Numerical Study of Nonstandard Trajectories of Celestial Bodies Invading Earth’s Atmosphere

Abstract

The interaction of meteoroids with Earth’s atmosphere is studied. Based on the physical theory of meteors, a mathematical model of the trajectories of celestial bodies that have invaded Earth’s atmosphere has been constructed. This model considers rarely observed cases of a change in the mode of the descending movement of meteoric bodies to ascending with their possible return back into outer space. The kinematic conditions and physical characteristics that these bodies must satisfy in order to achieve such extraordinary behavior are determined.

INTRODUCTION

Everything related to the study of movements of small bodies of the Solar System in the near-Earth space, with their penetration into Earth’s atmosphere and collisions with it, is not only one of the problems of cosmogony—the question of the origin and functioning of our planetary system—but also a vital problem for humanity. Moreover, even objects that are small by cosmic standards, ~50–100 m in size, pose a real hazard to Earth’s population (the estimated size of the body that caused the 1908 Tunguska disaster was ~50 m).

Let us note some aspects of the asteroid-comet hazard problem that are rarely mentioned in scientific publications due to their uniqueness. These are, first, cases of celestial bodies with end-to-end flight trajectories. For example, on August 10, 1972, a bright bolide was recorded flying through the atmosphere, detected not only by American Air Force satellites [1], but also observed by a large number of people. Experts noted the bolide’s unusually long path through the atmosphere, ~1500 km. Eyewitnesses even heard sounds like thunder, which indicated that the object was moving at a low altitude. The object probably should have fallen to Earth, but this did not occur. The reason, apparently, is that the body flew at a small angle to Earth’s surface and, ricocheting from the layers of the atmosphere, returned back into outer space [2]. That is why, even when some effects appear as though from the fall, but in fact are from the impact of shock waves of the air “explosion” of meteoroid fragments, search expeditions often find neither impact craters nor fallen meteorite material on Earth’s surface. After the 1972 event, there were several more reports in the scientific literature about the end-to-end trajectories of celestial bodies. A review of such trajectories is given in [3].

In this work, using a computational experiment based on the physical theory of meteors, similar flight trajectories of large meteoroids in Earth’s atmosphere, which differ from the standard “starfall” trajectories, have been studied. As well, the problem of destruction of meteoric bodies under the action of thermal and force loads in the atmosphere is considered. It is shown that in some cases, the invasion of cosmic bodies into the atmosphere does not necessarily end with their fall to Earth, and at small angles of entry into the atmosphere, they can fly several thousand kilometers through it and return to outer space, which is explained by the curvature of Earth’s surface (curve 1 in Fig. 1). Complex flight trajectories of meteoroids with alternating stages of descending and ascending motion are also analyzed (curve 2 in Fig. 1).

Fig. 1.

Nonstandard flight trajectories of celestial bodies: (1) flight trajectory; (2) trajectory with alternating descending and ascending modes of movement.

FORMULATION OF THE PROBLEM AND BASIC EQUATIONS

A model describing the movement of a meteoroid in Earth’s atmosphere is considered. One of the important aspects of this model is determination of the law of motion of the center of mass of the meteoroid; another is the study of the parameters of flow around a body taking into account the effects of heat transfer and disintegration. Changes in the velocity of a meteoroid V, mass M, and the angle of inclination of the velocity vector to the horizon θ are described by the equations of the physical theory of meteors:

$$\begin{gathered} M\frac{{dV}}{{dt}} = Mg{\kern 1pt} \sin {\kern 1pt} \theta - {{C}_{D}}S\frac{{\rho {{V}^{2}}}}{2} - fV\frac{{dM}}{{dt}}, \\ MV\frac{{d\theta }}{{dt}} = Mg{\kern 1pt} \cos {\kern 1pt} \theta - \frac{{M{{V}^{2}}\cos {\kern 1pt} \theta }}{{{{R}_{E}} + z}} - {{C}_{N}}S\frac{{\rho {{V}^{2}}}}{2}, \\ {{H}_{{{\text{eff}}}}}\frac{{dM}}{{dt}} = - {{C}_{H}}S\frac{{\rho {{V}^{3}}}}{2}, \\ \frac{{dz}}{{dt}} = - V{\kern 1pt} \sin {\kern 1pt} \theta , \\ \frac{{dL}}{{dt}} = V{\kern 1pt} \cos {\kern 1pt} \theta . \\ \end{gathered} $$

