RADAR DETECTABILITY STUDIES OF SLOW AND SMALL ZODIACAL DUST CLOUD PARTICLES. I. THE CASE OF ARECIBO 430 MHz METEOR HEAD ECHO OBSERVATIONS

The first task we address is to assess the ability of the Arecibo 430 MHz radar to detect small particles (m < 10 μg) traveling at slow velocities (V < 20 km s⁻¹). Although similar attempts have been made in the past (e.g., Janches et al. 2008, 2014), in the work presented here we introduce several improvements to our methodology that include our latest understanding of meteor ablation processes and the determination of the meteor head echo signal-to-noise ratio (S/N) that will be detected by the Arecibo 430 MHz, given a set of meteor dynamical parameters (i.e., mass, velocity, and entry angle). For this we utilize the radar equation given by

$$\begin{equation} {\rm S/N} = \frac{P_t \lambda _R^2 G^2\sigma }{(4\pi)^3 P_n R^4}, \end{equation} \tag{ 2 }$$

where in particular for the observational results we utilize here, P_t is the transmitted power (1 MW), λ_R is the radar wavelength (69 cm), G is one-way antenna gain pattern, R is the meteor range, σ is the meteor radar cross section (RCS), and P_n is the system noise power given by

$$\begin{equation} P_n = kT_{{\rm sys}}B, \end{equation} \tag{ 3 }$$

where k is the Boltzmann constant, B is the noise bandwidth (1 MHz), and T_sys is the system temperature (∼120 K).

In previous works we have utilized σ as a proxy of the sensitivity of the radar to the detection of meteor head echoes (Janches et al. 2008; Fentzke & Janches 2008; Fentzke et al. 2009; Pifko et al. 2013), which requires the assumption that the antenna radiation pattern is uniform and thus for a particle with a given σ, the resulting S/N will be independent of where the meteor is detected within the radar beam. In the present study we improve our previous treatment by modeling the meteor S/N, a task that requires understanding the relation between Equation (2) and the meteor entry parameters. In other words, we need to calculate how the S/N varies as a function of particle mass, velocity, and entry angle while considering all possible paths that the meteoroid trajectory can take while crossing the radar beam. For this we need to model both the Arecibo antenna gain and σ.

In order to model the gain (G) of the Arecibo antenna we adopt the same approach reported by Dyrud & Janches (2008), which characterizes the pattern of the radar main beam as a Gaussian with a 3 dB point at approximately 150 m from the center. The peak of the main lobe is given by

$$\begin{equation} G = 10\times \log _{10}\left (\frac{4\pi A_{{\rm eff}}}{\lambda _R^2}\right), \end{equation} \tag{ 4 }$$

where the A_eff is the approximate aperture given by

$$\begin{equation} A_{{\rm eff}} = \eta \pi 150^2, \end{equation} \tag{ 5 }$$

where η is the approximate aperture efficiency of 0.7 and 150 m is the Arecibo dish radius, resulting in G ∼ 61 dB. This is 2 dB lower than considered previously in Janches et al. (2014) because in that work we considered η to be equal to 1. However, this difference is within the expected uncertainty. We also consider the Arecibo first side lobe in which detection occurs frequently (Janches et al. 2004), assuming it to be also a Gaussian centered about 500 m from the center of the main beam and a 3 dB width of 50 m. Although the description of the Arecibo beam as two Gaussians is somewhat simplistic, it is in good agreement with more robust modeling of the antenna pattern described in the past (Breakall & Mathews 1982; Mathews et al. 1997). The peak of the side lobe is considered to be 17 dB lower than the peak of the main beam. This description resulted in good agreements with observations in previous modeling work of head echoes (Dyrud & Janches 2008). It is worth noting that the expected error in the S/N values resulting from Equation (2) introduced by errors in the measured variables of Equations (3) and (4) is approximately 3 dB (factor of two).

