2.1. Ballistic coefficient

The ballistic coefficient, $\alpha$ , represents the ratio between the aerodynamic drag and weight forces experienced by a falling meteoroid. If $\alpha$ is high for a particular object, then that object experiences high drag relative to its weight, and is rapidly decelerated by the dense lower layers of Earth’s atmosphere. If the value of $\alpha$ is low, the drag force is comparatively low, and atmospheric penetration is enhanced. Its formal definition is as follows:

(1) \begin{equation} \alpha = \frac{c_d \rho_0 H_0 S_e}{2 M_e \sin(\gamma)}\end{equation}

Here $c_d$ refers to the aerodynamic drag coefficient. It is typically assumed to be unity, which represents a spherical body in a hypersonic airflow regime. Next, $\rho_0$ and $H_0$ are respectively the critical density (1.29 kg/m $^3$ ) and critical altitude (7 160 m) of an exponential atmosphere. The variables $S_e$ and $M_e$ are respectively the cross-sectional area and mass of the meteoroid before atmospheric entry. Area and mass can be calculated in tandem if a particular shape and density are assumed. For instance, a modified formula that uses meteoroid density to calculate initial mass has been published by Sansom et al. (Reference Sansom2019), assuming a spherical shape. Finally, $\gamma$ is the angle between the meteoroid’s trajectory and the local horizon at the point of entry.

2.2. Mass loss parameter

The mass loss parameter, $\beta$ , is the ratio between the fraction of a meteoroid’s kinetic energy that is converted into heat, and its effective enthalpy of vaporisation (Gritsevich & Popelenskaya Reference Gritsevich and Popelenskaya2008). A high $\beta$ value indicates a meteoroid that rapidly loses mass due to ablation and fragmentation processes. Ablation is the continuous process by which matter at the surface is melted, vaporised, ionised and stripped away by the surrounding atmosphere. Fragmentation refers to discrete breakup events that are detectable as luminous flares, as well as subtle increases in deceleration. The meteoroids that are most likely to survive their descent long enough to deposit meteorites yield low $\beta$ values and experience a degree of mass conservation. The mass loss parameter is defined by this equation:

(2) \begin{equation} \beta = \frac{(1 - \mu) \sigma_b V_e^2}{2}\end{equation}

Here $\mu$ is the rotational shape factor, which is typically held constant throughout a meteoroid’s descent. It is typically set to $\frac{2}{3}$ , which corresponds with an object that is rotating rapidly enough for its entire surface to be evenly ablated. The bulk ablation coefficient $\sigma_b$ indicates the amount of mass that is removed per unit of heat energy transferred (see Section 2.4). Finally, $V_e$ refers to the earliest observed velocity of the meteoroid. High velocity results in higher dynamic (‘ram’) pressure across the leading face of the meteoroid, which accelerates mass loss (McMullan et al. Reference McMullan2024).

2.3. Calculation of dynamic parameters

For a given meteorite fall event, $\alpha$ and $\beta$ are traditionally determined analytically using normalised velocity (v) and normalised altitude (y) data (Figure 1). For GFO events, velocity is normalised relative to entry velocity, as determined by an Extended Kalman Smoother (EKS; Sansom et al. Reference Sansom, Bland, Rutten, Paxman and Towner2016). Altitude data are normalised following the method of Lyytinen & Gritsevich (Reference Lyytinen and Gritsevich2016), which allows NRLMSISE-00 (see Picone et al. Reference Picone, Hedin, Drob and Aikin2002) or an equivalent ‘real’ atmospheric model to be incorporated for fall site prediction purposes. We prefer a ‘real’ atmosphere when possible; differences between our results and those of earlier literature (e.g. Sansom et al. Reference Sansom2019) can be attributed to differences in atmospheric models.

A curve is fitted to the normalised velocity versus altitude plot using a global minimisation function. The curve includes an inflection point whose position is determined by $\alpha$ . The slope of the curve below this inflection point is controlled by $\beta$ . Figure 1 includes the deceleration curve fitted to trajectory data for Arpu Kuilpu, an ordinary chondrite tracked and recovered by the GFO (Shober et al. Reference Shober2022).

