Issue |
A&A
Volume 692, December 2024
|
|
---|---|---|
Article Number | A144 | |
Number of page(s) | 9 | |
Section | Planets, planetary systems, and small bodies | |
DOI | https://doi.org/10.1051/0004-6361/202451162 | |
Published online | 09 December 2024 |
Possible origin of Mars-crossing asteroids and related dynamical properties of inner main-belt asteroids
1
National Key Laboratory of Science and Technology on Aerospace Flight Dynamics,
100094
Beijing,
China
2
Beijing Aerospace Control Center,
100094
Beijing,
China
3
Shandong University of Science and Technology,
266590
Qingdao,
China
4
Chinese Academy of Surveying and Mapping,
100036
Beijing,
China
5
Nanjing University,
210023
Nanjing,
China
6
Wuhan University,
430079
Wuhan,
China
7
Bejing Normal University,
100875
Beijing,
China
★ Corresponding authors; shanhongliu@whu.edu.cn, hanyl@sdust.edu.cn
Received:
18
June
2024
Accepted:
14
October
2024
Context. The orbital element distribution of the inner main belt (IMB) provides clues to the origin of the main-belt asteroids. Mars-crossing asteroids (MCAs) and near-Earth objects (NEOs) can provide some references to validate and improve theoretical models of the IMB evolution.
Aims. With the updated Asteroid Families Portal database, we analyzed the distribution of orbital elements and the dynamic completeness limit of IMB asteroids. By incorporating larger and more diverse datasets, the study seeks to provide a more comprehensive understanding of the IMB and MCAs origin and evolution.
Methods. We fitted the completeness-limit magnitude for the IMB. The size frequency and albedo distribution were used to analyze the family characteristics. The role of chaotic effects in the dynamic evolution of IMB and MCAs is further quantified by simulations.
Results. An albedo analysis showed that some halo asteroids may have originated from family asteroids, whereas the remaining non-family asteroids (14%) are likely to be members of a potential ghost family. We estimated the chaotic diffusion of asteroid orbits considering 1M/2A mean motion resonance. The eccentricity diffusion rate is estimated to be 0.45 and the inclination diffusion rate is 0.4 for resonant asteroids. The loss rate of MCAs IIMC(17.6) = 24.13 Myr−1, while the loss rate of the IMB asteroids due to the chaotic diffusion is 0.2648 Myr−1, which represents only 1.1% of MCAs. This indicates that chaotic diffusion has a limited capacity to replenish MCAs. However, for the large MCAs, a loss rate of IIMC(12) = 0.2646 Myr−1 was observed. This suggests that the large MCAs (H < 12) are in the dynamic equilibrium, primarily evolving through chaotic diffusion.
Key words: minor planets, asteroids: general
© The Authors 2024
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1 Introduction
The origin and evolution of asteroids are crucial to understand the meteorite formation and the mechanics of planet formation in the solar system. To study long-term asteroid evolutionary models, it is essential to comprehend the impact factors or mechanisms of gravitational and non-gravitational effects. Gravitational perturbations typically result in chaotic diffusion of proper elements (Minton & Malhotra 2010; Carruba et al. 2003), while non-gravitational perturbations, such as the Yarkovsky and YORP effects, can drive the evolution of the semi-major axis and the spin of asteroids, respectively (Bottke et al. 2006, 2015).
Moreover, collisions play a crucial role in the evolution of asteroid families (Durda & Dermott 1997). Generally, asteroid families originate from catastrophic collisions with the same original parent body, and they have similar orbital elements. The hierarchical clustering method (HCM) is aimed at identifying asteroid families by defining a cutoff distance in the proper element space to classify the asteroid families (Nesvorny et al. 2015).
Non-family asteroids cannot be classified by the HCM and this type includes several large primitive asteroids formed by direct accretion of solids in protoplanetary disks, as well as the halo asteroids situated in the family’s edge region and the ghost families boasting dispersed orbital elements. The ghost family originated from collisional events in the early stage of the solar system. Due to the long-term Yarkovsky effect and gravitational effects, the orbital elements of these asteroids are excessively dispersed (Delbo et al. 2019). The birth of halo asteroids is understood to be similar to the family asteroids; however, the Yarkovsky effect and collisions ejecta processes cause some asteroids to disperse from the central region of the family toward the periphery (Nesvorny et al. 2015).