The meteoroid RCS depends on the production of electrons as the meteoroid ablates while entering the Earth's atmosphere. In this work, we adopt the description of the RCS reported by Mathews et al. (1997), which assumes that the radar return originates from coherent electron-scatter from the free electrons in the small volume surrounding the meteor and that the backscatter cross section of an ensemble of N_e electrons is given by

$$\begin{equation} \sigma (V,\alpha,m) = 4\pi N_e(V,\alpha,m)^2 r_e^2, \end{equation} \tag{ 6 }$$

where r_e is the classical electron radius (2.8179402894 × 10⁻¹⁵ m) and V, α, and m are the meteoroid entry velocity, zenith angle, and mass, respectively. In this scenario all electrons (single) scatter in-phase and thus the scattered electric fields add so that scattered power is a function of $N_e^2$. This is valid because the characteristic size of the ensemble is small compared with the radar wavelength (Mathews et al. 1997). We assume also that, at a given time, the diameter of the cloud of electrons producing the head echo is of the order of the atmospheric mean free path (MFP), which is in agreement to various head echo models (Close et al. 2002, 2004), given by

$$\begin{equation} {\rm MFP} (h) =\frac{R\times T_a(h)}{\sqrt{2}\pi \times d_a^2\times L \times P_a(h)}, \end{equation} \tag{ 7 }$$

where R is the gas law constant (8.314510 J (K mol)⁻¹), T_a(h) is the atmospheric temperature at a given altitude, d_a is the air molecule collisional cross section (3.57 × 10⁻¹⁰ m), and P_a(h) is the air pressure at a given altitude.

The production rate of electrons depends on the ionization probability β_ip (Jones 1997), which in turns depends only on meteoroid mass, composition, and velocity. Radar detectability of meteors has typically been estimated by using a crude average of this parameter (Close et al. 2002, 2005); in reality it can vary up to two orders of magnitude depending on the constituent under consideration (Vondrak et al. 2008). Thus, in order to overcome this limitation we further improve our radar detection sensitivity treatment by including results from the chemical ablation model (CABMOD) developed by Vondrak et al. (2008). CABMOD predicts differential ablation, i.e., the most volatile elements—Na and K—ablate first, followed by the main constituents Fe, Mg, and Si, and finally the most refractory elements such as Ca. The model considers the full treatment of the ablation and ionization of the individual chemical elements by including the following processes: sputtering by inelastic collisions with air molecules before the meteoroids melt; evaporation of atoms and oxides from the molten particle; diffusion-controlled migration of the volatile constituents (Na and K) through the molten particle; and impact ionization of the ablated fragments by hyperthermal collisions with the air molecules. Evaporation is based on thermodynamic equilibrium in the molten meteoroid (treated as a melt of metal oxides), and between the particle and surrounding vapor phase. The loss rate of each element is then estimated by applying Langmuir evaporation. Figure 2 illustrates the elemental injection profiles calculated by CABMOD for two different meteoroid speeds assuming the initial composition of the meteoroids is chondritic (Plane 1991). The two cases shown in this figure in particular have been validated against meteor head echo observations using the Arecibo radar in which the predicted small-scale temporal and spatial features have been observed (Janches et al. 2009).

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As shown in Figure 2 the entire ablation profile, and thus electron production, of a meteoroid with a given mass, entry angle, and velocity will occur over a prolonged altitude range (>60 km). However, only a relatively small part of the ablation curve of the meteoroid will occur within the same region of space that is being illuminated by a given radar. If we define the beam entry point (BEP) as the meteoroid altitude when the particle is at a predefined horizontal distance Δx from the beam center, then for meteoroid BEPs higher than a certain value, the particle will cross the radar beam at the beginning of the ablation process when N_e is small and the detected S/N will be below the radar detection threshold. On the other hand, if the BEP is in the range of typical meteor observe altitudes (∼80–120 km; Janches & ReVelle 2005; Sparks & Janches 2009a, 2009b), then the meteoroid will cross the beam during the time when most of the ablation occurs and result in an S/N well above the noise threshold and the event will be detected. Finally, if the BEP is lower than a certain value, most of the ablation of the particle would have occurred before it entered the radar beam and thus the electron production will be low, resulting in an undetected event. This is shown in Figure 3 where three examples of beam entry cases of a given meteor trajectory are displayed. In fact, within this altitude range, there are two different ablation regimes that can produce the detected signal as shown in Janches et al. (2009). The first one occurs at higher altitudes over a narrow altitude range producing very strong signals due to the rapid ablation of the meteoroid alkalis (Na and K), while the second regime produces lower signal intensity at lower altitudes and over a wider range and is produced by the ablation of the main elements. This effect will be discussed in more detail later in this section.

