Meteorites from meteor showers: A case study of the Taurids

Authors

  • Peter Brown,

    Corresponding author
    1. Centre for Planetary Science and Exploration, University of Western Ontario, London, Ontario, Canada
    • Department of Physics and Astronomy, University of Western Ontario, London, Ontario, Canada
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  • Valerie Marchenko,

    1. NASA Spring Intern Program, NASA Marshall Space Flight Center, Huntsville, Alabama, USA
    2. Department of Physics, Brandeis University, Waltham, Massachusetts, USA
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  • Danielle E. Moser,

    1. MITS/Dynetics Technical Services, NASA Marshall Space Flight Center, Huntsville, Alabama, USA
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  • Robert Weryk,

    1. Department of Physics and Astronomy, University of Western Ontario, London, Ontario, Canada
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  • William Cooke

    1. Meteoroid Environments Office, EV44, Space Environment Team, Marshall Space Flight Center, Huntsville, Alabama, USA
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Corresponding author. E-mail: pbrown@uwo.ca

Abstract

We propose that the Taurid meteor shower may contain bodies able to survive and be recovered as meteorites. We review the expected properties of meteorite-producing fireballs, and suggest that end heights below 35 km and terminal speeds below 10 km s−1 are necessary conditions for fireballs expected to produce meteorites. Applying the meteoroid strength index (PE criteria) of Ceplecha and McCrosky (1976) to a suite of 33 photographically recorded Taurid fireballs, we find a large spread in the apparent meteoroid strengths within the stream, including some very strong meteoroids. We also examine in detail the flight behavior of a Taurid fireball (SOMN 101031) and show that it has the potential to be a (small) meteorite-producing event. Similarly, photographic observations of a bright, potential Taurid fireball recorded in November of 1995 in Spain show that it also had meteorite-producing characteristics, despite a very high entry velocity (33 km s−1). Finally, we note that the recent Maribo meteorite fall may have had a very high entry velocity (28 km s−1), further suggesting that survival of meteorites at Taurid-like velocities is possible. Application of a numerical entry model also shows plausible survival of meteorites at Taurid-like velocities, provided the initial meteoroids are fairly strong and large, both of which are characteristics found in the Taurid stream.

Introduction

Sample return missions are a powerful means of bringing astromaterials with known parent body provenance to the lab. The Stardust (Brownlee et al. 2006) and Hayabusa (Yano 2006) missions returned the only materials from specific, known, small solar system bodies to date, namely 81P/Wild 2 and 25143 Itokawa. These samples have reshaped our understanding of asteroids and comets, with Hayabusa closing the long-standing controversy regarding space weathering (it exists) and linking Itokawa with LL-chondrites and Stardust, suggesting that some comets have mineral assemblages more reminiscent of chondrites than unprocessed, pristine material (Zolensky et al. 2006), implying that large-scale radial mixing was common in the early solar nebula (Ciesla 2011).

In addition to the foregoing direct sample returns, strong evidence suggests that the howardite-eucrite-diogenite (HED) meteorites originate from Vesta or its immediate family (Pieters et al. 2006), SNC meteorites originate from Mars (Treiman et al. 2000), while comparison of Apollo and Luna returned samples has shown convincingly that some achondrites are clearly lunar in origin (Papike 1998). Other meteorite associations with specific asteroids have been suggested (e.g., 6 Hebe and the H-chondrites—in Gaffey and Gilbert 1998), but the evidence in these cases is less clear. Clearly expanding the astromaterial–parent body connection beyond the presently sampled five specific parent objects (81P/Wild 2, Itokawa, Vesta/Vestoids, Mars, and the Moon) is among the most important of future planetary science goals. This has been explicitly acknowledged in the National Research Council Decadal report (2011) and is the motivation for several upcoming sample return missions such as OSIRIS-Rex (Lauretta et al. 2012), Marco Polo (Brucato et al. 2009), and Hayabusa 2 (Yoshikawa et al. 2008).

Meteoroids associated with meteor showers offer a unique window to the nature of small solar system bodies. Shower meteoroids are the only material that collides with Earth whose origin may be directly traced to individual small solar system bodies. The physical and chemical properties of daughter meteoroids are thus a direct proxy for known asteroids and comets. While some data on their parent bodies can be gleaned from study of shower meteor spectra (Borovička et al. 2005), lightcurves (Beech and Murray 2003), and ablation behavior (Trigo-Rodríguez and Llorca 2006; Kikwaya et al. 2011), these indirect techniques pale in comparison with the information that may be gathered from study of samples in the laboratory. Attempts have been made to collect interplanetary dust particles (IDPs) from specific meteor showers (e.g., Messenger 2002), but the association of collected dust with particular showers remains uncertain.

One way to expand the meteorite–parent body connection is to recover meteorites from known meteors showers. Could meteorites be recoverable at the Earth from meteor showers? Conventional wisdom is that this is not possible. This conclusion is reached on the basis of the generally high entry velocity of most meteor showers and the perception that cometary meteoroids are uniformly highly friable (e.g., Ceplecha et al. 1998). Reflecting this common wisdom, Campins and Swindle (1998) note: “Incidentally, one argument often given against cometary meteorites is the lack of falls during meteor showers (except for one coincidental iron meteorite fall). This argument is not valid because the entry velocity of most shower meteors is sufficiently high (approximately 28 km s−1) that even strong achondrite meteorites would not survive atmospheric entry.”

Clearly, meteorites from the fastest cometary showers (Perseids, Lyrids, etc.) can be safely ruled out. However, there are a handful of meteor showers whose meteoroids are (relatively) strong, such as the Geminids and Taurids (Babadzhanov 2002). But are such showers slow enough that a fraction of possibly stronger than average material has a chance to survive to Earth's surface?

As most meteor showers are derived from comets, this issue is synonymous with the possibility of cometary meteorites. In the past, a number of authors had suggested the possibility of many common (ordinary chondrite) types of meteorites originating from comets (e.g., Opik 1968; Wetherill 1971). This school of thought, however, was driven, in part, by the perceived dynamical difficulty of delivering meteorites from the main asteroid belt on time scales consistent with the cosmic-ray exposure (CRE) ages of meteorites. It entirely disappeared when the dynamical delivery time scale was reconciled with CRE ages through radiation forces (Vokrouhlický and Farinella 2000). The possibility that a small number of primitive meteorites (CI/CM) originate from comets has been suggested in the past (Campins and Swindle 1998), and this prospect is worthy of closer re-examination in light of the detection of some chondrite-like materials/mineral assemblages in returned samples from 81P/Wild 2 by Stardust (Zolensky et al. 2006). Analysis of Campins and Swindle (1998) of the expected characteristics of cometary meteorites concludes that statistically some cometary meteorites should already be in our collections and while no class of meteorite is unquestionably cometary, the primitive carbonaceous chondrites are the most likely candidates among macroscopic meteorites. They also suggest that, on physical and dynamical grounds, materials from Jupiter-family comets are the most likely cometary meteorite producers.

