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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, A09308, doi:10.1029/2004JA010950, 2005
A new method for determining meteoroid mass from head echo data S. Close,1,2 M. Oppenheim,3 D. Durand,1 and L. Dyrud3 Received 2 December 2004; revised 7 February 2005; accepted 4 May 2005; published 21 September 2005.
[1] A head echo is the radar reflection from the plasma immediately surrounding a
meteoroid upon its entry into the Earth’s atmosphere; analysis of these plasmas can help determine a parent meteoroid’s inherent properties, such as mass. In the past, meteoroid mass was calculated using head echo velocity and deceleration data by assuming momentum conservation between the meteoroid and air molecule. We refer to such masses as ‘‘dynamical masses.’’ This method, however, can only be used to determine meteoroid mass if either the meteoroid radius or density is assumed. In this paper, we expound upon a new method for determining a meteoroid’s mass by utilizing our new spherical scattering theory. This theory allows us to use head echo measurements to calculate head echo plasma density. Then, by using the plasma density in an established formula that estimates the ratio of unionized to ionized material produced by an ablating meteoroid, we can determine a meteoroid’s mass. We refer to such masses as ‘‘scattering masses.’’ We show that our new mass determination method applies to head echoes detected simultaneously at VHF and UHF and verify that the meteoroid mass is the same at both frequencies. We conclude with a comparison between dynamical and scattering masses and show that in general, these methods agree to within an order of magnitude. Citation: Close, S., M. Oppenheim, D. Durand, and L. Dyrud (2005), A new method for determining meteoroid mass from head echo data, J. Geophys. Res., 110, A09308, doi:10.1029/2004JA010950.
1. Introduction [2] When a meteoroid enters the Earth’s atmosphere, it ablates and forms plasma between approximately 140 and 70 km altitude. Because micrometeoroids are typically too small to directly detect using radar or optics, we measure and analyze the ablated meteoroid plasmas in order to determine the parameters of the parent meteoroid. One such radar measurement is called the head echo, which is the radar return from the plasma that immediately surrounds the meteoroid and moves approximately at the meteoroid’s velocity. [3] Analysis of head echoes can provide information about meteoroid populations. For instance, we determine meteoroid velocity and deceleration by assuming that a head echo travels at the meteoroid’s velocity. By applying interferometric techniques, we determine a meteoroid’s 3-D position and hence its orbital parameters. Finally, we can derive meteoroid masses by utilizing either the combination of velocity and deceleration or the measured radar cross section (RCS) of the head echo. [4] Scientists have been calculating meteoroid masses using radars [Elford et al., 1964; Verniani and Hawkins, 1965], optical instruments [Whipple, 1943; Ceplecha, 1 Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts, USA. 2 Now at Space and Remote Sensing Sciences, Los Alamos National Laboratory, Los Alamos, New Mexico, USA. 3 Center for Space Physics, Boston University, Boston, Massachusetts, USA.
Copyright 2005 by the American Geophysical Union. 0148-0227/05/2004JA010950$09.00
1988], and space probes [Grun and Zook, 1980] for over 40 years. The majority of these methods, however, invokes assumptions and uses coarse measurements that lead to sizable errors in the calculated masses [Ceplecha, 1992]. For instance, we can measure a meteor’s velocity and luminosity using optics, and then assume a luminous efficiency in order to convert these parameters to a meteoroid mass, but the luminous efficiency is difficult to determine. Likewise in specular trail radar data, we measure a meteor echo’s signal-to-noise ratio (SNR) and velocity, and then convert these parameters to meteoroid mass by assuming an ionization probability and a scattering model; here, the velocity is sometimes a coarse estimate, and the ionization probability must be assumed. [5] A ‘‘new’’ method for determining a meteoroid’s mass from precise head echo data was first proposed by J. Evans [Evans, 1966] using UHF data collected at the Millstone Hill Radar. He calculated a parameter called the radiusdensity product, which is equivalent to the ballistic parameter equation derived by Opik [Opik, 1958]. Both of these equations can be used to determine meteoroid mass if either the meteoroid’s radius or meteoroid’s density is assumed. Janches et al. [2000] and Mathews et al. [2001] used the ballistic parameter equation to calculate meteoroid masses, referred to as ‘‘dynamical masses,’’ using head echo data collected at Arecibo by assuming that the meteoroid densities are all 3 gm/cm3. In this paper, we propose a new method for calculating meteoroid mass using head echo scattering data and provide a comparison with the traditional dynamical mass method. [6] Section 2 gives a brief overview of the multifrequency ALTAIR head echo data set. Section 3 gives an overview of
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our spherical scattering model. Section 4 describes the two methods used to calculate meteoroid masses and a comparison of the methods. Section 5 summarizes and expands upon future work.