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For the purpose of this work, we use CABMOD to produce look-up tables of elemental injection profiles as well as the number of electrons per unit length along the meteor trajectory as a function of altitude (gray line in Figure 2) for meteoroids of specified mass, speed, and entry angle. The electron density is then multiplied by the MFP at the given altitude in order to obtain N_e. These results are then used in Equation (6), which together with G, is then used in Equation (2) to determine the S/N as a function of (V, α, m) for all possible BEPs. Given the results of ZoDy mentioned in Section 1, we first focused on determining Arecibo's ability to detect 10 μg particles traveling at 11 km s⁻¹. For this purpose we calculate the S/N along the trajectory of the meteor as it crosses the radar beam for BEPs ranging from 80 to 150 km, calculated every 100 m. If at some point along the trajectory the calculated S/N reaches a value above a predefined threshold, we assume that the meteor was detected. The threshold value is determined from Figure 4 where the distribution of detected S/N is shown for nearly 50,000 particles detected during a year-long observing campaign carried out in 2002 utilizing the Arecibo radar. The meteor head echo arises from the radar return scattered back from a plasma region that moves at the speed of the meteoroid. From the measurements the S/N is calculated as

$$\begin{equation} {\rm S/N} = \frac{{\rm Signal}-{\rm Noise}}{{\rm Noise}}, \end{equation} \tag{ 8 }$$

where the signal is the intensity of the radar return when the meteor is present and the noise is the intensity of the radar return from the background noise. Typical meteor head echo observations transmit radar pulses approximately every millisecond and since a meteor will take longer time to travel through the radar beam, this technique enables to measure the instantaneous meteor S/N along its trajectory, as long as enough electrons are produced. The details of these measurements have been reported in several investigations employing various radars. Some examples can be found in Janches et al. (2003, 2014); Chau & Woodman (2004); Sparks et al. (2009), and Pifko et al. (2013).

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Figure 4 demonstrates the large dynamic range in S/N of the Arecibo detections with a clear peak in the distribution located approximately at 5–6 dB. However, targets with a S/N as low as −30 dB are detected by the radar. In particular, 0.1% of the detections have recorded S/N of −10 dB and approximately one order of magnitude less for S/N values equal to −20 dB. These are significant percentages considering that only 10% of the detected meteors have S/Ns equal to the peak value. Note that although an S/N value of −10 or −20 dB may appear to be low, this value refers to an S/N measurement for a single pulse, in which not only the object is detected but the instantaneous Doppler shift due to its radial velocity can be obtained.

Results from our new methodology are shown in Figure 5, where the three rows represent particles entering at three different zenith angles (20°, 40°, and 60°). If we define Δx to be equal to 700 m from the radar beam center, which represents the outer edge of the first radar-side lobe (Mathews et al. 1997; Janches et al. 2004; Dyrud & Janches 2008), the three panels on the left of Figure 5 show the meteoroid S/N as a function of altitude for those meteors for which the BEP was such that their S/N exceeded the threshold value at some point during the ablation of the particle while entering the atmosphere (−20 dB for the examples shown in this figure). That is, particles with higher or lower BEPs would have resulted in meteors with undetected S/Ns because significant ablation would have occurred before or after crossing the radar beam. For reference purposes, we also show in these panels vertical lines representing Arecibo detection threshold of −30, −20, and −10 dB. The three panels on the right side of Figure 5 show the meteoroid trajectories (traveled altitude versus traveled horizontal distance; dash lines). The solid blue portion of the trajectories show the section during which the meteor was detectable (i.e., the S/N value was above the detectable threshold). Plotted on these panels is also G in dB (left vertical axis) to provide an indication of where in the radar beam these meteors become detectable.