(1)

Here, C_D, C_N, C_H are the coefficients of drag, lift, and heat transfer to the surface of the body, respectively; f is the reactive efficiency coefficient (–1 ≤ f ≤ 1); S is the cross-sectional area of the body; R_E is the radius of Earth; L and t are the flight range and time; ${{H}_{{{\text{eff}}}}}$ is the effective enthalpy of vaporization of the meteoroid material; z is the height of the meteor body above Earth’s surface. The reactive force in Eq. (1) can be neglected if the effective enthalpy of meteoroid disintegration ${{H}_{{{\text{eff}}}}} \geqslant 4.2$ kJ/g [4], as, e.g., for meteoroids with a chondritic structure ${{H}_{{{\text{eff}}}}}$ = 8 kJ/g.

The change in air density $\rho$ with height z is given by the formula

$$\rho = {{\rho }_{0}}\exp ( - z{\text{/}}h),$$

where ${{\rho }_{0}}$ is atmospheric density at z = 0 and h is the characteristic height scale. In Earth’s atmosphere for heights $z < 120$ km, the average value is h = 7 km.

The area of the midsection S in the general case is a variable, since the mass of the meteor body changes with height:

$$\frac{{{{S}_{e}}}}{S} = {{\left( {\frac{{{{M}_{e}}}}{M}} \right)}^{\mu }}.$$

Here and below, the subscript e corresponds to the parameters of the body entering the atmosphere. The parameter μ characterizes the effect of changes in body shape due to mass loss. At μ = 2/3, entrainment occurs uniformly over the entire surface and the body shape coefficient is maintained. A necessary condition for this is the rapid and random rotation of the meteoric body, ensuring uniform removal of mass from the entire surface. In another limiting case—oriented motion without rotation—maximum heating and, consequently, mass loss occur in the vicinity of the critical point of the body. This case is equivalent to the assumption of a constant midsection, i.e., $S$ = const, μ = 0.

To calculate the movement of meteoroids in the lower layers of the atmosphere, it is necessary to take into account changes in body mass. In a high-temperature gas flow, there are two mechanisms of heat transfer from gas to the surface of the body: convective heat transfer and radiative heat transfer. The formulas for calculating heat fluxes are given in [5].

Statistics of meteoroid falls show that most of them fell to Earth in fragments, so calculating mass loss requires taking into account the their fragmentation process. A celestial body can break up into several large fragments, which then travel autonomously, or break up into a cluster of small fragments, united by a common shock wave and traveling as a single whole. This cluster typically expands rapidly and decelerates during flight, causing a bright burst of radiation. When a large meteoroid breaks apart, both fragmentation scenarios can occur simultaneously.

In this study, the process of meteoroid fragmentation is considered using a progressive fragmentation model, taking into account the influence of the scale factor on the ultimate strength of the object. The statistical theory of strength is used [6], when it is considered that fragmentation occurs along defects and cracks that are inherent to such structurally heterogeneous bodies as meteoroids. As a result, fragmentation is represented as a process of sequential elimination of defects with increasing force load via breakup of the body along these defects. The fragments that appear in this way have greater strength than the original body. In this regard, the fragmentation process ends as soon as the velocity head begins to decrease. This model is presented in detail in [5, 7].

The problem of the movement of a fragmenting meteor body is solved in three stages. At the first stage, the movement of a single body from the height of entry into the atmosphere to the height of the beginning of fragmentation is considered; at the second, movement of a swarm of fragments from the height of the beginning of fragmentation to the height of maximum velocity head. At the third stage, when the fragmentation process is already completed, it is assumed that the fragments move independently, and the movement of one fragment is considered, since it is assumed that all the resulting fragments are of the same size.

