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Nature Communications volume 17, Article number: 6070 (2026) Cite this article
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The formation and evolution of the Solar System’s two major comet reservoirs, the Oort Cloud and the Scattered Disk, remain poorly constrained. Observations of comet flux imply a population ratio between these reservoirs that exceeds predictions from models of giant planet instability by more than two orders of magnitude. A major source of uncertainty lies in determining comet sizes, as the nucleus is often obscured by the surrounding coma. Here we derive independent nucleus sizes for 28 comets by activity modeling with artificial intelligence to infer sizes directly from variations in water emission. We find that long-period comets possess significantly larger nuclei than short-period comets with comparable absolute brightness. Our results indicate that the Oort Cloud is nearly three orders of magnitude more populated than the Scattered Disk, implying that processes such as stellar flybys could have played an important role in shaping the outer Solar System.
Our knowledge of the extent and structure of the Solar System beyond Neptune remains incomplete, primarily due to the observational challenges posed by its vast distance and the faintness of its constituents^1,2. The architecture of that remote region is conventionally inferred from the observed fluxes of various types of comets^3,4. Long-period comets (LPCs)–which approach the inner Solar System from nearly isotropic directions and have orbital periods exceeding 200 years–are believed to originate from the distant, spherical Oort Cloud (OC)⁵. Short-period comets (SPCs), particularly Jupiter-family comets (JFCs), are dynamically linked to the Scattered Disk (SD), a region consisting of Trans-Neptunian Objects with high eccentricity and inclinations⁶.
It is widely accepted that the present-day OC and SD were populated during the phase of giant planet instability in the early Solar System (see Nesvorný⁷ and references therein). Primordial planetesimals were dispersed primarily due to the migration of the giant planets, particularly the outward movement of Neptune^2,4. Numerical simulations stated that  < 1% of the objects in the outer protoplanetary disk were implanted into the SD, and up to about 5% into OC, yielding an OC/SD population ratio of approximately 5–20^2,4,8,9. However, this theoretical ratio is two to three orders of magnitude lower than that inferred from the observed comet flux. Observational data suggest that roughly 3–7 dynamically new LPCs per year have absolute magnitudes brighter than about 11 and perihelion distances less than 4 au^10,11,12, implying an OC population of 7−8 × 10^{11 13}. Estimates of the SD population, based primarily on the observed population of JFCs, are more uncertain due to the so-called “fading” problem^14,15. As a result, the prediction of SD population ranges broadly from 4.4 × 10⁸ to 6 × 10⁹ (see Kaib & Volk⁴ and references therein), leading to an OC/SD population ratio of approximately 100–2000.
Efforts have been made on both fronts to reconcile this discrepancy. From the modeling perspective, recent works incorporate interactions between planetesimals and the early solar nebula, as well as perturbations from stellar encounters in the Sun’s birth cluster, offering a more comprehensive picture of outer Solar System evolution^8,16,17,18. Meanwhile, observational interpretations have been revisited on the relationship between comet flux and the inferred populations of the OC and SD, for which a central issue is the poorly constrained sizes of cometary nuclei^4,9,19.
Determining the sizes of comets poses significant challenges due to their extreme faintness during inactive phases and the obscuration caused by dust comae during periods of activity^20,21,22,23. Existing methods include photometric techniques, infrared and thermal modeling, dynamical analysis, and direct imaging (a comprehensive review of these methodologies can be found in the literature^21,23). Photometric and infrared analyses are most widely applied. However, as most comets are observed in the active phase, both approaches face the challenge of subtracting signals from the dust coma. The contamination of residual coma, together with unconstrained surface albedo, could lead to large uncertainties in the estimated nucleus size²⁴. Dynamical modeling, which reconstructs the observed non-gravitational forces exerting on the comet by its gas emission, is less sensitive to coma-related uncertainties but is influenced by the diverse and often poorly understood activity patterns of individual comets²⁴. Direct spacecraft imaging provides the most accurate measurements of cometary nuclei, yet its applicability remains limited to the small number of comets that have been visited by space missions.
