The Shapes of Splash-Form Tektites: Their Geometrical Analysis, Classification and Mechanics of Formation

Abstract

Splash-form tektites are found with a wide range of sizes and in an intriguing array of shapes ranging from spheres to flat discs to dumbbells. Despite the considerable interest that exists in tektites, there has been relatively little effort to develop rational shape descriptors and to understand the origin of their shapes based on basic physics. Tektites represent a natural laboratory experiment that can be analyzed to better understand the physics of rotating fluid drops. In this paper, we propose a classification scheme based on the axial ratios of ellipsoids, and we analyze the frequency of tektite shapes using a database of over 1,000 measured tektites. We show that the shape distribution for tektites from Thailand and Vietnam are very similar and that the most common tektites are moderately deformed discs but there exist also a significant number of moderately deformed dumbbells, and we argue that this distribution comes about because fluid drops first deform as oblate forms and then undergo a non-axisymmetric instability to become prolate. We also find that the largest tektites are most likely to be weakly deformed oblate objects while the most strongly deformed and most highly prolate forms are considerably smaller. A numerical model for the evolution of an axisymmetric fluid drop, such as a tektite in its molten early stage, is presented which demonstrates that drops that deform relatively slowly over a longer period of time are likely to develop central thinning while those that deform more rapidly are more likely to retain the shape of an ellipsoid. For the numerical parameters used the characteristic time scale for deformation was less than 1 s.

1 Introduction

1.1 Purpose of This Paper

It is becoming recognized that an understanding of spinning liquid drops is important to the physics of a large number of systems (Cardoso 2008). As yet, however, little work has focused on tektites as a ‘natural laboratory’ of such drops, other than a study by Elkins-Tanton et al. (2003).

Tektites are small glass objects (Fig. 1) formed by the freezing of molten drops of material ejected from large, terrestrial, impact craters (Schnetzler 1971; Barnes and Barnes 1973, and papers therein; O’Keefe 1976; Koeberl 1994; Glass 1984; Dressler and Reimold 2001; McCall 2001, and numerous other authors referred to in McCall; Artemieva 2008). They are typically between about 0.1 cm and 20 cm in maximum dimension (a few grams weight to several hundred grams), although both larger and smaller (microtektites) are known. Readily recognizable tektites occur in four main strewn fields (Barnes 1963; O’Keefe 1976; McCall 2001) plus several other minor localities, varying in age from about 35 Ma (North American Strewn Field) to about 0.78 Ma (Australasian Strewn Field).

Fig. 1

Photographs of various shapes of Australasian splash-form tektites: a several common forms from Vietnam, including: from upper left to right, a disk, and a triaxial ellipsoid, and from lower left to right, a rod, a dumbbell, and a teardrop; b pseudo-torus, made from three possible torus fragments from Vietnam; c bread-crust bikol-type from the Philippines; and d two ablation-form, Australian buttons, seen from the top (or sky side) showing both whole and broken flanges

Splash-form tektites (Reinhart 1958) are a common variety which forms in a number of shapes as a result of spinning while cooling and solidifying, although the physical model of how these shapes develop is understood in only a general way at present.

The main purposes of this paper are fourfold: (a) to develop a rational and useful way to analyse the shapes of splash-form tektites, (b) to analyse the shapes of over one thousand Australasian tektites using these new descriptors, (c) to use these new descriptors to help understand some aspects of the mechanics of formation of splash-form tektites, and (d) set the stage for more thorough time-process physical modelling of splash-form tektites as natural examples of spinning fluid drops. We also include analyses of the shapes and sizes of a small group of tektites from the Czech Republic as part of this study. Our analysis should lead to a better understanding of the various internal and surface textures that decorate tektites, which will be the subject of a later study.

1.2 General Introduction

In regards to the mode of their origin, tektites occur in two main groups. Group I (spin or modified spin-induced forms) (Fig. 1)—tektites formed by the freezing of spinning drops of molten rock, but often modified by a succession of other processes. Group II (spin not involved) (Fig. 2a)—also known as Muong Nong-type tektites, which do not display features indicative of rapid, molten-state spin, and many of which are strongly layered and blocky in form (Schnetzler 1992)—although originally meant to refer only to tektites of this type which form a subset of the Australasian strewn field (Lacroix 1935), the term Muong Nong-type now refers to all similar tektites regardless of location (e.g. North American Strewn Field and possibly Libyan desert glass). They will not be discussed further in this paper.

Fig. 2

Photographs of several tektite features: a Group II, Muong Nong-type from Thailand; b disk with bubble-surface on the lower left one-third and ablation or spall surface on the right and upper two-thirds, from Vietnam; c oval disk with two classic spall scars; d detail of the lower scar for Fig. 2c from Vietnam; e three fragments of splash-form tektites from Vietnam; f bent teardrop form, from Vietnam

Group I tektites form in a wide variety of shapes, dependant on several different processes that occur during and after flight. Previous workers have tended to discuss these in terms of three stages of tektite shaping (summarized in McNamara and Bevan 2001; and McCall 2001); however we think that the processes are best discussed under a four-part division, as below.

• 1. Primary shaping into what have been termed splash-form tektites (Reinhart 1958). These include discs, dumbbells, ellipsoids, and other forms that can be considered to be either ‘bodies of revolution’, formed by spinning, or derived from them, such as apioidal forms (teardrop-shaped objects) (Fig. 1a), or torus-shaped objects (doughnut-shaped) (Fig. 1b). Elkins-Tanton et al. (2003) produced several splash-form tektite shapes experimentally, demonstrating that the bodies take on shapes controlled by centrifugal forces and surface tension acting on the spinning masses as they solidify. The shapes of splash-form tektites form the main basis of this paper. However, Bikol-type tektites (Fig. 1c) are not discussed (that is, splash-form tektites that have a bread-crust appearance, a fat discoid shape, and that are found mainly in the Philippines) (Beyer 1962b).

• 2. Ablation-form tektites (Fig. 1d) involve the secondary, aerodynamic shaping effects of ablation, mainly after their centrifugal force-generated splash-form shapes have been produced, but while they are still in flight (e.g. Baker 1958, 1963; Chapman and Larson 1963). Ablation occurs when a portion of the surface of the tektite is re-melted as a result of atmospheric friction, after primary solidification, and most of the new molten material is shed off. Historically, ablation has been considered to be the most important secondary process, but occurs to a significant degree only in the distal portion of large strewn fields, such as the Australian portion of the Australasian strewn field. O’Keefe (1976, p. 50) also interpreted its possible importance for the production of some bald spots on otherwise non-ablated splash-form indochinites (Fig. 2b); however, we believe that most such spots are more likely spall scars instead (see below, Fig. 2c, d and O’Keefe 1976 p. 49) (this point is part of ongoing research into tektite formation by the authors).

• 3. Other secondary modifications, much of which may have taken place in flight, include spalling (Fig. 2c, d), fragmentation (Fig. 2e), and degassing (Figs. 2b, c, 3a). It is our view that the majority of tektite surface texturing is produced by a combination of melt flow (the schlieren and its folding deformation), bubbling (internal bubbles and surface bubble pits), spalling, and cracking, although these features can be highly modified by groundwater corrosion, as in many of the tektites from the Czech Republic. A more thorough analysis of these processes is part of our ongoing investigations.