In this case, the strength of the fragment is written as

$$\sigma _{f}^{*} = {{\sigma }_{e}}{{\left( {{{{{M}_{e}}} \mathord{\left/ {\vphantom {{{{M}_{e}}} {{{M}_{f}}}}} \right. \kern-0em} {{{M}_{f}}}}} \right)}^{\alpha }},$$

(2)

where ${{\sigma }_{e}}$, $\sigma _{f}^{*}$, ${{M}_{e}}$, ${{M}_{f}}$ are the ultimate strength and mass of the meteoroid upon entry into the atmosphere and for the fragment; $\alpha$ is an indicator characterizing the degree of heterogeneity of the collapsing material [6, 7] (the more heterogeneous the material is, the greater is its value). It should be noted that when $\alpha \to 0$, the meteoroid is broken up into tiny fragments that move in hydrodynamic mode, like a large mass of a fragmented body, which is described, e.g., by the model [8]. For large values $\alpha$, no fragmentation occurs and the body moves as a single unit. The value of parameter $\alpha$ for stony meteoroids, as a rule, is 0.1–0.5 [9].

The behavior of a celestial body depends on the ratio of its strength characteristics (compression, tension, shear) and the magnitude of the velocity head, which monotonically increases to the maximum value with decreasing flight altitude.

The condition for the start of breakup of a bolide in the atmosphere is as follows:

$${{\rho }_{ * }}V_{ * }^{2} = \sigma \text{*},$$

(3)

where on the left is the magnitude of the velocity head, and $\sigma \text{*}$ is one of the strength characteristics of the meteoroid material. Parameters with an asterisk refer to the moment fragmentation begins. If relation (3) is not satisfied on the trajectory, then the celestial body passes through the atmosphere without fragmentation, as a single body.

The height of the onset of fragmentation ${{z}_{ * }}$, taking into account ρ = ρ₀exp(−z/h), is determined as

$${{z}_{ * }} = h{\kern 1pt} \ln \left( {{{{{\rho }_{ * }}V_{ * }^{2}} \mathord{\left/ {\vphantom {{{{\rho }_{ * }}V_{ * }^{2}} {{{\sigma }^{ * }}}}} \right. \kern-0em} {{{\sigma }^{ * }}}}} \right).$$

(4)

If we consider the progressive fragmentation model, then, starting from this height, instead of a single body, a swarm of breaking fragments falls with an ever-increasing number N, the strength of which depends on their mass ${{M}_{f}}$ according to law (2).

Assuming that the resulting fragments are spheres of the same mass ${{M}_{f}}$ (${{M}_{f}} = {M \mathord{\left/ {\vphantom {M N}} \right. \kern-0em} N}$), using Eqs. (2)–(4), we can find their number depending on the current values of the velocity head and the total mass of all fragments:

$$N = \frac{M}{{{{M}_{ * }}}}{{\left( {\frac{{\rho {{V}^{2}}}}{{{{\rho }_{ * }}V_{*}^{2}}}} \right)}^{{1/\alpha }}} = \frac{M}{{{{M}_{ * }}}}{{\left( {\frac{{\rho {{V}^{2}}}}{{\sigma \text{*}}}} \right)}^{{1/\alpha }}}.$$

If we consider the movement of a swarm of fragments, then the effective area of the midsection of this swarm depends on the number of pieces formed. If we assume that the resulting pieces of the same mass do not overlap, then we obtain the following formula for determining the effective midsection area of a swarm of fragments:

$$S = {{S}_{ * }}\frac{M}{{{{M}_{ * }}}}{{\left( {\frac{{\rho {{V}^{2}}}}{{{{\rho }_{ * }}V_{*}^{2}}}} \right)}^{{1/\left( {3\alpha } \right)}}}.$$

(5)

According to this model, starting from a height ${{z}_{ * }}$, a swarm of fragmented fragments moves, surrounded by a common shock wave, with a progressively increasing number of fragments. If we study the ballistics of a swarm of fragments, then we can use the equations of motion as for a single body, but with a variable midsection area determined by formula (5).

RESULTS AND DISCUSSION

Let us consider how the movement of a large stony meteoroid with a mass $M = {{10}^{9}}$ kg and density ${{\rho }_{b}} = 3 \times {{10}^{3}}$ kg/m³, which entered the atmosphere at height ${{z}_{e}} = 100$ km with velocity ${{V}_{e}} = 30$ km/s, at different angles of entry into it ${{\theta }_{e}}$. It is believed that the body has an ideal spherical shape, i.e., ${{C}_{D}} = 1$, ${{C}_{N}} = 0$.