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The results shown in Figure 5 assumes that all these trajectories will cross the radar beam at the center (maximum gain). This is represented in Figure 6, where a horizontal cross section of the radar main beam and first side lobe is shown. The dash lines represent the cases considered in this work, that is, meteors crossing the center and traveling through the 3 dB point (150 m off center). Figure 7 shows the same cases but assuming that the trajectory crosses the beam at the 3 dB point (Figure 6). Examining Figures 5 and 7, we can conclude that, according to our model, a 10 μg particle traveling at 11 km s⁻¹ will produce enough electrons and thus an echo radar return signal to be detectable by the Arecibo radar. However, this is true as long as the value of the meteor BEP falls within a certain range of altitudes, which for this case is very narrow, as can be seen in these figures. In addition, it is observed that the range of BEPs for which detectable S/N will be produced will depend on the entry angle and for the cases displayed in these panel the range is greatest for an entry angle around 40° and decreases at both larger and smaller angles. Furthermore, results not shown here indicate that for this particle mass and velocity the head echo S/N will be below the detectable level for entry zenith angles higher than 60°. This is due to the fact that particles entering at very large zenith angle never heat up enough to melt because the effective scale height experienced by the meteoroid upon entry is inversely proportional to the cosine of the entry zenith angle. This means that the rate of pressure increase encountered by the particle is relatively slow, allowing the particle to radiate heat and stay cool enough to avoid melting. This is in agreement with previous modeling efforts reported by Janches & Chau (2005), Janches et al. (2006), and Fentzke & Janches (2008), who required the assumption of an empirical atmospheric filtering effect, introduced to take into account how the meteor detection rate from a source region varies with elevation above the radar site local horizon. The authors assumed that, on average, meteors with radiant elevations below 20° (i.e., entry zenith angles greater than 70°) should be completely neglected.⁷ Our modeling results show that the effective angle at which this filtering effect will occur will depend on the physical properties of the meteoroids (i.e., velocity, composition, and mass) and will be discussed in more detail in the next section.

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Figures 8 and 9 show the importance of considering an accurate description of the ablation process that includes the dependence of the ionization efficiency as a function, not only of meteor entry parameters, but also its constituents rather than a crude mean value. Specifically, at low velocities the entry conditions become more limited as the particle mass decreases. For example, Figure 8 shows that a 5 μg traveling at 11 km s⁻¹ and with a zenith entry angle of 45° will also be detectable by Arecibo as long as the BEP is within a ∼0.5 km altitude range. Note that the electron production is completely due to the ablation of alkalis. The transition from becoming undetectable occurs very rapidly as a particle of mass equal to 4 μg entering at the same angle results in an S/N value below the −20 dB Arecibo threshold. Interestingly, this more comprehensive detection treatment that includes the Arecibo radar gain, the meteor S/N, and CABMOD shows good agreement with previously modeled results by our more simplistic approach that considered only the meteor RCS (Fentzke & Janches 2008; Pifko et al. 2013; Janches et al. 2014), which predicted that particles traveling at 15 km s⁻¹ would be detected only for masses equal or larger than 1 μg.

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On the other hand, Figure 9 shows that with a change of 1 km s⁻¹ in the incoming particle velocity the portion of the ablation profile produced by the main elements becomes detectable, significantly increasing the beam entry conditions required for the resulting meteor to be detected. Because the evaporation rates depend exponentially on the temperature (according to the Herz–Knudsen treatment utilized in CABMOD), the effect will be an apparently sudden change. This is shown in the four panels in Figure 9 where the S/N and trajectories for a 10 μg particle with speeds of 14 and 15 km s⁻¹ are displayed for a particle crossing the beam center. Comparing the top two panels of Figure 9, we can observe that the peak of the portion of the ablation curve, which is produced by the evaporation of the main elements increases over 10 dB (i.e., an order of magnitude) in signal strength with only 1 km s⁻¹ increase in velocity. The alkalis, on the other hand, ablated efficiently at both speeds and produced enough electrons to enable detection even at the much lower gain side lobes. Furthermore, the figure suggests that if the particle's composition would lack alkali elements at the moment it encounters the atmosphere, then it would have been completely undetected for velocities lower than 14 km s⁻¹. It is important to note that the meteoroid speeds and masses used in these examples are the typical values of the majority of meteoroids predicted by ZoDy, suggesting that Arecibo should detect at least a portion of the flux predicted by ZoDy. It is also worth noticing that if we instead take a more conservative detection threshold of −10 dB, some of the cases discussed would be undetected by the Arecibo radar. For example, this is the case for the a 10 μg particle traveling at a speed of 11 km s⁻¹ entering the beam at 60° off zenith and passing through the center of the beam (Figure 5; bottom panels) or entering at angles greater than 40° for the case of particles traveling off center (Figure 7; middle and bottom panels).

