS values from ground-based radars.

Figure 6 shows the absolute difference between the RCS calculated using the FDTD model developed by Marshall & Close (2015) and the coherent scattering model utilized thus far by our approach. It can be seen from this figure that the ensemble of electrons model is in reasonably good agreement with the FDTD simulations for the range of masses between 0.01 and 1000 micrograms, where at low velocities the RCS from the FDTD simulations is at most a factor of 0.1 smaller than that resulting from the coherent scattering approach. The FDTD model assumes a Gaussian electron density distribution (Close et al. 2012), and from these results it appears that coherent scattering works reasonably well as an approximation when the HE is underdense, that is, the peak plasma frequency (${\omega }_{{\rm{p}},\max }$) is lower than the radar frequency (${\omega }_{0}=2\pi {f}_{0}$). For this case, the head plasma should emulate coherent scattering as the radar wave is able to penetrate the plasma and be seen by all of the head plasma electrons, which each scatter the radar wave independently and coherently. On the other hand, when the HE is overdense, the radar wave hits a "wall" where ${\omega }_{{\rm{p}},\max }\gt {\omega }_{0}$, and the head plasma reflects the wave; only under those circumstances, Marshall & Close (2015) report an RCS $\sim \pi \times {r}_{{\rm{p}}}^{2}$, where r_p is the overdense radius. The calculated electron densities from CABMOD result in normalized plasma frequencies that are smaller than 3 and actually generally smaller than 1. Thus we observe that the relatively simple coherent scattering parameterization approximates reasonably well the results of the general FDTD model for the particular case of underdense HEs, which applies to the range of dominant ZoDy meteoroid masses and speeds. Therefore the choice of the HE scattering model used here is not expected to yield very different results.

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Finally, we explore in our model the validity of using the atmospheric MFP, scaled by Equation (10) as the characteristic size of the HE, as well as any potential angular dependence the detection of this target may have. We compare our results with those used by previous works, in particular that reported by Close et al. (2004), who also argued that the physical size of head plasma must scale approximately with the atmospheric MFP because the electrons detected by the radar are contained within a region of ionization produced by the ablation of the meteoroid resulting from collisions with neutral air molecules. Therefore, as the MFP decreases with decreasing altitude, so would the size of the HE. In addition, there is a transition region between the atoms and ions being released from a meteoroid and expanding largely without deflection to a point where the expansion occurs at a slower diffusive rate. Jones (1995) defined this distance as the initial trail radius and estimate it as

$$\begin{eqnarray}&&{r}_{i}=2.845\times {10}^{18}\times \displaystyle \frac{{V}^{0.8}}{n},\end{eqnarray} \tag{ 11 }$$

where V is the meteoroid velocity in km s⁻¹ and n is the background number density at the altitude at which the HE is detected. We refer hereafter to Equation (11) as the Jones formula or Jones radius. Below, it deviates from MFP and can be up to a factor of five smaller. In fact, Close et al. (2004), using simultaneous VHF/UHF detections obtained with the ALTAIR system, estimated the head radius to be $0.023\times {r}_{i}$. Figure 7 displays the ratio of the Jones radius to MFP. The Jones radius is calculated at the altitude at which ablation is maximum according to CABMOD. In this figure, the Jones radius is calculated with the 0.023 scale factor introduced by Close et al. (2004) and it can be seen that the MFP is a factor of ∼5 larger than the head radius at the lowest speed, and on average a factor of 3 times larger than the head radius below 20 km s⁻¹ at the altitude at which the ablation is maximum according to CABMOD. This comparison suggests that using the MFP or Jones radius does not change the results significantly. Furthermore, from Figure 7 of Marshall & Close (2015), it can be seen that the RCS derived from Close et al. (2004) analytical model for a given radius is larger than the one derived by the FDTD simulations. This suggests that the Jones radius correction (0.023) derived by Close et al. (2004) could be even smaller.

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Given these results, we assume the radius to be a free parameter of the model and explore two possibilities following previous work: head plasma radii equal to MFP and MFP/5. The results for Arecibo are shown in Figure 8, where the solid blue line represents the results for a head plasma radius equal to MFP and the red solid line represents the results for MFP/5. In this figure, it can be seen that for the case of the smaller HE radius, a degree of reconciliation between ZoDy and radar observations becomes possible. This is because a sufficient ammount of the ∼30 t day⁻¹ predicted by ZoDy becomes undetected, making the predicted rates and velocity distributions fall below the observed. This is expected because ZoDy does not include the higher speed particle populations introduced by long-period comets (i.e., HTCs and OCC; Nesvorný et al. 2011b; Pokorný et al. 2014). These results imply that the HE size must be smaller than the MFP in order to have a meteor flux that is mostly composed of JFCs but represents the minority of detections of sensitive radars. However, when looking at the velocity distributions displayed in the bottom panel of Figure 8, it can be seen that, although the disagreement between model and observations is reduced by an order of magnitude between MFP and MFP/5, the distributions are still overpredicted by a factor of ∼2.5.