An advantage of this method is that it requires no assumptions regarding the shape, size, density or rotation of the falling meteoroid. It is only when deriving other parameters from $\alpha$ or $\beta$ , such as the preatmospheric mass or bulk ablation coefficient, that such assumptions become necessary (Gritsevich Reference Gritsevich2009). The $\alpha$ - $\beta$ method provides a means of characterising objects captured by fireball networks, whether or not they survive descent. Sansom et al. (Reference Sansom2019) removed the dependence of $\alpha$ upon trajectory slope to eliminate a potential bias of entry angles first identified by Gritsevich et al. (Reference Gritsevich, Stulov and Turchak2012). This is achieved by plotting $\alpha \sin \gamma$ , where $\gamma$ is the entry angle measured from the local horizon. This method was applied to 273 fireballs (Sansom et al. Reference Sansom2019), including three from which meteorites have been recovered, to generate a ln( $\alpha \sin \gamma$ ) versus ln( $\beta$ ) plot and determine probabilistic thresholds for meteorite deposition (‘survivability curves’). Both studies agree that meteorite falls tend to yield low ballistic coefficients and low mass loss parameters when compared with the majority of fireball-producing meteoroids.

2.4. Ablation coefficient

Once $\alpha$ and $\beta$ are known, their defining equations can be rearranged and solved to determine linked parameters, given reasonable assumptions of shape, size, density and rotation. For our purposes, the most pertinent material parameter is the bulk ablation coefficient, $\sigma_b$ . It can be estimated by rearranging the equation for $\beta$ , then substituting a rotational shape factor ( $\mu$ ) and earliest observed velocity ( $V_e$ ):

(3) \begin{equation} \sigma_b = \frac{2 \beta}{(1 - \mu) V_e^2}\end{equation}

An exact value for $\mu$ can be derived from light curve analysis, as proposed by Gritsevich & Koschny (Reference Gritsevich and Koschny2011). When a sufficiently detailed light curve is not available, we assume $\mu = \frac{2}{3}$ to indicate uniform ablation across the entire surface.

Ablation coefficients are sometimes considered diagnostic of particular meteoroid types (Ceplecha Reference Ceplecha1975), and are expressed in either kg MJ $^{-1}$ or s $^2$ km $^{-2}$ ; these units are equivalent. We will provide $\sigma_b$ units as kg MJ $^{-1}$ .

2.4.2 Calculated bulk ablation coefficients for chondrite falls

In addition to GFO data, we have independently calculated mass loss parameters and bulk ablation coefficients for a global dataset of 36 meteorites with resolved atmospheric trajectories (Appendix B) using the traditional analytical fit method. Of the fallen meteorites, 35 are chondritic. As shown in Figure 2, $\sigma_b$ is highly variable among all chondrite types. The H-chondrite Košice (Kuznetsova et al. Reference Kuznetsova, Gritsevich, Vinnikov, Rault and Roggemans2014) provides a notable outlier. It is believed that this meteoroid produced an airburst and experienced extensive fragmentation throughout its descent, losing comparatively little mass via pure ablation (Tóth et al. Reference Tóth2015; Moilanen et al. Reference Moilanen, Gritsevich and Lyytinen2021). The majority of chondrites tend to break up in two distinct stages as they encounter dynamic pressure thresholds on the order of 0.1 and 1 MPa (Borovička et al. Reference Borovička, Spurný and Shrbený2020).

Figure 2.

Bulk ablation coefficients calculated for a selection of instrumentally observed meteorite falls, assuming uniform ablation ( $\mu = \frac{2}{3}$ ). $^*$ indicates a provisional name. Previously published values are shown for stony and carbonaceous $\sigma_b$ from Ceplecha (Reference Ceplecha and Milani1994), for iron $\sigma_i$ from ReVelle & Ceplecha (Reference ReVelle and Ceplecha1994), for stony $\sigma_i$ from Ceplecha & ReVelle (Reference Ceplecha and ReVelle2005) and for carbonaceous $\sigma_i$ from Ceplecha (Reference Ceplecha2007).