We note that velocity is the most crucial limiting factor in survival for all types of meteorites (Ceplecha et al. 1998); hence, lower velocity showers are necessarily where our search should be focused.

Among the approximately 17 well-described meteorite falls with instrumentally determined orbits, the highest initial entry velocity observed was the Morávka meteorite fall at 22.5 km s−1 (Popova et al. 2011). Past entry modeling suggests an approximate limit of 30 km s−1 as the extreme upper end of survivability for typical rocky, chondritic material (ReVelle 1979). The slower, stronger, more resistant to fragmentation, and smaller the initial zenith angle of entry for a particular meteoroid, the greater are the chances of material surviving intact to the ground. We suggest that among the major meteor showers, the Taurids (29 km s−1) plausibly meet most of these criteria and are therefore the most promising major meteor shower candidate for meteorite recovery.

The Taurids are known to contain very large meteoroids (Wetherill 1974; Spurný, unpublished data) at least in the hundreds of kilograms range—more than an order of magnitude larger than other showers (Ceplecha et al. 1998). The Taurid entry velocity of 29 km s−1 is tantalizingly close to the cut-off suggested by modeling for meteorite production and not far above the 22.5 km s−1 of the Morávka meteorite fall. The Taurids show great variations in strength including some very strong objects among fireballs (Borovička 2006). Trigo-Rodríguez et al. (2006) examined tensile strengths of average small shower meteoroids and found that next to the Geminids, the Taurids contained the strongest shower meteoroids. Taurid fireballs have inferred average bulk densities near 1.4 g cm−3, comparable to the Tagish Lake meteorite (Hildebrand et al. 2006). Most Taurids appear to be type II or III fireballs (as will be shown later), the same fireball-type category as found for the fireballs associated with the Tagish Lake (Brown et al. 2001) and Almahata Sitta meteorites (Jenniskens et al. 2009).

Here, we argue that some Taurid fireballs may be expected to produce meteorites, a notion previously suggested (e.g., Dergham et al. 2011), but without detailed development of supporting evidence. This opens the possibility of studying material from comet 2P/Encke in the laboratory without the expense of a direct sample return mission. On the basis of literature data, we argue that some Taurid fireballs are both large and strong enough to produce meteorites. We begin by examining the characteristics of fireballs necessary (although not strictly sufficient) to produce meteorites based on data in the literature, and establish the criteria that may be used to identify plausible Taurid fireballs capable of producing meteorites. Next, we analyze in some detail two specific Taurid fireballs that meet our necessary characteristics for a meteorite-producing fireball. One that occurred over Canada on October 31, 2010 was observed to penetrate to 35 km altitude and decelerate to approximately 9 km s−1, while the other, a possible Taurid fireball recorded in Spain in November 1995, is among the deepest penetrating fireballs associated with any major shower published to date. We further suggest that the recent Maribo meteorite fall, which is purported to have had a Taurid-like entry velocity of 28.5 km s−1 and produced a weak CM2 meteorite (Haack et al. 2012), supports our basic thesis. Indeed, it has already been suggested that Maribo may be linked to the Taurids (Haack et al. 2011), although the detailed analysis of the fireball trajectory and flight has not yet been published. This directly demonstrates that some bright Taurid fireballs can reach very low end heights and velocities, but are these values consistent with what is expected for fireballs just able to produce meteorites?

Limiting Characteristics of Meteorite-Producing Fireballs

Establishing which fireballs may produce meteorites has been a particularly pressing problem for many decades in meteor science, the question being driven by the establishment of photographic fireball networks in the 1960s (Halliday 1973), which had as their primary goal the recovery of meteorites with measured orbits. The question is a particularly difficult one to answer in the absence of a large and diverse set of instrumentally observed meteorite-producing fireballs, most particularly at the low mass/energy end where potentially modest fireballs may produce small (but still recoverable) meteorite fragments. Of particular note is the strong bias toward searching for meteorites only from the very brightest fireballs (Halliday et al. 1989a); this may be the reason that prior to the last decade, only three meteorites had been recovered among the >1000 fireballs recorded by the major networks. Here, we wish to address the related question of the endmember characteristics for fireballs just able to produce meteorites—i.e., conditions, which would be necessary, although not always sufficient, for fireballs to produce meteorites on the ground.

As a starting point, we examine the 18 fireballs with instrumental recordings that have produced meteorites. The details and characteristics of these meteorite falls are given in Table 1. These events span a wide range of sizes/energies and were measured with a diverse set of instruments. Nevertheless, some common characteristics are apparent: virtually all fireballs which have produced recovered meteorites are either type I or II fireballs (as expected—see later), most have low-ish (<22 km s−1) entry speeds, and most have low observed end heights (<25 km, many even below 20 km). Similarly, the observed terminal speeds are typically in the 3–5 km s−1 range. These parameter ranges can safely be associated with fireballs producing nearly certain meteorite falls; historically, these are also the events where ground recovery is attempted. We are interested, however, in the more marginal events associated with the generally smaller masses present in meteoroid streams; in this respect, the heterogenous data set comprising existing meteorite-producing fireballs is less applicable to our goal of identifying potential (but not certain) meteorite falls expected to have terminal masses far lower than the events presented in Table 1. Finally, as ground searches are only undertaken for events like those found in Table 1, there is a natural selection effect, which mitigates against the marginal cases as searching is rarely done in those instances.