The European Space Agency’s Rosetta mission, which conducted an in-depth investigation of comet 67P/Churyumov-Gerasimenko (67P) over more than two years, has significantly advanced our understanding of cometary activity^25,26,27. A key discovery was the ubiquitous presence of water ice in the shallow subsurface of the nucleus²⁵, giving rise to the release of water vapor controlled by insolation^26,27,28. Nucleus thermophysical models could reproduce 67P’s time-varying water production rates by optimizing physical properties of the nucleus^29,30. This modeling approach inherently avoids complications associated with dust coma, and could be in principle applied to any comet with reliable measurements of water production rates. However, producing complete water production profiles with sophisticated thermophysical models and vast parameter spaces remains computationally intensive–particularly for comets with limited prior information.
Recent advancements in deep learning (DL) offer promising solutions to the computational bottleneck encountered by traditional numerical approaches^{31,32,33,34,35,36,37}. Leveraging the deep operator neural network (DeepONet)³⁸, we developed ThermoONet–a DL-based thermophysical modeling tool for small bodies^39,40. Similar to conventional numerical methods, ThermoONet predicts the temperature at the subsurface ice front of a comet, which governs the sublimation rate of water ice. However, it achieves this with computational speeds several orders of magnitude faster than traditional models⁴⁰. This substantial performance advantage enables integration with computationally intensive global optimization techniques, such as simulated annealing (SA), to infer time-dependent water production rates and constrain key physical parameters, including nucleus size.
In this study, we utilize a unique dataset of water production curves (hereafter “water curves”) derived from Lyman-α observations by the Solar Wind ANisotropies (SWAN) ultraviolet spectrometer aboard the Solar and Heliospheric Observatory (SOHO)^41,42. SWAN’s all-sky coverage and consistent observational cadence have enabled continuous monitoring of comets throughout their perihelion passages, providing water curves for over sixty comets spanning nearly three decades^43,44. By applying ThermoONet to these water curves, we derive independent estimates of nucleus sizes across a diverse population of comets, which lead to updated estimates on the population of cometary reservoirs. We find that the Oort Cloud is significantly more populated than the Scattered Disk by almost three orders of magnitude, indicating that various dynamical processes could have contributed to shaping the architecture of our Solar System.
We analyzed water curves for a total of 28 comets: 26 derived from SWAN observations^45,46, and two–comet 1P/Halley and 67P–were derived from earlier HI Lyman-α observations⁴⁷ and in situ Rosetta/ROSINA measurements⁴⁸, respectively. This sample includes 7 SPCs, comprising 5 JFCs and 2 Halley-type comets (HTCs), as well as 21 LPCs that are further categorized as Old Long-period (OL), Young Long-period (YL), and Dynamically New (DN) comets, following the classification scheme in A’Hearn et al.⁴⁹.
We applied the ThermoONet-based fitting procedure to retrieve physical parameters of the nucleus that best reproduce the observed water curve (see “Methods”). To ensure the robustness of the derived size estimates, we further employed a post-optimization perturbation strategy (see “Methods”). Figure 1 displays the measured water curves of all 28 comets alongside the corresponding fits, and Fig. 2 shows the distribution of size estimation resulting from the perturbation analysis. The mean values (μ) of each distribution is adopted as the best-fit nucleus diameter, with the 1 − σ standard deviation representing the uncertainty, which is about 17% on average. Dynamical and physical properties of the comets and fitting results of their sizes are listed in Table 1.
Water production rates are shown as black points for HI Lyman-α measurements obtained by SOHO/SWAN^45,46, and by IUE for 1P/Halley⁴⁷. The water production rates of 67P, shown as rectangles, were obtained by Rosetta/ROSINA⁴⁸. The error bars indicate observational uncertainties reported in the original literature. For each comet, the red curve represents the best-fit water curve after global optimization of nucleus parameters, while the pink bands indicate solutions with a mean absolute percentage error below approximately 30%, obtained through random perturbations of the optimized solution. The vertical dashed line indicates the perihelion epoch of the corresponding comet. Source data are provided as a Source Data file.
Blue histogram bars represent probability densities across different diameter intervals for approximately 400 samples per comet, with the red dashed curves as the fitted normal distribution. The pink vertical line ithe mean value adopted as the effective diameter of the nucleus for the corresponding comet, and black vertical dashed lines indicate the 1-σ confidence interval adopted as the error bar. Source data are provided as a Source Data file.
