© The Authors 2025

1 Introduction

Near-Earth asteroid 2024 YR₄ has recently attracted significant attention from both the scientific community and the general public that is mainly due to its predicted close approach to Earth in 2032. Its initial probability of impacting Earth was estimated at 3% (Bolin et al. 2025; Rivkin et al. 2025). This ≈50 meter asteroid briefly reached a Torino scale rating of 3, which is the highest rating for any object since the rating for asteroid (99942) Apophis in 2004. However, subsequent observations and refined orbital calculations reduced the impact probability in 2032 to zero as of March–April 2025 and only confirmed a close approach to Earth and the Moon in 2032. The Torino scale rating is now zero. A low impact probability (7.8 · 10⁻⁶) remains for 2047¹.

Although the impact risk was removed, 2024 YR₄ remains a subject of considerable scientific interest as a result of its orbital characteristics and potential dynamical history. According to the NASA JPL website, its semimajor axis of 2.52 AU and eccentricity of 0.66 place it within the Apollo class of near-Earth asteroids, with a minimum orbit intersection distance of approximately 0.003 AU from Earth. More importantly, it is close to the Kirkwood gap 3:1 with Jupiter (Kirkwood 1867; Murray & Dermott 1999). This raises a question whether the asteroid dynamics are shaped by mean-motion resonances (MMRs).

Mean-motion resonances arise when the orbital frequencies of the involved bodies are commensurate. In the simplest scenario of a two-body resonance, only the asteroid and one planet are involved (e.g., Jupiter or Mars). Three-body resonances include two planets (e.g., Jupiter and Saturn) and an asteroid Nesvorný & Morbidelli 1998; Gallardo 2006; Smirnov & Shevchenko 2013). Both types of resonances can significantly alter the asteroid dynamics, contribute to chaotic diffusion in the main belt, and create pathways for asteroids to migrate toward Earth-crossing orbits (Nesvorný & Morbidelli 1998; Murray & Holman 1997; Wisdom 1983; Farinella et al. 1993). Detailed theoretical and dynamical atlases of both two- and three-body resonances in the Solar System were constructed (Gallardo 2006, 2014; Gallardo et al. 2016; Smirnov & Shevchenko 2013; Smirnov et al. 2017) that revealed numerous asteroids trapped in two- and three-body MMRs.

The 3J-1 mean-motion resonance with Jupiter (note that we adopt the universal form of the resonance definition (e.g., 3J-1 instead of 3:1. Writing the exact integers that appear in the resonant argument keeps the notation unambiguous for two- and three-body MMRs, which can include positive and negative terms, and it facilitates identifying the planets involved in the MMR. The planet symbols are E = Earth, M = Mars, J = Jupiter, and S = Saturn), which dynamically characterizes the Alinda asteroid group, has been identified as a source region for near-Earth asteroids that functions as an efficient delivery mechanism that can increase eccentricities to Earth-crossing values (Wisdom 1983; Farinella et al. 1993). The proximity of the current semimajor axis of 2024 YR₄ to this resonance location is noteworthy, in particular, when it combined with population model analyses that indicate a 73.9% probability of an origin from the 3J-1 MMR (Bolin et al. 2025). This has to be confirmed dynamically, however.

In addition to the well-known 3J-1, this region includes other two- and three-body MMRs (see Fig. 1). The asteroid might be trapped in one of them. Previous studies have demonstrated the phenomenon of resonance sticking, which is characterized by the temporary capture in multiple resonances. Several such studies were conducted for the scattered-disk and centaur populations (Lykawka & Mukai 2007; Bailey & Malhotra 2009) and for the Hungaria family in the main belt (McEachern et al. 2010). Furthermore, some trans-Neptunian objects (TNOs) can simultaneously be trapped in two- and three-body MMRs, which can be determined by double libration of the resonant angle (Smirnov 2025). For scattered TNOs, this mechanism can account for up to 38% of the dynamical lifetime of an object, and captures predominantly occur at semimajor axes smaller than 250 AU (Lykawka & Mukai 2007). Preliminary calculations based on initial orbital data suggested that 2024 YR₄ may have experienced a similar resonance sticking behavior and may frequently have become trapped in two-body MMRs with Jupiter and Mars and in three-body MMRs with Mars and Jupiter or Jupiter and Saturn.

While the upcoming close approach in 2032 introduces significant uncertainties in a forward propagation of the asteroid orbit because it is more sensitive to the initial conditions (Milani et al. 2000), a backward integration offers considerably more reliable results for investigating the previous dynamical state of the object. We therefore study whether asteroid 2024 YR₄ was trapped in the resonance 3J-1, and if the object was indeed trapped in the MMR, we determine the probability that it was in the resonance. We also examine whether asteroid 2024 YR₄ was trapped in other two- or three-body MMRs.