Theoretical studies have led to an improved understanding of the formation and evolution of the inner main belt (IMB). Dermott et al. (2018, 2021) concluded that both family and non-family asteroids were born from the collision of a few large primordial asteroids. The Yarkovsky effect causes family asteroids to exhibit a V-shaped distribution on the semi-major axis and the reciprocal of diameter (1/D), allowing to estimate the family age (Milani et al. 2014; Spoto et al. 2015). Delbo’ et al. (2017); Delbo et al. (2019) identified two unknown families of collisional asteroids in the IMB through the V-shaped distribution, supporting the existence of the ghost family. Dermott et al. (2021) suggested that the Yarkovsky effect is the primary mechanism for the evolution of asteroids ranging from 1 to 10 km in diameter. Chaotic diffusion has also been identified as a key factor for the evolution of main-belt asteroids into partially Mars-crossing asteroids (MCAs), near-Earth objects (NEOs), and meteorites (Morbidelli & Nesvornỳ 1999; Minton & Malhotra 2010; Dermott et al. 2021).
The number of the fully identified asteroids has dramatically increased thanks to new telescopes and space missions with a better statistical description of their population (Liu et al. 2023) Dermott et al. (2018, 2021) examined only 66451 IMB asteroids using the value 16.5 as the observational completeness limit This paper adopts a dynamic completeness limit description of the IMB and reanalyzes the distribution of orbital elements using the updated Asteroid Families Portal (AFP) data, aimed at adding further constraints on the origins of the IMB and NEOs, while optimizing theoretical models of IMB evolution.
Section 2 outlines the dataset and the method used to determine the completeness limit. Section 3 examines the distribution characteristics of the various asteroid types in the IMB and presents our conclusion on the origins and evolution models. In Sect. 4, we simulate the diffusion of orbital elements caused by chaotic effects and estimate the rate of chaotic diffusion loss for the IMB and MCAs.
2 Data processing
2.1 Databases
We collected the IMB asteroid orbital elements from the AFP, which was published in 20171. The AFP give access to proper element data for a total of 1 052 382 objects, including 596 526 numbered asteroids. To obtain a reliable analysis result, this study excluded asteroids observed during several oppositions, for which an accurate calculation of the orbital parameters remains challenging (Carruba et al. 2021). We also considered their physical properties, such as the albedo of the asteroids derived from the Minor Planet Physical Properties Catalogue (MP3C) of the Observatoire de la Côte d’Azur2.
2.2 Completeness limits
Small asteroids with higher absolute magnitudes are hard to observe by ground telescopes. This can result in biased IMB samples, as small asteroids still remain hidden. To mitigate this observation bias, the completeness limit of the observational data need be estimated. Additionally, the completeness limits should be dynamically changed as the number of observed asteroids continually increases. In this study, we adopted the method proposed by Hendler & Malhotra (2020) to estimate the completeness limit value of asteroids listed by the AFP. The detailed calculation process is shown in Fig. 1.
From Fig. 1, there are mainly two steps: (a) determining the empirical completeness limit value (Hlim) for each bin, with the bin width being about 0.02 au and H being 0.5, and identify the H bin with the greatest number of asteroids. We take the center of this bin as the value of Hlim at that radial distance; and (b) fit each Hlim to a nonlinear function of the semi-major axis as,
(1)
where a is the semi-major axis of an asteroid; the free parameter uses the Markov chain Monte Carlo (MCMC) algorithm. Here, C needs to be estimated from the posterior distribution, taking the median of the sampling results as the optimal estimation and using the standard deviation as the uncertainty3. The optimal process is shown in Fig. 2.
From Fig. 2, the fitted value of the completeness limit Hlim is 17.62 ± 0.039, corresponding to a free parameter C of 20.11 ± 0.039. Generally, increasing the sample size boost the statistical confidence level. Thus, the fitting method to determine asteroid sample completeness dynamically could yield more reliable and statistically sound outcomes.
![]() |
Fig. 1 Adjustment procedure of completeness limit based on AFP data (Hendler & Malhotra 2020). |
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Fig. 2 Completeness-limit fitting for AFP dataset. The black dot represent the empirical values Hlim and uncertainties within a fixed semi-major axis interval (Δa = 0.02 au). The value of fitting is in red, while the light blue shading is the confidence interval (1σ). The dashed blue line represents the upper limit Hlim = 18.3 and lower limit Hlim = 17.3 of the fitting. Dermott et al. (2018) also estimated the completeness limit in gray. The black vertical line corresponds to the median semi-major axis (at 2.344 au) of the samples, with a fitting value of 17.62. |
2.3 IMB families
For the IMB asteroid with a Sun distance of 2.1 au < a < 2.5 au, the numbered asteroid count is 183129, of which 72451 are family asteroids and 110 678 are non-family asteroids. The proper elements of typical families in the IMB are analyzed in Table 1.
From Table 1, the asteroid from major families, including Vesta, Flora, Massalia, and Nysa–Polana–Eulalia, takes up 85% of the IMB family asteroids, with the semi-major axis ranging from 2.16 au to 2.49 au, and a low-middle orbital inclination below eight degrees. The Vesta family is approximately 28% of those in the IMB, with a low overall eccentricity. The asteroids in the Nysa–Polana–Eulalia complex region is about 30%, with a low orbital inclination. Notably, the proper elements, including a, e, and i, of the Flora and Baptistina families are close and there may be an overlap between the two families. The asteroid count of IMB is 124 109 when the completeness limit value is 17.6, which approximately increases by 87% with respect to the results from Dermott et al. (2018).