The results presented in Figures 5 and 7–9 show once again that in fact, the Arecibo 430 MHz radar is sensitive to the detection of small/slow meteors and thus the optimal radar to constrain ZoDy with ground-based observations. However, these results also imply that in order to accurately model the ability of a system to detect a certain population of particles, it is not only crucial to understand if the population posses the physical characteristics required to produce enough electrons, but also the probability that a particle from this population with those characteristics will enter the radar beam within the adequate BEP in order for those electrons to produce a detectable signal. In the next section we develop a scheme to estimate this probability as well as to determine the amount of detected particles given an incoming flux.

In the previous sections we described a new method for estimating the sensitivity of the 430 MHz radar to detect meteor head echoes produced by small and slow extraterrestrial particles and used those results to assess the probability of detection as a function of particle mass, velocity, and entry angle. We then weighted the meteor flux predicted by ZoDy in order to examine what portion of the Arecibo observed distribution should be attributed to the ZoDy flux. With our current knowledge of meteor ablation and ionization, we have estimated that the Arecibo detections should be heavily dominated by the ZoDy flux, at least an order of magnitude higher than the actual observations. In principle, one could argue that such results suggest that ZoDy's description of the ZDC needs revision. However, besides the strong constraint by IRAS observations reported by Nesvorný et al. (2010), the meteoric mass input rate of 30–40 t d⁻¹ of slow particles predicted by ZoDy is able to explain several atmospheric phenomena related to the meteoric flux. For example, Marsh et al. (2013); Feng et al. (2013), and Langowski et al. (2014) recently described global modeling of the mesospheric Na, Fe, and Mg/Mg⁺ layers, respectively, using the meteor input function (MIF) developed by Fentzke & Janches (2008). The authors showed that in order to find agreement between the model and observations for these three metals, the MIF had to be artificially scaled because the larger average speed predicted by the MIF (∼25−30 km s⁻¹) does not produce the degree of differential ablation observed in these layers (Plane 2003). A larger and slower MIF, as predicted by ZoDy, would, in principle, produce the degree of differential ablation required to model the Fe, Mg, and Ca mesospheric layers at the same time as the Na layer, without artificially reducing their injection rates. In addition, the ZoDy flux produces a sensible input rate of cosmic spherules and the correct optical extinction of meteoric smoke in the mesosphere (these topics will be treated in a forthcoming publication). Thus, these arguments, together with the strong agreement found with IRAS spectral measurements, motivate a deeper exploration of possible reasons for the disagreement between ZoDy and Arecibo's observations.

In order to first understand these results, we show in Figure 18 a comparison between the characteristics of the ZoDy flux that travels through the Arecibo beam, which we refer here as the input distributions, and the portion that is predicted to be detected by Arecibo. The top panel shows the absolute velocity distributions while the bottom panel shows the mass distribution. As expected, the input distributions reflect ZoDy's main characteristics with a velocity distribution heavily weighted toward meteors with absolute speeds of 12–14 km s⁻¹. As can be seen in the top panel, less than 1 in 10⁵ of these particles are detected by Arecibo. In fact, the slow line-of-sight velocity distribution predicted to be detected by Arecibo (Figure 17, bottom) is comprised of particles with an absolute velocity distribution peak of ∼30 km s⁻¹, even though particles are detected with velocities as low as 11 km s⁻¹, which are the typical radar-detected velocities of meteors originating from the helion and anti-helion Sporadic Sources (Fentzke & Janches 2008). Furthermore, although ZoDy's input includes masses as low as 10⁻³ μg, the IRAS measurements are effectively constrained by particles with masses between 1 and 10 μg. Thus the distribution below about 1 μg is essentially unconstrained, except that it is known that these small particles do not contribute to the ZDC IR emission due to the featureless nature of its spectrum. The manner in which ZoDy mimics the continuous distribution of particles as a function of size, N(D), is by parameterizing this quantity as a power law given by

