Plasma distributions in meteor head echoes and implications for radar cross section interpretation

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Plasma distributions in meteor head echoes and implications for radar cross section interpretation ⁎

Robert A. Marshalla, , Peter Brownb,c, Sigrid Closed a

University of Colorado Boulder, Department of Aerospace Engineering Sciences, 1111 Engineering Drive, 431 UCB, Boulder, CO 80309, USA University of Western Ontario, Department of Physics and Astronomy, London, Ontario, Canada N6A 3K7 c Centre for Planetary Science and Exploration, University of Western Ontario, London, Ontario, Canada N6A 5B7 d Stanford University, Department of Aeronautics and Astronautics, 496 Lomita Mall, Stanford, CA 94305, USA b

A R T I C L E I N F O

A B S T R A C T

Keywords: Meteor Radar Head echo RCS Plasma Distribution

The derivation of meteoroid masses from radar measurements requires conversion of the measured radar cross section (RCS) to meteoroid mass. Typically, this conversion passes first through an estimate of the meteor plasma density derived from the RCS. However, the conversion from RCS to meteor plasma density requires assumptions on the radial electron density distribution. We use simultaneous triple-frequency measurements of the RCS for 63 large meteor head echoes to derive estimates of the meteor plasma size and density using five different possible radial electron density distributions. By fitting these distributions to the observed meteor RCS values and estimating the goodness-of-fit, we determine that the best fit to the data is a 1/ r 2 plasma distribution, i.e. the electron density decays as 1/ r 2 from the center of the meteor plasma. Next, we use the derived plasma distributions to estimate the electron line density q for each meteor using each of the five distributions. We show that depending on the choice of distribution, the line density can vary by a factor of three or more. We thus argue that a best estimate for the radial plasma distribution in a meteor head echo is necessary in order to have any confidence in derived meteoroid masses.

1. Introduction The problem of determining the meteoroid mass flux input to Earth's atmosphere has persisted for decades (von Zahn, 2005; Plane, 2012); two orders of magnitude separate the high and low ends of the commonly-cited estimates (Murad and Williams, 2002). These differences arise due to different observational methods, but also due to a large number of uncertainties in assumed model parameters for each method. This makes the accuracy of individual meteoroid mass estimates quite uncertain, a problem compounded for many techniques by the ubiquitous effects of meteoroid fragmentation (Ceplecha et al., 1998). We focus herein on plasma and meteoroid mass estimates derived from meteor head echoes observed with meteor radars. These measurements arguably suffer less biases from the effects of meteoroid fragmentation, though such effects are present in some head echoes (Kero et al., 2008). Meteor head echoes are frequently detected by High-Power Large-Aperture (HPLA) radars, and radar cross sections (RCS) can be determined in certain scenarios. Head echoes are radar reflections from the immediate dense plasma cloud surrounding the ablating meteoroid and appear to move with the meteoroid. Lower-

⁎

Power, Broad Beam (LPBB) radars, designed to detect meteor echoes through transverse scattering, can detect the radial scattering from head echoes when comparatively larger meteoroids occur in the beam (Baggaley, 2002). Such systems tend to detect head echoes at the rate of one per hour (in the case of the Southern Argentina Agile Meteor Radar for example (Janches et al., 2014)) as compared to several or more per minute or per second as is typical for narrow-beamed HPLA (Close et al., 2004). However, for either type of system, because of the plasma nature of the meteor head, the relationship between the measured RCS and the meteor plasma parameters is not straightforward. Close et al. (2004) and others relate the meteor head RCS to the electron line density q in the meteor, and then relate the line density to the meteoroid mass from:

m=

dt ∫ qvμ β

(1)

where the integration is taken over the duration of the meteor. Normally, the velocity v is measured, and the mean molecular mass μ is assumed. The ionization coefficient β is summarized by CampbellBrown et al. (2012) as a function of velocity, but is also a function of composition. A reliable description of β is still a topic of active research, though recent laboratory measurements (Thomas et al., 2016) are in

Corresponding author.

http://dx.doi.org/10.1016/j.pss.2016.12.011 Received 28 September 2016; Received in revised form 5 December 2016; Accepted 26 December 2016 0032-0633/ © 2017 Elsevier Ltd. All rights reserved.