Proper elements of the asteroid families.
3 Asteroid distributions
In this section, we study the distribution of major families and non-family asteroids in the proper elements space, their size frequency distributions (SFDs), and their plausible evolution scenarios. The observation samples were classified by their orbital elements distribution to reveal the ghost family and MCAs as well.
3.1 Family characteristics
3.1.1 Family analysis
The 124 109 numbered IMB asteroids were analyzed in Fig. 3. In Fig. 3a, the dashed line indicates the v6 secular resonance (SR), where the resonance position is related to the semi-major axis, and the shape of the v6 SR can be approximated by a second-order polynomial related to the proper inclination (I) (Delbo et al. 2019):
(2)
The green arrow indicates the routes of the Yarkovsky force-driven asteroids escaping to the v6 SR boundary or the Jupiter 3:1 mean motion resonance (MMR) boundary (a = 2.5au). When sin I > 0.15 (I > 9°), the number of asteroids decreases significantly. The major families show clustering in the low-inclination zone. Dermott et al. (2018) proposed that a limited number of huge primitive asteroids may have been at the origin of these asteroids. Thus, a different origin for the asteroids with dispersed orbital elements in the high-inclination zone could be inferred, indicating that non-family asteroids could be a supplementary source of asteroids in the v6 SR boundary. The v6 SR boundary is one of the main sources of NEOs as well, which implies that non-family asteroids have a higher probability of evolving into NEOs (Granvik et al. 2017).
In Fig. 3b, the dashed lines indicate the asteroid with the 4M (Mars)/7A (Asteroids) and 1M/2A MMR. Asteroids in the green shade region may cross the Mars orbit region; and the region width depends on Mars’s eccentricity. The change cycle of Mars eccentricity is about 2 Myr, ranging from 0.004 (the upper limit of the green shade) to 0.141 (the lower limit of the green shade), and is currently at a 0.093 value (green line) (Murray & Dermott 1999). If asteroids are located in or above the shadowed region, their perihelion is smaller than Mars’s aphelion and intersect the orbit of Mars. Michel et al. (2000) demonstrated that the spread of eccentricity is constrained and the high-eccentricity asteroids may be expelled from the IMB on account of close encounters with Mars, and subsequently evolve into NEOs.
Figure 3c shows that the major families are clustered in different e and I ranges, confirming again that these major families may have originated from several large asteroids. There is an overlap between the Flora and Baptistina families, due how the HCM algorithm works, where the average distance between the asteroids is similar, making it difficult to completely separate these two families with close proper elements.
Figure 3d plots the albedo distribution of the Vesta family and its neighboring non-family asteroids. The average albedo of the Vesta family is 0.35. The Vesta family is surrounded by asteroids, called the halo asteroids (in blue), and they could originate from the same large asteroids, which is also evidenced by their physical properties (Nesvorny et al. 2015). Furthermore, the correlation between the average proper elements and the absolute magnitude of the family asteroids is shown in Fig. 4.
In Fig. 4, the mean proper e and mean proper I of each major family tend to be flat because the members of each family are largely clustered in a smaller range of proper elements. However, the average proper elements differ greatly between each family (can also be observed from Table 1), which causes all families (orange line) to show a larger range of changes than major families. Furthermore, the curves for all families (orange line) and non-families (black line) exhibit greater variability and point to a similar trend, with a decrease in mean proper e and an increase in mean proper I as H decreases. This suggests that non-family asteroids are correlated with family asteroids in the IMB, which is consistent with the conclusions reported in Dermott et al. (2018).
![]() |
Fig. 3 Distributions of the major families, with non-family asteroids in black. The distribution of: (a) proper inclination and (b) proper eccentricity against the proper semi-major axis; (c) proper eccentricity and inclination and (d) albedo plots for the Vesta family and background asteroids. |
![]() |
Fig. 4 Changes in the (a) mean proper e and (b) mean proper I with absolute magnitude. The point represents an average value in a bin of width dH = 0.5, and the error bar indicates the standard error for each bin. The dashed line is the IMB completeness limit (H = 17.6). |
3.1.2 SFDs
The relationship between family and non-family asteroids was further analyzed by SFDs. The SFD of a known family, formed through catastrophic collisions of large celestial bodies, is influenced by the collision exponents, the size of the parent asteroid, and the evolutionary history of the family. Assuming that all asteroids have the same collision rate independently of the asteroid size and eventually reaches a dynamic equilibrium in a collision cascade, their size distributions can be described by the SFD equation (Durda & Dermott 1997):
(3)
where c is a constant and the slope parameter b can be determined by a polynomial fit. The SFD is plotted in Fig. 5.
