1 Introduction

Astronomical objects are usually studied through the light collected by telescopes, although, from time to time, we can get a more direct way to study the Universe through free samples that fall on Earth as meteorites. They can bring us information about the primitive solar system and even on the evolution of the Universe (Anders 1971; Amelin and Krot 2007; Bouvier and Wadhwa 2010). For instance, Allende’s meteorite allowed to determine the age of the solar system by use of its isotopic composition (Bouvier et al. 2007). In this sense, meteorites can be thought of as the fossils condensed from the primitive solar nebula. Moreover, they also serve as an indirect mean to study the Earth’s interior (Emiliani 1992) and composition (Allègre et al. 1995).

Meteorites are commonly divided in two major groups: chondrites and achondrites. The former have chondrules, which were formed by the melting and accretion of particles of dust and grit present in the ancient Solar System. These specimens, Allende being one of a kind, bring important clues not only for dating past events but understanding the origin of the Solar System, the synthesis of organic compounds, the origin of life or the presence of water on Earth. The latter, i.e., the achondrites, are those which do not develop chondrules. These consist of igneous or metamorphic rocks (melts, partial melts, melt residues), breccias of igneous rock fragments from differentiated asteroids and planetary bodies (Weisberg et al. 2006) (encompassing martian and lunar meteorites), and iron-type specimens.

On another classification system, meteorites are organized in three types: stony (92.8%), iron (5.7%), and stony-iron (1.5%), (see Ref. Emiliani 1992), where the percentages indicate their corresponding abundance. Here, the type reveals the section of the original celestial object they composed initially: the stony meteorites correspond to the near-to-the-surface zone, the iron ones to the core, and the stony-iron to the intermediate region. In what follows let us discuss in some detail the iron class, which will turn out to be important for our contribution.

The iron meteorites allow us to study not only the deep inside of their parent bodies, but also provide information about the core of the Earth. While currently there is no conceivable manner to reach this zone, the iron and stony- iron meteorites are our only available analogues to materials in the deep interiors of the Earth and other terrestrial planets (Davis 2003). On the other hand, on a historical context these meteorites where the main sources of iron before the man was able to produce this methal (Buchwald 1975). Several cultures used them to make ornamental or tool pieces for work, which are invaluable to museums or collectors.

This class of meteorites is divided into 12 groups, see Table 1 (Scott and Wasson 1975). A meteorite that is out of these groups is considered anomalous. Each category is basically defined by the percentage of nickel, gallium and germanium. In addition, the groups are separated on the basis of systematic variations in their chemical, mineralogical, and structural properties. In general, the specimens are primarily composed of an iron-nickel alloy, where Ni is particularly important; as a rule concentrations smaller than 5% excludes the body from being considered an iron meteorite.

Table 1 Structural and compositional properties of generic groups of iron meteorites
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Iron meteorites are characterized for being the largest so far on Earth (Table 2). In this category we briefly review five that lie at the top of the list. The huge Hoba, leading the list, was found in 1920 in Namibia, Africa. It has a shape similar to a box with approximate dimensions of 2.7 × 2.7 × 0.9 m, and an estimated mass of 60 tons, falling in the group IVB. The second most massive corresponds to a fragment of the Campo del Cielo, the Chaco, found in Argentina in 1967, with a mass of 37 tons (Cabanillas 2006), and belonging to group I. Its dimensions are not reported neither by Buchwald (1975), nor by the Meteoritical Bulletin (http://meteoriticalsociety.org/bulletin/database.html). The third and fourth ones are two specimens of the Cape York meteorites, Ahnighito and Agpalilik, respectively, both of which were found in Greenland and are part of group IIIA. Buchwald mentions the enormous impact the Cape York meteorites had on the populations in the nearby region; the skimos apparently would used the meteorite to producing tools for hunting and fishing. Ahnighito, having a mass of 30.8 tons, dimensions of 3.25 × 2.10 × 1.60 m, volume of 3.8 m3, and a density of 8.1 ± 0.4 g/cm3, whereas Agpalilik exhibits a mass of 20.14 tons, dimensions of 2.1 × 2 × 1.5 m, volume of 2.55 dm3, and a density of 7.9 ± 0.4 g/cm3. The fifth largest according to the Meteoritical Bulletin is Bacubirito. An anomalous meteorite (Gómez and Marquina 2016) found in 1863 in the Northwest of Mexico. It has a mass that has varied wildly among studies from 50 through 19 tons (see Table 3). Its dimensions, though hard to establish due to its complex structure, have been estimated by Buchwald at approximately 4.1 × 1.8 × 0.2 m. Notably different of our calculations.

