Uncontrolled reentry of Low Earth Orbit Decaying Objects: a hidden threat to global safety and legal frameworks

Abstract

This study investigates the dynamics of uncontrolled Low-Earth Orbit Decaying Objects (LEODOs) using a high-fidelity simulation framework that incorporates real-time atmospheric winds (NRLMSISE-00), variable drag from wind-relative motion, Coriolis effects, and Earth’s geopotential up to the sixth zonal harmonic (J₆). Unlike previous models, our approach resolves fine-scale thresholds in reentry behavior. We identify a narrow transition regime near escape velocity (e.g., 10.5 km/s), where small changes in elevation angle (Δγ ~ 0.1°) produce distinct outcomes: impact, atmospheric skipping, or orbital retention. For azimuths of 135° and 315°, the reentry threshold is γ = −2. 9°. In some cases, impact occurs in under 10 min, within the uncertainty margins of current forecast systems. These findings reveal limitations in deterministic reentry prediction models and support the need for probabilistic approaches to account for trajectory sensitivity. The results have implications for forecasting, space safety, and international response planning.

Introduction

Orbital debris (OD), or space junk, is a growing problem caused by the increasing amount of human activity in space, particularly satellite launches and space exploration. OD arises from launch artifacts, decommissioned satellites, and spacecraft breakups. More debris is created by the Kessler syndrome, which is the collisions between orbiting satellites and existing debris¹. This situation is exacerbated by the absence of a global consensus and regulation, and overexploitation of the space. This potentially stifles further space innovation. The World Economic Forum estimates that orbital debris currently exceeds 6.3 million kilograms, with projections indicating up to 60,000 satellites at risk of contributing to this problem by 2030^2,3,4. At present, human activity witnesses the heaviest launch traffic into low-Earth Orbit (LEO), occupying the region of space approximately from 160 km to 2000 km above the Earth’s surface. This region is home to Earth weather observation and imaging satellites, communication satellite constellations and networks such as Starlink and iridium, and space exploration missions such as the International Space Station. It is also used for military and reconnaissance purposes. The launch traffic to LEO, which poses 96% the debris reentry problem, is estimated to have more than tripled in the last decade⁵, with approximately 80% of the active payloads inhabiting LEO. OD is therefore an issue that requires urgent global attention^2,3,4,5. Attempts are currently being made to catalog and track objects that are in various orbits around the Earth⁶.

Most prior studies rely on global static atmospheric models like NRLMSISE-00 without accounting for real-time wind shear⁷. In contrast, our simulation dynamically integrates altitude- and latitude-dependent wind fields from NRLMSISE into trajectory solutions, revealing transient skip and rebound behavior in marginal entry geometries that is rarely captured in existing literature.

For context, suppose that a 1 kg LEO object is orbiting 160 km above Earth. It has an orbital speed (v = (GM_E/r)^1/2) ≈ 7.5 km/s, a kinetic energy of 30.5 MJ and a potential energy (mgh) of 61.1 MJ relative to the Earth’s surface. Initial reentry speeds, defined at approximately 100 km altitude (the Kármán line), have been reported to reach 12.5 km/s. The Kármán line is taken as the edge of the atmosphere and the beginning of space. For reference, 1 kg of TNT explosive releases approximately 4.2 MJ of energy. Thus, only the kinetic energy of this LEO is comparable to that of a small conventional air-to-surface missile carrying an explosive equivalent of 12 kg of TNT^8,9,10,11. At this speed, a LEO can traverse a complete orbit in 90–120 min, depending on its altitude. This means that any point on the Earth’s surface is within reach of the object in a short time. Assuming that there is no loss of speed due to its flight environment, such an LEO object releases an energy closer to 61.1 MJ when it crashes on the surface of the Earth. In reality, this kinetic energy is expended as friction in the atmosphere, which is responsible for the extremely high temperatures of reentry in the atmosphere.

In recent years, there has been a steady increase in the number of reports of suspected ODs crashing in countries, most without a space program of their own. Table 1 lists a handful of recent LEODO events. It highlights some challenges in LEODO classification. A persistent challenge in addressing the legal implications of LEODOs lies in the difficulty of identifying debris, often complicated by uncertainty and lack of definitive attribution. This underscores the relevance of proposed identification solutions in recent literature¹², e.g., improved tracking and object tagging strategies for post-reentry accountability. A debris field can be extensive, as seen in the case of western Uganda, where it spanned three districts, and with a Delta II PAM-D stage, which fell in the United States and Saudi Arabia. Identification is complicated because the debris is often charred during reentry. The sensitive nature of the objects involved adds a layer of complexity when attempting to access detailed information from their sources. Therefore, high-level international cooperation is essential for assigning responsibility in cases of damage and claims.