  Fig. 3

  

  a Photograph of stretched bubbles along a portion of the length of an elongate teardrop form, from Vietnam. bOval disk displaying schlieren, from Vietnam

It is clear, however, that degassing (and bubble formation) was a common process in tektites and has no doubt led to their well-know extremely low water content (Friedman 1958; McCall 2001). For example, Fig. 2e, upper right-hand photo, shows a portion of a large central bubble exposed by fragmentation of what was probably a fat, disk-shaped splash-form tektite, and Fig. 2e, lower left-hand photo, shows a complexly shaped fragment which may have been produced by either spalling or the explosion of large bubbles. The surface pitting of most splash-form tektites is considered by us to be due to small bubbles formed through degassing of the tektite melt, although some authors have considered such pitting to be due entirely to ground water corrosion, whereas others attribute pitting to be due to aerodynamic effects (see O’Keefe 1976, pp. 42–49 for a summary of many of the arguments). O’Keefe (1976, pp. 48–49) convincingly makes the case that ground water corrosion merely reinforces textures that were previously formed by other mechanisms. The fact that there exist smooth spall scars on otherwise pitted tektites, when both regions have the same composition and have been exposed to the same ground water for the same period of time, argues that these pitted textures were not caused by dissolution (Nininger and Huss 1967).

Spalling is a form of fragmentation that results from differential temperatures between the outer and inner portions of the tektite during either cooling and solidification, or reheating of the surface during ablation. O’Keefe (1976, pp. 49–50), and especially Chapman (1964) consider reheating during the ablation phase to be the most important cause of spalling. Regardless of the cause, the resulting temperature differential produces stresses that can cause fragments to ‘spall off’ the body, or perhaps even to cause the entire body to fragment. Spalling in flight has been considered important only for strongly ablating australites by most previous workers (e.g. Chapman (1964) and McNamara and Bevan (2001). However, many splash-form tektites have smooth patches with few if any bubbles marring their surfaces, and we feel that many of these patches could have formed through spalling (Fig. 2c, d). This contrasts with the interpretation of O’Keefe (1976, p. 49) who considered these bald spots to be mainly the result of ablation. Indeed, the explosion of gas bubbles may aid the spallation process, and it seems likely that spattering of some of a bubble’s surface material could also be an independent third way for mass to be lost, at either the primary or secondary shaping stages, although such spattering appears to have not been discussed in the tektite literature to-date.

• 4. Ground impact effects, such as plastic flattening or bending (Fig. 2f) (Nininger and Huss 1967), or impact fragmentation. Spalling and degassing may also continue to occur after landing.

Tektites formed by any of the above processes can, of course, become further modified by various geological events after the body lands on Earth, including especially chemical corrosion (Barnes and Barnes 1973, p. 247; Vand 1965), surface abrasion, and breakage during sedimentary reworking (e.g. redistribution by river flooding).

Regardless of the cause of breakage, broken fragments of splash-form tektites outnumber whole pieces by about 2:1 in all locations where data is available (e.g. of all tektites from Vietnam offered for sale on eBay in 2007, 2,124 were fragments, compared with 1,197 whole forms) Whole forms are defined as having more than an estimated 75% of their original size.

2 Splash-Form Tektite Shapes

2.1 Introduction

At present, the shapes of splash-form tektites are poorly defined, limiting the detail of physical modelling that can be applied to their study. The terms used to describe their shapes in the literature are imprecise, often vague, and frequently confusing, although in some cases poetic (e.g. Baker 1963, Table 1, p. 7). A partial list of terminology problems includes: two-dimensional and three-dimensional terms mixed together in the same list (such as ‘ellipsoidal’ and ‘oval’); the term ‘round’ used for what might be either a ‘disc’ or a ‘sphere’; no defined boundary between what is called short versus long for ‘ovals’, ‘rods’, ‘cylinders’ or ‘dumbbells’; no boundary between thin and thick for ‘disks’; the term ‘teardrop’ being used for thin, leaf-like objects as well as for ‘apioids’ (pear-like shape); the term ‘rod’ used in the same list as the word ‘ellipsoid’ (instead of using ‘prolate ellipsoid’); the term ‘dumbbell’ being used for almost any shape showing central thinning; and a variety of ‘homey’ terms used without definition, such as ‘pickle-shaped’ (presumably cucumber pickle), and ‘boat-shaped’ (canoe?).

The origin and development of different splash-form tektite shapes is dependant on the physics of liquid spinning drops, and a discussion of this is presented later in Sect. 2.3 of this paper. It is already well established, however, that the various tektite forms are transitional, beginning with a crudely spherical blob, and that this deforms through viscous fluid flow processes into other shapes depending partly on the rotational velocity (Fenner 1934, 1935; McNamara and Bevan 2001; Elkins-Tanton et al. 2003). As pointed out by Tobak and Peterson (1964), early workers did not include surface tension as one of the important forces, and did not properly utilize the known physics of spinning bodies, a state of affairs corrected by the combined theoretical and experimental study by Elkins-Tanton et al. (2003). Initially the material inside a tektite will be liquid, but as it cools it takes on the consistency of ever stiffening taffy, continuing to deform until it either becomes too stiff, ceases to spin, or lands. The result of such deformation will be a change in shape, of course, but also, any bubbles formed in the late stages of spin will become trapped and possibly stretched (Barnes 1963) (Fig. 3a), and the internal schlieren may become distorted. (‘Schlieren’ are planar streaks of glass with varying compositions whose origin is also thought to be related to flow of the liquid) (Barnes 1963; O’Keefe 1976; McCall 2001) (Fig. 3b). These three features all record different aspects of the internal flow the tektite is undergoing, but analyses of processes other than shape change will be left for later study. An appropriate tektite shape classification should facilitate the analysis of the ways in which splash-form tektite shapes evolve during spin and solidification.

At present, the evolution of splash-form shapes is considered to be dependant on spin and the spin rate (Fenner 1934; Beyer 1962a; Vand 1965; McNamara and Bevan 2001, p. 22), such that relatively rapid rates of spin will result in a progression from spheres to rods to dumbbells, and eventually enough central thinning will occur that dumbbells will separate into teardrop (apioidal) forms, and perhaps, if still sufficiently fluid, back to a sphere (Beyer 1962a). At slower spin rates, disks will develop, which will progress to bowls (including double-sided bowls or, and eventually undergo enough central separation to form doughnut-shaped objects (tori). Those that have undergone either no central thinning (such as perfectly ellipsoidal disks and rods), or insufficient central thinning to separate (such as bowls and dumbbells) are here grouped under the term main-field splash-form tektites, and their discussion constitutes the major part of this paper. Teardrop forms are discussed somewhat, but tori are not. However, as will be seen later, the simple relationship between relative spin rate and shape mentioned above is not an accurate description of the evolution of forms. There remains a large uncertainty in actual spin rates (Elkins-Tanton et al. 2003).

2.2 The Main-Field Tektite Shape Diagram

The shape classification diagram proposed here is based on comparing the axial ratios of splash-form tektites, other than teardrops and tori, with that of ellipsoids, where the major axis is the length (L), the intermediate axis is the width (W), and the minor axis is the thickness (T) (Figs. 4, 5). This is equivalent to a variety of shape diagrams used in geology, such as the Zingg diagram for sedimentary particle shapes (Krumbein 1941) and the Flinn deformation diagram used in structural geology (Flinn 1962). All of these shape descriptor diagrams are based on perfect ellipsoids, which tektites most certainly are not. However, the diagrams are still very useful, as long as we remember that a tektite’s position on such a diagram is approximate—rather in the way that an apple can be called spherical, or a pear can be said to have the shape of an apioid (as we did three paragraphs earlier). Despite this short-coming, the Tektite Shape Diagram turns out to be extremely useful for describing the evolution of splash-form tektite shapes, as will be seen throughout this paper.