The structure of the paper is as follows. Section 2 details the construction of the virtual-asteroid ensemble, the selection of candidate MMRs, and the numerical scheme we adopted for the 10⁵-yr backward integrations. The statistical outcomes of these integrations (capture probabilities, residence times, and representative examples of resonance sticking) are presented in Sect. 3. Section 4 discusses the dynamical implications of these findings in the wider context, and it summarizes our principal conclusions.

Fig. 1 Two- and three-body mean-motion resonances close to 2024 YR4 based on the theoretical value of the resonant semimajor axis. The red triangle indicates the position of asteroid 2024 YR4. The gray dots represent other asteroids in this region, for which proper elements (instead of osculating elements) are used.

2 Methods

In order to determine whether asteroid 2024 YR₄ has been trapped in a mean-motion resonance, a statistical approach considering orbital uncertainties was implemented. First, a set of potential MMRs was identified based on the current value of the asteroid semimajor axis. Specifically, all two-body resonances located within a ±0.1 au interval around the asteroid semi-major axis were surveyed, as well as all three-body resonances within a ±0.10 au interval. This initial assessment allowed us to shortlist resonances that might be relevant to the dynamical behavior of the asteroid. The full list of the MMRs selected at this stage is available in Appendix A.

To account for observational uncertainties, a set of 1000 virtual asteroids was generated by sampling from Gaussian distributions centered on the nominal orbital elements. The 1σ uncertainties for the semimajor axis (σ_a = 1.0147 × 10⁻⁵ au) and eccentricity (σ_e = 1.4469 × 10⁻⁶) were obtained from the NASA JPL database². For this analysis, the virtual asteroids were created within a 3σ range, which ensured a comprehensive coverage of the plausible orbital parameter space while maintaining other orbital elements at their nominal values. Following the suggestion by the referee of this paper, an additional analysis on a set of 100 virtual asteroids was performed using the full covariance matrix from the NASA JPL Small Body Database, which accounts for correlations between orbital elements.

Next, a dynamic identification of these candidate resonances was performed by integrating the orbits of all virtual asteroids for 10⁵ years backward. The dynamical model included perturbations from all eight planets and Pluto. Several preliminary simulations were run with the Moon and the Earth instead of the barycenter of the Earth–Moon system. The results showed no significant difference, however.

Because of the presence of close encounters with Earth, special attention was paid to the selection of the integrator, since this choice can be crucial (Smirnov & Timoshkova 2014). Multiple integrators, such as whfast (Rein & Tamayo 2015), SABA (Rein et al. 2019), ias15 (modified Everhart integrator; see Everhart 1974; Rein et al. 2019), and bs (Bulirsch-Stoer; see Bulirsch & Stoer 1966) were tested. As expected, the preliminary results demonstrated that ias15 outperformed other integrators in terms of the overall performance and robustness. Thus, it was selected as the primary integrator. The integration time was set to −10⁵ yr.

After completing the integrations, the relevant resonant argument (sometimes referred to as the resonant angle) was calculated for each candidate resonance across all virtual asteroids, $$\sigma =\sum _{i=1}^{\mathrm{N}}{\lambda }_{\mathrm{i}}{m}_i+{\varpi }_{\mathrm{i}}{p}_i,$$(1)

where N is the number of bodies (N = 2 for two-body MMRs, and N = 3 for three-body MMRs), λ_i and ϖ_i are mean longitudes and longitudes of perihelion of all involved bodies, and m_i and p_i are integers satisfying the D’Alembert rule, ∑m_i + p_i = 0. Only the leading subresonance in the resonant multiplet was considered (p_i = 0 for all bodies except the asteroid).

Normally, to determine whether an asteroid is trapped in resonance, the following criteria had to be met:

1. The resonant argument must librate for a sufficiently long interval, that is, 20 000 years (Murray & Dermott 1999; Smirnov & Shevchenko 2013).

2. The resonant argument and semimajor axis should have matching oscillation frequencies that are usually determined either visually or with Lomb-Scargle periodograms (see the detailed explanation in Smirnov & Dovgalev 2018; Smirnov 2023).

The eccentricity of 2024 YR₄ is high, however, and is located near the Kirkwood gap 3/1. We might therefore expect to see a noisy behavior of the resonant angle, with multiple overlapping frequencies. We therefore decided to only apply the first condition and then perform a visual inspection. This was implemented for each virtual asteroid and allowed us to statistically assess the resonance capture probability. The fraction of virtual asteroids satisfying both criteria provides a quantitative measure of the likelihood that 2024 YR₄ experiences resonant behavior, and it properly accounts for the observational uncertainties.

