Introduction

Near-Earth Objects (NEOs) refer to asteroids and comets with a perihelion distance of less than 1.3 astronomical units (AU). Among them, a subset may collide with Earth and are designated as Potentially Hazardous Asteroids (PHAs)1, defined as those with a Minimum Orbit Intersection Distance (MOID) of less than 0.05 AU and an absolute magnitude of 22 or lower. While most NEOs do not pose an immediate threat, asteroid impacts are not uncommon. In 2024 alone, four small NEOs were detected just hours before impact and subsequently entered Earth’s atmosphere(available: https://newton.spacedys.com/neodys2/NEOScan/index_past_imp.html). These objects, typically around 1 meter in diameter, did not cause damage. However, it is important to acknowledge that additional impacts likely occurred without prior detection and were only identified through atmospheric observations or meteorite recoveries.

NEOs are dynamically evolving populations, influenced by gravitational interactions with planets and non-gravitational forces. One of the most significant non-gravitational effects is the Yarkovsky effect, a thermal radiation force that causes gradual changes to an asteroid’s semi-major axis2. This effect depends on several parameters, including the asteroid’s size, spin state, and surface thermal properties. The transverse component of the Yarkovsky force plays a dominant role in modifying the semi-major axis over long timescales3. Consequently, it affects NEOs’ long-term orbital evolution and the prediction of collision probability4. Other mechanisms, such as resonances with planetary orbits, YORP-induced rotational changes, and planetary close encounters, further contribute to NEOs’ chaotic evolution5.

Accurate assessment of impact probability is essential for planetary defense. Various methods have been developed to quantify this probability, including the Line of Variations (LOV) method6, Monte Carlo sampling7, and approaches based on the State Transition Matrix (STM)8. The LOV method samples possible trajectories along the LOV direction and propagates them to assess impact probabilities, while Monte Carlo simulations generate a large number of virtual asteroid clones based on initial uncertainty distributions. In the STM-based approach, the matrix is propagated to the close approach time under the assumption that the final state probability density function (PDF) remains Gaussian, allowing the collision probability to be directly integrated. This approach is referred to as the linearized model, as it relies on the linearization of system dynamics. However, the Yarkovsky effect introduces additional uncertainty into these calculations, particularly for small asteroids with long observational arcs9. Therefore, we have to analyze the influence of the Yarkovsky effect on collision probability for small asteroids over long time scales.

In this study, we analyze the asteroid 2024 YR4, which was discovered in December 2024 and initially had a collision probability exceeding 1%. Its absolute magnitude is 23.95(available: https://minorplanetcenter.net/db_search/show_object?utf8=%E2%9C%93&object_id=2024+YR4), corresponding to an estimated diameter of 40–90 meters(available: https://neo.ssa.esa.int/search-for-asteroids?sum=1&des=2024YR4), assuming a geometric albedo pv between 0.05 and 0.2510. By February 24, 2025, updated orbit determinations from NASA’s Sentry(available: https://cneos.jpl.nasa.gov/sentry/), the Near-Earth Objects Dynamic Site (NEODyS)(available: https://newton.spacedys.com/neodys), and ESA’s Near-Earth Objects Coordination Centre (NEOCC)(available: https://neo.ssa.esa.int/) indicated that the impact probability had decreased below 0.01%.

The objectives of this communication are as follows: (1) to compare our precise orbit determination results with those from NEOCC, NEODyS, and JPL Horizons(available: https://ssd.jpl.nasa.gov/horizons/); (2) to evaluate our collision probability estimates against those provided by Sentry, NEOCC, and NEODyS; (3) to investigate the impact of different observational arc lengths on the estimated collision probability; (4) to assess the influence of the Yarkovsky effect on collision probability by incorporating non-gravitational forces into uncertainty propagation; (5) to propose a collision risk assessment framework that integrates population density distribution for a more comprehensive evaluation of potential impact scenarios.