Fig. 4

Natural-log, main-field, splash-form tektite shape diagrams (main-field shapes include those of all splash-form tektites except teardrops and tori) (the classification terms are placed on three separate diagrams for clarity of presentation): a basic type with first-order shape classification; b second-order shape classification; c classification based on the total deformation determined from L/T (P_r = atan(ln(L/W)/ln(W/T))

Fig. 5

Photographs of classic examples of main-field tektites on the natural-log shape classification grid: a rods to disks, with little to no central thinning; b dumbbells to bowls, examples with moderate to strong central thinning. Specimens on both diagrams include indochinites (from Vietnam and Thailand), and moldavites (from the Czech Republic)

Such a diagram can be used in linear or logarithmic versions (base 10 or natural log). The natural-log version (Figs. 4, 5, 6, 7 and 8) seems best for the detailed analysis of how shapes develop and change during spin, and is used throughout this paper, but a non-logarithmic form is probably easier to use as an adjunct to literal shape terms for those wishing only to classify the shapes of tektites.

Fig. 6

Plots of main-field tektite shapes on the natural-log shape classification diagram: a 649 vietnamites; b 492 thailandites; c 43 moldavites; and d 1,184 data from all three locations

Fig. 7

Two-dimensional shape histogram for all 1,141 main-field, splash-form vietnamites and thailandites, with shape classification boundaries as per Figs. 4, 5 and 6. The arrows represent the shape trajectory for an ellipsoid that first deforms from a sphere to an oblate ellipsoid with W/T = 2.46 and then evolves keeping T fixed while W and L decrease and increase, respectively, to become a prolate ellipsoid

Fig. 8

Plots of main-field tektite shapes on the natural-log shape classification diagram for all locations, comparing those with different amounts of central thinning. a Little to no central thinning (disks to oval disks to bars to rods (409 vietnamites + 404 thailandites + 29 moldavites); b marked central thinning (bowls to bowties to dumbbells) (240 vietnamites + 88 thailandites + 14 moldavites). The vertical solid line indicates the value of W/T at which equilibrium oblate forms start to show central thinning

A useful shape factor, here called the prolate factor (P), is based on the axial ratios, where (L/W) = R_LW, and (W/T) = R_WT. In natural log form this is:

$$ {\text{P}} = { \ln }{{\left( {{\text{R}}_{\text{LW}} } \right)} \mathord{\left/ {\vphantom {{\left( {{\text{R}}_{\text{LW}} } \right)} {{ \ln }\left( {{\text{R}}_{\text{WT}} } \right)}}} \right. \kern-\nulldelimiterspace} {{ \ln }\left( {{\text{R}}_{\text{WT}} } \right)}} $$

(1)

A prolate factor of ‘0’ refers to an oblate ellipsoid, and a factor of infinity refers to a prolate ellipsoid. This can also be written in angular form as Pr angles (2), and this is the form used throughout most of the rest of this paper.

$$ { \Pr } = {{{ \arctan }\left( {{\text{ln R}}_{\text{LW}} } \right)} \mathord{\left/ {\vphantom {{{ \arctan }\left( {{\text{ln R}}_{\text{LW}} } \right)} {\left( {{\text{ln R}}_{\text{WT}} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {{\text{ln R}}_{\text{WT}} } \right)}} $$

(2)

Our own Monte Carlo simulations showed that randomly generated L, W and T values will be uniformly distributed in Pr.

Also, the ratio of length to thickness (L/T) is a useful parameter to help describe the total deformation, or shape change from a sphere, without regard to the prolate factor. In this paper, when we discuss the degree of deformation, we are referring to the size of the L/T ratio. L/T ratios less than 3 indicate weak deformation, L/T ratios between 3 and 4.5 indicate moderate deformation, and those greater than 4.5 indicate high deformation.

Clearly it is possible to define the shape of a tektite by merely stating its (L/W) and (W/T), and perhaps by including the derived ‘P’ or ‘Pr’ values and L/T. (Pr values are of considerable use when describing the evolution of splash-form shapes as will be seen in Sect. 2.4). However, common names and literal descriptors tend to be preferred by many scientists and laypeople, such as tektite collectors. For this reason, a more literal shape terminology is utilized on the following shape diagrams.

Four major shape fields can be based on the Pr angles (or prolate factors) (Fig. 4a, b). At Pr = 0° (R_LW = 1; horizontal axis) objects are perfect oblate ellipsoids (P = 0) and can be called round disks; however, experience shows that the human eye is likely to call most objects ‘round’ provided the L/W ratio is less than 1.15, as shown on Fig. 4a. Between round disks and Pr = 45° (P = 1) objects are triaxial ellipsoids that we can call oval disks. At Pr = 90° (R_WT = 1; vertical axis), objects are perfect prolate ellipsoids (P = inf.), or what we can call rods, again experience shows that the human eye is likely to call most objects rods as long as W/T is less than 1.15, as shown on Fig. 4a. Between rods and Pr = 45°, objects are triaxial ellipsoids that we can call bars (a common term for tektites in the middle of the Pr > 45° field). Further subdivision of the Pr fields at 22.5° and 67.5° is also useful, dividing oval disks into slightly oval and highly oval; and dividing bars into oval bars and classic bars (the latter field contains most of the objects popularly called bars) (Fig. 4b). The terms in brackets on Fig. 4a, b are suggested names for tektites of various Pr factor that display marked central thinning, such as dumbbells, bowties, oval bowls, round bowls, and even doughnut-shapes (tori), most of which are already in common use, although without specific definitions.

Further subdivision of these shape fields can be based on L/T ratios (Fig. 4c). We use the terms elongate and flattened to describe the degree of deformation of prolate and oblate tektite shapes, respectively. Disks and bowls are named thick (L/T < 1.5), slightly flattened (1.5 < L/T < 3), flattened (3 < L/T < 4.5), highly flattened (4.5 < L/T < 6), and ultra-flattened (L/T > 6P). The more prolate forms (bars, rods, bowties, and dumbbells) can be called stubby (L/T < 1.5), slightly elongate (1.5 < L/T < 3), elongate (3 < L/T < 4.5), highly elongate (4.5 < L/T < 6), and ultra-elongate (L/T > 6).

Figure 5 displays photographs of tektites from many of these classification fields. Teardrop forms are not included in Figs. 4 or 5, but are discussed in Sects. 2.6 and 2.7.

2.3 The Physics of Rotating Fluid Drops

The important forces that affect the shape of a liquid drop in flight are surface tension, centrifugal forces, aerodynamic drag, inertial forces due to the mass of the fluid, and viscous forces. When analyzed in a reference frame moving with the drop, the latter two forces exist only while the drop is in the process of deforming. The magnitudes of surface tension forces are inversely proportional to the radius of curvature of the outer surface of the fluid. In the absence of other forces, surface tension pulls fluids into a spherical shape since a sphere has the same curvature everywhere on its surface. Surface tension is most important in small fluid drops and in those that are strongly deformed, since these have the smallest radii of curvature. Centrifugal forces arise due to the rotation of the fluid blob and act radially away from the rotation axis.

The only published study investigating the dynamical causes of the shapes of splash-form tektites is that of Elkins-Tanton et al. (2003). These authors employed scaling arguments to conclude that aerodynamic forces would overcome surface tension forces and cause drops to rupture for drop radii greater than 3 mm if drops are traveling at their terminal velocities through air. They also showed that centrifugal forces should dominate over aerodynamic forces provided that the rotational velocity of a fluid drop exceeds 0.01 of its translational velocity, a condition that is very likely to be the case for tektites. Centrifugal forces will also cause drops to rupture and hence tektites greater in size than 3 mm should not exist if these represent equilibrium shapes. The fact that tektites are found with sizes of order 10 cm indicates that either: (a) their relative velocity with the air through which they move is very low and that they are rotating very slowly, or (b) that they are not equilibrium shapes, but rather transient shapes that solidified before rupture could occur. The fact that tektites occur in shapes such as tori and biconcave ellipsoids (double-sided bowls) that are unstable forms of revolution is good evidence for the latter scenario.