Table 2 List of the largest meteorites of the world
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Table 3 Reported Bacubirito meteorite mass estimates until year 2001 (see Refs.)
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As we can see, there is a broad range of iron meteorites varying in size and composition. Bacubirito is a notorious specimen that appears among the largest meteorites known (Table 1). In fact, at the time of its finding, it was considered the biggest worldwide (Ward 1902). Nowadays, it is still maintained as the fifth largest. A peculiar feature being that it is categorized as anomalous, a unique body out of any group among the five largest (Grady 2000).

Bacubirito was found at 25°42′05″N, 107°54′19″W—verified on a personal recent expedition (Fig. 1), near to el Camichín, a small village about 10 km away from Bacubirito, located in the northern mountains of Sinaloa, Mexico. Its finding was first presented by Barcena (1876) (see Refs.). It remained from 1959 to 1992 on the outdoors of the museum, Centro Cívico Constitución, located in Culiacán Rosales, the capital city of Sinaloa. Currently, it still remains outdoors but located at the science museum, Centro de Ciencias de Sinaloa (24°49′44″N 107°23′05″W), of the same city.

Fig. 1
figure 1

Ruins of the building where the Bacubirito meteorite was found in 1863 and exhibited until 1959

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As stated formerly, the meteorite has suffered classification changes in size over time and is considered anomalous, which may be due to a lack of detailed investigations. Back in 1975, it was indicated by Buchwald that only few studies had been carried out on the meteorite. Little has been made 40 years later. In particular, there have only been rough estimates of properties such as its mass, dimensions, and densities (see Tables 1, 2, 3), as noted from the large mass variations reported. Moreover, for most of the estimations, uncertainties have not been provided.

In this paper we address the elaboration of a three dimensional model of Bacubirito and from this, accurate measures of geometrical parameters, the mass, and statistics of the regmaglypts are determined.

Indeed, one can ask about the scientific importance of studying the geometry of such a complex object. This can be understood for instance in terms of the paper it plays in the entry dynamics of the meteorite. If we consider the theoretical prediction given in Regan and Anandakrishnan (1993) for the speed, two important variables appear, namely, the drag coefficient C d and cross-sectional area A, that are determined from its geometry. Another important dynamic variable that can be investigated is the mass: using the ablation model in Revelle (1979) and the two parameters just mentioned, one can obtain an estimate of the object mass before entry. Finally, given that the meteorite shape is non-spherical and assuming that it was not greatly modified during entrance, we can use the model and study the drag force for such asymmetrical object as in Leith (1987). The mentioned studies are beyond the scope of the present investigation and we will limit ourselves with the model acquisition and parameters previously cited.

The rest of the paper is organized as follows. Section 2 describes the methodology used to make the model. Section 3 presents the results and discussion. Finally, conclusions are presented in Sect. 4.

2 Methodology

We obtain here a precise geometrical model of the meteorite. We retrieve and discuss some of its important features including geometric parameters, its mass and regmaglypts distribution. The investigation proceeds by first generating the model. Later, we define and calculate the geometric features. An analysis of reported densities to obtain a precise mass estimate is then performed. Last, we report the size and depth statistics of the regmaglypts.

The Bacubirito meteorite has a rather complex shape (see Fig. 1) and large weight. Under these restrictions we have chosen to use a portable scanner for the determination of the model. This instrument allows for a vastly more accurate estimation of the relative positions of surface points on a rigid object and is highly suited from a practical perspective—the outdoor location of the meteorite and it resting on an elevated position are pros here. This device has been used in civil engineering applications, in cases where detailed, precise three-dimensional representations are required. Being a highly sophisticated instrument, it allows us to provide the uncertainty in our results (Fig. 2).

Fig. 2
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Perspectives of the Bacubirito meteorite

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The scanner used was Leica Nova MS50. This equipment radiates and collects a laser beam which can either directly interact with the object to measure or by means of a prism; here the direct-interaction operation mode is mostly employed since it offers a higher resolution and precision to acquire the 3D points. According to an extensive characterization from the manufacturer (see Leica Nova MS50 Datasheet), it provides an angular accuracy of \(\delta \theta = 1^{\prime\prime}\) in horizontal and vertical angular measurements, a linear precision of l = 2 − 3 mm + 2 ppm, collects as much as 1000 pts/s, and can measure objects up to 1 km away. Our measurements were carried out at distances of about 15 m (see Fig. 3) in 5 different instrument positions around the meteorite, trying to reach most spots of its surface. For the inaccessible regions, a prism was employed. We end up with a model with 1,812,875 points on the meteorite surface.