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Previous modeling and observational work using HPLA radar results have shown that particles from the Helion and Antihelion sources combined should represent less than ∼45% of radar detections for the case of Arecibo and PFISR and (Fentzke & Janches 2008; Fentzke et al. 2009), and less than ∼20% for the case of MU (Pifko et al. 2013), independent of their dominance in the overall dust population. Similarly, for the case of meteor radars, this portion increases to where these populations do not represent more than ∼45% (see Table 2 in Janches et al. 2015a). Thus, even though the results presented in Figure 8 show that the predicted detection rates at Arecibo are, in general, below the actual detected rates, for the case of smaller HE radius, they are nevertheless significantly higher than expected (∼71% of the observed counts). In addition, the predicted rates do not currently reproduce the temporal separation between the detection of the Helion and Antihelion sources (i.e., dip in the rates at 6 a.m. LT). In fact, Janches et al. (2006) and Fentzke & Janches (2008) were able to reproduce this feature using a filtering effect by arbitrarily removing meteors with elevation angles between lower than 20°. Following that work, we introduce a similar effect here by multiplying the S/N estimated by our model by a Gaussian function with a standard deviation in zenith distance equal to 70. The dashed lines in Figure 8 represent the predicted rates taking into account this effect, and two results can be observed. First, the rates for the case of the smaller HE radius decrease to ∼43% of the observations, which, considering the accuracy of the astronomical model, agree with the expected rates, thus achieving reconciliation between observations and predictions. Second, the temporal separation between the detection of the Helion and Antihelion sources is clearly reproduced in the predictions.

It is clear from these results that this angular effect is the missing link required to obtain reconciliation between the ZDC model and HPLA observations, assuming that all other improvements described earlier are also included. The origin of such a filter is unclear. Janches et al. (2006) and Fentzke & Janches (2008) hypothesized that, because of the simple ablation and ionization model utilized in that work, there were unaccounted effects that may contribute to the lack of detection of low-elevation meteors. However, Kero et al. (2012) did not find that such filtering effect of meteor radiants with elevations <20° exists in the MU radar data. Kero et al. (2011) showed that for detections of Orionid meteors, the detection rate varied approximately as ${\sin }^{1.5}(\mathrm{el})$, where el denotes the elevation; and thus, they were required to apply that correction to their observed rates in order to agree with the expected shower Zenith Hourly Rates. The correction using this equation, which is derived from naked-eye visual observations of meteors (Zvolankova 1983), is essentially the same effect. This is because while Kero et al. (2011, 2012) corrected the observed rates in order to agree with the predicted incoming flux, Janches et al. (2006) and Fentzke & Janches (2008) used the filtering effect to correct the predicted incoming flux in order to agree with the observed rates. Nevertheless, since for our current work we utilized a detailed physical and chemical model of the ablation of meteoroids entering the atmosphere (e.g., CABMOD), in theory our new approach should account for all such effects without the need for additional filtering.

Another possible explanation is that the angular effect is introduced by the fact that the HE is not a spherical target and thus the S/N of the detected signal is angular dependent. However, Kero et al. (2008b), utilizing meteor HE observations with the European Incoherent Scatter (EISCAT) radar tristatic system, show a lack of angular dependance on the observed echoes. To shed light on this issue, we show the results of our model for the case of the less sensitive PFISR and MU radars in Figure 9. Once again, the solid blue and red lines in these panels represent the model without the proposed angular filtering for HE radius MFP and MFP/5, respectively, while the dash lines represent the modeled results taking into account the angular dependences. It can be seen that the effect is not as pronounced for these less sensitive radars, suggesting that the angular dependence becomes weaker as the particle mass increases. Specifically, while the angular effect decreases Arecibo predicted counts by 40%, for the case of PFISR and MU, the angular effect results in a 20% and 14% decrease, respectively. When operated in tristatic mode, the EISCAT system is also less sensitive due to both the much smaller common volume and larger distances of the remote antennas. In fact, Kero et al. (2008a) determined the observed masses by this system to be equal or larger than 10 μg and thus the expected angular effect on those observations will be negligible according to our modeling results. Thus, the results presented here in principle agree with the conclusion of Kero et al. (2008b).

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