Of the three ordinary chondrite classes, H-chondrites tend to experience the lowest degree of mass loss, which is reflected by their mean $\sigma_b$ of 0.0236 kg MJ $^{-1}$ (s.d. 0.0099 kg MJ $^{-1}$ ), excluding Košice. Based on averages, LL-chondrites (mean 0.0262 kg MJ $^{-1}$ , s.d. 0.0063 kg MJ $^{-1}$ ) and L-chondrites (mean 0.0289 kg MJ $^{-1}$ , s.d. 0.0133 kg MJ $^{-1}$ ) are progressively less resistant to mass loss when compared with H-chondrites. Among all three ordinary chondrite groups, median $\sigma_b$ is taken as 0.0250 kg MJ $^{-1}$ , which will be employed in our analysis of minimally decelerating asteroidal bodies (Section 3).

The friable carbonaceous chondrites Maribo and Tagish Lake produce $\sigma_b$ values of 0.0430 and 0.0540 kg MJ $^{-1}$ respectively. They fall within the upper range of possible values for ordinary chondrites. Winchcombe provides the lowest carbonaceous value known to date, 0.0276 kg MJ $^{-1}$ . This meteorite experienced relatively low dynamic pressures due to its slow entry velocity, which reduced the rate of mass loss and spared it from total destruction (McMullan et al. Reference McMullan2024).

The enstatite chondrites Raja (EH3) and Neuschwanstein (EL6) yield 0.0296 and 0.0353 kg MJ $^{-1}$ respectively. Neuschwanstein experienced a singular fragmentation event at the lower end of its trajectory, producing several large fragments of high mechanical strength (Oberst et al. Reference Oberst, Heinlein, Köhler and Spurný2004). The slightly lower value obtained for Raja implies a less destructive fragmentation regime and/or a difference in ablation behaviour due to high metal content in comparison with Neuschwanstein.

Particularly astute readers may note that our $\sigma_b$ value for Neuschwanstein is exactly three times that calculated by Gritsevich (Reference Gritsevich2008 b). This is because, whilst we have assumed uniform ablation across all surfaces due to meteoroid rotation, the earlier study assumed no rotation ( $\mu$ = 0) and produced $\sigma_b$ = 0.0117 kg MJ $^{-1}$ for the same entry velocity. In conjunction, these two approaches yield the full range of possible bulk ablation coefficients for a given fall. The values illustrated in Figure 2 are maxima; the corresponding minima are one-third in magnitude.

Our $\sigma_b$ calculations for chondrites are higher than the $\sigma_i$ values presented by Ceplecha et al. (Reference Ceplecha and Borovička1998) for equivalent meteoroid types, but generally agree with more recent models and plasma wind tunnel experiments (e.g. Matlovič et al. Reference Matlovič2024). It is clear that $\sigma_b$ can vary significantly among meteorites belonging to the same compositional group (Figure 2). We attribute this variation to differences in fragmentation and ablation behaviour primarily caused by differing sizes, shapes, entry angles and atmospheric conditions. Consequently, it is difficult to select a single bulk ablation coefficient to represent all meteoroids of the same type for the purposes of dynamic trajectory analysis. Accurate characterisation of fallen meteorites requires that each fall be considered independently of the others.

2.4.4. Ablation coefficients for meteor shower bodies

Ablation coefficients for meteor shower bodies (MSBs) have historically been difficult to determine due to material fragility and frequent lack of visible deceleration. Several MSBs captured by the GFO do show significant deceleration, allowing $\alpha$ and $\beta$ to be determined using the traditional method. $\sigma_b$ has been derived for these objects using Equation (3) and assuming uniform ablation ( $\mu=\frac{2}{3}$ ), which is the most common situation according to Bouquet et al. (Reference Bouquet, Baratoux, Vaubaillon, Gritsevich, Mimoun, Mousis and Bouley2014).

Table 1 displays mean $\sigma_b$ values for significantly decelerating objects associated with meteor showers for which multiple such objects have been detected. MSB $\sigma_i$ values are available in Buccongello et al. (Reference Buccongello, Brown, Vida and Pinhas2024); we recommend consulting their paper for a thorough review of meteor shower properties and provenance.

Table 1.

Ablation coefficients of significantly decelerating objects associated with selected meteor showers. Bulk coefficients ( $\sigma_b$ ) calculated from GFO detections between 2014 and 2024. Intrinsic coefficients ( $\sigma_i$ ) sourced from Buccongello et al. (Reference Buccongello, Brown, Vida and Pinhas2024). s.d.: standard deviation; n: number of objects.