2. ALTAIR Head Echo Data [7] The ARPA Long-Range Tracking and Instrumentation Radar (ALTAIR) is a 46-m diameter, high-power, twofrequency radar operating at 160 MHz (VHF) and 422 MHz (UHF), which resides in the central Pacific at 9N and 167E (geographic) on the island of Roi-Namur in the Kwajalein Atoll, Republic of the Marshall Islands. ALTAIR transmits a peak power of 6 MW simultaneously at the two frequencies with right-circularly (RC) polarized signal energy in a half-power beam width of 2.8 and 1.1 at VHF and UHF, respectively. ALTAIR receives both right-circular and left-circular (LC) energy and has four additional receiving horns for the purpose of angle measurement, which gives the position of an object in three dimensions. [8] Radar meteor data were collected by ALTAIR on 18 November 1998, during a 4-hour period in 2-min segments, which was designed to span the predicted peak of the Leonid storm (07:30 AM local time). The ALTAIR data showed a peak detection rate of 1.6 VHF head echoes every second [Close et al., 2002], although we believe that less than 1% of our detections were actually Leonids (P. Brown, personal communication, 2003). Amplitude and phase data were recorded for each frequency and four receiving channels for altitudes spanning 70 to 140 km at VHF and 90 to 110 km at UHF. The UHF altitude extent was smaller because of the very small (7.5 m) range sample spacing and the corresponding limit on disk space. The two ALTAIR waveforms used to collect the data were a 40 msec VHF chirped pulse (30 meter range spacing), and a 150 msec UHF chirped pulse (7.5 meter range spacing). A 333 Hz pulse repetition frequency (PRF) was utilized for its high sampling rate, which allows the calculation of 3-D speed as a function of altitude. Using these waveforms, ALTAIR can detect a target as small as 55 decibels relative to a square meter (dBsm) at VHF and 75 dBsm at UHF at a range of 100 km. [9] To obtain head echo speeds and decelerations, we extract each head echo and fit an exponential to its range data, since we expect the head echo deceleration to be proportional to the atmospheric density. Specifically, the fit is structured such that range ¼ ao þ a1 t þ a2 et
ð1Þ
where t is time. The derivative in fitted range with respect to time provides range rate, and the derivative in range rate gives head echo deceleration along the radar line of sight. We then process the data from the two angle difference channels (azimuth and elevation) to determine the angular offset of the detections from the radar boresite (in units of radians). We apply fitted angular offset data, referred to as monopulse data, to the fitted range data to obtain 3-D position and subsequently 3-D speed and 3-D deceleration.