Table 1. Meteorite-producing fireballs. The following table lists recovered meteorite falls for which instrumental records of the associated fireball produced a preimpact orbit. The table lists the official meteorite fall name, date of fall (UT), meteorite type (Met. type), total recovered mass (Mass [kg]) initial velocity (V), final observed velocity (Vfinal), initial mass (Minit), peak fireball magnitude (Magp), observed begin height (Hbegin), observed end height (Hend), zenith angle of local apparent radiant (Zenith angle), equivalent total fireball source energy in kilotons TNT equivalent (1 kT = 4.185 × 1012 J) (E [kT]), fireball type (PE [type]), observed fireball duration (Duration) and the type of instruments used to measure the fireball (Technique). Under the latter category, these techniques include P = dedicated photographic network, CV = casual video, V = dedicated video network, I = infrasound, S = seismic, Sa = satellite, R = radar
NameDate of fall (UT)Met. typeMass (kg) V V final M init Magp H begin H end Zenith angle E (kT)PE (type)Duration (s)Technique Reference
km s−1km s−1kgmagkmkm°
  1. 1. Ceplecha (1961), 1a. Borovička and Kalenda (2003), 1b. Ceplecha (1977); 2. McCrosky et al. (1971), 2a. Ceplecha et al. (1996), 2b. Ceplecha and ReVelle (2005); 3. Halliday et al. (1978), 3a. Halliday et al. (1981); 4. Spurný (1994), 4a. Borovička et al. (1998); 5. Brown et al. (1994), 5a. Ceplecha et al. (1996); 6. Brown et al. (2001), 6a. Hildebrand et al. (2006), 6b. Brown et al. (2002); 7. Borovička et al. (2003); 8. Spurný et al. (2003), 8a. Spurný et al. (2002); 9. Brown et al. (2004); 10. Trigo-Rodríguez et al. (2006), 10a. Llorca et al. (2005); 11. Spurný et al. (2012a); 12. Jenniskens et al. (2009), 12a. Welten et al. (2010); 13. Milley et al. (2010), 13a. Milley (2010); 14. Keuer et al. (2009), 14a. Haack et al. (2010), 14b. Haack et al. (2010); 15. Spurný et al. (2010); 16. Brown et al. (2011), 16a. Cartwright et al. (2010); 17. Borovička et al. (2013); 18. Spurný et al. (2012b).

Příbram4/7/1959H55.820.891300−199813470.068−4.4(I)6.8P1, 1a, 1b
Lost City1/4/1970H51714.23.5163−12.48619520.004−4.4 (I)9.0P2, 2a, 2b
Innisfree2/6/1977L54.5814.542.730−12>6220220.001−4.3 (I)4.09P3, 3a
Benesov5/7/1991H5/LL3.50.01121.124000−19.597.716.79.50.213−4.6 (I)5.2P4, 4a
Peekskill10/9/1992H612.414.7255000−16>47<33870.129N/A>25CV5, 5a, 1a
Tagish Lake1/18/2000C2 ~1015.860000−22>4029721.790−5.4 (II)>10CV,I,S,Sa6, 6a, 6b
Moravka5/6/2000H51.422.541500−208021700.091−4.1 (I)5CV, I,S,Sa7, 1a
Neuschwanstein4/6/2002EL66.1920.952.4300−17.28516410.016−4.6 (I)5.3P,I,S8; 8a
Park Forest3/27/2003L51819.511000−21.782<18290.500−4.9(II)5CV,I,S,Sa9
Villalbeto de la Pena1/4/2004L63.516.98600−18>4722.2610.020−4.4 (I)>4CV,I,S10, 10a
Bunburra Rockhole7/20/2007Euc0.32413.45.822−9.66329.6590.001−4.5(I)5.7P11
Almahata Sita (2008 TC3)10/7/2008Ure-Anom3.9512.4250000−20>65<33700.921−5.7 (IIIa)>6CV,I,Sa12, 12a
Buzzard Coulee11/21/2008H4>50188000 8112.7230.310−4.5(I)> 5CV,I13, 13a
Maribo1/17/2009CM20.02528.574−20120<32600.007−4.5(I)4.5P,CV,V,R14, 14a, 14b
Jesenice4/9/2009L63.613.8170−158818310.004−4.3(I)6.6P,V,I,S15
Grimsby9/25/2009H4-60.21520.913.1100−14.510019.6350.005−4.1(I)6.04V,I, R16, 16a
Kosice2/28/2010H54.3154.53500−18>5517.4300.094−4.9(II)4.5P,V,I,S17
Mason Gully4/13/2010H50.02514.534.140−9.4842436.10.001−4.4 (I)6P18

For the above reasons, we examine earlier studies that explored more homogenous datasets to try and establish characteristics likely to be more applicable to the marginal meteorite-producing cases of interest to our study. Ceplecha and McCrosky (1976) analyzed over 200 fireballs recorded by the Prairie photographic network (PN) with the goal of distinguishing populations of larger meteoroids differing in physical characteristics. While they did not focus on the characteristics of meteorite-producing fireballs explicitly in their analysis, by using the PE criteria (see Equation (1) later), they suggested a link between the fireball classes having the strongest atmospheric penetration ability, namely fireballs of types I and II in their nomenclature with ordinary chondrites and carbonaceous chondrites, respectively. A consequence of their work is that if all other initial parameters are equal (initial velocity, initial mass, and entry angle), those meteoroids with the greatest resilience against ablation (due to high bulk density, strong internal structure, aerodynamic shape, or all of the above) will penetrate to the lowest end height. Hence, fireballs that reach low end heights have the greatest chance of producing meteorites on the ground and these will be distinguished by high PE values.

Wetherill and ReVelle (1981) also examined PN fireballs and suggested four criteria for the selection of fireballs likely to produce ordinary chondrite-like meteorites. These included the following:

  1. The deceleration of the fireball to <8 km s−1 before the end of luminous flight.
  2. The initial meteoroid mass as determined by fireball deceleration (termed dynamic mass, md) must agree with the photometric mass, mph (calculated by integration of the light curve) to within a factor of two.
  3. The fireball end-height must agree within 1.5 km with that found for the Lost City meteorite fall and as determined through single-body theory, scaling for velocity, mass and entry angle.
  4. The shape of the lightcurve must be qualitatively similar to the lightcurve of other meteorite-producing fireballs.

Criterion 1 reflects the fact that once a fireball decelerates to a velocity of approximately 8 km s−1 or less, ablation virtually ceases; provided some measurable mass remains at this point, survival of meteorites is likely.

Criterion 2 underscores that fragmentation (a common process for meteoroids) will tend to artificially lower md as only the main fragment is being followed. mph better represents all mass lost during flight, so fireballs which have comparable dynamic and photometric masses probably undergo less fragmentation and hence have lower total ablation favoring meteorite production, in addition to being more likely to be physically similar. The value of mph/md found for the Lost City meteorite was defined for ease of relative comparison to be unity, the Innisfree fireball was found to have mph/md = 1.4. This criterion, being physically equivalent to having a lower ablation coefficient, is generically true, but as shown in Table 1, it is clear that many meteorite-producing fireballs show extensive fragmentation. Hence, as a stand-alone criterion, this is not likely to be useful for our purposes.