$$\begin{equation} {dN}(D) = N_0\times D^{-a}{dD}, \end{equation} \tag{ 15 }$$

where N₀ is a normalization constant and a is the mass index of the distribution. ZoDy assumes a = 2.9. Because of Arecibo's sensitivity, it could be argued that the excess in predicted detections is at least in part due to the large amount of small unconstrained particles in the model. This is explored in the bottom panel of Figure 18, where the number of meteors as a function of their mass is displayed for the input and detected distributions. While particles greater than 1 μg suffer a decrease of about an order of magnitude between input and detection, the difference increases significantly for lower masses. In fact, while the number of particles with masses lower than 1 μg represent 99% of the input distribution, their contribution is reduced to 67% and 54% in the detected distributions for the −20 and −10 dB thresholds. In terms of the mass input, these numerous particles provide 50% of the input and only 4% and 7% of the detected mass for both thresholds (Table 1). Thus, a potential solution to reach better agreement would involve reducing the exponential in the size distributions of small particles. Although not shown here, we have performed this sensitivity study introducing a shallower distribution (a = 2) for particles with diameter lower than 100 μm. Because the whole population had to be re-calibrated to match IRAS fluxes, this change results in an accretion rate slightly larger than in the original distribution and the overall result is an increase in the discrepancy between Arecibo observations and ZoDy. A simpler approach is to invoke the stated 50% uncertainty on ZoDy's prediction (Nesvorný et al. 2010, 2011a). However, it is evident from the results presented in Section 4 that reducing the influx by a factor of two would not suffice to obtain agreement and most likely the solution must be found in our treatment of the detectability.

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Table 1. Percentage of Number and Mass Input Contributed by Particles with Masses Smaller than 1 μg

βip	Threshold (dB)	$\frac{{\rm Number }\; (m\,<\, 1\,\mu {\rm g})}{{\rm Total}}$	$\frac{{\rm Mass }\; (m\,<\, 1\,\mu {\rm g})}{{\rm Total}}$
Jones (1997)	−10	0.54	0.04
	−20	0.67	0.07
Revision 1	−10	0.27	0.01
	−20	0.38	0.02
Revision 1	−10	0.3	0.02
	−20	0.67	0.07

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In this section, we will explore the origins of the extensively used β_ip value derived by Jones (1997),⁸ which is based on experiments involving the firing of high-velocity Fe particles into a chamber of air at low pressure and measuring electron production along the particle track. These experiments reported in Friichtenicht & Becker (1973) are a rather indirect way of measuring ionization efficiency because the rate of ablation of Fe from the particles has to be inferred from the deceleration of the particles. Nevertheless, Jones (1997) used these experiments combined with data on meteor luminosity and radar scattering to derive expressions for the ionization efficiency of the major elements. As stated by the author, however, the expressions reported in that work overpredicted the ionization efficiency by an order of magnitude⁹ (Jones 1997).

In this work, we re-estimate β_ip as a function of collision energy by utilizing measurements of the ionization cross section of K atoms in collision with O₂ and N₂ over the full range of collision energies reported by Cuderman (1972). This is the only metal atom for which there is experimental data with both collision partners over a wide collision energy and a detailed description of how the absolute cross sections were measured. The results show that charge transfer with O₂ is much more important than with N₂: the K + N₂ cross section is at least one order of magnitude less than that for O₂, so in-air N₂ collisions cause a 5% contribution to the total ionization at a collision velocity of 11 km s⁻¹, increasing to a 16% contribution at 72 km s⁻¹. Although about 70% of the product is O$_2^-$ rather than free electrons at the relatively low maximum impact velocities employed in a study by Moutinho et al. (1971), the O$_2^-$ will be produced with sufficient translational speed (and internal vibrational excitation) to auto-detach the electron since the electron affinity of O₂ is only 0.45 eV. Thus we assume that the final products of these collisions are K⁺ and e⁻.