Table 1 Some recent LEODO events reported by the indicated authority or published in various international media

LEO objects naturally reenter Earth’s atmosphere as a result of atmospheric drag, which slows them down over time. At the orbital heights characterizing LEO, the atmosphere is greatly diminished but not zero. As altitude decreases, friction with the atmosphere generates heat, causing most objects to burn up, though larger debris may survive and crash onto the Earth’s surface. Factors such as initial orbital altitude, velocity, and atmospheric drag influence when and how objects reenter. Although some objects may reenter under control to avoid falling in populated areas, most reenter on their own volition. Advances in space-age materials impart some amount of temperature resilience to many of these objects, making them less prone to burning up on reentry. In sum, the scale of the debris reentry problem implies increased threats to inhabited areas, installations, and infrastructure. A further hazard is the potential for catastrophic mid-air collisions between a reentering object and an airborne aircraft. This is a real possibility given the amount of air traffic at any given time around the Earth. A significant number of space launch failures occur within the first 8 min of flight. To reach Low Earth Orbit (LEO), a rocket must accelerate rapidly during its first stage, shedding mass as fuel burns while combating atmospheric drag. Reaching an orbit requires covering 1500–2500 km of ground track through a gravity turn, shifting from vertical ascent to horizontal motion. In the first 8 min, the rocket accelerates past 7.5 km/s while gaining altitude, influenced by the thrust-to-weight ratio, trajectory, and staging. Some debris reenters the atmosphere during this phase.

The legalities surrounding damage caused by LEODOs in physically remote areas involve space law, liability, and international agreements. The Outer Space Treaty of 1967 holds launching states responsible for any harm caused by their space activities, whether from debris, malfunctioning satellites, or uncontrolled reentry¹³. The Liability Convention of 1972 further specifies that any damage to another country by space objects requires compensation from the responsible state¹⁴. Damage to remote areas may be more consequential, as fallen objects can introduce concentrated materials harmful to ecosystems such as oceans, rivers, and other sensitive environments, with environmental costs often exceeding direct impact damage.

This article investigates and highlights the destructive potential posed by uncontrolled LEODOs on inhabited parts of the planet, highlighting the rarely discussed OD hazard from which all points of the planet are not immune from the impact of OD. The physics of atmospheric reentry has been extensively modeled using both analytical approximations and high-fidelity simulations. Traditional reentry models have focused on controlled spacecraft, considering well-defined parameters for capsule or satellite return^15,16,17. A significant body of work, including that by the European Space Agency and NASA, focuses on estimating reentry windows for objects with known orbital states^18,19. However, many existing models rely on averaged atmospheric profiles and exclude finer nonlinearities associated with shallow-angle entries and high-speed nonlinear transition regimes.

The methodology relies on a plausible computer simulation to calculate the trajectory and impact coordinates under various reentry conditions. Specifically, the simulations take into account the initial conditions of the object, atmospheric and rotational effects, such as wind speed and the Coriolis effect^20,21 to determine the possible impact coordinates. The Python simulation program runs on Google Colaboratory²². The calculations highlight that the high crash site variability of LEODOs subjects the surface to considerable hazards. Uncertainty in predicting the location of space debris reentry is often high due to the complex and variable conditions during descent. A prominent example is Cosmos 482, a failed Soviet Venus probe launched in 1972. Upon launch, the vehicle fragmented. Two pieces reentered and disintegrated within 48 h, while a third entered a highly elliptical Earth orbit, awaiting interplanetary transfer. However, the failure of its propulsion system left it stranded in a slowly decaying Earth orbit for decades. The exact landing site of Cosmos 482 remains uncertain, although the best estimates place it near Jakarta, Indonesia. The European Space Agency (ESA) continually updated its expected reentry at 06:16 UTC on 10 May 2025, with a temporal uncertainty of ±22 min²³. This case illustrates that LEODOs can persist in orbit far longer than expected and eventually reenter. Notably, Cosmos 482 was engineered to withstand the comparatively more extreme atmospheric conditions of Venus, increasing the probability that it survived reentry intact, with a potential for greater damage upon impact. Launched during the Cold War, Cosmos 482 likely used radioisotope thermoelectric generators (RTGs) for power and nickel-hydrogen batteries for energy storage, though details remain undisclosed. The presence of such systems raises potential environmental concerns at the site of their eventual impact. Furthermore, the simulations below show that the objects can reach the surface in 10 min, well within the predicted uncertainty window in the reentry of Cosmos 482, further complicating damage mitigation strategies.