Both the Lost City and Innisfree fireballs had luminous trajectories ending at heights consistent with the theoretical values computed using single-body theory, under the assumption that md computed early in flight reflected the true initial mass. This empirical criterion is assumed to apply to all fireballs that might produce ordinary-chondrite-like meteorites.

The final criterion is qualitative and meant to reflect a general sense that the lightcurve of known meteorite-producing fireballs, particularly Lost City in the Wetherill and ReVelle (1981) analysis, should be another proxy for their fragmentation behavior (and hence physical structure). The type of fireball lightcurves, which would not behave as expected for chondritic meteorite producers according to Wetherill and ReVelle's (1981) analysis, includes fireballs showing sudden flares or other major irregularities in their lightcurve or peaks in brightness very early in flight.

It is important to recognize that the analysis by Wetherill and ReVelle (1981) was meant to identify likely ordinary chondrite-like producing fireballs and not all meteorite-producing fireballs, such as those associated with carbonaceous chondrites. In this sense, their conservative criteria are likely sufficient, but almost certainly not strictly necessary to identify all fireballs capable of producing all types and sizes of meteorites. It should also be emphasized that their analysis was heavily influenced by the two meteorite-producing fireballs then known, Lost City and Innisfree.

The most comprehensive empirical study of the characteristics of meteorite-producing fireballs to date was presented by Halliday et al. (1989a) based on photographic fireball records from the Meteorite Observation and Recovery Project (MORP). In examining 44 fireballs, which they state were identified to have produced meteorites with surviving main masses between 0.1 and 11 kg, a number of distinguishing characteristics emerged. These included terminal velocities below 10 km s−1 and end heights below 35 km. Significantly, Halliday et al. (1989a) found that the typical meteorite-producing fireball in their sample had peak absolute magnitude near −9, but can range as low as −7, emphasizing that meteorites may be recovered from relatively modest fireballs, in contrast to earlier perceptions that only very bright fireballs are likely meteorite producers. Indeed, McCrosky et al. (1971) noted that prior to recovery of the Lost City meteorite in 1970 that “… most fireballs were not produced by meteorites.” Halliday et al. (1989a) also found that the typical visible flight duration of a meteorite-producing fireball was between 3 and 5 seconds, with longer durations unsurprisingly correlating with larger meteorite falls. Other notable results from the work of Halliday et al. (1989a) include the following:

  1. Three events, which probably produced meteorites having initial velocities of 26–28 km s−1, including one event that probably produced multiple-kilograms of meteorites based on md at the end of the trail.
  2. Fireballs with terminal masses >1 kg always remained luminous <32 km, while events with terminal masses <0.5 kg always disappeared above 27 km.
  3. The highest mass survival fraction, independent of initial velocity, is associated with initial masses between 1 and 10 kg; initial masses of only a few kilos are sufficient to produce small meteorite falls.
  4. The survival mass fraction tends to be higher at shallower zenith entry angles.

In summary, earlier studies of meteorite-producing fireballs, using different methodologies and somewhat different data sets find a number of common characteristics for possible (but not necessarily definite) meteorite-producing fireballs. These include: a) relatively low end heights (<35 km), b) low terminal velocities (10 km s−1 or less), and c) measurable (>0.1 kg) terminal masses as estimated from md. We adopt these as an operational definition for our work to define possible meteorite-producing fireballs, noting that criterion (c) is rarely available as it requires high astrometric precision at the terminal point. Hence, a very simple necessary (but not strictly sufficient) criteria we use as a coarse initial filter (a) and (b) with the understanding that many such fireballs may not produce meteorites or only very small masses (gram-sized).

Properties of Taurid Fireballs

The Taurids are a well-known and extensively studied stream originally linked to comet 2P/Encke (Whipple 1940). Subsequent work has suggested that a large family of near-Earth asteroids (NEAs) may be part of a broader “Taurid” complex (Steel et al. 1991; Asher et al. 1993; Steel and Asher 1996). Clube and Napier (1982) proposed that the Taurid complex was formed by the breakup of a large comet some 20,000 yr ago and ongoing hierarchical fragmentation has produced both the Taurid stream, several related showers, and a family of associated NEAs. They further postulated that large debris from this complex have collided with Earth producing airbursts causing damage at ground level. The Tunguska event, for example, is proposed to be part of the broader Taurid complex via its linkage with the Beta Taurid meteor shower in late June. While this history for the Taurid complex remains controversial (e.g., Valsecchi et al. 1995), that the stream is massive, long-lived (arguably the longest of all major recognized showers), may consist of several related streams, and is a significant total contributor both to the sporadic meteor complex at the Earth and of the total dust flux received by the Earth has been recognized for decades (Whipple 1967; Stohl 1986) and found in more recent meteoroid modeling studies (e.g., Wiegert et al. 2009).

The relatively old age of the Taurids (>5–10 ka) has led its component meteoroids to differentially precess by at least one complete cycle in the argument of perihelion (Steel et al. 1991). This makes it possible for the Earth to intersect the stream at least four times. The pre- and postperihelion leg of the Taurid orbit (e.g., Babadzhanov and Obrubov 1991) contains twin daytime/nighttime showers, respectively, each with a North and South branch. The postperihelion leg of the stream, which is associated with the classical or core Taurid shower detected in the October–November time period each year, is further divided into North and South branches, with preperihelion twins to these showers being the Daytime Beta Taurids and Zeta Perseids, respectively (Steel et al. 1991). Several authors have proposed associating the Taurid-Encke complex with additional minor streams such as the N. and S. Piscids of September–October, the S. Arietids, and the N and S. Chi Orionids (in December) among others (see Babadzhanov and Obrubov 1991 for a review).

In an effort to gauge the possibility of meteoritic material surviving from the Taurid stream, we focus on the fireball component of the shower (initial masses >10 g). We begin by constructing a data set of instrumentally recorded Taurid fireballs previously published in the literature. In total, we have extracted 33 Taurid fireballs, out of a total of 714 fireballs, including orbital and trajectory information together with mass estimates. Masses from various fireball surveys were corrected to a common mass scale using the mass conversion proposed by Ceplecha (1988). The literature sources, consisting of data from the Prairie Network (PN), European Network (EN), and MORP, are shown in Table 2.