Please cite this article as: Marshall, R.A., Planetary and Space Science (2017), http://dx.doi.org/10.1016/j.pss.2016.12.011

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general agreement with earlier theoretical estimates (Jones, 1997), providing some confidence that values are known to modest accuracy, notably at speeds >20 km/s. The line density q is thus left to be determined. In prior work (Close et al., 2002, 2004), the plasma radial distribution is assumed to be either Gaussian or Parabolic Exponential (defined later in this paper), and a direct relationship between q and RCS is thus provided. However, these assumptions at the core of the technique used to invert meteoroid masses from the plasma radial distribution are worth further observational validation, a process which requires multi-frequency simultaneous observations of head echoes. In this paper, we use results of numerical simulations of meteor head echo plasma radar (radial) scattering together with triple frequency radar data to constrain the plasma distributions of these meteor head echoes. The results show that the estimate of the line density q used to derive meteoroid mass is more complicated than previously thought, and that multi-frequency measurements of RCS are necessary to provide a best estimate of the plasma distribution.

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We analyze meteor head echo data from the Canadian Meteor Orbit Radar (CMOR), a triple frequency transverse scattering radar located near London, Ontario, Canada (Jones et al., 2005). CMOR has three identical radars operating simultaneously at 17.45, 29.15, and 38.45 MHz. The broad, all-sky beam has a transmit beam width of 30° and a receive beamwidth of 45° to the 3 dB points. The total system directivity is 14 dBi, with interferometric precision of order 1° for signals more than 10 dB above the noise floor (Brown et al., 2008). CMOR detects 1–2 head echoes per day at 38 MHz to as much as 10 per day at 17 MHz. However, only of order one head echo per week has three frequency detection at high enough signal to noise (at least 6 dB above the noise floor) at all three frequencies for measurements to be feasible and shows no significant signal interference or pulsations associated with fragmentation [e.g., (Kero et al., 2008)]. A total of 63 meteor head echoes observed simultaneously on all three frequencies are used in this analysis covering the period May 2006–December 2007. Fig. 1 shows an example head echo range-time-intensity plot from one of these events detected by the CMOR 17.45 MHz system. Measurements were made of peak returned head echo amplitude at a common range and time where the returned power was at maximum for the 17 MHz frequency. The power calibration procedure follows the methodology given in Weryk et al. (2013). The resulting radar crosssection for our backscatter radar systems were found using the standard radar equation (eg. Skolnik, 2001):

10 5 0 20

25

30

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RCS (dBsm) Fig. 2. RCS values derived from CMOR meteor head echo data at three frequencies. The shift to higher apparent RCS, in particular the minimal detectable RCS, at lower frequencies is in part due to the higher noise floor at 17 MHz than 38 MHz. This, in addition to wavelength dependent considerations, determines the final observed RCS distribution.

σ=

4 (4π )3PR r 2 PG t tGrλ

(2)

where σ is the equivalent radar cross sectional area, Pr is the received power from the head echo, R is the range to the head echo, Pt is the transmitter peak power (in our cases these are all measured and ∼6 kW), Gt and Gr are the transmit and receive gain in the direction of the echo and λ is the wavelength of the radar. CMOR measures transmit power on a continual basis, with an accuracy of 5%, while the gain uncertainty is driven mainly by the uncertainty in interferometric direction, which is typically less than 2 − 3° for SNR >10 , corresponding to a product accurate to 10%. The range is interpolated following the procedure in Weryk and Brown (2012) and based on direct comparison with simultaneous optical measurements is typically accurate to < 1 km, though in some cases this range error can be as large as 5–6 km for poor SNR echoes. It is worth noting that for this study the head echo ranges at 17, 29 and 38 MHz were estimated independently and the average deviations were < 0.3 km. The observed head echoes had very large RCS values compared to typical HPLA radar data, as expected for a LPBB system. Fig. 2 shows the peak RCS values for the 63 meteors analyzed here; some meteors show RCS above +40 dBsm at 17 MHz. As we will show, these high RCS values imply either very large meteor plasma radii or very high peak plasma density. If the meteor radius is related to the mean free path in the atmosphere, then it must be the case that the peak plasma density is high.