In Fig. 5a, the SFDs of non-family and all-family asteroids are highly consistent. The trend of the Vesta family (green line) and the Nysa-Polana-Eulalia asteroids (red line) SFD is similar to the trend of all-family asteroids, as they take about 57% of all-family asteroids. The Flora family SFD curve is lower than others at H = 17.5 due to the v6 SR. For some families that reached collisional equilibrium, the production rate of small asteroids is lower than the loss rate. The SFD of the Massalia family displays a linear trend when H > 15.5, indicating that Massalia family has not yet achieved collisional equilibrium.
Dermott et al. (2021) also pointed out that the Massalia family, which is approximately only 160 Mys old, is a significant source of small asteroids. Combined with Fig. 3b, the Massalia family is near a 2:1 resonance with Mars. The Massalia family asteroids may have entered 2M/1A MMR through collisional evolution and subsequently transformed into MCAs.
Once the equilibrium cascade is established, Dohnanyi (1969) showed that the value of b is around 0.5, which is similar to the slope at low H for the major families in Fig. 5b. The slope of the SFD approaches zero at high H, indicating that the SFD of smaller asteroids does not support the equilibrium distribution, because Dohnanyi (1969) only considered a closed system and ignored losses. That is to say that considering only a single collision mechanism is insufficient to explain the SFDs of small asteroids.
![]() |
Fig. 5 IMB SFDs: (a) cubic polynomial fit adopted for the asteroids’ SFD. The Poisson error is the square root of the asteroid count in each interval, and the range of error bars is the reciprocal of the Poisson error; (b) same cubic polynomial in (a) but with its slope b plotted as a function of H. The error bars of b are determined by Monte Carlo methods. |
3.2 Ghost family
Non-family asteroids with dispersed orbital elements are shown in Figs. 3a and d. These asteroids have unique physical properties compared to the family asteroids, and their presence may provide insights into the potential existence of the ghost family. The question of how to distinguish the ghost family remains open.
Dermott et al. (2018, 2021) demonstrated that asteroids with high inclinations can be distinguished in simulations. The Yarkovsky effect likely caused dispersion along the semi-major axis, but their inclinations remained relatively unchanged. The albedos can be related and used to classify the physical properties of IMB asteroids as well, as plotted in Fig. 6.
From Fig. 6, the asteroid count in regions with I < 8° decreases significantly after the removal of family members; for I > 8°, there is no obvious change. Thus, the majority of asteroids with I > 8° are non-family asteroids, while most of the family asteroids are in low-inclination regions. The red rectangle (Erigone family) points to a typical region where the asteroid albedos are less than 0.13 (black points). The family asteroids and non-family asteroids (included halo-family) display a similar distribution, suggesting a shared origin for the family halo and the major families. This significantly influences the distribution of low-inclination asteroids. Furthermore, from Fig. 6b, the albedos of most non-family asteroids with high inclinations are below 0.13, while the albedos of most halo family asteroids with low inclinations are above 0.13. This suggests that non-family asteroids with inclinations greater than 8° have physical characteristics and orbital elements that differ from most family asteroids, thus, they could belong to the potential ghost family.
To study the ghost family and eliminate the impact of family asteroids, some thresholds were established in orbital element space to effectively differentiate ghost family asteroids from family asteroids and associated halo asteroids. According to Dermott et al. (2018), for non-family asteroids in IMB, if the asteroids with an eccentricity in the range from 0.07 to 0.23 or have an inclination in the range from 0° to 8°, they are called halo asteroids; otherwise, they may belong to the potential ghost family asteroids, as shown in Fig. 7.
From Fig. 7, 102 690 asteroids (86% of samples) can be classified as family and halo asteroids. The remaining 16 688 asteroids (14%) can be classified as non-halo asteroids. The distribution of family asteroids and non-family asteroids is visualized in Fig. 8.
In Fig. 8a, the family asteroids display a bimodal distribution, which be observed in the halo asteroids as well. This indicates that halo asteroids and the major families have the same origin. The non-halo asteroids (potential ghost asteroids) with high inclinations notably decrease. Evolution on the semi-major axis of non-halo asteroids, due to the Yarkovsky effect, is the main mechanism transferring them to the v6 SR, thus, giving the depletion value during their long-term evolution (Dermott et al. 2021). The mean proper elements and SFDs for the family asteroids and the halo asteroids are plotted in Fig. 9.
From Figs. 9a and b, we can see the mean orbital elements of family and halo asteroids are similar, but non-halo asteroids have distinctive mean orbital elements. Figures 9 c and d show a high similarity between the SFD of halo and family asteroids. This strongly supports the idea that halo asteroids and major asteroid families have a common origin, and furthermore that the non-halo asteroids and family asteroids have not a common origin but probably come from the ghost family (Dermott et al. 2018).