Methods

In this work, we perform precise orbit determination using the least-squares method11, incorporating a dynamic removal scheme. The automatic outlier rejection routine follows the approach used by NEODyS, discarding observations when \(\chi =\sqrt{10}\) and recovering them at \(\chi =\sqrt{9.21}\). The parameter χ is defined as:

$$\chi =\sqrt{{\left(\frac{({\alpha }_{o}-{\alpha }_{c})\cos {\delta }_{o}}{{\sigma }_{\alpha }}\right)}^{2}+{\left(\frac{{\delta }_{o}-{\delta }_{c}}{{\sigma }_{\delta }}\right)}^{2}}$$
(1)

where α and δ denote the right ascension and declination, respectively. The subscripts ‘o’ and ‘c’ indicate observed and calculated values, while σα and σδ represent the 1σ uncertainties of the observations. The weighting rules follow the ‘vfcc17’ file in OrbFit (available at: https://adams.dm.unipi.it/~orbmaint/orbfit/). The star catalogue is considered in our work and follows12. The orbit determination software used in this work, developed by us13,14, is referred to as ‘NJU_OD’.

In this study, collision probability assessment is performed using the linearized model8, the Line of Variations (LOV) method6, and Monte Carlo (MC) simulations7. The linearized model propagates the State Transition Matrix (STM) to the time of close approach, assuming that the final state probability density function (PDF) remains Gaussian, thereby allowing direct integration of the collision probability. This method is referred to as the linearized model because it relies on the linearization of system dynamics. In the LOV method, we generate 1,200 virtual orbits along the LOV direction within the range of [− 6σ, 6σ]. For the MC method, a total of 10,000 sample orbits are generated.

Preliminaries

The force model used in this study incorporates the eight major planets, the Sun, the Moon, Pluto, 16 minor asteroids and general relativity (refer to Fig. 1). Ephemeris DE441 is employed15. All observation data were obtained from the Minor Planet Center (MPC)(https://minorplanetcenter.net/).

Fig. 1: Relative strength of different forces on 2024 YR4 in the dynamical model.
figure 1

The J2 represents the main component of Earth’s non-spherical perturbation. FSun in the ordinate is the strength of the monopole acceleration of the Sun.

Full size image

Comparison on precise orbit determination results

We first present our precise orbit determination results in Table 1. Table 1 shows the state vector relative to the solar system’s barycenter and its 1σ uncertainty of 2024 YR4, determined by NJU_OD and given in the equatorial coordinate system under the International Celestial Reference Frame (ICRF)16.

Table 1 The state vector relative to the solar system’s barycenter and its 1 σ uncertainty of 2024 YR4
Full size table

We then present our precise orbit determination results and compare them with those from NEOCC, NEODyS, and JPL. As new observations continue to be incorporated, our analysis is based on data collected up to February 21, 2025. This dataset consists of 452 observations obtained from the MPC, spanning approximately 58 days. The asteroid’s ephemeris is determined on January 20, 2025 (MJD 60695.14841712 in TDB), which is close to the midpoint of the observation arc. Table 2 compares our precise orbit determination results with those from JPL and NEODyS, presenting the positional differences in kilometers (km). Since NEOCC provides equinoctial elements at MJD 60695.03849798 (on January 20, 2025), we specifically examine the differences between our results and those from NEOCC. The table indicates that the positional differences are minimal, and the 1σ positional uncertainty is approximately 140 km. This ensures that the differences fall within the uncertainty ellipsoid, reinforcing the reliability of our results. It is important to note that different organizations use different observational datasets and weighting strategies, which can lead to variations in the determined trajectories. For instance, as of February 21, 2025, 452 observations have been collected from the MPC, whereas JPL and NEODyS utilize a different number of observations.

Table 2 Positional differences (in kilometers) between our results, JPL, and NEODyS at MJD=60695.14841712 are shown below, along with the positional difference between NEOCC and our results at MJD = 60695.03849798 (TDB)
Full size table

Based on the trajectory information, the close approach epoch and distance for December 22, 2032, are presented in Table 3. This table includes the close approach data from NEOCC and NEODyS, collected on February 21, 2025, as indicated in the table. In addition to the results from NEODyS, NEOCC, and our own (denoted as ‘NJU_OD’), the close approach distance and epoch are also computed using our software, with initial orbit information provided by JPL, NEOCC, and NEODyS. Notably, NEOCC does not provide second-level information. Our computations use these initial conditions to determine the close approach distance and epoch for the specified close approach. The close agreement between our predicted close approach parameters and those reported by NEODyS confirms the reliability of our orbit propagation model and methodology. A similar level of agreement is observed when comparing our results with those from NEOCC. Although the initial orbits used by different systems are quite similar, with positional differences within 100 km (see Table 2), the predicted close approach distance to Earth can vary significantly, reaching the order of O(104) km. For example, the initial positional difference between our trajectory and JPL’s is only 13 km, but the predicted close approach distance differs by O(104) km (see Table 3). These discrepancies arise primarily from differences in the propagation process and force models used by different systems.