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echo’s RCS to the properties of the parent meteoroid. Close et al. [2004] derived a new scattering theory in order to convert head echo RCS to head plasma density, and finally meteoroid mass. In this section, we provide only a brief overview of the scattering theory and refer the reader to Close et al. [2004] for a more complete description. [11] We stipulate that head echo RCS depends upon the density and the size of the meteoroid plasma distribution. In order to apply our spherical scattering model, we must therefore define one of these parameters in order to derive the other. For our purposes, we have defined the head echo radius, rmax, to be 0.023 times the Jones formula [Jones, 1995], or rmax ¼ ð:023Þ * 2:845 1018 v0:8 =n
ð2Þ
where v is the 3-D speed of the head echo in km/s and n is the background number density at the head echo detection altitude in m3; we refer to this equation as the ‘‘modified Jones formula.’’ The value 0.023 was determined by utilizing our dual-frequency observations [Close et al., 2004]. At high altitudes (>105 km), the modified Jones formula is approximately one atmospheric mean free path; at lower altitudes, it will produce head radii that are slightly smaller than a mean free path. In conjunction with the head echo size, we must also define a plasma density distribution. While we have explored many such functions, we use a Gaussian distribution for the analysis contained in this paper. This formula is given by nðrÞ ¼ nmax exp ðr=rmax Þ2
ð3Þ
where nmax is the maximum plasma density for each head plasma (near the meteoroid’s position at the center of the head plasma), and r is the radial distance from the center of the head plasma. [12] By definition, our spherical model uses a spherical coordinate system to derive scattering solutions for head plasmas, since we expect the leading edge of head plasmas to be approximately spherical in nature. We base our theory on the equations derived for scattering from a sphere with a uniform dielectric constant, which is given by Mie [1912], and Stratton [1941], as well as the equations for scattering from a cylindrical meteor trail with a nonuniform dielectric constant, which are given in Kaiser and Closs [1951], Jones and Jones [1991], and Poulter and Baggaley [1977]. We combine these analyses to derive a new equation for scattering from head plasma and assume the following: (1) The head plasma is approximated as a sphere with a peak plasma frequency that can be either smaller or greater than the radar frequency; (2) the density and dielectric constant depends only on r; (3) the head plasma is a nonabsorbing medium, (4) the head plasma radius depends upon altitude and scales with the atmospheric mean free path and meteoroid speed; and (5) the meteoroid plasma is typically smaller than a wavelength, which limits the accuracy of our model at high altitudes (large head echo radii). [13] The reflection coefficient, Rn, of the head echo plasma, derived in Close et al. [2004] is
3. Overview of Spherical Scattering Theory [10] A fundamental head echo measurement is radar cross section (RCS), yet in the past, it was difficult to tie a head 2 of 6
2nþ1 1 nh2n ðkrmax ÞAn rmax ffi2 ðn þ 1Þjn ðkrmax ÞBn Rn
ð4Þ
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where h2n is the nth-order Hankel function of the second kind, jn is the nth-order Bessel function of the first kind, and k is the wave number. We must still solve for the constants An and Bn that are coefficients determined by boundary conditions and the peak plasma density, nmax. These coefficients can be determined either numerically or analytically. Also note that if the head echo plasma is overdense (peak plasma frequency greater than the radar frequency), we need to integrate around the singularity at e = 0, since the dielectric constant varies from a negative value (overdense) to a positive value outside the head echo plasma (underdense). Finally, we relate the scattering cross section, smeas, to the reflection coefficient using smeas ¼
X l2 ðn þ 1=2 Þ2 p
n
j Rn j 2
ð5Þ
where l is the radar wavelength [Jackson, 1975; Morse and Feshbach, 1953] and smeas is the head echo RCS measured by the ALTAIR system. We continue to sum over n until our series converges. In summary, the measured head echo RCS converts to a reflection coefficient, Rn with coefficients An and Bn. These coefficients relate to the peak plasma density, nmax, which allows us to convert RCS to nmax.
4. Meteoroid Mass Determination Methods [14] In the past, the only way to determine meteoroid mass from head echo data was by assuming that the head echo travels at the velocity of the meteoroid and using the measured head echo velocity and deceleration in the dynamical mass equation [Janches et al., 2000]. This equation, which determines the ratio of the meteoroid mass to its cross section, is derived by conserving momentum between the air molecule and the meteoroid and assuming a drag coefficient. The dynamical equation can then be converted to meteoroid mass (i.e., ‘‘dynamical mass’’) by assuming either the meteoroid’s radius or the meteoroid’s density. [15] An alternative technique to determine meteoroid mass is by utilizing our new spherical scattering model to convert measured RCS to line density and meteoroid mass. In the next subsections, we will outline each of the meteoroid mass determination methods and give a comparison between the two methods using the ALTAIR data set. 4.1. Mass From Spherical Scattering Theory (‘‘Scattering Mass’’) [16] For the first time, we calculate the mass of a meteoroid using head echo data without assuming a meteoroid density or radius. We achieve this new result by converting head echo RCS to peak plasma density, and thence to electron line density, q, using our new spherical scattering theory. For meteor trails, q is constant at a given altitude. For head echoes, however, q depends strongly on r and varies as a function of r up to its maximum radius, rmax. We therefore use the integrated line density for subsequent use in our calculations, which is given by Zrmax q ¼ 2p
nðrÞrdr 0
ð6Þ
Figure 1. A single head echo, detected simultaneously at VHF and UHF on 18 November 1998, showing RCS as a function of altitude.