In order to compute β_ip, the ionization cross section is first divided by the cross section for momentum-changing collisions to produce a single collision ionization probability β₀; β₀ is then increased to allow for ionization through subsequent collisions of the metal atom as it loses momentum (Jones 1997). In order to estimate the momentum-changing collision cross section, we employed quantum chemistry trajectory calculations. The hybrid density functional/Hartree–Fock B3LYP method was employed from within the Gaussian 09 suite of programs (Frisch et al. 2009), combined with the 6-311+G(2d) triple zeta basis set. This is a large, flexible basis set that has both polarization and diffuse functions added to the atoms. Classical trajectories were performed using the atom centered density matrix propagation molecular dynamics model (Schlegel et al. 2002). Trajectories were initiated at relative velocities of 11–72 km s⁻¹. One definition of a momentum-changing collision is that it is inelastic, i.e., there is a transfer of collisional kinetic energy into internal vibrational energy of the O₂. The maximum impact parameter b_max was then determined for collisions in which the O₂ after the collision possessed just one vibrational quantum. On the doublet surface for K + O₂, the average over the range of collision velocities is b_max = 2.9 Å (b_max increases by only 20% when V increases from 11 to 72 km s⁻¹). For the less reactive quartet surface, b_max = 2.0 Å. Statistically, reaction on the quartet surface is twice as likely as on the doublet surface, so overall for K + O₂, b_max = 1/3× 2.9 + 2/3× 2.0 = 2.3 Å. For K + N₂, b_max = 1.8 Å. These impact parameters are treated as independent of collision velocity. The momentum-changing cross section is then given by $\pi b_{{\rm max}}^2$. Figure 19 (top panel) illustrates β₀ for O₂ and N₂ as a function of collision velocity, calculated from the ionization cross sections of Cuderman (1972). Also shown in this figure are fits to the experimental data points using the analytic expression of Jones (1997):

$$\begin{equation} \beta _0(V) = \frac{c(V-V_0)^2V^{0.8}}{1+c(V-V_0)^2V^{0.8}}, \end{equation} \tag{ 16 }$$

where V₀ is the threshold velocity given by

$$\begin{equation} V_0 = \sqrt{\frac{2(M_e+M_a)e\psi }{M_eM_a}}, \end{equation} \tag{ 17 }$$

where M_e and ψ are the mass and ionization potential of the atom (K, in this case), respectively, e is the electronic charge, and M_a is the molecular mass of O₂ or N₂. c is a fitted parameter.

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In the atmosphere, K has a 20% chance of collision with an O₂. β₀ for air is then obtained as 0.2β₀ (O₂) + 0.8β₀ (N₂). The composite fit of Equation (16) for K in air is then achieved with c = 5 × 10⁶ (km s⁻¹)^−2.8, V₀ = 7 km s⁻¹ (an average between 6.9 km s⁻¹ for K + O₂ and 7.1 km s⁻¹ for K + N₂). β_ip is then given by Jones (1997):

$$\begin{equation} \beta _{\rm ip}(V) = \beta _0(V)+2\int _{V_0}^{V}\frac{\beta _0(V^{\prime })}{V^{\prime }}{dV}^{\prime }. \end{equation} \tag{ 18 }$$

The resulting curve of β_ip versus V₀ for K + air collisions is also shown in Figure 19.

There is some experimental data available for Na + O₂ and N₂ collisions (Bydin & Bukteev 1960; Moutinho et al. 1971; Kleyn et al. 1978), but the absolute cross sections are not consistent, although the Na ionization cross sections are clearly smaller than the corresponding K cross sections. Therefore, in order to determine β_ip for Na and other meteoric metals, we adopt the following approach. The maximum interaction distance between a metal atom and a collision partner is given by the curve-crossing (or harpoon) distance (Smith 1980),

$$\begin{equation} R_{c} = \frac{e(\psi -\gamma)}{4\pi \epsilon _0}, \end{equation} \tag{ 19 }$$

where γ, the vertical electron affinity of O₂ and N₂, is close to zero. The ionization cross section is likely to scale as $R_c^2$, particularly at high V where threshold effects are small. We therefore estimate c for Na, Mg, Fe, Si, and O by dividing c = 5 × 10⁻⁶ (km s⁻¹)^−2.8 for K by factors of 1.4, 3.1, 3.3, 3.5, 9.8, respectively. V₀ for each element from its respective ψ (Equation (17)). The values of c and V₀ for calculating β_ip(V) using Equations (16) and (17) are listed in Table 2.

The resulting β_ip for the six elements are plotted against V in Figure 19 (bottom panel). This shows that, compared to previous estimates (Jones 1997; Vondrak et al. 2008), the ionization efficiencies are about two orders of magnitude lower for Na and K at speeds below 20 km s⁻¹, and slightly less than one order of magnitude lower for the main elements (Fe, Mg, Si, and O) at higher speeds.