The package “resonances” (Smirnov 2023) was used to calculate the resonant arguments, and “rebound” (Rein & Liu 2012) was used to integrate the orbits. The NASA Horizon system (through the API) was the source of the initial data for the planets and the asteroid. The virtual asteroid set was generated using custom Python routines implementing Gaussian sampling within specified uncertainty bounds.

Fig. 2 Resonant angles (left) and semimajor axis (right) for various virtual asteroids trapped in various resonances. The resonance in which 2024 YR4 is engaged in each case is specified in the legend.

3 Results

First, we conducted a pilot study on a sample of ten virtual asteroids. The preliminary results confirmed that it was almost impossible to extract the frequency of the resonant angles and the semimajor axis accurately (see Figs. 2, 3). This implied a necessity for visual inspection of the images because the package resonances cannot process edge cases like this.

To reduce the number of MMRs considered (and hence, those that were to be analyzed visually, i.e., manually), another preliminary study was conducted on a sample of 100 virtual asteroids. The results demonstrated that the resonant angle showed nonchaotic behavior in four resonances: 3J-1,1 M-2, 2M+3J-5, and 1E-4. We therefore decided to run the full simulation with 1000 virtual asteroids for these MMRs alone.

The statistical analysis revealed that asteroid 2024 YR₄ was frequently trapped in the 3J-1 mean-motion resonance, with 721 out of 1000 virtual asteroids demonstrating libration for significant time intervals. The behavior of the resonant angle displayed notable variability in the temporal characteristics and dynamical stability, contingent upon the initial orbital parameters. Panels 1 and 2 in Fig. 2 illustrate cases in which the asteroid is temporarily trapped in the 3J-1 resonance before it transitions to a nonresonant state. Panel 3 in Fig. 2 shows that resonant capture events close to the current epoch are possible. The analysis of the nominal orbit solution (Fig. 3) indicates that 2024 YR₄ experienced three distinct episodes of resonant capture: a primary interval from −100 000 to approximately −55 000 years, a secondary period from −51 000 to approximately −24 000 years, and a tertiary period from −11 000 to −7000 years.

For other MMRs (1M-2,2 M+3J-5, and 1E-4), the behavior is different: While 2024 YR₄ can sometimes be trapped in these resonances, the time in resonance is usually brief. For example, the virtual asteroid in Panel 4 of Fig. 2 is trapped twice in 1M-2, and Panel 5 shows a few transient captures in 2M+3J-5. According to the results, the resonant capture time is often short, however, from 2000 to 10 000 yr. Nevertheless, these captures are frequent: for 1M-2, there is at least one capture in 16% of the cases, and for 2M+3J-5, there is at least one capture in 12% of the simulations. For 1E-4, the results appear to be false positives: Despite the occasional librations of the resonant angles, the corresponding semimajor axis values are far from the theoretical resonant values.

Following the suggestion of the referee of this paper, we performed an additional analysis using the full covariance matrix from the JPL Small Body Database to assess whether the generation of virtual asteroids using correlations between orbital elements affects our results: The covariance matrix approach may represent the uncertainty structure better than independent Gaussian sampling. This supplementary analysis was conducted on a subset of 100 virtual asteroids. The covariance-based results confirm the primary findings: 70% of the virtual asteroids show capture in the 3J-1 resonance (compared to 72% in the original analysis), 29% in 1M-2 (compared to 16%), and 9% in 2M+3J-5 (compared to 12%). The slight variations, in particular, for 1M-2, may be attributed to the smaller sample size or to the correlations that are captured by the covariance matrix. These results clearly demonstrate that the main conclusions about the high probability of multiple resonant captures in 3J-1 and other MMRs are robust to the choice of the error-propagation method.

Fig. 3 Resonant angle σ, filtered resonant angle σf, filtered values of the semimajor axis af, Lomb–Scargle periodograms for the resonant angle and semimajor axis, and eccentricity e for the nominal initial data and the MMR 3J-1.

4 Discussion and conclusion

As previously suggested (Wisdom 1983; Morbidelli & Nesvorný 1999), chaotic diffusion near 3J-1 can lead to the ejection of asteroids from the main belt, which results in significant orbital modifications. The results of our study indicate that asteroid 2024 YR₄ was likely ejected from the 3J-1 MMR as a result of chaotic diffusion, which led to a higher eccentricity and its subsequent transition into a near-Earth asteroid. This scenario is consistent with existing models (Nesvorný & Morbidelli 1998; Morbidelli & Nesvorný 1999), in which chaotic diffusion that arises from the overlap of MMRs (two- and three-body resonances) and secular resonances drives objects from the main asteroid belt into Earth-crossing orbits.