Table 2. Literature sources used in defining the historical dataset of instrumentally recorded Taurid fireballs
SourceNetworkNumber of fireballs
Ceplecha (1971)EN 1
Ceplecha (1977)EN 42
Ceplecha and McCrosky (1976)PN234
Ceplecha et al. (1973)EN 3
Ceplecha et al. (1976a)EN 1
Ceplecha et al. (1976b)EN 1
Ceplecha et al. (1979)EN 1
Ceplecha et al. (1980)EN 1
Ceplecha et al. (1983a)EN 1
Ceplecha et al. (1983b)EN 29
Ceplecha et al. (1987)EN 15
Halliday (1988)MORP 14
Halliday et al. (1989b)MORP 2
Halliday et al. (1996)MORP259
IAU Meteor Data Center Photographic Database (2003)EN 1
McCrosky et al. (1976)PN 58
Spurný (1994)EN 16
Spurný (1997)EN 32
Spurný and Borovička (2001)EN 1
Spurný et al. (2002)EN 1
Spurný and Porubčan (2002)EN 1

To isolate Taurid fireballs, we have used the radiant ephemeris for the core North Taurid and South Taurid stream found by Brown et al. (2010a) from radar data. We use this ephemeris as it employs a 3-D wavelet approach to identify the shower center relative to the long-term background activity and is based on typically more than 1000 radar Taurid radiants/orbits per solar longitude interval of stacked data from about a decade of radar observations. The large data set is particularly useful in accurately identifying the center of the long-lived/diffuse Taurid radiants. Ideally, we would like to use the radiant ephemeris generated from brighter fireballs alone as we might expect the radiant location and size to change with particle size, but studies that use photographic or fireball data alone to isolate the Taurids have far fewer radiants to gauge the drift for this long-lasting shower. As a result, different authors find different shower associations/radiant drifts (Steel et al. 1991; Porubčan and Kornoš 2002; Porubčan et al. 2006), so we have chosen for our operational definition of the shower the tighter ephemeris produced by the large number of radar orbits, recognizing that some difference may be present compared with larger Taurids. We have used the annual time period identified by Porubčan and Kornoš (2002) as showing noticeable Taurid fireball activity.

Table 3 summarizes our selection criteria for radiant drift, speed, and shower duration for the core of the Taurid stream. While the Taurid complex is certainly broader than this core stream and probably involves many minor showers as noted earlier, we conservatively adopt this criterion in our search for Taurid fireballs to avoid contamination; as a result, we believe that our fireball data set consists mainly of Taurids alone. Previous works (e.g., Steel et al. 1991) have adopted somewhat different criteria to identify Taurids, but generally these have been less stringent than we have chosen to use.

Table 3. Adopted criteria for identifying fireballs with the Taurid stream
 NTASTAReference
  1. 1: Porubčan and Kornoš (2002).

  2. 2: Brown et al. (2010).

Vg (km s−1)28.328.01
λmax (°)2191962
αg (°)48.930.92
Δαg (°/day)0.840.8172
δg (°)17.78.12
Δδg (°/day)0.250.29052
    
Vg criteria25–3125–31This work
Date criteriaOct 16–Nov 29Oct 4–Nov 241
Radiant criterion<=5°<=5°This work

As a means to compare the Taurid fireball data set with other showers having known parent bodies, we have also selected a suite of Geminid and Perseid fireballs from our literature database and analyzed these in the same way. The criteria used to identify these shower fireballs are shown in Table 4.

Table 4. Adopted criteria for identifying fireballs from the Perseid and Geminid stream
 PERGEMReference
  1. 2: Brown et al. (2010).

  2. 3: International Meteor Organization, www.imo.net

Vg (km s−1)59352
λmax (°)1402612
αg (°)48112.52
Δαg (°/day)1.391.122
δg (°)57.232.12
Δδg (°/day)0.29−0.172
    
Vg criteria57–6133–37This work
Date criteriaJul 17–Aug 24Dec 7–Dec 173
Radiant criterion<=5°This work

We have calculated the PE strength criterion for each fireball using the Ceplecha (1988) common mass scale conversion normalized to the original mass scale used by Ceplecha and McCrosky (1976). We note that this systematically decreases the PE for MORP fireballs by −0.21 over the published mass values in Halliday et al. (1996). The PE value is a one-dimensional index that gauges the relative strength (bulk density, fragmentation propensity, ablation rate) of a meteoroid based primarily on the luminous fireball end height. As defined originally in Ceplecha and McCrosky (1976), the PE value is given by:

display math(1)

where ρE is the atmospheric mass density (in units of g cm−3) at the height of the fireball end point; m is the entry mass in grams, computed from the total light production; V is the entry speed in km s−1; and ZR is the entry angle from the zenith. Larger PE values (less negative) correspond to stronger material displaying less ablation. Ceplecha and McCrosky (1976) further proposed that specific “types” of fireballs might be distinguished as four distinct taxonomic classes via the PE criteria. These strength groups and their probable material association are given as (Ceplecha and McCrosky 1976):

type I: PE > −4.60 ordinary chondrite-like

type II: −5.25 < PE ≤ −4.6 carbonaceous chondrite (CI/CM)

type IIIa: −5.7 < PE ≤ −5.25 short period cometary

type IIIb: PE ≤ −5.7 weak cometary material

Note that particularly the association of type I fireballs with ordinary chondrites (or comparable strength stony meteorite material) has been largely validated by recorded meteorite falls (see Table 1 and Popova et al. 2011), but that mixing of comparable material between adjacent types certainly occurs; the PE limits should not be strictly interpreted to correspond rigorously with the physical character of meteoroids (Ceplecha and McCrosky 1976).

Figure 1 shows the distribution of PE values for Taurid fireballs from the historical database. The mass range covered in this diagram is 0.020–15 kg. From the figure, it is clear that Taurid fireballs comprise a heterogeneous set of strengths, ranging from weak cometary to chondrite-like strength, a conclusion reached separately by Borovička (2007). The mean and standard deviation of the PE values for the Taurids are PEσ = −5.39 ± 0.34. In contrast, the Perseids show a lower PEσ = −5.52 ± 0.24 and the Geminids, a much larger PEσ = −4.5 ± 0.3, as illustrated in Figs. 2 and 3.