Fig. 1. An example of a head echo range (y-axis), time (x-axis), intensity (color coding) plot at 17 MHz. The horitzontal arrow along the time axis represents 1 s in time or 532 pulses. The horizontal lines at constant range represent transverse meteor echoes, while the sloping line running from a range of 96–54 km is a head echo. The individual echo returns have a range extent of approximately 12 km, equal to the width of the gaussiantapered transmit pulse. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

3. Numerical modeling We use the Finite-Difference Time Domain (FDTD) meteor plasma scattering model of Marshall and Close (2015) to provide the relationship between meteor plasma distribution and RCS. The FDTD model simulates a broadband pulse scattering from an arbitrary meteor plasma 2

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distribution. RCS values are computed at discrete frequencies through a near-to-far field transformation. Critical to this paper, the FDTD model allows for arbitrary meteor plasma distributions. In Marshall and Close (2015), we used Gaussian and Parabolic Exponential distributions, and found for both that the RCS was given by the “overdense area”, i.e. the cross-section area of the meteor plasma where the plasma frequency exceeds the radar frequency. Inside this area, the meteor plasma behaves like a conducting sphere, and the RCS is given by πrp2 , where rp is the “overdense radius”. This result was found to hold for either plasma distribution. The result is not necessarily surprising, as plasmas tend to reflect when the plasma frequency ωp exceeds the wave frequency ω0; however it may be surprising that the result still holds for very small meteor plasma, with radii considerably smaller than the wavelength of the radar wave. Nonetheless this overdense scattering has been proposed before (Pellinen-Wannberg et al., 1998; Close et al., 2002); our FDTD modeling has simply confirmed it with rigorous, physics-based modeling. We can apply this simple result to any overdense meteors; in the case of underdense meteors, the FDTD model provides an empirical relationship between the plasma density and RCS, which is only slightly more complicated that the overdense case. However, for the CMOR data used here, we have high confidence that the very high RCS values reflect overdense meteors. Observational validation of this comes from comparison of head echoes observed by CMOR with a simultaneously operating all-sky camera system (Brown et al., 2010) which shows that many of the weaker head echoes in the main beam are associated with meteors with optical magnitudes of ≈ − 4 or brighter. The optical magnitude transition to overdense transverse scattering occurs near magnitude ≈+5 (Weryk et al., 2013) so we conclude that most of the events in our dataset qualify as fireballs and are atypical of HPLA head echoes. Were they underdense, the FDTD model predicts that the meteor plasma radii would have to be at least 50 m, which is extraordinarily unlikely at the altitudes of these meteors (primarily 90–110 km). At each of the three frequencies for which we have RCS values, we can use the overdense area result to compute data pairs of plasma density versus radius as follows. Consider a measured RCS value of 25 dBsm (314 m2) at 17 MHz. The RCS is given by the overdense area, or the area where the plasma frequency is equal to the radar frequency. The plasma frequency at 17 MHz is 1.07 × 108 rad/s, equivalent to an electron density of 3.6 × 1012 m−3. The 314 m2 RCS implies an overdense radius of 10 m. Thus, we have a data point of ne = 3.6 × 1012 m−3 at a radius of 10 m. Repeating this simple calculation at 29 and 38 MHz, we now have three data points of ne versus r. At this point, we have made no assumptions about the plasma distribution: only that the overdense scattering result applies to any distribution. Next, we will fit different distributions to the resulting data points Fig. 3.

peak density (normalized)

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/ r02

Parabolic Exponential: ne = ne0

2e πr / r0 e +1

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-2

-1

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radius (r ) Fig. 3. Distributions used to fit CMOR data using overdense scattering model.