Table 1. Long description

A table comparing ablation coefficients of meteor showers with their likely parents and standard deviations. The table has 8 rows and 6 columns. Column headers are Meteor shower, Likely parent, σb (kg MJ^-1), σi (kg MJ^-1), and n. Row 1: Geminids, 3200 Phaethon, 0.020; s.d. 0.011, 0.028; s.d. 0.002, n = 15, n = 5. Row 2: α-Capricornids, 169P/NEAT, 0.026; s.d. 0.014, 0.017; s.d. 0.004, n = 3, n = 3. Row 3: η-Virginids, 2003 FB5, 0.031; s.d. 0.018, , n = 2. Row 4: October α-Virginids, 1998 SH2, 0.015; s.d. 0.001, , n = 2. Row 5: Northern Taurids, 2004 TG10, 0.049; s.d. 0.028, 0.027; s.d. 0.002, n = 5, n = 2. Row 6: Southern Taurids, 2P/Encke, 0.050; s.d. 0.027, 0.027; s.d. 0.002, n = 6, n = 7. Row 7: Other showers, , 0.043; s.d. 0.029, , n = 12.

For the Geminids, the mean value of $\sigma_b$ is lower than that of $\sigma_i$ , contrary to the pattern observed for stony meteorite falls and objects associated with the other meteor showers in Table 1. Objects originating from the Geminids shower are believed to approximate carbonaceous chondrites in bulk composition (Babadzhanov Reference Babadzhanov2002; Jewitt & Li Reference Jewitt and Li2010). There are numerous theorised differences between these and meteorite progenitors that could contribute to the apparent reduction of $\sigma_b$ compared to $\sigma_i$ , such as smaller sizes and densities, higher porosities, and faster entry velocities. A deeper discussion of MSB structure lies beyond the scope of this study.

2.5. Results of decelerating meteoroid types

The reduced GFO 2014–2024 dataset contains 1 118 events that showed sufficient deceleration for the standard $\alpha$ - $\beta$ method to be applied. Of these, 45 are associated with meteor showers and another 16 with fall trajectories of recovered meteorites from participating nations (https://dfn.gfo.rocks/meteorites.html). The remainder have been analysed and classified as either asteroidal or cometary on the basis of orbital properties (Shober et al. Reference Shober2021). They include many suspected meteorite droppers from which solid fragments have not yet been recovered. Since all of these objects experienced a significant degree of deceleration, their $\alpha$ and $\beta$ values have been calculated using the method described in Section 2.3. They are displayed in Figure 3, an update of Figure 3 from Sansom et al. (Reference Sansom2019). We have adopted the convention proposed by the same study to remove the influence of entry angle by plotting $\ln(\alpha \sin\gamma)$ instead of $\ln(\alpha)$ . This removes potential bias against objects with a comparatively shallow entry angle and consequently long flight time. We also include the survivability curves devised by Sansom et al. (Reference Sansom2019) for a terminal mass of 50 g or more. Figure 3 lends credibility to these survivability curves, showing recovered meteorites are indeed only found in the ‘Likely’ and ‘Possible’ fall zones.

Figure 3.

Ballistic coefficient (with slope dependence removed, $\ln(\alpha \sin\gamma)$ ) versus logarithmic mass loss parameter ( $\ln\!(\beta)$ ) for significantly decelerating meteoroids captured by the GFO between March 2014 and December 2024. Non-GFO meteorites Chelyabinsk, Košice, Tagish Lake and Ådalen* are included for comparison. Survivability curves of Sansom et al. (Reference Sansom2019) are given for 50 g meteorites. *indicates a provisional name. Note that ‘asteroidal’ versus ‘cometary’ classifications are defined by orbital parameters only.

Figure 3. Long description

A scatter plot representing the relationship between the ballistic coefficient (with slope dependence removed) and the logarithmic mass loss parameter for significantly decelerating meteoroids captured by the GFO between March 2014 and December 2024. The horizontal axis represents the logarithmic mass loss parameter, while the vertical axis represents the ballistic coefficient. The plot includes data points for recovered meteorites, shower cometary, geminids, sporadic cometary, and asteroidal meteoroids. The plot is divided into three zones: unlikely fall zone, possible fall zone, and likely fall zone. Specific meteorites such as Chelyabinsk, Košice, Tagish Lake, and Ädalen are highlighted. Survivability curves for 50 g meteorites are also shown. The plot indicates clusters and patterns within these zones, with some meteorites falling into the likely fall zone and others into the possible or unlikely fall zones.