Fig. 3
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Angular and linear deviations of the surface points due to intrinsic measurements uncertainty

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Before proceeding to computing the geometric parameters, here we propose a Monte Carlo simulation to obtain yet more precise positions and evaluate the uncertainty in our results from the single series of measurements—a classical reference to the Monte Carlo method and its applications in many areas is found in Rubinstein and Kroese (2016). With this goal in mind, we first develop a simple probabilistic model and later explain its practical implementation. We start assuming that the uncertainties in our measured positions are random, namely, that our calibrated instrument presents negligible systematic deviations as supported by the small parts per million on its linear accuracy. Furthermore, it is well known that both distance and angle measurements present systematic-free errors that follow a Gaussian distribution. Now, the propagation error in position due to the angular uncertainty is negligible (≪1 mm) for our short separation distances and angular precision, and we conclude that this will follow a Gaussian distribution. From this discussion, the deviation \(\delta \overrightarrow {{r_{i} }}\) of a measured position with respect to the instrument \(\overrightarrow {{r_{i} }}\), can be written as \(\delta \overrightarrow {{r_{i} }} = \lambda \widehat{{r_{i} }}\), with \(\widehat{{r_{i} }}\) a unit vector (see Fig. 2), where the parameter \(\lambda\) is the random variable in the Monte Carlo simulation with a Gaussian distribution centered at zero and \(\sigma = 6{\text{mm}}/2\)—this accounts for both the characterization provided by the manufacturer (see datasheet) and the measurement technique in which the instrument was positioned in different locations with uncertainties lower than 4 mm.

The application of the method consists in taking the i-th position measurement \(\overrightarrow {{r_{i} }}\), then a particular value for \(\lambda\) is generated based on the distribution described above, we then add \(\delta \overrightarrow {{r_{i} }}\) accordingly, and repeat the process for each of the measurements. This amounts for one experiment from which the desired parameters can be extracted. The experiment is repeated many times for a simulation, averaging over the intermediate results leading to a convergence.

We then start determining Bacubirito’s dimensions. We define its length as the separation between the two farthest-apart points. An axis is fixed onto these two points. We next define the width as the distance between the two farthest-away points that are both perpendicular to the axis and lie on a plane containing the axis. Similarly, the thickness is defined as the largest distance between two points that define a line orthogonal to that plane and that passes through the axis. A sketch is shown in Fig. 4.

Fig. 4
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Sketch of the Bacubirito meteorite and its dimensions

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In the next step we estimate the meteorite volume by combining the Monte Carlo simulation with Gauss’s Theorem. For this, a Monte Carlo experiment is first run. Gauss’s Theorem is then used to estimate the volume enclosed by the simulated surface defined by a Delaunay triangulation. The associated volumes are obtained for each simulation run.

We continue with the analysis of the meteorite’s densities reported so far and its mass determination. The percentages of its chemical elements are shown in Table 4. Those values were measured in different samples from Bacubirito. On average, the measurements should be an approximation to the full meteorite’s concentrations and densities. We obtain the average density from the table and with the aid of the volume calculate the mass.

Table 4 Reports of elements concentrations for the Bacubirito meteorite
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The diameter and depth of the regmaglypts were the next determined quantities using the model. We define a regmaglypt’s diameter as that of the best-fit circle for the corresponding pit mouth. The depth is the difference between the bottom and top.

Finally, in order to characterize the distribution of regmaglypts, three zones around the meteorite were selected for presenting contrasting structures as shown in Table 6. The diversity in structures is assessed statistically using the Kolmogorov–Smirnov test—see Durbin (1973) for a thorough and rigorous treatment of the technique. To this end we retrieve diameters and depths for each region from the model and take each as a random variables. The null hypothesis that any two of the regions originate from a same distribution for either variable is then evaluated.

3 Results and Discussion

Renders of our tridimensional model are given in Fig. 5. It can be observed the high level of detail achieved. A precise retrieval of any geometrical parameters and its derivatives is possible. We focus here in the dimensions, mass, and statistics of the regmaglypts.

Fig. 5
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Renders of the Bacubirito meteorite model

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Let us now discuss the basic defining parameters of Bacubirito. The dimensions and their uncertainties are depicted in the last column of Table 5. Contrasting differences are observed with respect to one of the first studies performed by Ward, and the latest one by Buchwald.