Figure 20 shows the same results as Figure 17 but utilizing the revised values of β_ip shown in the bottom panel of Figure 19 which we refer to as Revision 1 in Table 1. As can be seen in these figures, although improvement is achieved, ZoDy still predicts a flux that should dominate the Arecibo meteor rate and velocity distributions. In particular, at a detection threshold of −20 dB, ZoDy predicts that Arecibo should detect three times more particles that is actually observed, while at −10 dB they would represent 1.3 times the detected rates. Once again we emphasize that the final disagreement will be even greater because ZoDy does not include the populations that would produce most of the detected particles by Arecibo (high-speed). In terms of the radial velocity distributions, the ZoDy flux should provide a peak at ∼7 km s⁻¹ that is 10–80 times larger than those currently detected.

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As discussed above, β_ip is calculated by dividing the ionization cross section by the cross section for momentum-changing collisions between the metal atom and the air molecule (O₂ or N₂). The requirement that a momentum-changing collision has to be inelastic (i.e., arising from translational–vibrational energy transfer) may be too restrictive, since elastic collisions at large impact parameters will makes small changes to the momentum of the metal atom. The upper limit to the cross section for interaction between a metal atom and O₂ or N₂ molecules is given by $\pi R_c^2$ (see Equation (19) for the definition of R_c). For K + O₂/N₂, R_c is about 48% larger than the distance required to impart vibrational excitation (b_max), which would correspond to a decrease of β_ip by a factor of 1.48² ∼ 2.2 for K (and the other metal atoms, which are scaled to it). We refer to this lower limit of β_ip as Revision 2 in Table 1. The comparison between ZoDy predictions and Arecibo observations utilizing this second revision are displayed in Figure 21, where it can be seen that a much better agreement is found, particularly for a detected threshold of −10 dB. According to these results, if ZoDy provides most of the incoming flux, it would represent about 60% of the actual detected rates at the lower higher threshold. However, if we invoke the higher sensitivity (i.e., detection threshold of −20 dB), ZoDy should provide 1.6 times more meteors than are actually detected. Note that even at the −10 dB results, ZoDy predicts a rate of particles in the evening before midnight that is significantly larger (a factor of four) than those detected by the radar. If we take the lower limit to the ZoDy flux (50% uncertainty), then a reasonable agreement is obtained between model prediction and observations, in particular for the −10 dB detection threshold where they would represent about 30% of the observed rates and agree well with the diurnal distributions. Interestingly, ZoDy's particles originating from JFCs will have radiants mostly concentrated around the helion and anti-helion sporadic meteor sources (Nesvorný et al. 2010) and thus the 30% rate in the detections of particles coming from these sources is in good agreement with predictions of our earlier MIF model reported by Fentzke & Janches (2008). In terms of the radial velocity distribution, however, the disagreement is still significant.

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To explore potential sources of improvement, we display in Figure 22 a comparison between the input and detected distributions predicted by ZoDy, similar to Figure 18, but for both revised estimates of β_ip. The solid lines in Figure 22 represent the results derived with Revision 1, while the dashed lines represent Revision 2. These panels show, as expected, that the revised estimates of β_ip contribute even more to filter out particles with masses lower than ∼1 μg. However, for higher masses, the reduction of detected particles is not significant when compared to the results utilizing the original values of β_ip. Looking at the top panel of Figure 22, particles with velocities lower than 15 km s⁻¹ were completely removed from the detected distributions. As discussed in Section 2.3, once the "larger" particles have speeds higher than 15 km s⁻¹, the S/N is much higher than the detection threshold, so that even if the ionization efficiency is close to its lower limit, these particles will be detected. So while these results do support ZoDy's main hypothesis that most of the 12–14 km s⁻¹ particles with mass equal to 1–10 μg could remain undetected, even by the most sensitive radar utilized for meteor observations, the number of particles with higher velocities entering the beam at larger zenith angles is too high and thus continues to dominate the predicted distributions. One potential solution is perhaps to explore the possibility to revised ZoDy such that the IRAS constraint is still met, but with a reduced contribution from particles with velocities larger than 15 km s⁻¹.

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