Although it is unlikely that tektites represent equilibrium forms of revolution, it is likely that during the time evolution of a spinning fluid drop, the shape of the fluid drop will be similar to equilibrium forms. Elkins-Tanton et al. (2003) summarized the analytical work of Chandrasekhar (1965) and the numerical calculations of Brown and Scriven (1980) concerning the equilibrium forms of fluid drops under the influence of centrifugal forces and surface tension. These workers considered only the equilibrium shapes, when pressure and centrifugal forces are exactly in balance everywhere inside the drop (surface tension imparts a discontinuity in pressure at the fluid interface), they did not consider the transient time evolution of fluid drops. In equilibrium, the shape of a fluid drop is uniquely parameterized by the Bond number,

$$ {\text{B}}_{\text{o}} = {{\rho \Upomega^{ 2} {\text{ R}}^{ 3} } \mathord{\left/ {\vphantom {{\rho \Upomega^{ 2} {\text{ R}}^{ 3} } {\left( { 8 { }\sigma } \right)}}} \right. \kern-\nulldelimiterspace} {\left( { 8 { }\sigma } \right)}}, $$

(3)

where ρ, Ω and σ are the density, rotational frequency, and surface tension of the fluid drop. The variable R represents the radius of a sphere with the same volume as the drop.

At B_o = 0 centrifugal forces are absent and surface tension pulls the drop into a perfectly spherical shape. As the Bond number increases up to 0.09, centrifugal forces push fluid away from the rotation axis leading to increased curvature near the equator and flattening near the rotation poles leading to equilibrium forms that are roughly oblate ellipsoids. For 0.09 < B_o < 0.31 fluid drops can either assume the shape of oblate ellipsoids or two-lobed “dumbbell” shapes as stable equilibrium forms. Dumbbells have one axis significantly longer than the other two and presumably come into being when one of the equatorial axes of the initial roughly spherical blob is slightly larger than the other two. Above a Bond number of 0.31, axisymmetric equilibrium forms continue to exist but these are unstable to non-axisymmetric perturbations. These axisymmetric forms become bi-concave for B_o greater than 0.5 and they achieve the topology of a torus for B_o = 0.57. Beyond B_o = 0.57 no equilibrium forms were found to exist. Stable dumbbells can occur up to Bond numbers of 0.57. If the Bond number exceeds 0.57, no stable shapes exist and the fluid drop will eventually pinch-off into two or more drops.

The shapes of equilibrium rotating fluid drops were also investigated numerically by Heine (2006) and experimentally in microgravity experiments by Wang et al. (1986) and using magnetic levitation by Hill and Eaves (2008). Brown and Scriven (1980) also found three and four lobbed unstable equilibrium forms but there are no known tektites of such forms. Inertial and viscous forces do not play a role in determining equilibrium shapes.

Using the equilibrium forms of fluid drops as a guide to the types of shapes that one should expect for the transient forms of fluid drops under rotation, we should expect then to find two populations of tektite morphologies. One group is predicted to be roughly oblate while the other is predicted to be highly prolate. It is reasonable to assume that since most tektites represent quenched non-equilibrium forms, their shape distribution should reflect the time evolution of a rotating fluid drop. We will investigate the time evolution of oblate fluid drops using a numerical model later in this paper.

2.4 The Shapes of Main-Field Splash-Form Tektites

Nearly 1,200 main-field, splash-form tektites from Vietnam (649; vietnamites), Thailand (492; thailandites) and the Czech Republic (43; moldavites) were studied (Figs. 6, 7). Tektites from Vietnam and Thailand form part of the Indochinese portion of the Australasian strewn field, which is about 780,000 years old. Those from the Czech Republic represent the Eastern European strewn field, which is about 15 million years old.

Size measurements and weights for most of these were obtained from eBay sellers (Vietnamminerals for the vietnamites, The Rock Billionaires for the thailandites, and several sellers of moldavites from the Czech Republic). About 150 of these were purchased and used to confirm the eBay data. Several additional vietnamites, moldavites, and pieces from the Philippines and several locations in China were also purchased, some from eBay sellers and some from non-eBay sellers. Vietnamite data were obtained from the Yen Bai field about 120 km northwest of Hanoi, thailandite data comes from scattered localities throughout north-eastern Thailand, and moldavite data comes from scattered localities in the south-central Czech Republic. In all of these localities tektites are mined as business ventures, and, except for the moldavites, almost every piece found reaches eBay sellers. The only selectivity that occurs is that some rare, special pieces do not appear on eBay, but were often available through non-eBay sources, and, more importantly for our sampling, small pieces weighing less than 10–15 g may be missed during mining, thus small tektites are under-sampled. Other than this, the sampling of the various shapes and sizes is probably as close to statistically random as one could wish. Moldavite sampling on the other hand is not random; first, it is too small a sample, and second, at least one-third of the data comes from cherry-picked, museum-quality specimens that were purchased from non-eBay sources; non-the-less the data displays broad similarities to the Australasian data.

Only whole, well-formed, splash-form tektites were chosen for this study, which is about half the total that was available. Those rejected had either numerous chips or displayed some sort of post-formation modification that would influence size measurements.

There were several protocols followed for the measurement of dimensions of those tektites in our collection, and these are consistent with measurements made by sellers whose data we used. These protocols are necessary because few tektite specimens are perfect ellipsoids. First, length (L), width (W), and thickness (T), were measured in mutually perpendicular directions. Second, in every case, the largest measurement possible in a given direction was used, even if the line of measurement did not pass through the center of the other lines of measurement. Finally, for those tektites that display central thinning, such thinning was ignored for our present purposes (but will be dealt with in a later study). Dumbbells, for example, were easy to measure along their length, but the width and thickness were measured from the largest end, as they were frequently not symmetrical. Likewise, the (L) and (W) of bowls was typically easy to measure, but thickness measurement often required strict adherence to a consistent protocol, such that for a bowl with a convex side, (T) was measured from the outside arc to a plane resting on the rim of the bowl (and keeping the first two protocols intact). For a bi-concave bowl, (T) was measured between planes tangent to the rims on both sides. Measurements could be made easily to 1 mm accuracy, reproducible between several measurers, even with a simple ruler (occasionally callipers were used). As a check, we could compare our measurements with those of the sellers for the pieces we purchased (we found very few errors in the sellers’ measurements). In a few cases, where mass and size measurements provided on eBay did not appear consistent for specimens that we did not purchase, we asked them to re-measure—which they were always willing to do.

Several features are readily seen on the shape-diagram plots for all main-field tektites from each of the three locations (Fig. 6a–d), including: (a) there is broad similarity in the distribution of shapes of vietnamites and thailandites (cf. Fig. 6a, b and d); (b) the most common shape of vietnamites and thailandites is that of a flattened to slightly flattened round disk/bowl; (c) there is a secondary concentration of data in the classic bar/bowtie to rod/dumbbell fields; (d) there is a paucity of vietnamite and thailandite data in the stubby to slightly elongate rod fields, centered at a (L/W) of about 1.75, although this is more apparent with the vietnamite data; (e) only a few vietnamite and thailandite specimens have L/T ratios greater than 6; and (f) the few moldavite data (Fig. 6c) display a broadly similar range of shapes, but with somewhat higher axial ratios in the oval and classic bar fields. It is unlikely that the moldavite data indicates an overall greater degree of fluid-flow during flight than does the Indochina data—it is more likely that this is due to poor statistical sampling of the moldavites, and to the fact that the seven specimens concerned are high market-value, museum-quality specimens, chosen for sale and subsequent purchase by us partly because they are so rare. Despite this problem with moldavite sampling, the paucity of low (L/W) rod specimens is still clear (‘d’ above). This paucity, along with the secondary concentration in the bar/bowtie to rod/dumbbell fields (‘c’ above) are explained by the physics of rotating fluid drops.