![]() |
Fig. 6 Albedo distribution of asteroids: (a) family and non-family asteroids and (b) only non-family asteroids. Assuming an average IMB albedo of 0.13 (Dermott et al. 2021), the asteroids with albedos above 0.13 are plotted in blue, while those with albedos below 0.13 are shown in black. |
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Fig. 7 Definition of halo and non-halo asteroids. |
![]() |
Fig. 8 Proper element distributions of halo and non-halo asteroids. |
![]() |
Fig. 9 Average proper elements distribution and SFD of halo and non-halo asteroids. |
3.3 MCAs
The MCAs are considered one of the main sources of NEOs and meteorites. Michel et al. (2000) estimated that at least 50% of large NEOs from MCAs play a key role in maintaining the stability of the NEOs. That is to say, MCAs must have continuous replenishment to maintain their abundance. Dermott et al. (2019, 2021, 2023) proposed that Mars inject main-belt asteroids into the inner solar system, potentially causing them to evolve into NEOs; especially, IMB asteroids are the primary source of MCAs (Gladman et al. 1997; Granvik et al. 2017, 2018). The absolute magnitude and proper inclination distribution of MCAs are plotted in Fig. 10.
As seen in Fig. 10a, most of MCAs have a diameter less than 1 km (H > 17.6). The YORP effect and collision evolution are responsible for the escape of these smaller asteroids from IMB. From Fig. 10b, the percentage with I > 8° in MCAs is 16.6%, while the percentage with I > 8° in IMB is only 9.7%. This result can be explained by the fact that high-inclination asteroids are more likely to escape from the IMB due to the Yarkovsky effect (Dermott et al. 2019, 2021, 2023). The Mars-crossing region also includes a number of asteroids with I < 8°, as most major families and halo asteroids have similar inclinations. Such asteroids originate from these major families and their halo asteroids in the IMB.
Average initial proper elements of test groups.
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Fig. 10 Normalized distribution of MCAs and IMB asteroids. |
4 Chaotic orbital evolution
We recall that Fig. 3b shows that asteroid eccentricity in the 1M/2A resonance displays a diffusion phenomenon, possibly caused by chaotic effects. In this section, we quantify this chaotic diffusion rate through numerical simulations and further investigate the complement from the IMB to MCAs.
4.1 Simulated particles
To study how an orbit evolves over time, we generated test particles with different initial orbital elements inside and out-side the 1M/2A MMR. The test particles included three groups in resonance and one non-resonance group; each group had four hundred particles, which is consistent with the findings of Dermott et al. (2018). The particles in the resonance group display a fixed 2.4184 au osculating semi-major axis, while those in the non-resonance group have a fixed 2.4230 au. The initial osculating longitudes of perihelion, ascending node, and mean anomaly of particles were assigned uniformly random values between 0 and 360°. The resonance groups are defined with (e = 0.15, I = 6.5°), (e = 0.15, I = 10°) and (e = 0.19, I = 6.5°), while the non-resonance group was assigned values, with (e = 0.20, I = 6.5°).
We explored the dynamics of the resonance by a series of numerical integration using the Orbit9 software (Milani & Nobili 1987; Knežević & Milani 2001). We used T = 100 Myr as the integration period and a multi-step predictor scheme with automatic step size control combined with a symplectic fixed-step scheme as a starter. The maximum integration step size was set to 0.2 yr and the output step size was set to 10 000 yr. The solar system dynamics model included seven planets except Mercury, as we also referred to the Sun-Mercury barycenter Dermott et al. (2018). The initial planetary state vectors were taken from the JPL DE431 ephemeris at the epoch JD 2460200.5 (September 13, 2023).
4.2 Results and analysis
The initial particles did not intersect the Mars orbit, but they were in the resonance region so their eccentricity was expected to gradually grow converting them into MCAs (Christou et al. 2022). We can eliminate the impact of MCAs, according to the following equation:
(4)
where n = 2, q is the asteroid perihelion distance, QM is the Mars aphelion distance, and RH is the Hill radius of Mars. Only three particles from the high-eccentricity resonance group transitioned into Mars-crossing orbits.
The asteroid osculating elements experience periodic changes due to gravitational force, while the proper elements were affected by chaotic evolution. Therefore, the focus was on tracking the spread of proper elements exclusively. For each particle, the average of the previous 10 Myrs of output was chosen as its initial proper elements. The mean and standard deviation of the initial proper eccentricity and inclination for each test particle group are listed in Table 2.