Results

The simulations in this article solve a system of initial-value ODEs numerically in Python code for the object reentering the Earth’s atmosphere. The outputs describe the most likely impact coordinates for an object in a slowly decaying low-earth orbit (LEO). The initial conditions of the ODEs are the entry coordinates relative to the center of the Earth, and simulation speeds of 7.5 km/s and 11.5 km/s. The lower speed is near the threshold for low Earth orbit at a higher altitude. The higher speed is slightly above the escape velocity of approximately 11.2 km/s near the surface of the Earth. In general, several simulations showed that different initial latitudes and longitudes produced similar tracks, particularly at lower reentry speeds, but 0° latitude, 0° longitude and azimuth angles of 315° and 135° were used for clarity of illustration on the geodesic plots. Figure 1 plots object trajectories, each starting from a unique initial coordinate, but at 100 km altitude and 7.5 km/s velocity. The nearly symmetrical trajectories for diametrically opposite points suggest that the application of the ECIF modeling abstraction is robust. Subsequently, all objects had initial azimuths of 135° and 315°, and 0° latitude and longitude. Crucially, the simulations suggest that no region on Earth’s surface is inherently unreachable by a returning object at these speeds and entry angles. This underscores the theoretical global accessibility of reentry vehicles from orbital or suborbital trajectories, given sufficiently accurate control of initial conditions.

Fig. 1: Geodesic plots generated by the Python simulation showing trajectories with different initial coordinates.

The similarity of the trajectories from diametrically opposite initial starting points suggests robustness of the application of the ECIF.

The simulations below examine the altitude profiles of objects that reenter the atmosphere of the Earth from an initial altitude of 100 km, with entry velocities of 7.5 km/s, 10.5 km/s and 11.5 km/s. These velocities correspond, respectively, to speeds below the gravitational escape threshold, near the threshold, and slightly above it. The resulting profiles reveal distinct and noteworthy characteristics associated with each velocity regime. The simulation incorporates aerodynamic drag, atmospheric density variations, wind dynamics, and the rotational influence of Earth, all of which play crucial roles in shaping the trajectory.

Velocities below escape threshold

Figure 2a, b illustrates the evolution of altitude-time-varying elevations for two azimuth angles at an initial velocity of 7.5 km/s. Figure 2a highlights the primary deceleration trends, emphasizing how aerodynamic resistance governs descent. Figure 2b provides a comparative view, showing the effects of wind dynamics on the trajectory. For γ ≳ 0°, the object initially gains height because the elevation effectively points it away from Earth.

Fig. 2: Altitude-time profile of reentering object with initial velocity of 7.5 km/s obtained from the Python simulation.

The azimuth angles are a 315° and b 135° over a wide range of elevation angles. In both cases, the object initially gains height when γ ≳ 0°.

As the object begins its descent, gravitational acceleration dominates, resulting in a rapid initial loss of altitude. However, as the atmospheric density increases with decreasing altitude, aerodynamic resistance becomes a significant factor. Table 2 shows the flight parameters of the object. At high positive elevation angles, such as γ = 20°, the initial velocity vector is pointed steeply upward, resulting in a trajectory that may not descend within the simulation time frame. In such cases, the object can reach altitudes well beyond 1000 km, where the atmospheric density models used (NRLMSISE-00 and COESA76) become unreliable or undefined. This explains the absence of a landing event for these conditions and reflects the limits of the reentry-focused simulation framework, rather than indicating a numerical or computational error. In this case, the program’s landing event is not triggered, but the simulation runs to the full 7200 s time span, whereupon the object speed reported is at that time. This explains the absence of a landing event and reflects the limits of the reentry-focused model rather than a computational error.

Table 2 The simulated object flight parameters from the given azimuth (θ) and elevation (γ)

This resistance leads to a gradual reduction in velocity, resulting in a more controlled descent. The effect of atmospheric wind introduces further complexity into the trajectory. Localized wind variations cause slight oscillations in the altitude-time profile, leading to minor fluctuations in the descent rate. Additionally, the Earth’s rotation exerts Coriolis forces that subtly influence the trajectory, causing lateral deviations that are more pronounced at higher altitudes. Figure 3a, b shows the ground tracks of both these scenarios on the geodesic map.