3.1. Minimally decelerating objects in the Global Fireball Observatory dataset

The standard $\alpha$ - $\beta$ method relies upon an analytical fit to normalised trajectory data. A reasonable degree of deceleration must be present for the fit to converge upon a reliable solution, such that the final observed velocity of an object is less than 80% of its entry velocity. If this is not the case, we consider it a ‘minimally decelerating object’ (MDO) upon which the standard method is unreliable. Figure 1 illustrates the practicality of fitting a curve to a trajectory characterised by ‘significant’ versus ‘minimal’ deceleration, even when smoothed data are employed. After a meteoroid has slowed to below 80% of its entry velocity, as in the case of Arpu Kuilpu, the inflection point in its trajectory is fully visible and the slope of the lower portion can be tightly constrained. By contrast, if a meteoroid does not reach the 80% threshold, then the inflection point cannot be fully resolved and the lower portion of the trajectory cannot be constrained with any degree of accuracy. This is illustrated by the iron meteoroid DN240406_02 (Figure 1).

Over one-third of all fireballs tracked by the GFO can be considered MDOs. Of the 671 MDO falls, 418 represent sporadic impactors, and 253 are associated with known meteor showers. We expect the majority of MDOs to be small, sub-cm bodies that are completely vaporised at relatively high altitudes. Total vaporisation is typical of fast-moving, poorly consolidated cometary remnants. Since atmospheric density is low at high altitude, these objects are destroyed before the atmosphere can exert significant deceleration upon them. Terminal altitudes ( $H_t$ ) and velocities ( $V_t$ ) are always high, and terminal masses ( $M_t$ ) are always zero.

3.2. Novel formulae for the ballistic coefficient and mass loss parameter

To better characterise MDOs, the following logarithmic equation from Gritsevich & Popelenskaya (Reference Gritsevich and Popelenskaya2008) is employed to estimate $\alpha$ and $\beta$ , wherein the variable $y_t$ represents the normalised terminal altitude. This formula was derived from fundamental meteor physics equations under the assumption of zero terminal mass ( $M_t = 0$ ). It effectively states that the terminal altitude of a totally vaporised meteoroid is proportional to the product of its ballistic coefficient and mass loss parameter:

(4) \begin{equation} y_t = \ln\!(2 \alpha \beta)\end{equation}

Moreno-Ibáñez et al. (Reference Moreno-Ibáñez, Gritsevich and Trigo-Rodríguez2015) validated this formula by applying it to small meteoroids detected by MORP (Halliday et al. Reference Halliday, Griffin and Blackwell1996; Campbell-Brown & Hildebrand Reference Campbell-Brown, Hildebrand, Hawkes, Mann and Brown2005). We have applied the same formula to meteoroids within the GFO dataset and found the same correlation displayed in Figure 1 of Moreno-Ibáñez et al. (Reference Moreno-Ibáñez, Gritsevich and Trigo-Rodríguez2015). Interestingly, this statement holds true for many past events in which a non-zero terminal mass was deposited. Figure 4 compares observed terminal altitudes with those predicted by Equation (4) for GFO decelerating bodies with known compositions and independently calculated $\alpha$ and $\beta$ values. Both the recovered meteorites and vaporised MSBs fall within a 10% error margin of the 1:1 equivalency line. Note the position of the 2024 Iberian Superbolide (Peña-Asensio et al. Reference Peña-Asensio, Grèbol-Tomàs, Trigo-Rodríguez, Ramírez-Moreta and Kresken2024), which falls exactly on the equivalency line and is believed to have been totally reduced to dust.

Figure 4.

Observed versus predicted terminal altitudes for GFO-detected meteoroids of known composition showing significant deceleration. Predicted terminal altitudes are calculated using $H_0 \ln \!(2 \alpha \beta)$ , where $H_0$ is the standard altitude normalisation factor of 7.16 km. Chelyabinsk, Tagish Lake and the 2024 Iberian Superbolide are included for comparison.