To estimate the chaotic diffusion of proper elements, we used
(5)
where x is one of the proper elements (ap, ep, ip) and N denotes the count of particles in each group. The time series was computed by averaging the integration output every 0.32 Myr, and the average of the orbital elements over the preceding 10 Myr was utilized as the initial proper element Dermott et al. (2018). The time dependence of the variable σx(t) was described by a power law, as proposed by Christou et al. (2022),
(6)
where the exponent b and constant C are estimated by fitting the log linear relationship between σx and t, expressed as log10 σx(t) = b log10 t + log10 C. The results for eccentricity are displayed in Table 3 and the inclination fittings are presented in Table 4.
From Tables 3 and 4, for the resonant asteroids, the eccentricity diffusion rate, be, ranges from 0.419 to 0.474 and the inclination diffusion rate, bi, ranges from 0.403 to 0.432, following the random walk theory (Carruba et al. 2003). This suggests that the chaotic evolution is a random process. The average value for bi is 0.419 ± 0.012 and for be is 0.450 ± 0.023, setting be = 0.45 and bi = 0.4 appear therefore to be effective approximations in fitting the slopes of these power laws. Christou et al. (2022) proposed that b should be about 0.4 in either eccentricity or inclination, so they may have underestimated the diffusion rate of eccentricity.
By extrapolating the fitting results, we can more intuitively observe the long-term diffusion of orbital elements. Fig. 11 shown the log-linear fit, where the slope corresponds to the diffusion exponent b. The extrapolation of these results extends to 2 Gyr. The standard deviation and the slope b of non-resonant asteroids (in black) are significantly lower than those of resonant asteroids (in color).
From Fig. 11, the standard deviation of eccentricity for the resonant asteroid is close to 0.1 over a 2 Gyr evolution, but the inclination only spreads for about 1°, indicating that chaotic orbital evolution can lead to a significant diffusion of eccentricity, but not in inclination. The orbital inclination can retain its original characteristics, however, due to the smaller diffusion distance (Dermott et al. 2018, 2021).
Log-linear fitting of σe and t.
Log-linear fitting of σi and t.
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Fig. 11 Chaotic orbital evolution of resonant and non-resonant asteroids. The standard deviations of the proper eccentricities (a) and inclinations (b) increase with time. The scatter represents the standard deviations of different test particle groups within each period (32 000 yr). The resonant groups are depicted in color, while the non-resonant group is shown in black. The scatter between two black vertical dashed lines is used to fit the log-linear function (discarding the output of the previous 10 Myr). |
4.3 MCAs
The chaotic diffusion causes the IMB asteroids to spread, and some evolve into MCAs (Morbidelli & Nesvornỳ 1999). To accurately quantify the complement of IMB among MCAs, we employed a resonant fitting model. Christou et al. (2022) confirmed that after 4 Gyr of chaotic orbital evolution, the eccentricities of resonant asteroids can diffuse to be 0.25, allowing close encounters with Mars and subsequent removal from the IMB. This enables the estimation of the escape fraction of asteroids in the 1M/2A resonance. Assuming an initial uniform distribution of asteroid eccentricities, after an evolutionary period δT, the loss function is,
(7)
where Δn0 represents the loss quantity and n0 denotes the initial count of bodies. The term Φ(z) is the scaling factor of the number reduction, where z = (e0 − ecrit)/σ(δT) and Φ is the cumulative normal distribution. Then, ecrit represents the eccentricity threshold for escaping asteroids, which is 0.23. Considering the wide diffusion of eccentricity in the resonance, the lower integration limit eLB is set to 0, and for non-resonant asteroids is set to 0.19 (Christou et al. 2022). The diffusion rate of eccentricity selected is 0.45. Compared to the results of Christou et al. (2022) (fRES = 0.0312, fNON–RES = 0.0044), we calculated the losses for resonant asteroids as fRES = 0.0373 (3.73%) and for non-resonant asteroids as fNON–RES = 0.0052 (0.52%).
Furthermore, based on the loss functions, we estimated the total loss in population for IMB. In the simulation, we defined the resonance range as 2.4174 < a < 2.4194 au (Gallardo et al. 2011) and the count of resonant asteroids as 3317, with a loss of 124. At the same time, the count of asteroids in the non-resonance was 179 812, with a loss of 935. Considering that the 1M/2A resonance is one of the primary routes for IMB asteroids evolution into MCAs (Morbidelli & Nesvorny 1999), we assumed that the efficiency of other resonant escape mechanisms for supplementing MCAs is comparable to the 1M/2A resonance. Therefore, the count of asteroids escaping via chaotic orbital evolution after 4 Gyr is estimated to 1059 with a loss rate of 0.2648 Myr−1.