Fig. 3: Ground track trajectories showing the Coriolis effect.

The reentering objects all have an initial velocity of 7.5 km/s but reenter at azimuths of either 315° or 135°.

In conclusion, the descent profile confirms that aerodynamic resistance plays a dominant role in shaping the trajectory, while atmospheric wind and rotational effects introduce additional variations. These factors must be taken into account in future reentry modeling to ensure precise trajectory predictions.

Escape threshold dynamics

The descent trajectories presented in Fig. 4a, b illustrate the pronounced sensitivity of atmospheric entry profiles to initial elevation and azimuth for an initial speed of 10.5 km/s. The range of elevation angles considered in Table 3 is narrower than that in Table 2, as it targets the critical transition region near the reentry-orbit boundary. Due to the strong sensitivity of the results in this regime, a finer sampling over a limited span was required to accurately identify the elevation threshold for orbital escape. The simulations incorporate a comprehensive set of physical effects, including wind shear, variable atmospheric density, and Coriolis forces owing to Earth’s rotation, to emulate realistic entry conditions. In Fig. 4a, which corresponds to an azimuth of 315°, the descent profiles for elevation angles ranging from −3.5° to −3.2° exhibit a systematic progression in both the deceleration rate and the terminal altitude. Trajectories at lower elevation angles (for example, −3.5° and −3.4°) show a more rapid altitude decay, leading to ground impact within approximately 750 s. As the elevation angle increases toward −3.2°, the deceleration becomes less abrupt and the object retains higher altitudes for longer periods of time, indicating increased atmospheric glide and the potential for an extended ground track. Figure 4b, corresponding to an azimuth of 135°, shows a similar trend with one notable distinction: the trajectory at −3.2° undergoes a temporary re-ascension at approximately 200 s, reaching an altitude of about 83 km before descending again. This rebound effect suggests that the dynamic coupling of the initial flight-path angle, atmospheric lift, and Earth’s rotational frame can lead to quasi-bouncing behavior in marginally shallower entries. The lift-induced delay in descent observed here highlights the potential for controlled skip reentry or extended atmospheric braking in mission designs targeting specific landing footprints.

Table 3 Simulated object flight parameters at the given θ and γ

Fig. 4: Descent Altitude-Time Profiles from 100 km at 10.5 km/s.

Simulated profiles from a Python model for azimuths of a 315° and b 135°, showing the effect of varying low initial elevation angles. Curves represent different elevation angles; lower angles yield shallower descents.

In general, these results emphasize the critical dependence of descent dynamics on the interplay of initial flight configuration and atmospheric interaction. Small elevation changes Δγ ~ 0.1° can substantially alter the descent profile, with implications for the distribution of the thermal load, the requirements of aerodynamic control, and the location of the impact. These findings reinforce the necessity of precision reentry targeting, particularly when navigating the complex transitional regime between exoatmospheric motion and dense atmosphere flight.

In the last few seconds before impact, the altitude-time profiles have near-constant gradients from roughly 10 km above the Earth. Table 3 shows the simulated flight parameters of the object.

Figure 5 shows the ground tracks of these objects. The longer ground tracks correspond to lower initial entry elevations. The simulations show that below −3.4° for lower initial speeds, the object does not land, but can either remain in low-earth orbit or fly off into space. The map plot illustrates a range of simulated reentry and orbital trajectories. Colored lines trace the ground tracks of space objects as they evolve under the influence of gravity, atmospheric drag, and Earth’s rotation. In cases where the object reenters the atmosphere, the trajectory terminates at an altitude of 100 meters, with the predicted landing location marked by a cross symbol (‘X’) in the same color as its trajectory. These markers link each descent path and its impact point. In contrast, when the initial conditions favor orbital retention, the trajectories appear as sinusoidal curves, reflecting the periodic ground tracks of objects that remain in a low-Earth orbit. To refine the transition boundary between reentry and orbital retention, we performed additional simulations in the elevation range −3.4° ≤ γ ≤−2.6° at an initial speed of 10.5 km/s, for azimuths of 135° and 315°. In both cases, we find that the upper elevation limit beyond which the object remains in orbit is γ = −2.9°. This threshold represents a sharp boundary between atmospheric reentry and orbital escape, which confirms the sensitivity of the model to small variations in initial conditions near the critical regime. These curves result from the projection of continuous orbital motion onto the two-dimensional geographic map.