If we consider the problem in the single body approximation without taking into account ablation and fragmentation, then the data in Fig. 2 show how the height of the meteoroid changes depending on time for different angles of its entry into the atmosphere. Flight trajectories depend significantly on the parameter ${{\theta }_{e}}$. From the above results, it is clear that when ${{\theta }_{e}} > 9^\circ$, a meteoroid with such parameters will fall to Earth, and if ${{\theta }_{e}} \leqslant 9^\circ$, starting from a certain height, its trajectory becomes ascending. At ${{\theta }_{e}} \leqslant 9^\circ$, the angle of inclination of the trajectory changes sign over time and its final segment becomes ascending. We call such trajectories flight trajectories (see curve 1 in Fig. 1). The smaller the value of this parameter, the sooner the sign changes, so for ${{\theta }_{e}} = 5^\circ {\text{,}}\,\,{\text{7}}^\circ {\text{,}}\,\,{\text{9}}^\circ$, the transition to the ascending branch of the trajectory occurs at time values $t = 20,\;30,\;40$ s, respectively. At the initial atmospheric angles of entry ${{\theta }_{e}} > 10^\circ$, a “hard” collision of meteoroids with Earth’s surface occurs at a sufficiently large angle, which can lead to tragic consequences for the environment, infrastructure, and people, depending on the fall velocity.

Fig. 2.

Flight altitude $z$ on flight time t at different angles of entry into atmosphere ${{\theta }_{e}}$.

In cases of flight trajectories at small angles of entry into the atmosphere (${{\theta }_{e}}$ = 5°‒7°), meteor bodies do not enter the dense layers of the atmosphere at all but penetrate the upper atmosphere in free-molecular flow mode, without experiencing hardly any resistance and, as calculations show, without losing their velocity; i.e., for the body under consideration at angles of entry ${{\theta }_{e}} \leqslant 7^\circ$, one can analyze the motion trajectory using the single body model. The critical entry angle for the single body model ${{\theta }_{e}} = 9^\circ$ at which the trajectories become flight trajectories, is also confirmed by the estimates given in [10].

However, if we take into account the effects of surface ablation and meteoroid fragmentation, the results for determining the critical angle of entry of a body into the atmosphere will be different. Let the body under consideration, entering the atmosphere at an angle of 9°, have a critical strength parameter of $\sigma \text{*}$ = 10⁵ N/m², then its fragmentation, according to (4), begins at a height of 67.4 km, and for the degree of material heterogeneity $\alpha$ = 0.25, the maximum number of fragments formed is N ~ 7.9 × 10⁵. In the case of a more strong meteoroid ($\sigma \text{*}$ = 10⁷ N/m²) fragmentation begins at a height of 35.3 km and at the same value of parameter $\alpha$, the maximum number of fragments N = 126. Figure 3 shows how the flight altitude changes $z$ in time t at the given values of the strength parameter $\sigma \text{*}$ and taking into account ablation losses of meteoroid mass.

Fig. 3.

Dependences of flight altitude $z$ on time t at angle of entry into atmosphere ${{\theta }_{e}}$ = 9°: (1) $\sigma \text{*}$ = 10⁵; (2) 10⁷ N/m², (3) without taking into account fragmentation.

Figure 3 also shows the flight altitude as a function of time (curve 3), obtained without taking into account fragmentation of the body, but taking into account the loss of mass under the action of heat fluxes. All three trajectories are not flight trajectories; i.e., in cases with ${{\theta }_{e}} = 9^\circ$, a celestial body falls to Earth’s surface. Note that in the approximation of a single body model taking into account ablation, the mode of motion of the body temporarily switches from the descending to ascending stage of flight (curve 3), but significant deceleration of the body in dense layers of the atmosphere makes the trajectory descending again. As well, the flight range of the meteoroid L along Earth’s surface, calculated from the projection onto it of the point of entry of the body into the atmosphere to the point of impact of the fragments for the calculation options presented in Fig. 3, is quite large (~1000 km).