In Fig. 7 we display a two-dimensional histogram of all of the measured main-field, splash-form tektites from Vietnam and Thailand. The histogram uses square bins of dimension 0.1 log units on a side. Along with the large number of oblate tektites and somewhat smaller number of prolate tektites with L/W > 3, it can be seen that there is a “pathway” of slightly elevated tektite frequency connecting these two clusters. We can interpret this distribution to come about because tektites first deform from roughly spherical drops along the W/T axis to become deformed oblate shapes before undergoing the previously described non-axisymmetric instability to become prolate. The pathway represents tektites that solidified while they were in the process of undergoing the change from oblate to prolate. The paucity of weakly deformed prolate objects can be explained if tektites do not deform from spheres directly to prolate shapes but rather deform to become oblate before undergoing the instability. The arrows indicate such a possible shape evolution with the upper arrow following the trajectory of an ellipsoid of constant volume where the T axis remains fixed while L increases and W decreases. As was mentioned in the discussion of the physics of spinning fluid drops (Sect. 2.3), for a range of Bond numbers, the equilibrium forms of tektites have both oblate and prolate stable forms, and for a further range, there exist unstable oblate forms and stable prolate ones together. It is thus plausible that during the time-evolution of a liquid drop, it could become unstable to a non-axisymmetric perturbation once a given degree of deformation is achieved. Also, Elkins-Tanton et al. (2003) showed laboratory results in which oblate forms transformed into dumbbell shapes. A three-dimensional numerical simulation of the time evolution of a rotating fluid drop would be helpful in order to test this hypothesis.

A comparison of the shapes of tektites from all locations that display no, or only slight, central thinning (Figs. 5a, 8a) with those displaying marked central thinning (Figs. 5b, 8b) shows the rather surprising result that there is no large increase in the axial ratios concomitant with the thinning; there is, however, a marked decrease in those with small axial ratios (L/T < about 2.5). In particular, there are very few oblate tektites with W/T < 2 among the centrally thinned tektites when compared with those showing no central thinning which can be explained by the fact that equilibrium shapes, which we will demonstrate show greater central thinning than transient shapes, do not start to become centrally thinned until W/T > 2.3 (Chandrasekhar 1965; Brown and Scriven 1980). We indicate a vertical solid line with W/T = 2.3 on Fig. 8b. There seems also to be a relative increase in the proportion of high Pr-value tektites (especially dumbbells); but no such trend is apparent for moldavites, indeed, rods and dumbbells are almost absent from the marketplace, and are considered to be rare in the non-scientific literature (e.g. Heinen 1998). We do not know the reason for this rarity.

2.5 The Sizes (Mass) of Main-Field Splash-Form Tektites

Although it is well known that splash-form tektites range in weight over at least five orders of magnitude, from less than 0.01 gram for microtektites (<1 mm diameter) to around 1 kilogram for the largest rare disks, no one previously seems to have carried out a detailed size analysis, and there is little in the way of descriptions of sizes in the literature in general. However, it is clear that the splash-form tektites from the Australasian strewn field in south-eastern Asia are considerably larger than those from other strewn fields (McCall 2001) (the reason is unknown), and the largest splash-form tektite recorded (Beyer 1962b) is a 1,069 gram sphere from the Philippines, although significantly larger Muong-Nong types are known (Barnes 1971).

The weights of all the main-field tektites discussed previously are displayed on the mass-frequency diagrams of Fig. 9a–c. The mass distributions for vietnamites and thailandites are crudely similar (cf. Fig. 9a, b), with both distributions peaking at around 50 g, although the vietnamite population contains proportionally more heavy pieces (>about 150 g), and the largest vietnamite in our sample is 480 g compared with the largest thailandite at 380 g (these were the largest available over a 3 year period of sampling). Moldavites are significantly smaller (cf. Fig. 9a, b with c). In our moldavite sample there are numerous pieces weighing less than 25 g, and no pieces greater than 106 g. The paucity of Indochina tektites less than about 25 g is likely a result of the way the sampling is carried out, and small pieces can be easily missed during mining, thus the peak in their distributions is an artefact, not a function of their mass distribution. Such is not the case for moldavites, however. The significantly higher prices obtained for these pieces result in more care being taken to find even the smallest material.

Fig. 9

Mass frequency histograms of main-field tektites—the heavy black curve on a and b is an exponential fit for vietnamites and thailandites combined, based on specimens between 50 and 350 g: a mass/frequency bar histogram for 649 main-field vietnamites; b mass/frequency bar histogram for 492 main-field thailandites; c mass/frequency bar histogram for 43 main-field moldavites

There is also some difference in mass with shape (Fig. 10). In all 3 populations studied, the largest splash-form tektites with low Pr values (round disks and bowls) are around three times the weight of those with the highest Pr values (rods and dumbbells) (Fig. 10a–c), although the moldavite sample is too small to draw independent conclusions. Both vietnamites and thailandites have bimodal mass distributions, with clusters at low and high Pr values (Fig. 10a, b), with the greatest density of data in the round to slightly oval disks and bowls (Pr values less than 22.5°). However, the centers of these clusters occur at similar masses, indicating that the difference in mass between the high and low Pr values occurs only for the larger specimens. That is, the larger a specimen is, the more likely it is to have a low Pr value, but only if it is greater than about 150 g. The paucity of indochinite data with low mass (Fig. 10a, b) has already been discussed as a problem with mining the small, low-value pieces. There is insufficient data for the moldavites, although most specimens in our sample also have low Pr values.

Fig. 10

Main-field tektite masses versus shape. Mass/Pr (degrees) diagrams: a 649 vietnamites; b 492 thailandites; c 43 moldavites

The amount of deformation appears to be partly a function of size as well, at least for the indochinites (Fig. 11a, b); there is insufficient data for the moldavites to make an interpretation (Fig. 11c). Most indochinites have L/T ratios between 1 and 6, and masses less than about 150 grams. However, for pieces weighing more than about 150 grams there tends to be an increase in deformation with size, as indicated by the paucity of high mass and low L/T value specimens (Fig. 11a, b). This relationship is not strong, and is due mainly to most large pieces being moderately deformed, whereas smaller pieces vary across the deformation spectrum from low to high. Furthermore, the very largest specimens (>200 g) are mostly disks (round to slightly oval) with L/T ratios greater than about 2 (cf. Figs. 10a, b with 11a, b). (Dumbbells greater than 200 g are extremely rare; only 3 vietnamites and no thailandites were offered for sale on eBay over a 3-year period during which thousand of splash-form specimens were available). The paucity of data in the large mass, high L/T portion of these plots (Fig. 11a, b) is likely a result of three things that have nothing to do with the relationship between mass and deformation: first the mutual rarity of both very large and highly deformed specimens; second, the fact that highly deformed specimens of any size may readily divide into two or more segments while in flight (i.e. teardrops from dumbbells and segments of tori from bowls); and third, the terminal velocity of large pieces is likely to be greater than that of small pieces (less frictional velocity retardation in the atmosphere) and these with high L/T values are more likely to break upon landing.

Fig. 11

Main-field tektite mass versus deformation diagrams. a 649 vietnamites, mass vs L/T; b 492 thailandites, mass vs. L/T; and c 43 moldavites, mass vs. L/T

The Bond number increases as R³ so one might expect larger tektites to be more strongly deformed than smaller ones. However, the rotational frequency would be expected to decrease with the size of the fluid drop since the moment of inertia of a sphere of uniform density increases as R⁵.

2.6 The Shapes of Teardrop Tektites

Teardrop-shaped tektites, along with their analogues from the disk field, the tori, represent the extreme cases of splash-form tektite evolution. This paper discusses their classification and presents a preliminary analysis of indochinite and moldavite samples. However, an analysis of the physics of their formation is left for later study.