where n(r) is the plasma density at radius r defined by our Gaussian function. Note that the plasma density n(r) can only be calculated by using our spherical solution to derive nmax. [17] The line densities, calculated using the spherical method, are then input into the standard meteoroid mass loss equation to determine meteoroid mass dm qmv ¼ dt b
ð7Þ
where m is the meteoroid mass, m is the mean molecular mass, which is approximately 20 amu for stony meteoroids dominated by 60% oxygen and 25% silicon, v is the head echo 3-D speed and b is the ionization probability, which depends upon the speed and scales as 4.91 106v2.25 [Jones, 1997]. Alternatively, we could use the ionization probability equation given by Lebedinets et al. [1973] or Bronshten [1983]; however, these alternatives for b typically only change the meteoroid mass by less than a factor of 5. Note that in order to obtain the scattering mass, we must estimate the meteoroid mean molecular mass, as well as the ionization probability. [18] We examine a single head echo detected simultaneously by ALTAIR at VHF and at UHF during the Leonid 1998 shower. First, we correct the RCS for position within the ALTAIR beam using the monopulse offset values. Specifically, we find the position of the head echo within the beam using the monopulse data, and then determine the reduction in SNR at that position by approximating the SNR distribution of the ALTAIR beam pattern. We then adjust the head echo RCS accordingly to remove any beam pattern effects. The corrected RCS is plotted as a function of altitude in Figure 1 for both the VHF and UHF detected head echo. On average, the VHF RCS is 20 dBsm higher than the corresponding UHF RCS. We must stipulate that our approximation of the SNR distribution of the ALTAIR beam pattern takes the SNR as a function of elevation and maps it to azimuth, which
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Figure 2. Electron line density calculated using the spherical model applied to the RCS data contained in Figure 1. creates a symmetric grid in both azimuth and elevation, which will introduce a slight error. [19] We now use the spherical model to calculate the maximum plasma density of each head echo pulse using this head echo streak at both VHF and UHF. Next, we input these densities, one per altitude bin, into equation (6) to calculate electron line density, and then input the line density as well as the head echo 3-D speed into equation (7) to calculate meteoroid mass, using both the VHF and UHF data. In order to calculate meteoroid mass, we assume no mass loss below the visible signal (98.8 km) and integrate up along the path to get the maximum mass. Therefore we are inevitably underestimating the meteoroid’s mass, since we cannot determine how much mass loss is occurring outside of the ALTAIR beam. However, the slow rate of change at the highest altitude may indicate that we have observed the majority of the meteoroid mass loss.
Figure 3. Meteoroid mass calculated using the electron line density shown in Figure 2.