Fig. 5: Ground track trajectories of objects with initial azimuths of 315° (landing left of 0° longitude) and 135° (landing right of 0° longitude), for varying initial low-elevation angles, simulated in the Python program.

The elevation threshold between reentry and orbital retention was refined through additional simulations at 10.5 km/s for azimuths of 135° and 315°. In both cases, the upper elevation limit for reentry was confirmed to be γ = −2.9°. The horizontal blue line is a simulation artifact that arises from a documented but cosmetic issue of GeoPandas (via Matplotlib) when handling cyclical longitude.

Beyond escape velocity

Figure 6 presents the altitude-time profiles of four trajectories resulting from launches executed with an initial speed of 11.5 km/s, at varying θ and γ. These simulations were conducted to assess the sensitivity of orbital outcomes to launch geometry at velocities near Earth’s escape threshold.

Fig. 6

Simulated altitude-time profile around the known escape velocity of Earth using the Python program.

The trajectories labeled (135°, 0°) and (315°, 0°) exhibit a steep and sustained increase in altitude, with no observable return toward the Earth within the simulation window. Both trajectories exceed an altitude of 10⁶ m, indicating that the objects have overcome Earth’s gravitational potential well. These results are consistent with hyperbolic escape trajectories, where the total mechanical energy of the system is positive. The purely horizontal elevation angle (0°) appears to favor the conversion of kinetic energy to form an effective escape trajectory. In contrast, the trajectories corresponding to (135°, −3.5°) and (315°, −3.5°) show markedly different behavior. These objects ascend to a finite maximum altitude before reversing course and descending, demonstrating the classical signs of gravitational trapping. Despite being launched at the same initial speed, the slight negative elevation angle imposes a critical limitation: a portion of the velocity vector is directed downward or insufficiently outward, resulting in insufficient kinetic energy to achieve gravitational escape. The total mechanical energy in these cases remains zero or negative, characterizing either ballistic suborbital arcs or closed elliptical trajectories that remain bound to Earth.

This analysis suggests that launch geometry is important in achieving orbital escape, even at velocities exceeding the canonical escape speed of 11.2 km/s. The findings reaffirm that escape depends not only on speed but also on the vector directionality in the gravitational field.

Discussion

This study contributes to the growing discourse on orbital debris by integrating high-fidelity reentry simulations with global safety and legal considerations and highlighting the complex and nonlinear dynamics that govern the reentry of LEODOs. Although the underlying physics of reentry is well known, our work distinguishes itself by demonstrating how small variations in initial conditions can lead to significant uncertainties in impact location, often within timeframes too short for effective mitigation. By openly sharing a Python-based simulation framework that incorporates atmospheric drag, wind dynamics, and gravitational perturbations, we provide a reproducible and accessible tool for researchers and policymakers alike. Our findings underscore the global vulnerability to uncontrolled reentries and highlight the urgent need for coordinated international response protocols and legal accountability mechanisms.

In contrast to traditional analytical models that treat drag and gravitational perturbations in simplified frames^18,23, our simulation integrates perturbations up to the J6 zonal harmonics and includes empirical wind models and Coriolis corrections, providing a more realistic profile of possible LEODO behavior under varying initial conditions. Compared to ESA’s deterministic predictions (e.g., Cosmos 482 with ±22 min uncertainty), our results emphasize a critical blind spot: A trajectory that can result in a surface impact in less than 10 min is not only plausible, but likely under certain reentry conditions. Such short time-to-impact scenarios are not well explored in current predictive frameworks.

