On the other hand, meteoroid rates on the Moon can be related to rates on Earth by taking into account the different gravitational focusing effect between both bodies. The gravitational focusing factor Φ is given by (6)\begin{equation} \Phi=1+\frac{V^{2}_{\rm esc}}{V^{2}}, \end{equation}Φ=1+Vesc2V2,where Vesc is the escape velocity of the central body and V the meteoroid velocity. The different gravitational focusing effect for the Moon and the Earth is given by the quotient γ between the gravitational focusing factors for both bodies. For sporadic meteoroids, with an average velocity of 20 km s-1 (Brown et al. 2002), this velocity-dependent focusing effect is higher for the Earth by a factor of 1.3 (Ortiz et al. 2006), and so γSPO = 0.77. According to this we have \begin{eqnarray} &amp;&amp;{\rm HR}^{\rm SPO}_{\rm Moon} =\gamma^{\rm SPO}{\rm HR}^{\rm SPO}_{\rm Earth}, \\[3mm] &amp;&amp;{\rm ZHR}^{\rm ST}_{\rm Moon} =\sigma\gamma^{\rm ST}{\rm ZHR}^{\rm ST}_{\rm Earth}. \end{eqnarray}HRMoonSPO=γSPOHREarthSPO,ZHRMoonST=σγSTZHREarthST.For the average hourly rate of sporadic events we have HR\hbox{$^{\rm SPO} _{\rm Earth} =10$}SPOEarth=10 meteors h-1 (Dubietis & Artl 2010). In Eq. (8) an additional factor σ is included to take into account that the distances from the Earth and the Moon to the meteoric filament will in general be different, and this would give rise to a different density of stream meteoroids for both bodies. If we assume a simple situation where this filament can be approximated as a tube where the meteoroid density decreases linearly from its central axis, the following definition can be adopted for σ, (9)\begin{equation} \sigma=\frac{d_{\rm Earth}}{d_{\rm Moon}}, \end{equation}σ=dEarthdMoon,where dEarth and dMoon are the distance from the center of the meteoric tube to the Earth and the Moon, respectively. On the other hand, the ZHR on Earth at solar longitude λ (which corresponds to the time of detection of the impact flash) can be related to the peak ZHR by means of (Jenniskens 1994) (10)\begin{equation} {\rm ZHR}^{\rm ST}_{\rm Earth}={\rm ZHR}^{\rm ST}_{\rm Earth}({\rm max})10^{-b|\lambda-\lambda_{\rm max}|}, \end{equation}ZHREarthST=ZHREarthST(max)10−b|λ−λmax|,where ZHR\hbox{$^{\rm ST} _{\rm Earth}$}STEarth(max) is the peak ZHR on Earth (corresponding to the date given by the solar longitude λmax). The values for the peak ZHR for different meteoroid streams and the corresponding solar longitudes for these maxima can be obtained, for instance, from (Jenniskens 2006). For streams with non-symmetrical ascending and descending activity profiles or with several maxima, Eq. (10) should be modified according to the expressions given in Jenniskens (1994). By putting all these pieces together in Eq. (2), we can write the following expression for the probability parameter: (11)\begin{equation} p^{\rm ST}=\frac{\gamma^{\rm ST}\cos(\phi)\sigma {\rm ZHR}^{\rm ST}_{\rm Earth}({\rm max})10^{-b|\lambda-\lambda_{\rm max}|}}{\gamma^{\rm SPO}{\rm HR}^{\rm SPO}_{\rm Earth}+\gamma^{\rm ST}\cos(\phi)\sigma {\rm ZHR}^{\rm ST}_{\rm Earth}({\rm max})10^{-b|\lambda-\lambda_{{\rm max}}|}}\cdot \end{equation}pST=γSTcos(φ)σZHREarthST(max)10−b|λ−λmax|γSPOHREarthSPO+γSTcos(φ)σZHREarthST(max)10−b|λ−λmax|·However, this formula does not take into account the fundamental fact that only those meteoroids capable of producing impact flashes detectable from Earth should be included in the computations. In fact, by employing only the hourly rates measured on Earth in Eq. (11), which measures the flux of meteor brighter than mag. +6.5, it is implicitly assumed that meteoroids producing meteor events on Earth can also produce detectable impact flashes on the Moon. However, this assumption is incorrect. Thus, for a given meteoroid stream (i.e., for a given meteoroid geocentric velocity), the mass mo of meteoroids giving rise to mag. +6.5 meteors on Earth can be obtained from Eqs. (1) and (2) in Hughes (1987). For instance, this mass yields 5.0 × 10-8 kg for Perseid meteoroids (Vg = 59 km s-1), 2.4 × 10-8 kg for Leonids (Vg = 70 km s-1), and 5.0 × 10-6 kg for sporadic meteoroids with an average velocity of 20 km s-1 (Brown et al. 2002). However, the masses corresponding to impact flashes recorded on the Moon are several orders of magnitudes larger than mo (see, e.g., Ortiz et al. 2006; Yanagisawa et al. 2006; Swift et al. 2011). This means that the minimum kinetic energy Emin to produce a detectable impact flash on the Moon is much higher than the kinetic energy of a portion of the