Figure 4 shows the results of calculating the meteoroid’s flight altitude depending on time at an angle of the meteoroid’s entry ${{\theta }_{e}} = 8^\circ$ and the value of the strength parameter $\sigma \text{*}$ = 10⁶ N/m². The results were obtained for two values of the heterogeneity index of the meteoroid material: $\alpha$ = 0.5, 0.25. Figure 4a also shows the trajectory for a nonfragmenting meteoroid, but with allowance for ablation of its surface (curve 1). If the number of fragments is relatively small (for $\alpha$ = 0.5, N = 70) or no fragmentation of the body occurs at all, then the trajectories turn out to be flight trajectories. At $\alpha$ = 0.25, the number of fragments formed reaches 2.3 × 10³; the result is a trajectory with alternating descending and ascending flight stages and debris falls to Earth’s surface. The number of fragments formed, associated with the structural heterogeneity of the body, significantly affects its ballistics.

Fig. 4.

Dependences of flight altitude $z$ on time t for angle of entry into atmosphere ${{\theta }_{e}}$ = 8° and at $\sigma \text{*}$ = 10⁶ N/m²: (a) 1, without taking into account fragmentation; 2, taking into account fragmentation, $\alpha$ = 0.5; (b) taking into account fragmentation, $\alpha$ = 0.25.

As the calculation results show, for bodies smaller than the meteoroids considered above, the flight mode of motion is achieved at lower angles of entry into the atmosphere. In addition, an irregular geometric shape can have a significant impact on the trajectory of the meteoroid; i.e., its trajectory will curve up or down depending on the sign of the lift-to-drag coefficient.

In [11], several more reasons for changes in the flight direction of celestial bodies are considered. Thus, rapidly rotating bodies can move along curved trajectories due to the Magnus effect: the rotation of the meteoroid in the oncoming flow creates additional force, while the axis of rotation of the body describes a cone, like a rotating top.

Interesting configurations of trajectories can arise in cases where the speed of entry of a meteoroid into Earth’s atmosphere is significantly lower than those considered above. In such cases even at small angles of entry ${{\theta }_{e}}$ (when flight trajectories are achieved at high velocities), the body can reach the dense layers of the atmosphere, then change the mode of motion to ascending, then slow to a velocity less than the second cosmic velocity, and ultimately still fall to Earth. Figure 5 shows example of calculating such movement at an initial velocity ${{V}_{e}} = 12.8$ km/s and atmospheric angle of entry ${{\theta }_{e}} = 7.2^\circ$ for a meteoroid with a mass $M = 4 \times {{10}^{5}}$ t. Curve 1 corresponds to a strong meteoroid that moves through the atmosphere without fragmentation. The meteoroid flight range L along Earth’s surface in this case is 3369 km, and the flight time is ~400 s. Curve 2 corresponds to the critical value of the strength parameter at which the fragmentation process begins, $\sigma \text{*}$ = 10⁷ N/m² and parameter of heterogeneity of the body material $\alpha$ = 0.25. In this case, an ascending–descending mode of movement of the body is also observed, and when $\sigma \text{*}$ = 10⁶ N/m² (curve 3) the usual “starfall” trajectory occurs. In this case, a less strong meteoroid produces a larger number of fragments than for $\sigma \text{*}$ = 10⁷ N/m², and they decelerate faster in the atmosphere.

Fig. 5.

Dependence of flight altitude $z$ on time t of body with mass $M = 4 \times {{10}^{5}}$ t at ${{V}_{e}} = 12.8$ km/s and ${{\theta }_{e}} = 7.2^\circ$: 1, without taking into account fragmentation; taking into account fragmentation: 2, $\sigma \text{*}$ = 10⁷ N/m²; 3, 10⁶.

Thus, the calculation results show that the mode of motion of a meteoroid depends on many factors: its size, velocity, strength characteristics, and angles of entry into the atmosphere.

CONCLUSIONS

Mathematical modeling was carried out based on the equations of the physical theory of meteors and the movement of large celestial bodies in Earth’s atmosphere. In the course of a numerical experiment, certain physical characteristics were determined that meteoroids must satisfy in order for their mode of motion to change from descending to ascending in the atmosphere at some point in time. In this case, surface ablation and mechanical fragmentation of meteoroids under the influence of thermal and force loads were taken into account. A defining criterion has been identified for which trajectories are realized for meteoroids without collision with Earth. It turned out to be the angle of entry into the atmosphere. The movement of the body at angles less than the critical value makes it possible to explain the previously incomprehensible results of searches at the “inferred impact site”: the absence of traces of an impact crater and any remains of meteorite material.