Through extensive numerical simulations at 7.5 km/s, 10.5 km/s, and 11.5 km/s, we demonstrate that small variations in the angle of entry elevation can lead to drastically different results, including direct reentry, atmospheric skipping, extended glide or escape. The transition between these regimes is especially sensitive near critical angles, where perturbations as small as 0.1° produce multi-million-meter variations in apogee or flight time. Critically, several high-speed trajectories result in surface impact in less than 10 min—an interval that falls well within the typical uncertainty windows reported for known events such as the reentry of Cosmos 482. This reveals a fundamental vulnerability in our global reentry prediction infrastructure. Even when reentry timing is forecast with high accuracy, the short time window between atmospheric entry and impact poses a significant challenge for mitigation, response, or evacuation. By quantifying these nonlinear sensitivities across a broad range of initial conditions, this work exposes key limitations in current modeling and tracking strategies and underscores the urgent need for more robust forecasting systems and coordinated international response protocols to address the growing risk posed by uncontrolled orbital debris. Conventional reentry prediction models generally assume that impact occurs with a predictable deceleration profile with long warning times. However, our simulations demonstrate that for entry speeds between 10.5–11.5 km/s and shallow angles (−3.5° < γ < −3.1°), objects skip, orbit, or impact depending on minute perturbations, showing characteristics similar to deterministic chaos. This limits actionable response time and challenges assumptions used in global risk forecasting tools such as ESA’s DRAMA. These findings imply the need for a probabilistic, rather than deterministic, reentry risk model for uncontrolled objects, particularly in shallow reentry corridors. The use of ensemble simulations and Monte Carlo methods could better capture the uncertainty range in future predictive algorithms.

To contextualize the contributions of this study within the broad landscape of reentry modeling, we compare our simulation framework with established tools and methodologies such as NASA’s Object Reentry Survival Analysis Tool (ORSAT)¹⁹, ESA’s Debris Risk Assessment and Mitigation Analysis (DRAMA)¹⁸, and recent academic models^17,24. Although the fundamental physics of reentry, such as atmospheric drag, gravitational acceleration, and sensitivity to initial conditions, is well understood, our work distinguishes itself through its integrative modeling approach, accessibility, and policy relevance. Unlike ORSAT and DRAMA, which are proprietary or institutionally restricted, our simulation is fully open source and implemented in Python, making it globally accessible to researchers, educators, and policymakers. This is particularly important for institutions in the Global South, where access to commercial-grade aerospace software is limited. The computational workload is offloaded to cloud-based GPU and CPU resources provided by Google Colaboratory, enabling easy invocation and execution while ensuring minimal local hardware requirements and globally consistent performance. Our model incorporates real-time atmospheric wind dynamics, Coriolis effects, and gravitational perturbations up to the J₆ harmonic, offering a more nuanced and physically realistic trajectory prediction than many simplified models. In contrast to Bernelli-Zazzera et al.¹⁷, who focus on air traffic management and debris breakup modeling using Monte Carlo simulations, our study emphasizes the global unpredictability of reentry events and the legal and environmental implications of impact dispersion. Although fragmentation events are not currently modeled, our modular framework is designed for future integration of standard breakup models. Table 4 summarizes the main contributions of our model against other standards.

Table 4 Comparison of key features across reentry modeling tools

This benchmarking clarifies the unique contributions of our work and highlights areas for future enhancement, including the integration of breakup dynamics and probabilistic risk estimation.

Methods

The simulation model considers gravitational forces, atmospheric drag, the effects of the rotation of the Earth, and wind patterns in both the higher and lower atmospheres. The state vector is defined in an Earth-centered inertial (ECI) frame as:

$${\bf{X}}=\left[\begin{array}{ll}{\bf{r}}\\ {\bf{v}}\end{array}\right]={\left[\begin{array}{c}x\,y\,z\,{v}_{x}\,{v}_{y}\,{v}_{z}\end{array}\right]}^{T},$$

with the time derivative:

$$\frac{d}{dt}{\bf{X}}=\left[\begin{array}{c}\dot{{\bf{r}}}\\ \dot{{\bf{v}}}\end{array}\right]=\left[\begin{array}{c}{\bf{v}}\\ {\bf{a}}\end{array}\right].$$

The final system of equations governing the motion is:

$$\frac{d}{dt}{\bf{X}}={\left[\begin{array}{c}{v}_{x}{v}_{y}{v}_{z}({a}_{x,{\rm{grav}}}+{a}_{x,{\rm{drag}}})({a}_{y,{\rm{grav}}}+{a}_{y,{\rm{drag}}})({a}_{z,{\rm{grav}}}+{a}_{z,{\rm{drag}}})\end{array}\right]}^{T}$$

The velocity of the object relative to the wind (v_rel) can be estimated by subtracting the vector components of the wind speed, that is:

$${{\bf{v}}}_{{\rm{rel}}}={\bf{v}}-{{\bf{v}}}_{{\rm{wind}}}.$$

(1)