Exponential: ne = ne0e−r / r0 1/ r distribution: ne = ne0

1 1 + r / r0

1/ r 2 distribution: ne = ne0

1 1 + (r / r0 )2

(5)

(6)

(7)

Each of these distributions has two parameters to be determined: the peak plasma density ne0 and the radius parameter r0. The latter is not equivalent for each distribution, and thus we cannot directly compare the values of r0 for each distribution. Note that the 1/ r and 1/ r 2 distributions have been defined in such a way as to provide a maximum value; the definition thus implies that the 1/ r or 1/ r 2 fall-off is valid only for r > r0 . Fig. 4 shows these distributions, plotted with the same parameters r0 = 1 and ne0 = 1. We observe that the Gaussian and Exponential profiles both have ne drop to ne0 / e at r = r0 , while the 1/ r and 1/ r 2 distributions drop to ne0 /2 at r = r0 . The Parabolic Exponential profile is the outlier, dropping much faster to less that ne0 /10 at r = r0 . We also observe that the 1/ r 2 distribution closely follows the Gaussian for small r, and then decays considerably more slowly for r > r0 . For each meteor event, we use the peak RCS observed during the duration of the event at 17 MHz, and then use this same time to measure the RCS at 29 and 38 MHz; this way the RCS values used are at the same time and thus the same altitude. These three RCS values at three frequencies provide three data points in electron density and radius, as described above. The distributions are fit to these data points using a linear least squares fitting procedure. Fig. 4 shows the resulting distribution fits for all 63 meteors. The figure panels have been constrained to the same axis limits for easy comparison. The data points from CMOR are easily evident, where three discrete values of ne are evident, corresponding to plasma frequencies of 17, 29, and 38 MHz. Furthermore, for the most part the distributions look reasonably similar, except that the exponential, 1/ r , and 1/ r 2 distributions suggest much higher peak densities than the other two distributions. From the distribution fits, we can extract the two key parameters for each distribution, namely the peak density ne0 and the radius parameter r0. These are shown in Fig. 5, along with the adjusted R2 value, providing a goodness-of-fit estimate for each case. These parameters are plotted as a function of altitude because we expect to see an altitude dependence on the radius and peak density. The radius is expected to be related to the mean free path, and so should decrease with decreasing altitude, while the peak density increases as the plasma is confined to a smaller volume. However, no such trends are observed in this dataset. The adjusted R2 values provide a quantitative means to compare the different distributions. For the five different distributions used here, the median adjusted R2 values are 0.894 for the Gaussian; 0.885 for the

(3)

2πr / r +0

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0.1

Previous work (Close et al., 2002, 2004) has focused on either Gaussian or Parabolic Exponential (PE) distributions for the meteor head plasma, in part because they result in tractable solutions in analytical models. It is the primary result of this work to show that the choice of plasma distribution has a significant effect on estimates of q and mass m; thus here we investigate different plasma distributions and fit these distributions to the CMOR data above. In particular, we focus on the following five distributions: 2

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4. Plasma distributions

Gaussian: ne = ne0 e−r

Gaussian Parabolic Exponential 1/r Exponential 1/r

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(4) 3

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Fig. 4. Distribution fits to meteor head echo data, for 63 meteors and five different distributions. The data points determined from CMOR data and FDTD model results are shown as circles; fits are colored curves. Colors are arbitrary denoting each meteor. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Altitude (km)

Parabolic Exponential; 0.895 for the Exponential; 0.531 for the 1/ r fit; and 0.931 for the 1/ r 2 fit. From these values we can definitively rule out the 1/ r fit, which furthermore shows very small meteor sizes (less than 1 cm) and very high peak densities. The 1/ r 2 fit provides the highest adjusted R2 values, providing first evidence that it may be the best fit to this data. Compared to the other distributions, we observe that the 1/ r 2 fit has radii in the range from 0.1 to 1 m, and peak densities about 1016 m−3. The other distributions result in much larger radii of tens of meters and peak densities much lower, in the range from 1013– 1015 m−3. According to several previous authors (Ceplecha et al., 1998), we

expect the meteor plasma radius to be on the order of a mean free path, or slightly smaller. Fig. 6 shows the mean free path in the upper mesosphere and lower thermosphere, along with the meteor radii for the 1/ r 2 distribution, identical to Fig. 5. The mean free path from 100 to 110 km altitude ranges from 10 cm to 70 cm; for this figure the mean free path λ is derived using the neutral atmosphere number density nd taken from the MSIS-E-90 model (Hedin, 1991) and is defined as:

λ=

μ nd

π 2mkBT

(8)

where μ is the viscosity of the atmosphere and m is the mass of a neutral

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Fig. 5. a) Radius parameter r0 and b) peak electron density ne0, extracted for each meteor and distribution. c) Adjusted R2 values for each fit case.