meteoroids included in the computation of hourly rates on Earth. This means that the velocity and mass distribution of meteoroids must be somehow included in Eq. (11) to take into consideration for the computation of the probability parameter p only those meteoroids with a kinetic energy above the threshold kinetic energy given by Emin (the method for the computation of Emin will be explained below). According to Eq. (2) in Bellot Rubio et al. (2000a,b), this can be accomplished by including in Eqs. (2) and (11) the factor (12)\begin{equation} \nu=\left(\frac{m_{\rm o}V^{2}}{2}\right)^{s-1}E^{1-s}_{\rm min}, \end{equation}ν=moV22s−1Emin1−s,where V is the impact velocity, mo is the mass of a shower meteoroid producing on Earth a meteor of magnitude +6.5, and s is the mass index, which is related to the population index r (the ratio of the number of meteors with magnitude m + 1 or less to the number of meteors with magnitude m or less) by means of the relationship (13)\begin{equation} s=1+2.5\log(r). \end{equation}s=1+2.5log(r).According to the definition of ν, this parameter is different for each meteoroid stream (and for sporadic meteoroids, of course). By taking this into account Eq. (11) should be modified as follows: (14)\begin{equation} p^{\rm ST}=\frac{\nu^{\rm ST}\gamma^{\rm ST}\cos(\phi)\sigma {\rm ZHR}^{\rm ST}_{\rm Earth}({\rm max})10^{-b|\lambda-\lambda_{\rm max}|}}{{\nu^{\rm SPO}\gamma^{\rm SPO}{\rm HR}^{\rm SPO}_{\rm Earth}}{+\nu^{\rm ST}\gamma^{\rm ST}\cos(\phi)\sigma {\rm ZHR}^{\rm ST}_{\rm Earth}({\rm max})10^{-b|\lambda-\lambda_{\rm max}|}}}\cdot \end{equation}pST=νSTγSTcos(φ)σZHREarthST(max)10−b|λ−λmax|νSPOγSPOHREarthSPO+νSTγSTcos(φ)σZHREarthST(max)10−b|λ−λmax|·If by the time of detection of the impact flash n additional meteoroid streams with significant contributions to the impact rate (and with compatible impact geometry) must be considered, the denominator in Eq. (14) must be modified in the following way, (15)\begin{equation} p^{\rm ST}=\frac{\nu^{\rm ST}\gamma^{\rm ST}\cos(\phi)\sigma {\rm ZHR}^{\rm ST}_{\rm Earth}({\rm max})10^{-b|\lambda-\lambda_{\rm max}|}}{{\nu^{\rm SPO}\gamma^{\rm SPO}{\rm HR}^{\rm SPO}_{\rm Earth}}{+\nu^{\rm ST}\gamma^{\rm ST}\cos(\phi)\sigma {\rm ZHR}^{\rm ST}_{\rm Earth}({\rm max})10^{-b|\lambda-\lambda_{\rm max}|}+\kappa}} , \end{equation}pST=νSTγSTcos(φ)σZHREarthST(max)10−b|λ−λmax|νSPOγSPOHREarthSPO+νSTγSTcos(φ)σZHREarthST(max)10−b|λ−λmax|+κ,where (16)\begin{equation} \kappa=\sum_{i\,=\,1}^{n}\nu^{\rm ST}_i\gamma^{\rm ST}_i \cos(\phi_i)\sigma {\rm ZHR}^{\rm ST}_{i,{\rm Earth}}({\rm max})10^{-b_i|\lambda_i-\lambda_{i,{\rm max}}|} \end{equation}κ=∑i = 1nνiSTγiSTcos(φi)σZHRi,EarthST(max)10−bi|λi−λi,max|accounts for these n additional streams. The minimum kinetic energy Emin defined above corresponds to the minimum radiated energy Er_min on the Moon detectable from observations on Earth, which in turn is related to the maximum visual magnitude for detectable impacts (mmax). And these values depend on, among other factors, the experimental setup employed. The kinetic energy of the impactor and the radiated energy are linked by the luminous efficiency: (17)\begin{equation} E_{r\_{\rm min}}=\eta E_{\rm min}. \end{equation}Er_min=ηEmin.With our experimental setup, the maximum visual magnitude for detectable impact is mmax ~ 10. The radiated energy can be obtained by integrating the radiated power P defined by the equation (18)\begin{equation} P=1.36949\times 10^{-16}10^{(-m+21.1)/2.5}f \pi \Delta \lambda R^2, \end{equation}P=1.36949×10-1610(−m+21.1)/2.5fπΔλR2,where P is given in Joules, m is the magnitude of the flash, 1.36949 × 10-16 is the flux density in W m-2 μm-1 for a magnitude 21.1 source according to the values given in Bessel (1979), Δλ is the width of the filter passband (about 6000 Å for our devices), and R is the Earth-Moon distance at the instant of the meteoroid impact. The factor f is related to the degree of anisotropy of light emission. Thus, for those impacts where light is isotropically emitted from the surface of the Moon f = 2, while f = 4 if light is emitted from a very high altitude above the lunar surface. As expected, according to Eq. (18) the minimum radiated power for flash detectability (and hence the minimum radiated energy and the minimum meteoroid kinetic energy), which is obtained by using m = mmax, is higher when the distance between Earth and Moon is greater. So, the detectability limit is time-dependent.
Citation: Swift et al. 2011
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