We further verified the correctness of the equilibrium hypothesis of the MCAs proposed by Bottke et al. (2002). Asteroids first migrate from the IMB to the Mars-crossing region and then evolve into NEOs. If the MCAs remains in dynamic equilibrium, the outflow asteroids from the IMB should be equivalent to the loss of MCAs. The loss rate of MCAs with an absolute magnitude lower than H can be calculated as follows (Bottke et al. 2002),
(8)
where CNEA is the count of objects with 13 < H < 15 in the NEOs, αMC is the ratio of MCAs in the NEOs, and is the average evolutionary time of MCAs in the NEO region. These numerical conditions are based on the model values from Bottke et al. (2002): CNEA = 13.26, αMC = 0.27, and
= 3.75 Myr. The estimated the loss rate value of MCAs is about 24.13 within the completion limit (H = 17.6).
The loss rate of the IMB asteroids via chaotic orbital evolution was 0.2648 Myr−1, which is only 1.1% of MCAs. Thus, chaotic orbital evolution has a limited capacity to replenish MCAs, but non-gravitational effects may play a dominant role, which can drive smaller asteroids to evolve into MMR and further be MCAs. However, for the large MCAs, where IIMC(12) = 0.2646 Myr−1 is observed, consistent with the loss rate of the IMB. This suggests the large MCAs (H < 12) are in a dynamic equilibrium, primarily evolving through chaotic orbital evolution.
5 Conclusions
This study analyzes the absolute magnitude completeness of the IMB population based on the latest AFP database. Compared to the findings of Dermott et al. (2018), the number of bodies considered in this study has increased by 87%, offering new insights for investigating the origin of the IMB and its evolution. Our specific conclusions are as follows:
As the count of asteroids increases, the completeness limit of the observational sample changes accordingly. The completeness limit value is calculated to be 17.6 for the IMB based in AFP, which was determined by a fitting model and effectively reduced uncertainty.
The orbital and physical analysis for IMB say that the major families could have originated from a small number of large primitive objects with different orbital elements, and most non-family asteroids could have originated from the major families as well. This confirms the conclusions drawn by Dermott et al. (2018). More detailed conclusions on this aspect are given below:
- (a)
The SFDs of smaller asteroids do not align with the equilibrium distribution found by Dohnanyi (1969) since Dohnanyi ignored the asteroid loss. Including only the collision mechanism is insufficient to explain the SFDs of small asteroids.
- (b)
About 86% of the asteroids in the IMB originated from a few major families, and the remaining 14% non-family asteroids may be members of potential ghost families.
- (c)
The IMB asteroids are the primary source of supplementation for MCAs, with non-gravitational effects as the dominant mechanism of supplementation.
- (a)
Through long-term numerical integration, we confirmed that the chaotic effect can result in the significant diffusion of orbital elements on timescales. The MMR accelerates the chaotic orbital evolution of asteroids, while the diffusion of orbital inclination is very limited, which is relatively consistent with the results presented in Dermott et al. (2018).
Chaotic diffusion is a random process and the diffusion distance is sensitive to the initial proper elements. This result is consistent with Christou et al. (2022). The diffusion rates of eccentricity and inclination in our simulation were estimated to be 0.45 and 0.4, respectively, and close to the expectation of a random walk (b = 0.5).
Estimations of diffusive loss indicates that large objects (with H < 12) within MCAs adhere to the equilibrium hypothesis (Bottke et al. 2002). The lost rate of asteroids from MCAs is equal to the supplement rate from the IMB. For these large asteroids, chaotic diffusion is the primary evolutionary mechanism.
Acknowledgements
This study was supported by the National Natural Science Foundation of China (Grant Nos. 42241116 and 12203002) and National Key Research and Development Program of China (No. 2022YFF0503202).
Appendix A Robustness of the equilibrium hypothesis
Tables 3 and 4 show that the diffusion rate depends on the initial proper orbital elements and the estimation of the equilibrium hypothesis of the MCAs is based on the eccentricity diffusion rate of the asteroid in the resonance region. To verify the robustness of the equilibrium hypothesis, we generated another four groups of test particles with different initial proper orbital elements in the resonance region for simulation.
These particles had a fixed 2.4184 au osculating semi-major axis. The initial osculating longitudes of perihelion, the ascending node, and the mean anomaly of particles were assigned uniformly random values between 0 and 360 degrees. The initial proper eccentricity and inclination of these resonant particles are (e = 0.13, I = 2.5°), (e = 0.15, I = 2.5°), (e = 0.15, I = 6.5°), and (e = 0.17, I = 2.5°), and they were analyzed with two other sets of resonance simulations (e = 0.13, I = 6.6° and e = 0.17, I = 6.6°, respectively). We fitted the diffusion rate of these six groups of resonant asteroids again and the results for eccentricity are displayed in Table A.1 and the inclination fittings are presented in Table A.2.
Log-linear fitting of σe and t.
Log-linear fitting of σi and t.