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5. Line densities Eq. (1) shows how the line density q relates to the parent meteoroid mass. We follow the method of Close et al. (2004) to determine the line density from the plasma distribution, but with a minor modification. Close et al. (2004) used a discrete summation from r=0 to r = r0 , where r0 is the Jones radius of the meteor plasma:

Altitude (km)

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r = rmax

∑

ne(r )πr 2

(9)

r =0

However, this method neglects meteor plasma at radii larger than r0, which is not a good assumption for large, overdense meteors where r0 may fall inside the overdense radius. We modify Eq. (9) to an integral form that covers radii from 0 to an arbitrarily large radius:

95

90 1 cm

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10 m

q = 2π

∫0

∞

ne(r )rdr

(10)

By integrating the 3D distribution over its cross section area, the resulting line density q in el/m corresponds to the number of electrons created per meter along the meteor trajectory. We numerically integrate our distributions to a radius large enough that additional contributions do not add significantly to the line density; for the results shown below we integrate to 500 m, or roughly ten times the largest derived radius. Fig. 7 shows the resulting line densities q for each meteor and each distribution. The 1/ r distribution results are not shown, as the resulting q values are two orders of magnitude smaller than those shown here. We observe that the 1/ r 2 fits provide larger q values than the other distributions. If we use the Gaussian distributions as the reference, we can find the average fractional q for each distribution as follows:

Fig. 6. Mean free path in the upper atmosphere (pink) along with the meteor radii derived using the 1/r 2 distribution fits. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

particle, taken as the mass of N2; kB and T are Boltzmann's constant and the atmospheric temperature, respectively, and the latter is provided by the MSIS-E-90 model. The viscosity is temperature dependent and given by Sutherland's formula. The median radius for the 1/ r 2 fit is 50 cm, corresponding to roughly one mean free path. However, there does not seem to be an altitude dependence to the meteor radii, and the scatter about the mean free path is about an order of magnitude. A similar spread in measured optical trail width from various literature sources in this height range also with a weak height dependence was noted in Stokan et al. (2013). In general, we expect optical radii to be larger than the corresponding ionized radii. From Stokan et al. (2013), measured optical radii (for much fainter meteors than considered here) range between 0.1 and 10 m at 100 km altitude and 1–100 m at 110 km altitude. Our values are physically consistent within these loose constraints. The lack of a clear altitude trend in Fig. 6 can be attributed to other unknowns in these meteor observations. For the meteors themselves, we have no information about the composition of the parent meteoroid, which may affect the plasma density and thus the derived radii. Similarly, while Fig. 6 plots the radius against the assumed mean free path based on a single MSIS-E-90 profile, in reality the atmospheric density profile may not be smooth at these altitudes, and may change with time between events. Finally, we have investigated any dependence on speed with the meteor observations (altitude and RCS) and with derived parameters (density, size, and adjusted R2 error). No significant correlations were observed, except a weak correlation between meteor altitude and speed, which is not unexpected; slower meteors ablate lower in the atmosphere and vice versa. However, the meteors observed in this dataset have speeds primarily between 50 and 70 km/s, and so do not represent a uniform distribution of possible meteor speeds. We can make a simple physical argument is favor of the 1/ r 2 distribution. The meteor plasma is generated by ablated meteor particles colliding with neutral atmosphere particles, ionizing the meteor particles (which have lower ionization threshold, i.e. 7.9 eV for iron vs 15.6 eV for N2). The ablated meteor particles are moving away from the meteoroid over an expanding area of 4πr 2 ; thus we expect the density of ablated particles to decrease as 1/ r 2 in the absence of collisions (i.e., inside one mean free path). Once collisions become frequent, the plasma density likely falls off even faster. Furthermore, this 1/ r 2 distribution has recently been validated through analytical modeling (Dimant and Oppenheim), which shows that inside a few mean free paths the distribution follows this 1/ r 2 dropoff.