The statistical analysis revealed that ≈70–72% of the population becomes trapped at some point in the 3J-1 resonance. These results are consistent with the population model analyses conducted by Bolin et al. (2025), who reported a probability of 73.9%. The simulations appear to be relatively robust based on a comparison of the results of the first 100, 200, and 1000 members and based on the results obtained by different error-propagation methods. The asteroid can sometimes be captured in the 1M-2 MMR (in 16–29% of the simulated cases) or 2M+3J-5 (in 9–12% of the cases), where it typically remains for approximately 2000 to 10 000 years.

The phenomenon of resonance sticking, demonstrated by 2024 YR₄, may play an important role in understanding the dynamical evolution of scattered objects. The phenomenon of resonance sticking, as described by Lykawka & Mukai (2007), occurs when objects experience one or more temporary captures in resonances throughout their dynamical evolution. The findings for 2024 YR₄ reveal clear evidence of this process, with many virtual asteroids experiencing multiple captures in various resonances (3J-1, 1M-2, and 2M+3J-5) throughout their evolution. This agrees with results from the trans-Neptunian region (Lykawka & Mukai 2007).

To summarize, these findings agree with the current understanding of near-Earth asteroid production, according to which chaotic diffusion that arises from overlapping mean-motion and secular resonances transports objects from the main belt into near-Earth orbits. 2024 YR₄ represents a valuable case study in resonance sticking dynamics. It shows the contribution of this mechanism to the complex orbital evolution of small bodies in the Solar System.

Several limitations affect these conclusions, however. First, the relatively short observational arc for 2024 YR₄ introduces uncertainties in its orbital elements, which limits the dynamical analysis when it is combined with multiple close approaches with planets. As noted in Section 3, the supplementary analysis using the JPL covariance matrix confirmed that our conclusions are robust to the choice of the error propagation method. Second, the model we used does not account for nongravitational forces, such as the Yarkovsky effect, which can influence small objects over long timescales. Third, there might be other high-order resonances (e.g., 9M+19), which are beyond the scope of the current study and can affect the dynamical behavior of the asteroid. Fourth, the design of the main simulation (virtual asteroids) was simplified, and it might therefore be improved. Fifth, the analysis was performed manually by the author because the software package used was limited. While the resonances package already uses Lomb-Scargle periodograms to extract frequencies and a Butterworth digital filter to remove short-periodic oscillations, extreme cases, such as that of 2024 YR₄, produce many false positives and false negatives due to instability and overlap of the MMRs. It might therefore be reasonable to configure and use large language models to improve the accuracy and inter-rater reliability, as suggested by Smirnov (2024).

Acknowledgements

The author thanks an anonymous reviewer for the helpful comments suggesting statistical research. Statement on the use of generative AI tools: The author reports that he used generative AI (ChatGPT 4o and Claude 3.5, 3.7, and 4) to check and correct some paragraphs. After using these tools, the author reviewed and edited the content as needed and takes full responsibility for the content of the publication.

Appendix A The full list of MMRs close to 2024 YR₄

1E-4, 1E+4, 1M-2, 1M+2, 3J-1, 3J+1, 1M+3J-3, 1M+6J-4, 2M-4J-3, 2M-1J-4, 2M+2J-5, 2M+3J-5, 2M+5J-6, 3M-4J-5, 3M-1J-6, 3M+1J-7, 3M+2J-7, 3M+3J-7, 1M-1S-2, 1M+6S-3, 2M-3S-4, 2M-1S-4, 2M+4S-5, 2M+5S-5, 2M+6S-5, 3M-2S-6, 3M-1S-6, 3M+3S-7, 3M+4S-7, 3M+5S-7, 3M+6S-7, 1J+5S-1, 5J-5S-1, 5J+2S-2, 5J+3S-2.

The first integer represents the integer coefficient for the mean longitude of the planet in Eq. 1. The second integer: for three-body MMRs, it is the integer coefficient for the mean longitude of the second planet; for two-body MMRs, it is the integer coefficient for the asteroid. The third integer appears only for three-body MMRs and represents the integer coefficient for the asteroid’s mean longitude. The letters designate planets: E — Earth, M — Mars, J — Jupiter, S — Saturn.

References

All Figures

	Fig. 1 Two- and three-body mean-motion resonances close to 2024 YR4 based on the theoretical value of the resonant semimajor axis. The red triangle indicates the position of asteroid 2024 YR4. The gray dots represent other asteroids in this region, for which proper elements (instead of osculating elements) are used.
In the text

	Fig. 2 Resonant angles (left) and semimajor axis (right) for various virtual asteroids trapped in various resonances. The resonance in which 2024 YR4 is engaged in each case is specified in the legend.
In the text

	Fig. 3 Resonant angle σ, filtered resonant angle σf, filtered values of the semimajor axis af, Lomb–Scargle periodograms for the resonant angle and semimajor axis, and eccentricity e for the nominal initial data and the MMR 3J-1.
In the text