Table 3 The close approach distance and epoch from NEOCC, NEODyS, and our results (denoted as ‘NJU_OD’) are shown below, along with results computed using initial inputs from NEODyS, JPL, and NEOCC
Full size table

Comparison on collision probability assessment

Additionally, Table 4 presents the collision probability data from Sentry, NEOCC, and NEODyS, collected on February 21, 2025, which is denoted with a ‘(website)’ note in the Table 4. In addition to our MC simulation results and the results published on the website, the collision probabilities are recalculated using our software but with the initial orbit and covariance matrix published on the website. The collision probabilities are approximately the same (see Table 4), further demonstrating the reliability of our software.

Table 4 The collision probability from Sentry, NEOCC, and NEODyS is shown below, with a “Website” note included
Full size table

Comparison of the three methods

Moreover, Table 5 presents the results for collision probability, the Palermo Scale17, and the Torino Scale18, computed using the MC simulation, LOV method, and the linearized model with observational data up to February 18, 2025. The collision probabilities obtained from the three methods exhibit only minor differences. We attribute this small difference to the relatively short propagation period, which constrains the growth of uncertainties. Additionally, the only significant close encounter before 2032 occurs in 2028, but at a relatively large distance from Earth. As a result, this encounter does not significantly amplify the trajectory’s uncertainty. Consequently, the uncertainty ellipsoid derived from all three methods remains nearly identical. To validate this assumption, Fig. 2a, c respectively illustrates the sample orbits obtained from the LOV method and MC simulation. The sample orbits during the close encounter on December 22, 2032, are projected onto the x − y plane for clarity. The reference trajectory and the 1σ uncertainty ellipsoid from the linearized model are also shown. It is evident that, on the same scale, the sample orbits and the 6σ uncertainty ellipsoid align in a same direction within the x − y plane. This indicates that the linearized model effectively captures the uncertainty in this scenario. Furthermore, both the linearized model and the LOV method provide reasonable approximations of uncertainty evolution in this nonlinear propagation. In conclusion, the relatively short propagation period limits the growth of uncertainties, resulting in comparable collision probability estimates across the three methods.

Table 5 The collision probability, Palermo Scale ("PS”), and Torino Scale ("TS”) calculated using the Monte Carlo (MC) simulation, LOV method, and linearized model are shown below, along with comparison results from NEOCC and Sentry as of February 18, 2025
Full size table
Fig. 2: Heliocentric positions during the close encounter on December 22, 2032, projected onto the x−y plane for clarity.
figure 2

a and c Display sample orbits obtained from the LOV and MC methods, respectively. b Illustrates the reference trajectory along with the 6σ uncertainty ellipsoid derived from the linearized model.

Full size image

Impact of the observation arc length

The uncertainty in the trajectory arises from observation error and decreases as the observation arc length increases. A longer arc and more precise measurements result in improved orbit determination. Therefore, in Fig. 3, we compute the collision probability using the MC method with varying observation arc lengths. The x-axis represents the observation period, with the starting epoch set to 2022/12/25 and the final epoch corresponding to the value on the axis. The figure shows that the initial collision probability exceeds 1% but drops below 0.01% after 2025/02/23, at which point it is no longer displayed. Given the negligible probability, the object has been removed from the urgent risk list.

Fig. 3: The influence of observation arc length on MC simulation results.
figure 3

The abscissa represents the last observation date in 2025. The collision probability drops below 0.01% after 2025/02/23, at which point it is no longer shown in the figure.

Full size image

Impact of the Yarkovsky effect

The Yarkovsky effect is a thermal radiation force that gradually alters an asteroid’s orbit. This effect depends on several factors, including the asteroid’s size, orbital parameters, rotation state, and surface thermal properties2. The force induced by the Yarkovsky effect can be decomposed into three components: radial, transverse, and normal. According to Gauss’ planetary equations, the transverse component has the most significant long-term impact on the semi-major axis. The direction of this effect depends on the asteroid’s spin state3. The obliquity, defined as the angle between the spin axis and the asteroid’s orbital angular momentum vector, determines whether the semi-major axis increases or decreases. If the obliquity is less than 90°, the Yarkovsky effect results in a gradual increase in the semi-major axis, classifying the asteroid as a prograde rotator. Conversely, if the obliquity exceeds 90°, the semi-major axis decreases over time, and the asteroid is considered a retrograde rotator.