From Table A.1, when the initial proper inclination is 2.5°, with the initial proper eccentricity increasing from 0.13 to 0.17, the diffusion rate of particles decreases from 0.4816 to 0.3798, which further verifies that the diffusion rate of eccentricity depends on the initial proper eccentricity, while Table A.2 does not observe the correlation between the inclination diffusion rate and the initial proper orbital elements. The average diffusion rate of eccentricity is 0.4313, and the average diffusion rate of inclination is 0.3921, which is close to the expected value of random walk law (b = 0.5), indicating that the evolution of chaotic orbit is a random process. The new simulation shows that be = 0.43 and bi = 0.4 are used to estimate the diffusion rates of eccentricity and orbital inclination, while the initial diffusion rates are be = 0.45 and bi = 0.4, respectively. Christou et al. (2022) proposed that the diffusion rates of eccentricity and inclination ought to be estimated by 0.4. The fitting results of the diffusion rate of eccentricity are reasonable in the range of 0.4 to 0.5, which is close to the random walk law.
Through different diffusion models, we can further explore the influence of different initial proper orbital elements on the equilibrium hypothesis of the group of small asteroids. We used the three different fitting values of be 0.40 (Christou et al. 2022), 0.45 (the average value of the three groups of test particles in the resonance region in Table 3), and 0.43 (the average value of the six groups of test particles in Table A.1) to re-estimate the replenishment of IMB to MCAs through chaotic diffusion. In the simulation, the range of resonance region is defined as 2.4174 < a < 2.4194au (Gallardo et al. 2011), the number of asteroids in resonance region is 3317, and the number of asteroids in non-resonance region is 179812. For different diffusion models, the results are listed in Table A.3.
Equilibrium hypothesis estimation.
The results coming from different dispersion models are similar, and the loss of asteroids in resonance region is between 3% and 4%, while in a non-resonance region, it is about 0.5%. For different diffusion models, the cutoff value of absolute magnitude is close to 12.0 when the equilibrium assumption is true, indicating that the initial proper orbital elements will not significantly affect the equilibrium hypothesis of large-scale small objects (H < 12). Even under different initial conditions, the main mechanism is chaotic diffusion and the equilibrium hypothesis is verified.
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All Tables
All Figures
![]() |
Fig. 1 Adjustment procedure of completeness limit based on AFP data (Hendler & Malhotra 2020). |
In the text |
![]() |
Fig. 2 Completeness-limit fitting for AFP dataset. The black dot represent the empirical values Hlim and uncertainties within a fixed semi-major axis interval (Δa = 0.02 au). The value of fitting is in red, while the light blue shading is the confidence interval (1σ). The dashed blue line represents the upper limit Hlim = 18.3 and lower limit Hlim = 17.3 of the fitting. Dermott et al. (2018) also estimated the completeness limit in gray. The black vertical line corresponds to the median semi-major axis (at 2.344 au) of the samples, with a fitting value of 17.62. |
In the text |
![]() |
Fig. 3 Distributions of the major families, with non-family asteroids in black. The distribution of: (a) proper inclination and (b) proper eccentricity against the proper semi-major axis; (c) proper eccentricity and inclination and (d) albedo plots for the Vesta family and background asteroids. |
In the text |
![]() |
Fig. 4 Changes in the (a) mean proper e and (b) mean proper I with absolute magnitude. The point represents an average value in a bin of width dH = 0.5, and the error bar indicates the standard error for each bin. The dashed line is the IMB completeness limit (H = 17.6). |
In the text |
![]() |
Fig. 5 IMB SFDs: (a) cubic polynomial fit adopted for the asteroids’ SFD. The Poisson error is the square root of the asteroid count in each interval, and the range of error bars is the reciprocal of the Poisson error; (b) same cubic polynomial in (a) but with its slope b plotted as a function of H. The error bars of b are determined by Monte Carlo methods. |
In the text |
![]() |
Fig. 6 Albedo distribution of asteroids: (a) family and non-family asteroids and (b) only non-family asteroids. Assuming an average IMB albedo of 0.13 (Dermott et al. 2021), the asteroids with albedos above 0.13 are plotted in blue, while those with albedos below 0.13 are shown in black. |
In the text |
![]() |
Fig. 7 Definition of halo and non-halo asteroids. |
In the text |
![]() |
Fig. 8 Proper element distributions of halo and non-halo asteroids. |
In the text |
![]() |
Fig. 9 Average proper elements distribution and SFD of halo and non-halo asteroids. |
In the text |
![]() |
Fig. 10 Normalized distribution of MCAs and IMB asteroids. |
In the text |
![]() |
Fig. 11 Chaotic orbital evolution of resonant and non-resonant asteroids. The standard deviations of the proper eccentricities (a) and inclinations (b) increase with time. The scatter represents the standard deviations of different test particle groups within each period (32 000 yr). The resonant groups are depicted in color, while the non-resonant group is shown in black. The scatter between two black vertical dashed lines is used to fit the log-linear function (discarding the output of the previous 10 Myr). |
In the text |
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