As demonstrated by Figure 4, Equation (4) is appropriate for relatively small meteoroids. It does not hold for objects with large preatmospheric masses, including the airburst-producing meteoroids Chelyabinsk (Trigo-Rodríguez et al. Reference Trigo-Rodríguez, Dergham, Gritsevich, Lyytinen, Silber and Williams2021) and Tagish Lake (Ceplecha Reference Ceplecha2007). We are confident that it provides reasonable $\alpha$ and $\beta$ estimates for MDOs, which tend to be much smaller than meteorite droppers and airburst generators.

Further derivation (see Appendix A for full details) produces the following formulae for $\alpha$ and $\beta$ , given the additional assumption of uniform ablation across the entire meteoroid surface:

(5) \begin{equation} \alpha = \frac{3 e^{y_t}}{\sigma_b V_e^2}\end{equation}

(6) \begin{equation} \beta = \frac{e^{y_t}}{2 \alpha} = \frac{\sigma_b V_e^2}{6}\end{equation}

Note that the value of $\beta$ depends upon the value of $\alpha$ , which in turn depends upon $\sigma_b$ , the bulk ablation coefficient. For any given value of $\sigma_b$ , both $\alpha$ and $\beta$ can be derived under the assumption of zero terminal mass. The better $\sigma_b$ values are constrained, the more accurate our dynamic characterisation of MDOs becomes. 253 of our MDOs are associated with meteor showers, thus we have used the $\sigma_b$ values recorded in Table 1 to estimate their dynamic parameters. The 418 sporadic MDOs have been classified as asteroidal or cometary on the basis of their calculated preatmospheric orbital elements, namely Tisserand’s parameter and eccentricity (Tancredi Reference Tancredi2014). Asteroidal MDOs have been assigned a bulk ablation coefficient of 0.025 kg MJ $^{-1}$ ; the median value for chondritic meteorites. For sporadic cometary MDOs, $\sigma_b$ = 0.050 kg MJ $^{-1}$ . This is the mean bulk value obtained from a subset of 6 significantly decelerating objects with highly elliptical orbits but no shower association.

We focus on a set of 10 MDOs detected over Australia and continental Europe that have been assigned iron compositions, based upon spectra analysed by the AMOS and EFN teams (Vojácek et al. Reference Vojácek and Borovička2015; Rudawska et al. Reference Rudawska, Tóth, Kalmancok and Zigo2016). The spectra have been matched with accurate fall trajectories provided by the GFO, or by Vojácek et al. (Reference Vojácek and Borovička2015, Reference Vojácek and Borovička2019), Borovička et al. (Reference Borovička and Spurný2022 a) for European cases. Entry velocities and terminal altitudes are well-constrained in all 10 cases. Each of these iron meteoroids has an estimated entry mass of 150 g or less, and no surviving terminal mass. We use Equations (5) and (6) to determine minimally decelerating $\alpha$ and $\beta$ for these objects, and in accordance with the work of ReVelle & Ceplecha (Reference ReVelle and Ceplecha1994) and references therein, we assume a melting-dominated ablation coefficient of 0.0743 kg MJ $^{-1}$ .

3.3. Results of minimally decelerating objects

Figure 5 shows the positions of all GFO-detected MDOs in $\alpha$ - $\beta$ space. The position of each object relies on an estimated bulk ablation coefficient, $\sigma_b$ , as explained in the previous section. This visualisation can be used as a tool for understanding meteoroid dynamic properties and the likelihood of meteorite deposition from a given fall event.

Figure 5.

$\ln(\alpha \sin\gamma)$ versus $\ln(\beta)$ for minimally decelerating meteoroids captured by the GFO between March 2014 and December 2024.

Initially, this study hoped to identify spatial variation among these data that could be attributed to variation in material properties, namely density. To investigate this hypothesis, MDOs of known composition can be used to test whether or not meteoroids’ distribution in $\alpha$ - $\beta$ space depends upon their density. For this purpose, detailed emission spectra are invaluable for determining bulk compositions, as in the case of the 10 AMOS/EFN iron meteoroids highlighted in Figure 6.

Figure 6.

Figure 5 with minimally decelerating iron meteoroids highlighted. The iron meteorite Ådalen is included for comparison.