In this study, we adopt the simplified Yarkovsky force model proposed by10, which accounts only for the transverse component. The acceleration along this direction is given by

$${a}_{t}={A}_{2}{\left(\frac{{r}_{0}}{r}\right)}^{2},$$

where A2 is the Yarkovsky parameter to be determined, r0 is a reference distance (set to 1 AU in this work), and r is the heliocentric distance of the near-Earth asteroid (NEA). The unit of A2 is AU/day2. To integrate the orbit with the Yarkovsky effect, we first estimate the Yarkovsky parameter A2 for 2024 YR4. Since A2 is inversely proportional to the asteroid’s diameter D10,

$${A}_{2}\propto {D}^{-1},$$
(2)

we use the well-characterized value of Bennu (101955) as a reference to ensure reliability. The Yarkovsky parameter for 2024 YR4 is then computed as

$${\left({A}_{2}\right)}_{\exp }={({A}_{2})}_{{\rm{Bennu}}}\frac{{D}_{{\rm{Bennu}}}}{D}.$$
(3)

Given that the estimated diameter of 2024 YR4 is approximately 55 meters—about 10% of Bennu’s size—the Yarkovsky effect is expected to be significant. The absolute value of A2 is set to 4.615 × 10−13 AU/day2. In this study, we analyze the close approach epoch and distance while incorporating the Yarkovsky effect. Since the spin direction of 2024 YR4 remains unknown, both retrograde and prograde rotation cases are considered, as shown in Table 6. Observations are included up to February 8, 2025, and the collision probability is computed using the Line of Variations (LOV) method. As shown in Table 6, while the collision probability remains unchanged, the Yarkovsky effect influences both the timing and distance of the close approach. Moreover, the difference in the close approach time with and without the Yarkovsky effect is only about 10 seconds, which is quite small. Although the magnitude of 2024 YR4’s Yarkovsky effect is relatively large compared to that of larger asteroids, the relatively short orbital propagation time (approximately 7 years) limits its cumulative impact. The transverse component of the Yarkovsky force induces a semi-major axis drift rate given by

$$\left\langle \frac{da}{dt}\right\rangle =\frac{2}{n{a}^{2}(1-{e}^{2})}{A}_{2},$$
(4)

where a is the semi-major axis, e is the eccentricity, and n is the mean motion. Over a span of seven years, this effect results in a semi-major axis displacement on the order of O(10−9) AU, which is small. Consequently, over such a short timescale, the Yarkovsky effect primarily acts as a perturbation on the mean anomaly, effectively shifting the asteroid’s position along its trajectory. Specifically, in the prograde case, the Yarkovsky effect causes a delayed close approach time and a reduced close approach distance, whereas in the retrograde case, it results in an earlier close approach time and an increased close approach distance.

Table 6 Collision probability, close approach time, and close approach distance under different A2 values
Full size table

Possible impact locations and population scale

For the orbits with potential impacts, the possible impact locations are shown in Fig. 4, based on observations collected on February 21, 2025. The computation of the impact location does not account for the influence of Earth’s atmosphere. As illustrated in Fig. 4, the possible impact locations are situated in central Africa, northern South America and south Asia. A population scale is introduced for the collision probability assessment, with global population density data sourced from19. The population scale is defined as:

$${S}_{p}=\left(\mathop{\sum }\limits_{i=0}^{N}{p}_{i}{\rho }_{i}\right)S/N$$
(5)

where N represents the number of potential impact trajectories, pi is the probability associated with the ith impact trajectory, and ρi is the population density within the impact area of the ith orbit. S represents the impact area of the asteroid. In this simplified model, we approximate it as

$$S=\pi {\left(\frac{D}{2}\right)}^{2}$$

where D is the asteroid’s diameter. This model assumes that the entire cross-sectional area of the asteroid contributes to the impact and does not account for potential mass loss due to atmospheric ablation. The unit for19 is persons per square kilometer. In this case, the population scale is approximately 190,613 people, indicating that an impact from this asteroid would result in catastrophic damage.

Fig. 4
figure 4

The possible impact location of 2024 YR4 based on the observations.

Full size image