The name ‘teardrop’ tends to be applied to any splash-form tektite with a triangular, tapered, or bulbous shape as seen from the side (in two-dimensional space), regardless of the thickness relative to the width. They are the most common form of whole, unbroken tektites (fragments of splash-form tektites are much more common than whole tektites from all locations studied). For shape classification purposes ‘W’ and ‘T’ have been measured from the thick end (Fig. 12a–c).

Fig. 12

Natural-log tektite shape classification diagrams for teardrop-shaped tektites: a the classification grid with suggested names for the various shapes; b photographs of tektites from all three locations; c plots of teardrop shapes from all three localities (202 vietnamites, 103 thailandites, and 22 moldavites)

The shape-classification of teardrop tektites suggested here subdivides the shapes into categories based on a grid of L/W vs. W/T lines (Fig. 12a) where stubby, slightly elongate, and elongate are defined with values of L/W < 2, 2–4, and >4 respectively. Apioids are considered to have W/T < 1.15 and oval teardrops to have W/T > 1.15. Ovals are bounded at W/T of 1.15 and 3. Many thin ovals have a ‘leaf-like’ form (Fig. 12b). Many of the stubbiest apioids have shapes commonly referred to as ‘onion- shaped’ (Fig. 12a, b) (to continue the common use of vegetative analogy); slightly more elongate forms could be called ‘pear-shaped’ (Fig. 12a, b). Even more elongate apioids could be called ‘classic teardrops’ (Fig. 12a, b) to ‘club shaped’, and many of the most elongate apioids look much like ‘carrots’ (Fig. 12a, b).

Vietnamite and thailandite teardrops have about the same shape variations (Fig. 12c), with the majority lying between L/W ratios of 1.25 and 3, and W/T ratios of 1–2. Moldavites, however, seem to be more oval (Fig. 12c), with ‘leaf-like’ forms being relatively common, and with a lack of stubby apioids; but whether this difference is real or due to a problem with sampling is unknown.

2.7 The Mass Distribution of Teardrop Tektites

There are vastly more small teardrop-shaped tektites than large ones (Fig. 13a–c), similar to the mass frequency of the main-field populations, and a similar drop-off in very small pieces, especially for the indochinites, although the small numbers of specimens make detailed analysis difficult, especially for the thailandites and moldavites.

Fig. 13

Teardrop mass/frequency histograms: a 202 vietnamites; b 103 thailandites; c 22 moldavites

As described previously, it is generally considered that teardrops originate by the central thinning and eventual separation of dumbbells. From this it might be expected that a particular dumbbell mass population would have about the same frequency as the same mass population derived by multiplying teardrop masses by two. Comparisons between dumbbell mass frequencies and double teardrop mass frequencies are shown on Fig. 14a for vietnamites and 14b for thailandites. Only one dumbbell exists in our moldavite sample so no such comparisons can be made.

Fig. 14

Mass/frequency diagrams comparing the masses of dumbbell-shaped tektites with that of double the masses of teardrop-shaped tektites: a Vietnamites (306 teardrops—black bars, and 160 dumbbells—gray bars), the mean mass of dumbbells is 77.0 g (standard deviation = 42.0 g), and that of teardrops is 45.7 g (SD = 27.5 g), the mean of double teardrop masses is thus 91.4 (SD = 55 g); b Thailandites (103 teardrops—black bars, and 47 dumbbells—gray bars), the mean mass of dumbbells is 73.3 g (SD = 34.9 g) and that of teardrops is 41.0 g (SD = 27.4 g), the mean of double teardrop masses is thus 82.0 g (SD = 54.8 g)

Comparisons between dumbbell data, shown on Fig. 14a, b, and that for double teardrops indicate that there was a somewhat greater likelihood for larger dumbbells to form teardrops. For the 160 vietnamite dumbbells (Fig. 14a), the largest are in the 225–250 g range, and the mean mass is 77.0 g, whereas for the 306 vietnamite teardrops (Fig. 14a), the largest double teardrops are over 300 g, and the mean double teardrop is 91.4 g, considerably larger than the dumbbell data. The same situation exists for thailandites, although there is less data. The largest of 47 thailandite dumbbells are in the 125–150 g range, and the mean mass is 73.3 g. (Fig. 14b). For the 103 thailandite teardrops, however, the largest double teardrops are over 300 g, and the mean double teardrop is 82.0 g. (Fig. 14b), again, considerably larger than the dumbbell data. Furthermore, the frequencies of double teardrop masses exceed that of dumbbell masses for all mass categories above 125 g for vietnamites and 100 g for thailandites.

The above interpretation is clouded by the irregularity of frequencies in the lighter mass categories for both vietnamites and thailandites (Fig. 14a, b), with dumbbells having higher frequencies than double teardrops in some categories and less in others. However, this variability is thought to result from the fairly small sample sizes, such that many of the mass categories do not have enough data to produce consistent differences in frequencies, should they exist, but be fairly small. However, the overall sample sizes are probably large enough for valid comparisons to be made between their means, as we did in the above paragraph. The values of these means are mainly due the more frequent smaller tektites and are hardly affected by the few large tektites in the sample.

3 Numerical Simulations

3.1 Model Description

Two-dimensional simulations of rotating fluid drops with surface tension were undertaken. The geometry is assumed to be cylindrical axisymmetric and hence the non-axisymmetric instability could not be investigated. The following discussion thus applies to oblate forms [the round disks, bowls and tori (Figs. 4, 5 and 6)], although some extrapolation into the more prolate fields may be possible. The incompressible, constant property, Navier-Stokes equations describing fluid flow within a rotating drop with surface tension are as follows:

$$ \rho \left( {{{\partial {\mathbf{u}}} \mathord{\left/ {\vphantom {{\partial {\mathbf{u}}} {\partial {\text{t}}}}} \right. \kern-\nulldelimiterspace} {\partial {\text{t}}}} + {\mathbf{u}} \cdot \nabla {\mathbf{u}}} \right) = \eta \nabla^{2} {\mathbf{u}} - \nabla {\text{p}} + \rho\Upomega^{2} {\text{r}}\,{\hat{\mathbf{r}}}, $$

(4)

$$ \nabla \cdot {\mathbf{u}} = 0. $$

(5)

The first equation gives the force balance within the drop while the second equation represents conservation of mass for an incompressible fluid. The dependent variables are u which represents the fluid velocity and p which represents pressure while t represents time and r is the radial distance from the rotation axis and ${\hat{\mathbf{r}}}$ is a unit normal in this direction. The parameters describing the properties of the fluid are the density, ρ, viscosity, η, while Ω is the angular rotation frequency and ${\hat{\mathbf{n}}}$ represents an outward unit normal at the outer surface and ∇ is the nabla or del operator of vector calculus. The left hand side of (4) represents inertial forces while the terms on the right hand side represent viscous stresses, pressure gradient stresses, and centrifugal forces.

Surface tension imparts a discontinuity in the normal stress on the surface of the drop that is proportional to the curvature of the drop ($\nabla \cdot {\hat{\mathbf{n}}}$ ) with proportionality constant given by the surface tension coefficient, σ. In order to allow for the evolution of the surface of the fluid drop, the level set method is used which introduces a function φ whose value is greater than 0.5 inside the fluid drop and less than 0.5 outside of it, and which evolves following the fluid motion. The surface tension force is determined by calculating the curvature of the 0.5 contour of φ. The domain has dimensions of 4R by 4R and the region outside of the fluid drop is filled with a second fluid of much lesser viscosity and density representing air.