Figure 1.

The distribution of PE values for all Taurid fireballs published in the literature. The PE mass scale has been normalized to the luminous efficiency used in Ceplecha and McCrosky (1976) following the mass conversions suggested by Ceplecha (1988). Note that the majority of these fireballs (22 of 33) are from the MORP network. Bin sizes are chosen according to the statistical definition of population significance given in Scott (1979).

Figure 2.

The distribution of PE values for all (a) Perseid and (b) Geminid fireballs published in the literature. The PE mass scale has been normalized to the luminous efficiency used in Ceplecha and McCrosky (1976) following the mass conversions suggested by Ceplecha (1988). Note that the majority of these fireballs are from the MORP network. Bin sizes are chosen according to the statistical definition of population significance given in Scott (1979).

Figure 3.

Ground path of the SOMN 101031 event (arrow) showing the three stations that detected the event.

Taurid fireball strengths show clearly that some strong material exists within the stream, increasing the prospect for meteorite recovery; however, among all our Taurid fireballs in the historical database, the lowest end height is only 48 km with a terminal velocity of 15 km s−1 associated with an 8 kg initial mass. This is not consistent with the necessary end height/speed expected for a meteorite-producing fireball as described earlier. Indeed, among the much stronger Geminid fireballs, we find at least three fireballs with end heights of 39 km, making these the record holders for lowest shower end height. In all Geminid cases reported in the literature, however, the end velocity is >15 km s−1, virtually ensuring that no macroscopic material survives in the form of meteorites.

Among our relatively small data set of 33 historical Taurid fireballs, we find evidence for strong material within the stream, a necessary but not sufficient condition for meteorite production. Similar conclusions were reached by Shrbený and Spurný (2012) who find evidence for strong fireball material in the Taurids. From these data alone, we have yet to locate a Taurid fireball, which shows both a low end height (<35 km) and a low (<10 km s−1) end velocity needed for meteorite production as defined in our earlier summary.

Southern Ontario Meteor Network Event #101031

The first potential Taurid fireball meteorite producer we examined was recorded on October 31, 2010 at 04:43:33 UT by the Southern Ontario Fireball Network (SOMN) all-sky video camera system in Canada. Details of the network and hardware specifics are given in Weryk et al. (2008) and Brown et al. (2010b). This event was detected by three stations of the network. Figure 3 shows the locations of cameras that recorded the Taurid fireball and its ground path, while Fig. 4 shows stacked video images of the fireball as seen from the two closest stations.

Figure 4.

Stacked video image from (a) station 3 and (b) station 6 of SOMN 101031.

A more distant third station (station 7) recorded only the early portion of the fireball flight (to 60 km height), the remainder of the path being obscured by trees. Station 6 had the fireball only 30° from the zenith, but the earliest portion of the trail was partially obscured by clouds. Station 3 had a complete, unobstructed view of the entire event, and all subsequent photometric and speed measurements are from this site alone. The solution geometry was nearly ideal with the observing planes containing the great circle path of each meteor as seen from stations 3 and 6 being at 88º angles. We found that the large range and low elevation from station 7 compromised three station solutions, so trajectory and orbital information is based on the geometric solution using stations 3 and 6 alone. We note that solutions including station 7 showed strong systematic residuals, although the end height was still within 1 km of our two-station solution and the radiant within 1.1°. Tables 5 and 6 show the geometric and orbital solution for this Taurid. The standard deviation of the residuals for this two station solution is approximately 50 m.

Table 5. The atmospheric trajectory and physical data for SOMN 101031 based on all-sky camera solutions. Geographic coordinates are referenced to the WGS84 geoid. Log L is the equivalent 0mag-sec total energy of the entire light curve in the photographic bandpass (see Halliday et al. 1996 for a full description)
 BeginningEnd
Height (km)88.8 ± 0.235.2 ± 0.1
Velocity (km s−1)31.9 ± 0.49 ± 1
Latitude (N)43.554° ± 0.001°43.794° ± 0.002°
Longitude (W)79.886° ± 0.001°80.092° ± 0.001°
Radiant zenith angle30.72° ± 0.15°
Azimuth of radiant147.94° ±0.22°
Trail length/duration62 km/2.3 s
Peak brightness (Mpan)−10.8
Log L4.1
m model ~2 kg
m ph ~4 kg
Table 6. Heliocentric orbit for SOMN 101031. The NTA and STA shower mean orbital elements from Brown et al. (2010) are shown for comparison referenced to the same epoch. Here, V is the estimated speed at the top of the atmosphere, VG is the geocentric velocity (i.e., the speed it would have relative to a massless Earth), (αG, δG) are the right ascension and declination of the geocentric radiant (corrected for zenithal attraction, diurnal aberration, etc.), a is the semi-major axis of the orbit, e the orbital eccentricity, i is the orbital inclination, ω is the argument of perihelion, Ω the longitude of the ascending node, q the perihelion distance, and Q the aphelion distance. All angular coordinates are referenced to J2000.0
 SOMN 101031NTASTA
V 31.9 ± 0.4 km s−130.630.2
V G 29.8 ± 0.4 km s−128.328.0
α G 46.2 ± 0.2°47.6°48.5
δ G 15.5 ± 0.1°17.4°14.4
a 2.92 ± 0.21 AU2.12
e 0.878 ± 0.010.830.82
i 2.2 ± 0.2°0.4°4
ω 111.8 ± 0.3°116°122
Ω 37.52737.5°16
q 0.355 ± 0.004 AU0.3440.35
Q 5.48 ± 0.42 AU3.83.7

The most remarkable aspect of this fireball is its low end height (35.2 km) and relatively low terminal velocity (average of 9 ± 1 km s−1 across both stations) at the end of luminous flight. This is among the very lowest (if not the lowest) end height and end velocity reported for any major shower meteor. These observed end point/terminal values are just barely within the range of possible meteorite-producing events based on our earlier adopted criteria of <35 km height and <10 km s−1 velocity.