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[20] Unlike the RCS data, which show a strong dependence on radar frequency, the line density and meteoroid mass should be the same regardless of radar frequency, since the object creating both the VHF and UHF head plasma, i.e., the meteoroid, is the same. The line density and meteoroid mass as a function of altitude are plotted in Figures 2 and 3 for both the VHF and UHF data. While the average measured RCS shows a 20 dBsm difference between VHF and UHF, the electron line densities, as well as the meteoroid masses, are approximately the same at VHF and UHF, which confirms our hypothesis. There is a slight offset between the VHF and UHF line densities and masses at the highest altitudes. We attribute this offset to either error in the monopulse data, which affects both the RCS correction and 3-D speed, or to error in the spherical model, which may arise from one of the approximations that we use, such as head plasma radius size. Nevertheless, the difference between VHF and UHF is extremely small (less than a factor of 2), which gives confidence in the spherical scattering solution. [21] We proceed to calculate meteoroid masses for 20 head echoes. These data are shown in Figure 4 and reveal the meteoroid mass as a function of altitude. Note how the meteoroids lose mass as altitude decreases, as expected, and that the highest mass loss rates occur at the lowest altitudes where the air densities are greatest. We also see a trend such that the highest-mass meteoroids are located in the upper left portion of the plot, which corresponds to low altitudes. Since only the most massive meteoroids can penetrate to low altitudes, our new mass determination method appears consistent. Using the initial mass from each meteoroid, the median mass over all 20 meteoroids is 3.2 106 grams. We must be cautious in using the initial mass, however, as there exists a discrepancy between the low- and high-mass meteoroids. Specifically, the initial mass of the high-mass meteoroids, located at the lower altitudes, is probably correct since we see that the mass is hardly changing as a function of altitude, i.e., the curve is flat where we calculate the initial mass. The low-mass meteoroids, however, show a
Figure 4. Meteoroid mass as a function of altitude for 20 VHF head echoes detected during the Leonid 1998 shower. The 3-D speeds vary from 45 km/s to 72 km/s.
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as m(dv/dh)(dh/dt) = m(dv/dh)v cos c, where h is the altitude and c is the zenith angle. Equating the two momentum expressions we have m vgrair sec c ¼ : A dv=dh
ð8Þ
The left hand side of this equation is the ratio of the meteoroid mass to its physical cross section and is defined as the ‘‘ballistic parameter.’’ We note that for roughly spherically shaped bodies, Evans [1966] chose A as 2.pr2. As with many other authors, we set g = 1. [24] The ballistic parameter will sometimes produce unphysical results. For instance, if we examine a single head echo streak, the dynamical mass will often increase and then decrease, instead of consistently decrease as it penetrates further down in altitude, as we would expect [Evans, 1966]. Also, for a small portion of head echoes detected by Arecibo Observatory, the change of the ballistic parameter (final – initial) is negative, instead of positive like it is for most of the detections (D. Janches, private communication, 2004). We believe this anomalous behavior is caused by the following: (1) errors in the head echo speed, which are introduced by either using just the range rate, which is not the ‘‘true’’ 3-D speed of the particle, or errors in the monopulse data, which causes an error in the 3-D speed of the meteoroid; (2) fragmentation processes, which we ignore in the single-body theory used to derive the dynamical parameter; and (3) excluding the secondorder terms in the equation, which may be important. Keeping in mind these problems, we now include a comparison of mass determination methods.
Figure 5. UHF head echoes detected on 18 November 1998, showing scattering and dynamical mass. Each plot contains five head echoes. The average ratio between the scattering and the dynamical mass ranges between (a) 1.4 to 4.9 and (b) 5.6 to 50. distinct curvature at our first detected altitude. Therefore we are undoubtedly underestimating the mass of the low-mass/ high-altitude meteoroids. 4.2. Mass From Dynamical Methodology (‘‘Dynamical Mass’’) [22] As the meteoroid descends through Earth’s atmosphere, it encounters an air mass of Arairvdt in the time interval dt, where A is the meteoroid’s physical cross section, v is the meteoroid speed, and rair is the mean atmospheric density, which we calculate here using the MSIS-90 atmospheric model. We ignore any fragmentation of the meteoroid body. If the drag coefficient of the meteoroid’s encounter with air molecules is g, then the momentum gained by the air molecules per second is gArairvdt. [23] The rate of loss of momentum of the meteoroid during the same event is given by mdv/dt, where m is the mass of the meteoroid. This momentum loss can be written
4.3. Comparison of ‘‘Scattering Mass’’ With ‘‘Dynamical Mass’’ [25] We extract 10 UHF head echoes detected during the Leonid 1998 shower and compute mass using both methods; these data comprise the five best and the five worst matches between the scattering and dynamical masses. In order to compute dynamical mass, we must assume a density. We choose the meteoroid densities to be 1.4 gm/cm3, which is consistent with the density of a cometary particle. These data are plotted in Figures 5a and 5b and are grouped according to agreement between the two mass determination methods. Figure 5a contains head echoes that agree, on average, to within a factor of 5, and Figure 5b contains head echoes that agree, on average, to within a factor of 50. Using all 10 head echoes, the mean ratio of scattering to dynamical mass is 9.6, with the scattering method typically producing the higher mass. Two general points are worth noting when examining Figures 5a and 5b. The first is that the dynamical masses often do not follow our physical intuition, where we believe the highest mass loss rates should occur at the lowest altitudes. Specifically, many of the dynamical mass plots are straight lines; this may arise, in part, because the range data have been fitted to equation (1). Second, the dynamical masses do not follow the same trend with altitude as the scattering masses, where higher-mass meteoroids are detected at progressively lower altitudes. On average, however, the two methods agree to within an order of magnitude. For comparison, an order of magnitude difference also exists when comparing dynamical masses with masses
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derived using photometric measurements, where the photometric masses are typically larger [Bronshten, 1983]. It has been argued (D. ReVelle, personal communication, 2004) that this order of magnitude discrepancy is due to meteoroid fragmentation, which is ignored in the dynamical methodology. Fragmentation could also account for the discrepancy between the scattering and dynamical masses shown herein.