Two independent dimensionless parameters can be formed from ρ, η, σ, Ω and R. These include the previously discussed Bond number which represents the ratio of centrifugal to surface tension forces and the Ohnesorge, O_n number,

$$ {\text{O}}_{\text{n}} = {\eta \mathord{\left/ {\vphantom {\eta {\left( { 2 {\text{R}}\rho \sigma } \right)^{0. 5} }}} \right. \kern-\nulldelimiterspace} {\left( { 2 {\text{R}}\rho \sigma } \right)^{0. 5} }} $$

(6)

which represents the ratio of the time needed to deform to pinch off when viscosity dominates to the time needed to deform to pinch-off when inertial effects dominate. Specifying B_o and O_n is enough to specify the time evolution of a particular drop.

We assume, for these calculations, that a rotating fluid drop in flight will not experience significant external torques and its angular momentum will be conserved as a result. When a droplet deforms so that mass is distributed further from the rotation axis, the moment of inertia of the drop will be increased and the rotation rate must decrease as a result. This effect is included in the model and, as a result, the effective bond number changes with time since the rotation rate decreases. The Bond number stated is the initial Bond number, before the fluid drop has deformed. All simulations were initiated from a spherical shape at time t = 0.

In order to test the model, some simulations with sufficiently low B_o were undertaken that the fluid drop attained an equilibrium form. The dimensions of these were compared with the published values of Brown and Scriven (1980) and were found to be in reasonable agreement for the same final Bond number. As stated previously, equilibrium forms are uniquely determined by the final value of B_o. O_n affects the time required to achieve an equilibrium state if the final value of B_o is less than 0.57 and an equilibrium state is achieved and for some low values of O_n for which inertial effects dominate viscous effects, the drops were seen to oscillate about the equilibrium shape. For higher values of B_o, O_n affects the time evolution of the drop.

Elkins-Tanton et al. (2003) estimated the range of values of 10⁻⁹ < B_o < 10⁶ and 10⁻² < O_n < 10³ for tektites. The values of density and surface tension are constrained to a reasonably small range: 2,700 kg m⁻³ < ρ < 3,200 kg m⁻³, and 0.2 kg s⁻² < σ < 0.5 kg s⁻² respectively (Elkins-Tanton et al. 2003), although Chapman et al. (1964) report density values close to 2,420 kg m⁻³.

So for a given size of tektite, we can think of B_o as a dimensionless rotation rate and O_n as a dimensionless viscosity. The equations were solved using the Chemical Engineering Module of the commercial finite element package COMSOL and our model was created by modifying the oscillating fluid droplet model that is included in the package as an example. The numerical method was able to calculate the evolution of the shape of a fluid drop for B_o up to 100. When O_n is increased, the time necessary for deformation increases and the numerical method becomes unstable after very long integration times. As a result, the maximum O_n that could be investigated increased with the B_o used.

In the following section we discuss the implications of the results of the simulations in terms of understanding the mechanism resulting in central thinning and in terms of the time-scale required for deformation.

3.2 Mechanism Resulting in Central Thinning

In Fig. 15 we display the cross-section of a liquid drop from the results of simulations with initial B_o = 10 and O_n = 0, 0.1 and 1 (dotted, solid and dashed line respectively) where z is the vertical height above the equatorial plane of rotation. The dimensions have been normalized by the initial radius, R, so that all lengths are in units of R. Since the simulations assume symmetry about a vertical axis at r = 0, these shapes represent oblate forms in three-dimensions. At an initial B_o = 10, the drops will never reach an equilibrium form and will eventually pinch off at the origin and become tori. We show results when each of the simulations reached a maximum radius of 1.73. Also shown are the cross-section of an oblate ellipsoid with a semi-major axis of 1.73 (dotted line) and the equilibrium shape of a fluid drop at final B_o = 0.57 with radius 1.67 (solid line with circles) which is the maximum radius that an equilibrium form can have. The solution is taken from Chandrasekhar (1965). At higher final B_o, pinch-off occurs at the origin and the drop becomes a torus. It can be seen that at very low Ohnesorge numbers (0 and 0.1), corresponding to low viscosities, the drop shape stays very close to that of an ellipsoid. However, as viscosity is increased, the degree of central thinning increases although none of our simulations showed central thinning approaching the extent that it occurs for the equilibrium form shown.

Fig. 15

“Snapshots” of 5 tektite cross-sections with similar maximum radii. Numerical simulation results with an initial Bond number of 10 and Ohnesorge numbers of 0 (dotted line), 0.1 (solid line), and 1 (dash line) demonstrate the progression of central thinning. Also shown are an oblate ellipsoid with the same semi-major axis (dash-dot line) and an equilibrium form (instantaneous Bond number of 0.57) showing extreme central thinning (solid line with circles)

If we take characteristic values for the melt density, surface tension coefficient and radius to be 2,500 kg/m³, 0.5 N/m and R = 0.02 m, which we will use throughout the rest of this section, then B_o = 10 corresponds to an initial rotation rate of Ω = 45 rad/s while O = 0, 0.1 and 1 correspond to viscosities of 0, 0.71 and 7.1 Pas. These viscosities are on the low end of the possible values for these quantities (Elkins-Tanton et al. 2003). Using the physical parameters above, the time to reach a maximum radius of 1.73 for calculations with O_n = 0, 0.1 and 1 is 0.038, 0.04 and 0.094 s, respectively. Clearly for such low viscosity tektites, the deformation time is extremely short but will increase if the viscosity of the fluid is increased. For a drop of radius 0.02 m, we can estimate the characteristic time to cool as R²/κ where κ is the thermal diffusivity, typically about 10⁻⁶ m²/s, which gives 400 s. So it is likely that the tektites will have more time to deform. It is thus probable that tektites of this size are considerably more viscous than the values used here.

As might be expected from considering the radial flow during spin, equilibrium shapes bulge at the equator and flatten at the poles. This is displayed in Fig. 16, which shows the changing radius and height (normalized again by R) of a deforming drop, defined as the distance from the origin to the edge of the tektite along the radial and vertical axes as they vary with time throughout the simulations with B_o = 10 and O_n = 0, 0.1 and 1 (dotted, dashed-dot and dashed lines respectively). Also included on Fig. 16 are the radius and height for an oblate ellipsoid of constant volume (upper solid line) and the locus of points for equilibrium oblate shapes taken from Brown and Scriven (1980) (lower solid line). Since mass at the equator is distributed over a much larger area, the flattening at the pole is a bigger effect and it causes the equilibrium shapes to plot to the left of the ellipsoidal shapes. As can be seen, simulations with lower viscosity and with all other parameters held fixed evolve in a manner more similar to ellipsoids while higher viscosity drops are closer to equilibrium forms indicating that more viscous tektites are more likely to show significant central thinning.

Fig. 16

Shows the radius and height, defined as the distance from the origin to the edge of the tektite along the radial and vertical axes normalized by the radius of a sphere of equal volume as they vary with time throughout the simulations with a B_o = 10 and O_n = 0, 0.1 and 1 (dotted, dashed-dot and dashed lines respectively). Also included are the radius and height for an oblate ellipsoid of constant volume (upper solid line) and the locus of points for equilibrium oblate shapes taken from Brown and Scriven (1980) (lower solid line). b A plot similar to Fig. 16, but where O_n = 1 and B_o = 1, 10 and 100 (plus symbols, dotted and dashed line) while the equilibrium shapes and ellipsoid of constant volume are again plotted as the upper and lower solid lines. These Bond numbers correspond to rotational frequencies of 14.2, 45 and 142 rad/s. c Figure 16a overlain by data for the 119 vietnamites (open circles) and 75 thailandites (solid squares) that have Pr < 5°. The normalized tektite radius and height were calculated by dividing the measured L and T values by 2 R1 = 2 (3 M/(4 π ρ))^1/3 where M is the tektite mass and ρ is the mean density of the tektites (2,420 kg/m³)