In the simulation, the initial reentry parameters of the object are the position vector r (latitude, longitude, altitude) and velocity magnitude v. The parameters are transformed into an ECIF to obtain the vector components. The two key parameters are the elevation angle, γ, which indicates steepness of reentry, and the azimuth angle, θ, which denotes the direction relative to the true north, as shown in Fig. 7. Reentry is considered shallow when −0.1° ≤ γ ≤ −5° and steep when −5° ≤ γ ≤ −20°. The eastward and northward velocity components, v_E and v_N, are related to the azimuth by $\theta ={\tan }^{-1}({v}_{E}/{v}_{N})$. The final coordinates of LEODOs entering the atmosphere were simulated under varying conditions. The initial altitude, latitude/longitude coordinates, speed, object mass, elevation, and azimuth angles are the input of the simulation. The ground track trajectories of the object were automatically visualized on a low-resolution geodesic map.

Fig. 7: A not-to-scale illustration of the object trajectory and reentry angles in ECIF.

An azimuth of 0°, 90°, 180° and 270° denotes an object reentering directly from the east, north, west and south, respectively. A low elevation angle indicates a gentle descent from orbit.

The angle of reentry and the direction of an object are critical initial conditions that influence its trajectory, heating, fragmentation, and impact point. The initial velocity vector using components aligned with latitude, longitude, and altitude is explicitly defined by:

$${{\bf{v}}}_{{\rm{initial}}}={v}_{r}\,\hat{r}+{v}_{\phi }\,\hat{\phi }+{v}_{\lambda }\,\hat{\lambda },$$

where v_r, v_ϕ, and v_λ are radial, latitudinal, and longitudinal velocity components. The full spherical harmonic expansion of the gravitational potential of Earth U(r) with zonal, tesseral and sectorial terms can be written as²⁵:

$$U(r,\phi ,\lambda )=-\frac{G{M}_{E}}{r}\left[1-\mathop{\sum }\limits_{n=2}^{N}\mathop{\sum }\limits_{m=0}^{n}{\left(\frac{{R}_{E}}{r}\right)}^{n}\left({C}_{nm}\cos (m\lambda )+{S}_{nm}\sin (m\lambda )\right){P}_{nm}\sin (\phi )\right],$$

(2)

where

$${P}_{nm}(x)={\left(1-{x}^{2}\right)}^{(m/2)}\frac{{d}^{m}}{d{x}^{m}}{P}_{n}(x),\quad {\rm{for}}\,{0}\le m\le n,$$

(3)

are the associated Legendre polynomials²⁶ of degree n (= 0, 1, 2, …), and order m, and $x=\cos (\phi )$. The form of U(r, ϕ, λ) above accounts for Earth’s longitudinal mass variations and improves the fidelity of the simulation by accounting for latitudinal and longitudinal variations within Earth’s gravitational field. This improves trajectory modeling accuracy for objects reentering at low altitudes or over regions with significant non-uniform mass distributions, such as over mountains and deep trenches. This aligns the model with established geophysical models such as EGM96 or EGM2008^27,28. For most applications, J₂ provides sufficient accuracy, but higher-order refinements improve precision. J₃ introduces a latitude-dependent perturbation in polar orbits, but has minimal impact on equatorial orbits. The J₄ term improves object location to sub-meter precision. It is valuable for studying objects in high-altitude trajectories and those experiencing long-term orbital decay, such as space debris. The term J₄, though significantly smaller than J₂, mainly affects mid-latitudes, induces long-period inclination oscillations in polar orbits, and has minimal impact on equatorial orbits. Our simulations include up to J₆ perturbations for a more precise model of Earth’s gravitational field by accounting for subtle asymmetries in Earth’s mass distribution^29,30. The gravitational acceleration is calculated from:

$${{\bf{a}}}_{{\rm{grav}}}=-\nabla U(r,\phi ,\lambda ).$$

(4)

The gravitational acceleration components in spherical coordinates are then:

$${a}_{r}=-\frac{GM}{{r}^{2}}\left[1+\mathop{\sum }\limits_{n=2}^{N}\mathop{\sum }\limits_{m=0}^{n}(n+1){\left(\frac{{R}_{E}}{r}\right)}^{n}\left({C}_{nm}\cos (m\lambda )+{S}_{nm}\sin (m\lambda )\right){P}_{nm}\sin (\phi )\right],$$