Figure 5 shows the lightcurve from station 3 for this Taurid. The inferred photometric mass assuming a 4500 K blackbody following Ceplecha et al. (1998) and using a fixed color term of −2 between our bandpass (roughly V) and the photographic (McCrosky 1968) together with the luminous efficiency of Ceplecha and McCrosky (1976) results in mph approximately 4 kg and corresponding PE = −4.4, suggesting a type I fireball. To better refine this estimate, we use the lightcurve and observed speed together with the luminous efficiency values adopted in Ceplecha and ReVelle (2005) to apply their FM entry model to attempt a numerical model match to the atmospheric dynamics and ablation behavior of this Taurid. Note that we assume single body ablation of a sphere and adopt a bulk density of 2200 kg m−3 (comparable to carbonaceous chondrites) and average ablation coefficient for type I fireballs (Ceplecha et al. 1998) of 0.014 s2 km−2. The ablation and shape density coefficient (0.72) were kept constant throughout the flight. The best fit to the lightcurve and dynamics are shown in Figs. 6a and 6b. This solution corresponds to an initial mass of approximately 2 kg and an initial velocity of 31.9 km s−1. The residual mass in this model is of the order of <10 g, suggesting that while a meteorite may have been produced, the mass would be quite small. We note that the lightcurve also suggests that some fragmentation may have occurred, but we have not attempted to model this aspect of the event. As the deceleration is rather poorly defined and the luminous efficiency is likely uncertain to a factor of approximately 2, the initial mass estimate could easily be changed by a factor of 2 and still reasonably fit the coarse observations.

Figure 5.

Lightcurve of SOMN 101031 derived from station 3 data. The absolute magnitude is in the photographic bandpass as discussed in the text.

Figure 6.

Application of the fragmentation model (FM) (Ceplecha and ReVelle 2005) to SOMN 101031 showing the model fit to (a) the observed speed and (b) the lightcurve.

In summary, fireball SOMN 101031 meets our criteria for association with the Taurid stream, and exhibited most of the main quantitative characteristics expected (although not necessarily sufficient) for a meteorite-producing fireball: it decelerated to below 10 km s−1; terminated at an end-height of 35 km; and showed a small, terminal mass.

Spanish Fireball—November 17, 1995

This fireball was recorded photographically at three stations in Spain on November 17, 1995. The basic information on the event is given by Betlem and Spurny (1998). The fireball reached a peak magnitude of −12 to −15, a terminal speed of 7 km s−1 and a terminal height of 29.5 km. It was identified by Betlem and Spurny (1998) as likely a type I fireball based on the low apparent ablation coefficient and low end height. Although no formal estimate of the dynamic mass/terminal mass is provided and no photometric mass could be calculated, it seems very probable that this event produced meteorites as it almost certainly met all our criteria for a meteorite-producing fireball outlined earlier.

The main uncertainty with including this fireball in a discussion of the Taurids is the ambiguity in identification of the event with the Taurid stream. The fireball occurs just inside our adopted activity and velocity interval for the Taurids (see Table 3), but the radiant is just more than 5° outside our nominal NTA or STA locations, which would formally remove it from our selection as a probable Taurid. As noted by Wetherill (1974), the Taurids occur in a region of orbital phase space where many asteroidal bodies are also present, so contamination by asteroidal meteoroids unrelated to the stream is a significant issue; this was a major driver for our conservative strategy in identifying Taurids at the outset. Nevertheless, although it falls formally outside our conservative boundary for the Taurid stream, it is very possible that this is a Taurid fireball and as such would be even more convincing than the SOMN 101031 event that Taurids produce meteorites. The event is also noteworthy for our discussion if for no other reason than it appears to have produced meteorites and had an initial velocity near 33 km s−1—almost as high as the Geminid shower.

The Maribo Meteorite Fall

The Maribo meteorite fall occurred on January 17, 2009 in Denmark (Haack et al. 2010). The recovered meteorite was 25 g and has been classified as a CM2 carbonaceous chondrite (Haack et al. 2012). CM2 chondrites have bulk porosities of 15–20%, among the highest of all meteorites (Britt and Consolmagno 2003). We therefore expect CM2 meteorites to be among the weakest, as porosity and tensile strength are inversely related (e.g., Greenberg et al. 1995). The associated fireball was recorded by video systems, and its radar head echo was directly detected by a meteor radar allowing a precise speed estimate (Haack et al. 2010). The entry velocity measured with radar was found to be 28.5 km s−1. Video records show an end height for the fireball below 30 km altitude. From cosmogenic nuclide measurement and modeling, the initial pre-atmospheric mass has been conservatively estimated to be 9–74 kg (Haack et al. 2012).

Although the detailed records of this event have not yet been published, the preliminary results offer tantalizing evidence supporting the notion that fireballs with Taurid-like velocities can produce meteorites. More remarkable is the fact that the recovered material is of a relatively friable CM2 composition. If the detailed fireball metric analysis can convincingly show that the entry velocity truly was above 28 km s−1, this fireball demonstrates the very real prospect of meteorites from a meteor shower like the Taurids. This would seem to be the case, provided some stronger material is present in the stream, as seems to be indicated by the identification of several type II fireballs from the literature and the two previous case examples, which are potential type I Taurid fireballs.

Discussion

In addition to the above evidence suggesting that survival of meteorites from Taurid fireballs is possible, recently, Shrbený and Spurný (2012) examined many previously unpublished Taurid records from the European Network and note at least one Taurid fireball with a terminal height of 30 km. Although the end velocity is not given, this result again points to the probable survival of meteorites from Taurid fireballs.

The largest observed meteoroid in the Taurid stream from among the 33 published Taurid fireballs (see the Properties of Taurid Fireballs section) is approximately 15 kg. Spurný (unpublished data) describes Taurid fireballs during the 1995 outburst with initial masses of 100–200 kg. Using the same FM model (Ceplecha and ReVelle 2005) and ablation/flight characteristics as applied to SOMN 101031, Figs. 7 and 8 show the expected observed terminal masses, speeds, peak brightness, and end heights for SOMN 101031-like Taurid fireballs having initial masses from 1 to 200 kg.

Figure 7.

FM model estimate of terminal mass and terminal height, defined as the point where the model predicted a terminal brightness below –2.5, approximately the detection limit of most all-sky camera systems. All other parameters are the same as used for Fig. 6.

Figure 8.

FM model estimate of terminal velocity, defined as the point where the model predicted a terminal brightness below –2.5, approximately the detection limit of most all-sky camera systems and peak absolute magnitude.

We note that these models are approximate and do not explicitly take into account fragmentation. Nevertheless, these theoretical results suggest that at Taurid velocities, bodies at the high end of the strength scale could be expected to produce macroscopic meteorites. We have here previously shown that there exists a wide spread in Taurid meteorite physical strengths and have presented details of two probable Taurids, which are likely type I (strong chondritic) fireballs. We emphasize that there is little physical meaning to the concept of an “average” Taurid meteoroid. Each meteoroid is quite different, a broad conclusion applicable to most meteoroid populations recently emphasized by the unusually strong Carancas ordinary chondrite meteorite fall (Borovička and Spurný 2008).