Table 5 Geometrical parameters of the Bacubirito meteorite
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While estimates of geometric parameters of Bacubirito had been reported previously, different experiments exhibit noticeable changes. Here we make their formal definitions for consistency among studies and determine them precisely from the model. Once the convention has been established we can confirm the old claim that Bacubirito is the longest meteorite in the world, with a total length of 4.130 ± 0.005 m (see Table 5). The remaining two parameters give the reader the idea of the aspect ratio of the body.

The volume calculated from a number of simulations through the Monte Carlo method is

$$V = 2.5151 \pm 0.0005\,{\text{m}}^{3} .$$

This represents a novel rigorous calculation for the volume of the meteorite which we later use to determine its mass—a previous estimate was based upon photographs by Buchwald (1975) resulting in 2.8 ± 10% m3, a very good approximation considering the prevalent instrumentation and data analysis techniques of the time.

We emphasize that the density of points considered here (a total of 1,812,875) reflects an upper bound for an instrument with uncertainties in the millimeter range. An increase in the number of points would result in separation distances along the meteorite surface in the range of m, which is well beyond the maximum precision of the device.

Next, the mass is calculated as a function of the density and the volume. The density is calculated from Table 4. A look at it reveals variations in the reported estimates that can be large in the case of concentrations. The net result is an overall density of 7.72 g/cm3 with an uncertainty of roughly 3%. It is worth noting this estimate is based on the available measurements, which do not provide the uncertainty for the Fe concentration (the most abundant element). More accurate measurements in the future should allow a better characterization. Finally, our mass is then calculated as

$$m = 19.43 \pm 0.51 \,{\text{tons}}.$$

We can see that this is consistent with the estimations made by Buchwald and Sánchez-Rubio (See Table 3). We obtained a mass uncertainty of around 3%. Notice here, that the volume misestimation is negligible when compared to that of the density, and therefore, it is the latter parameter that must be improved if better results are expected.

Finally, an important feature of iron meteorites is their regmaglypts (Fig. 6). Those meteorite traits provide important information related to its fall. For example, according to Lin and Qun (1987), larger pits correspond to zones where the surface vector is more aligned with the meteorite velocity vector. The statistics of the regmaglypts are depicted in Table 6. As observed in the table, different areas onto the meteorite exhibit contrasting average diameters and depths. To confirm those differences, a Kolmogorov–Smirnov test show a p value <0.01 among any regions. Consequently, each region has a characteristic structure and imply different origins.

Fig. 6
figure 6

a Render and b photography of Bacubirito

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Table 6 Diameter and depth of the meteorite’s regmaglypts
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Looking at these values and from our previous discussion on the importance of these characteristics, it is likely that during the peak ablation altitude, typically ranging from 25 to 40 km (see Revelle 1979), the face orthogonal to the fall was mainly the rear. If one is interested in obtaining an estimate of the dynamics of the speed as in Regan and Anandakrishnan (1993) one can use the cross-sectional area of this region (and probably average it with that for the below zone).

4 Conclusions

In this work it was obtained a geometrical model that found application on determining the mass of the Bacubirito meteorite and other of its geometrical features. Our results are listed as follows:

  1. 1.

    The mass has a value of 19.43 ± 0.51 tons. The results obtained here for both the geometric dimensions and mass are achieved through state-of-the-art analysis techniques and equipment. A considerable improvement on previous estimations was thus obtained.

  2. 2.

    The volume of the meteorite is 2.5151 ± 0.0005 m3. We have used the Monte Carlo method to simulate a series of measurements for a tridimensional rigid object. This also helps reduce time and expenses related to repetition of in-field measurement series.

  3. 3.

    The maximum length of the Bacubirito is 4.130 ± 0.005 m. Estimations of the meteorite’s main dimensions were also obtained. This establishes Bacubirito as the longest known in the world.

  4. 4.

    The depth and width of the regmaglypts of three regions are retrieved. The regions are found to exhibit different structures. The zone labeled as below is most likely to have been exposed more directly to the atmosphere in the ablation altitudes.

This present investigation opens a path to interesting research questions relating Bacubirito that can be addressed in the future thanks to the level of detail in the model. One can envision the study of the drag force that the atmosphere exerts on such a complicated geometry following a treatment as in Revelle (1979). The results can then be applied to estimate the mass loss and fall velocity as was mentioned in the introduction.