In Fig. 16b we show a plot similar plot to that on Fig. 16a, but where O_n = 1 and B_o = 1, 10 and 100 (+ symbols, dotted and dashed line) while the equilibrium shapes and ellipsoid of constant volume are again plotted as the upper and lower solid lines. These Bond numbers correspond to rotational frequencies of 14.2, 45 and 142 rad/s. As can be seen, the simulation with initial B_o = 1 goes to an equilibrium form because the rotation rate decreases as the drop deforms and the time-dependent Bond number drops below 0.57 which is the critical value for pinch-off. The locus of shapes that the drop goes through are nearly equivalent to the equilibrium shapes. As the Bond number increases, the drop deforms more quickly and its shape becomes closer to that of an ellipsoid. The times taken to reach a radius of 1.73 for the simulations with B_o = 10 and 100 is 0.04 and 0.0147 s while the simulation with B_o = 1 and O_n = 1 took 0.44 s to reach its final form. It was found that for values of B_o of 10 or greater, that drops with the same ratio of B_o/O_n evolve in a very similar manner through radius-height space. As a result, we can conclude that fluid drops with B_o > 10 O_n will maintain an almost elliptical cross-section while those with 10 O_n > B_o may have significant central thinning.

The fact that high viscosity, low rotation rate drops that deform more slowly are more likely to show central thinning than low viscosity, high rotation rate drops, may at first seem counterintuitive. It will be recalled that unstable equilibrium shapes show a significant degree of central thinning and the force balance within equilibrium shapes is between centrifugal, pressure gradient and surface tension forces as viscous and inertial forces do not occur when velocity is 0. The region of central thinning around the rotation axis result, when it occurs, has negative curvature, which results in surface tension forces that are pulling up and toward the rotation axis to counter the centrifugal force. Slowly deforming drops have small inertial and viscous forces compared with centrifugal, pressure gradient and surface tension forces and so their shapes remain similar to equilibrium shapes. In lower viscosity, rapidly rotating drops, the effects of the inertia of the fluid in the drop are large and overwhelm the effects of surface tension near the rotation axis, preventing central thinning.

In Fig. 16c we repeat Fig. 16a and include a scatter plot using data for the 119 vietnamites (open circles) and 75 thailandites (solid squares) that have Pr < 5°. The normalized tektite radius and height were calculated by dividing the measured L and T values by 2 R₁ = 2 (3 M/(4πρ))^1/3 where M is the tektite mass and ρ is the mean density of the tektites (2,420 kg/m³). As can be seen, most of the measured tektites lie between the limits of an ellipsoid and the equilibrium shapes. The outliers to the right and above the ellipsoid line have a very slight prolate character. The largest concentration of tektites lies close to the evolution line for B_o = 10 and O_n = 1 indicating that for most of the tektites the B_o was similar to 10 O_n . If we assume that the deformation time scales roughly with O_n for large O_n we can estimate that the deformation time for these tektites is in the range of 0.1–100 s.

The shapes of real tektites almost certainly depend also on the variable viscosity throughout a tektite since the colder outer layer will likely be significantly more viscous. However, the rate of deformation provides a mechanism to explain the difference between tektites that are centrally thinned and those that are not and explains why there is not a significant correlation between tektites that are centrally thinned and their total degree of deformation. Tektites that deform rapidly but for a short time are likely to have little central thinning while those that deform more slowly but for a longer time are more likely to have undergone significant central thinning. As indicated earlier these calculations are limited to the oblate field, but we can conjecture that the more slowly deforming prolate tektites are also more likely to become centrally thinned and form dumbbells while the more rapidly deforming prolate tektites will form rods. Using the characteristic values for density, surface tension coefficient and drop radius stated previously, the characteristic time scale below which central thinning takes place appears to be roughly 0.1 s.

4 Discussion and Conclusions

We have proposed a new quantitative way to study the shapes of splash-form tektites based on the ratios of their largest (L), intermediate (W) and smallest lengths (T). By taking ratios of these lengths we can calculate diagnostics such as P_r and L/T which indicate the “prolateness” and degree of deformation for a given tektite. Using these diagnostics, we are able to give quantitative definitions for what were previously only descriptive terms and we have proposed a set of improved descriptors. This approach proves useful in modeling tektite formation as natural examples of spinning liquid drops.

Our analysis of a large number of main-field tektites from Vietnam and Thailand and a smaller sample from the Czech Republic reveals that the shape distribution of vietnamites and thailandites is very similar while the moldavite sample, which is much smaller, is broadly similar. This leads us to reason that splash-form tektites from other strewn fields will also have similar shape distributions, such as those from the North American and Ivory Coast fields (Koeberl 1994; McCall 2001). However, the details of the shape distribution may differ somewhat because tektites from different strewn fields may vary in distance from the points of impact, and may differ in composition and degree of heating, which will result in different viscosities. Three dimensional modeling of rotating fluid drops is underway and will help to elucidate the effects of viscosity on the shape distribution.

We have also shown that most splash-form tektites are flattened to slightly flattened round disks and bowls but there is also a reasonable concentration of classic bar/bowtie and rods/dumbbells with a “path-way” of slightly elevated tektite frequency connecting the flattened round discs to the classic rods/dumbbells and there are very few stubby prolate tektites. We can explain this aspect of the diagram if tektites first deform from a roughly spherical shape to a flattened round disc before undergoing a non-axisymmetric instability to become a prolate shape. Elkins-Tanton et al. (2003) showed an experimental example of a fluid drop that changed from an oblate shape to a prolate shape while in flight, and the equilibrium theory of Brown and Scriven (1980) demonstrated that for a range of drop angular momenta, there exist unstable oblate equilibrium forms and stable prolate forms, both of which argue for the likelihood of this mechanism. Three-dimensional simulations of fluid drops with surface tension will also help to test this hypothesis.

We have shown that the mass distribution of vietnamites and thailandites is also broadly similar with a roughly exponential decrease in tektite number with mass for masses greater than roughly 75 g. We suspect that the decrease in tektite numbers for masses less than this is a result of incomplete sampling of the smallest tektites. The largest tektites all have low P_r and intermediate L/T while the most strongly deformed tektites and those with the highest P_r are all small (<100 g).

For the teardrop tektites, the shape and mass distributions are again similar for the vietnamites and the thailandites. We have also compared the mass distribution of double teardrops with the dumbbell mass distribution on the assumption that teardrops are separated dumbbells, and we have determined that the largest dumbbells are most likely to separate into teardrops.

Our axisymmetric numerical simulations have demonstrated that liquid drops that deform slowly, such that 10 O_n > B_o, are most likely to remain close to equilibrium forms and hence have a significant degree of central thinning if their W/T ratio is greater than 2, while more rapidly deforming tektites are more likely to have a roughly ellipsoidal shape. The models presented in this paper are all axisymmetric and hence prolate shapes could not be investigated so we can only speculate at this point that dumbbell shapes with significant central thinning deform more slowly while rods, with little central thinning, deform more rapidly. However, the real shape of tektites is likely strongly affected by variable viscosity throughout the fluid drop caused by variations in temperature.

Our simulations were limited to relatively low viscosities, so the typical deformation time for the fluid drops was of the order of 0.01–0.1 s, while the characteristic cooling time for a 2 cm drop is likely to be of order a few 100 s. As such, it would be useful to develop a new numerical method that would be stable for longer periods of time and that could successfully integrate models with much larger values of the Ohnesorge number. The fact that drops can deform rapidly compared with the time required to cool may also imply that some drops have undergone deformation, separated, and then deformed again. It would also be useful to combine the fluid mechanical model with a thermal model that could calculate the evolving temperature inside the tektite.

In the future, we intend to continue our analysis of tektites by analyzing the patterns of bubbles and schlieren in order to gain further insight into their deformation history.