5. Discussion and Future Work [26] Head echo observations collected at VHF and UHF by ALTAIR were analyzed in order to illuminate a new method for computing meteoroid mass by utilizing the head echo RCS data. This new method allows us to calculate meteoroid mass without assuming the meteoroid’s density, which is the first time this has been achieved with head echo data. The new method is also insensitive to fragmentation processes, which might be responsible for some of the anomalous behavior observed in dynamical masses. We have subsequently compared our new mass determination method, which we term ‘‘scattering mass,’’ with the standard ‘‘dynamical mass.’’ Our results indicate that the scattering masses produce more physically plausible results. Specifically, (1) the scattering masses always decrease as altitude decreases, contrary to dynamical masses which sometimes increase along a portion of the trajectory, (2) the scattering masses show an exponential decay consistent with atmospheric drag modeling, whereas dynamical masses often show a linear decrease, and (3) the scattering masses show a trend with altitude where the highest-mass meteoroids are located at the lowest altitudes as physical theory suggests, whereas the dynamical masses do not always follow this trend. [27] When comparing the methods on a meteoroid by meteoroid basis, we find that for the most part, the methods agree to within an order of magnitude. Some results, however, differ by almost two orders of magnitude, where the scattering mass is often (but not always) higher than the dynamical mass. This is an interesting result, because we believe that the scattering methodology for determining mass may actually underestimate a meteoroid’s mass. This arises because a meteoroid may be losing mass both before it enters the radar beam, and after it leaves the beam; the scattering mass equation does not account for this extraneous mass loss. [28] Our future work includes calculating more precise beam patterns to correct our RCS data, as well as calculating more precise ionization probabilities. We also hope to refine our spherical scattering theory to eliminate some of the assumptions, as well as develop a more sophisticated mass loss equation that we will extrapolate in order to account for mass loss outside of the radar beam. [29]
Acknowledgments. The authors gratefully acknowledge the contributions from the following people: Stephen Hunt, Peter Brown, Kristine Drew, Rob Suggs, and Bill Cooke for wonderful collaboration; Fred McKeen and Michael Minardi for data collection and analysis; and Gary Bust, Yakov Dimant, and Harry Petschek for theory and analysis. This work was sponsored by the SEE program under Air Force contract F19628-
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00-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the United States Government. This material is also based upon work supported by the National Science Foundation under grant 0334906. [30] Arthur Richmond thanks W. G. Elford and Diego Janches for their assistance in evaluating this manuscript.
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S. Close, Space and Remote Sensing Sciences, Los Alamos National Laboratory, Mail Stop B244, Los Alamos, NM 87545, USA. (sigrid@lanl. gov) D. Durand, MIT Lincoln Laboratory, Group 91, 244 Wood Street, Lexington, MA 02420-9108, USA. L. Dyrud and M. Oppenheim, Center for Space Physics, Boston University, Boston, MA 02215, USA.
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