(5)

$${a}_{\phi }=\frac{GM}{{r}^{2}}\mathop{\sum }\limits_{n=2}^{6}\mathop{\sum }\limits_{m=0}^{n}{\left(\frac{{R}_{E}}{r}\right)}^{n}\left({C}_{nm}\cos (m\lambda )+{S}_{nm}\sin (m\lambda )\right)\frac{d{P}_{nm}\sin (\phi )}{d\phi },$$

(6)

$${a}_{\lambda }=\frac{GM}{{r}^{2}\cos \phi }\mathop{\sum }\limits_{n=2}^{N}\mathop{\sum }\limits_{m=1}^{n}m{\left(\frac{{R}_{E}}{r}\right)}^{n}\left(-{C}_{nm}\sin (m\lambda )+{S}_{nm}\cos (m\lambda )\right){P}_{nm}\sin (\phi ).$$

(7)

The Cartesian gravitational acceleration vector a_grav is then obtained by projecting the spherical components onto the basis:

$${{\bf{a}}}_{{\rm{grav}}}={a}_{r}\,\hat{r}+{a}_{\phi }\,\hat{\phi }+{a}_{\lambda }\,\hat{\lambda },$$

where

$$\begin{array}{ll}\hat{r}=(\cos \phi \cos \lambda ,\cos \phi \sin \lambda ,\sin \phi )\\ \hat{\phi }=(\sin \phi \cos \lambda ,\sin \phi \sin \lambda ,-\cos \phi )\\ \hat{\lambda }=(-\sin \lambda ,\cos \lambda ,0)\end{array}$$

The simulation and geodesic visualization are carried out using a Python program that runs in Google Colaboratory. The approach is to numerically solve a system of dynamical equations that represent the flight of the LEODOS to determine the instantaneous position and velocity. The program implements the spherical harmonic expansion up to N = 6. The coefficients are normalized for compatibility with the EGM96 model. For simplicity of implementation and to avoid rescaling, the associated Legendre functions ${P}_{nm}\sin (\phi )$ are unnormalized. We reduce computational load and numerical instability by omitting the full derivative of ${P}_{nm}\sin (\phi )$ from a_ϕ by assuming that the latitudinal component contributes minimally compared to the radial and longitudinal terms. The relative velocity of the object with respect to the atmosphere must include the rotational effect of the Coriolis force²¹: v_rel = v − ω ∧ r. The atmospheric drag force is calculated using the relative velocity:

$${{\bf{F}}}_{{\rm{drag}}}=-\frac{1}{2}{C}_{d}\rho \,A\,| {{\bf{v}}}_{{\rm{rel}}}| {{\bf{v}}}_{{\rm{rel}}}.$$

Figure 8 shows the atmospheric wind model used in the simulation.

Fig. 8

A schematic flowchart illustrating the numerical wind model for the Python simulation approach.

The drag accelerations are based on the object’s velocity relative to the atmosphere, which includes contributions from Earth’s rotation and wind velocities. Wind velocity is modeled in the simulation as an empirical function of altitude and latitude and used to modify the relative velocity in the drag force calculation. Figure 8 illustrates the wind model. An object falling through the atmosphere experiences increasing drag as the density of the air increases exponentially with descent^15,17. The simulation models ρ(h) using an altitude-dependent factor H(h):

$$\rho (h)={\rho }_{0}{e}^{-h/H(h)},$$

(8)

where

$$H(h)={H}_{0}{\left(1+\frac{h}{{H}_{0}}\right)}^{k}.$$

(9)

The parameter k adds flexibility to the model. A smaller k keeps the scale height closer to a constant value, while a larger k allows for more pronounced variation at higher altitudes. Both H₀ and the scaling factor k can be tuned based on observational data. In the present simulations, k ~ 0.25. It is possible to use inverse kinematic calculations to calculate the initial state of an LEODO object that eventually lands at a specific location. However, this is a complex and highly iterative problem. To solve the problem, define an instantaneous objective, or ‘cost’, function f(ϕ), in terms of the simulated (sim) and final (f) coordinates, for example,

$$f(\phi )=\sqrt{{({\phi }_{{\rm{sim}}}-{\phi }_{{\rm{f}}})}^{2}+{({\lambda }_{{\rm{sim}}}-{\lambda }_{{\rm{f}}})}^{2}},$$

where the landing coordinates are ϕ_f (latitude), λ_f (longitude). Then, an optimization method finds an initial (i) x_i that minimizes the objective function, as suggested by an accompanying Python pseudo-code.