Thus, it appears that a small fraction of Taurids have nearly ordinary chondrite-like strength. We may roughly estimate this fraction by examining the SOMN archive of Taurid fireballs, consisting of events with peak absolute magnitude greater than roughly −4, corresponding to Taurid's with mass >10 g detected between 2007 and 2011. We find that SOMN 101031 is by far the deepest penetrating/lowest terminal velocity Taurid fireball in the SOMN data set (and nominally type I, although it is on the border with type II), which consisted of 116 Taurid fireballs in total. The next deepest penetrating fireball ended at 52 km height with an end velocity >20 km s−1, while the average end height was close to 75 km. SOMN 101031 was also the brightest/most massive Taurid meteoroid in our SOMN data set, so comparison is difficult. Crudely, the SOMN numbers suggest that approximately a few percent of Taurid fireballs are strong enough (type I material) to produce meteorites. This is consistent with our examination of 33 larger Taurid fireballs from literature data where only MORP 973 is plausibly just barely a type I fireball and all other Taurids are either type II or type IIIa/b. As the SOMN estimate is not normalized by mass, it should therefore be considered a lower limit when applied to kilogram-sized Taurids.

One immediate test of our thesis would be more aggressive attempts to recover material associated with Taurid fireballs from camera network observations. Historically, only the most energetic fireballs have had significant ground recovery efforts (e.g., Wetherill and ReVelle 1981) and this has seriously biased the type of material being recovered from meteorite-producing fireballs. Moreover, the recent proliferation of camera networks covering larger areas, with many more stations (e.g., Trigo-Rodríguez et al. 2004; Sonotaco 2009; Brown et al. 2010b; Tóth et al. 2011) makes it more probable that modest fireballs (like SOMN 01031) will occur very close to a camera station providing high fidelity astrometry near the endpoint as an aid in narrowing down search areas on the ground. Detection of actual meteoritic debris from fireballs using Doppler weather radars (Fries and Fries 2010) has become common, further emphasizing the utility in follow-up of every fireball event with adequate end height/terminal velocity combinations, independent of the event energy. Finally, the much more intensive spirit of meteorite search and recovery now permeating the ranks of amateur meteorite collectors is potentially the greatest single factor that may aid in finding smaller meteorite masses from modest fireballs, precisely the sort of situation which led to the Maribo meteorite being recovered. In this regard, early and public, widespread dissemination of potential fall locations estimated from observations by camera network fireballs may be the most efficient means to guarantee recovery for smaller meteorites.

We note parenthetically that while our focus has been the Taurid stream, the next most probable major shower that might (in very rare instances) produce meteoritic material are the Geminids. Compared with the Taurids, the Geminids are even stronger material, and as evidenced by the Spanish (possible) Taurid fireball (with an initial velocity only 2 km s−1 less than the Geminids), large, strong meteoroids in the Geminid stream might be expected to deposit small meteorites. Mitigating against meteorite production from the Geminids is a lack of evidence for extremely massive objects in the stream and the lack of any recorded Geminid fireball reported in the literature meeting even our very broad meteorite-producing criteria (e.g., Halliday 1988). Unlike the Taurids, however, a recovered Geminid meteorite would not probably be confused with background sporadic contamination and hence linkable to 3200 Phaethon with near certainty.

A final caveat to our study is to acknowledge that following the work of Valsecchi et al. (1995), some contamination of the Taurid orbital phase space by Main Belt interlopers is possible. This is also reflected by the fact that the Taurid radiant is buried within the antihelion sporadic meteor source (Stohl 1986), and hence a recovered meteorite with an orbit consistent with the Taurids is a necessary but probably not sufficient condition to conclude with absolute confidence that such a meteorite is genetically related to 2P/Encke. While we have attempted to mitigate this issue by adopting strict selection criteria for Taurid fireballs, we cannot rule out contamination from chondritic material emanating from the Main Belt. Recovery of meteorites from well-measured Taurid-like orbits followed by detailed dynamical studies of their individual past orbital evolution would be needed to provide greater confidence in an association with 2P/Encke. One circumstance where much stronger confidence could be placed on a fireball linkage to the Taurid stream would be during the occurrence of a Taurid “outburst,” such as documented by Spurný (unpublished data) in 1995. From an extension of the Asher et al. (1993) model of the Taurid swarm, which has been quite successful at predicting recent enhanced Taurid activity as shown by Dubietis and Arlt (2007), the next years for which enhanced Taurid fireball activity can be expected are 2015, 2022, and particularly 2032. Observations of the Taurids in these years may provide the best possibility of meteorite recovery with clear Taurid provenance.

Conclusions

We suggest that meteorites may be recoverable from Taurid fireballs. A summary of our lines of evidence includes the following:

  1. Taurid fireballs show a range of strengths, including a fraction on the order of a few percent of some strong material bordering on the strength expected from ordinary chondrite (type I)-associated fireballs.
  2. Some Taurid fireballs have been recorded with initial masses approaching or exceeding 100 kg (Spurný, unpublished data), potentially sufficient for survival of subkilogram-sized fragments.
  3. Two instrumentally documented events (SOMN 101031 and the 1995 Spanish [possible] Taurid fireballs) explicitly show the end height and terminal velocity characteristics expected of meteorite-producing fireballs.
  4. The Maribo meteorite fall appears to have been derived from a fireball with an entry velocity of 28.5 km s−1. It produced at least one fragment (mass 26 g) and had an initial mass of order a few tens of kilograms.
  5. Application of a numerical entry model predicts that small surviving masses may be associated with meteoroids having entry velocities up to 30 km s−1. Variations in ablation behavior and meteoroid strength might easily increase the meteorite survival mass in rare cases by factors of several.

Acknowledgments

We thank J. Borovička and Z. Ceplecha for use of their entry modeling and meteor trajectory-solving software. All authors thank the NASA Meteoroid Environment Office for funding support under co-operative agreement NNX11AB76A. PGB thanks the Canadian Natural Sciences and Engineering Research Council and Canada Research Chairs program for additional funding support. Helpful reviews by J. Borovička and J. Toth of an earlier version of this work greatly improved the manuscript.

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