Γi =

1 N

j=N

∑ j =1

qi, j qG, j

(11)

where qi, j is the q value for meteor j using distribution i, qG, j is the corresponding q value from the Gaussian distribution, and the summation is taken over N meteor events. The resulting ratio Γi tells us the average q relative the Gaussian distribution. Computing this ratio for the remaining four distributions, we find that ΓPE =1.28; Γexp =1.27; Γ1/ r =0.04; and Γ1/ r 2 =2.32. This tells us that the Gaussian, Parabolic Exponential, and Exponential distributions result in q estimates that mutually agree within 30%; but using the 1/ r 2 distribution yields q estimates a factor of 2–3 larger than the 115

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Fig. 7. Calculated line densities for each head echo. Note that using the magnitude – line density relation in Weryk et al. (2013) for CMOR, these line densities correspond to meteors with optical magnitudes between +1 at the lowest values to −4 at the high end, consistent with our magnitude range estimate from simultaneous head echo - optical meteor estimates (see text).

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Gaussian distribution. Thus, the choice of distribution makes a significant impact on the calculation of q, the error in which flows down to the calculation of mass in Eq. (1).

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6. Discussion and conclusions In this paper we have used triple frequency meteor head echo observations from the CMOR radar to estimate the plasma distributions and line densities associated with these meteors. We have found that the 1/ r 2 distribution provides the best fit to observations, with a median adjusted R2 value of 0.93, significantly higher than the other distributions. This fit yields meteor radii in the range of one mean free path, and very high peak electron densities. There are a few caveats worth noting in regard to this dataset. First, the meteor head echoes observed by CMOR are unusually large: previous head echo data sets from HPLA radars such as ALTAIR and Arecibo give head echo RCS values in the range from −90 dBsm to −40 dBsm, while these are +20 to +40 dBsm for CMOR. The corresponding optical brightess estimates show these CMOR events to be very bright, nearly fireball-class meteors. By comparison, the head echo producing meteors detected by ALTAIR or Arecibo are at least 10– 15 astronomical magnitudes (or more) fainter. As such, the CMOR dataset may be unusual events and not representative of typical head echoes from much smaller HPLA measurements. On the other hand, using the 1/ r 2 distribution provides meteor radii similar to the mean free path, as expected for head echoes. Second, it is not clear from the CMOR data that these meteoroids have not fragmented. While efforts were made at the measurement stage to eliminate head echoes showing obvious signal pulsations or large scale range-spread scattering, the low range gate resolution for CMOR (3 km) implies significant fragmentation may still be taking place for many of these events. The simple relationship between RCS and meteor plasma distribution is only valid for unfragmented meteoroids; if the particle fragments, the RCS will be complicated by the new arrangement of mass and the plasma distribution will no longer be spherical (Campbell-Brown and Jones, 2003; Mathews et al., 2010). The complex plasma distribution will then lead to interference of multiple scattering returns and an RCS measurement that is not straightforward. As such, fragmentation may create scatter in the RCS values presented here, leading to errors in the distribution fits. Solace is taken in the fact that the RCS values are monotonically decreasing with frequency in all cases, as would be expected for a spherical plasma distribution and so fragmentation cannot be an overwhelming dominant effect. Third, it is clear that we are fitting a two-parameter distribution to three data points, which is the absolute minimum to provide a fit and a measurement of the goodness of fit. It is highly likely that the error in the RCS measurements exceeds the error in the distribution fits. Nonetheless, the 63 meteors analyzed here provides a significant dataset to assess the relative goodness of different distribution fits. Indeed, when the process is applied to a random subset of these data, we find similar results, that the 1/ r 2 distribution is the best fit, the Gaussian and Exponential are next best, and so forth. Given the fit of two parameters to three data points, it is clear that improved estimates would be provided by data with more frequencies. Ideally, a broadband radar covering over an order of magnitude in frequency could provide five or more RCS measurements, which could then be reliably fit with these distributions. However, the development and application of such a radar to meteor head echoes is not likely to appear in the near future. Thus, a more confident assessment of the best-fit distribution to meteor head plasma is likely to rely on analytical and/or numerical ablation models.

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