We compare size estimates for six comets, four SPCs and two LPCs, derived using various data sources and methods (Table 2). For the SPCs, which were all spacecraft mission targets, we consider direct imaging results as ground truth, and found size estimates by water curve fitting achieve a mean relative error of about 20% with respect to the actual size (Fig. 3). Since no LPC has yet been visited by a spacecraft, most reliable estimates of their nucleus sizes are inferred from extremely distant observations, for example the observation of comet C/1995 O1 (Hale-Bopp) at 32 au⁵⁰. Using this benchmark, water curve fitting yielded a relative error of about 8%. Dynamical analyses provide reasonable size estimates for SPCs, but exhibit larger errors when applied to LPCs, likely due to the difficulty in accurately constraining the non-gravitational effects of LPCs with limited astrometric data^51,52.
Color coding corresponds to individual comets, and symbol notation distinguishes different estimation techniques as listed at the top of the figure. The error bars indicate the uncertainties of the estimations as stated in Table 2. References of the data points are listed in Table 2. Source data are provided as a Source Data file.
A widely accepted theory posits that LPCs are intrinsically more active than SPCs due to their more “primitive” nature⁹. Consequently, an LPC with the same absolute magnitude as an SPC is expected to have a smaller nucleus^{3,4,9,53,54,55}. We re-examine this hypothesis in light of the updated size estimation. Unlike asteroids, a comet’s absolute magnitude–defined as its apparent magnitude when located at one au from both the Sun and the Earth–reflects the combined brightness of its nucleus and coma, and is therefore dependent on its activity level^56,57. To ensure consistency throughout our analysis, we adopt the absolute magnitude values reported in the Jet Propulsion Laboratory (JPL) Small-Body Database (https://ssd.jpl.nasa.gov/tools/sbdb_lookup.html).
The majority of LPCs in our sample have effective diameters in the range of 4 to 20 km, whereas most SPCs have diameters between 1.5 to 5 km (Fig. 4). The largest LPC is C/1995 O1 (Hale-Bopp), with a diameter of 68.13 ± 10.42 km, while the largest SPC is 1P/Halley, with a diameter of 11.64 ± 2.13 km. To investigate the potential systematic size differences between LPCs and SPCs, we incorporated additional size estimates from Bauer et al.¹² and Knight et al.²³, and compared the average diameters of LPCs and SPCs within equivalent intervals of absolute magnitude. As shown by the green bars in Fig. 4, all LPC/SPC size ratios, except the one in the 11–11.5 magnitude bin, are greater than one, indicating that, on average, LPCs possess larger nuclei than SPCs of the same absolute magnitude over a broad range of brightness between 4 to 13. This result contradicts the prevailing expectation that LPCs should have smaller nuclei than equally bright SPCs.
Absolute magnitudes are sourced from the JPL Small-Body Database. Data for LPC with absolute magnitudes larger than 14 are excluded to avoid observational biases. Filled circles with error bars represent water curve fitting results. Triangles denote data complied by Knight et al.²³, and squares denote data estimated by Bauer et al.¹². Red symbols represent LPC and blue symbols represent SPC. The error bars of the filled circles represent the 1-σ confidence intervals of the size inversion results reported in Table 1. All other error bars are adopted from the corresponding literature. Light green bars (referenced to the right y-axis) show the ratio between the average effective diameter of all LPC and all SPC within each absolute magnitude bin of 0.5 magnitude. Source data are provided as a Source Data file.
We subsequently derived updated correlations between the absolute magnitudes and effective diameters using newly estimated nucleus sizes. For LPCs, we find a relationship of
where H_T is the JPL absolute total magnitude and D_LPC the effective diameter in kilometers (Fig. 5a). Compared to previous correlations found based on non-gravitational effect analyses⁵³, our updated relationship predicts larger nuclei for the same absolute magnitude. We also note that DN comets and YL comets follow a more confined linear trend than OL comets, possibly reflecting enhanced erosion of OL over repeated perihelion passages. We did not classify 1P/Halley and 55P/Tempel-Tuttle as OC objects because HTCs may originate from both SD⁵⁸ and OC⁵⁹, and the grouping has limited impact on the statistics (Supplementary Fig. 1).
a Linear correlations between absolute magnitudes and effective diameters of LPCs. The color coding of the filled circles represents water-curve-fitted sizes of subgroups of LPCs: red for DN comets, green for YL comets, and blue for OL comets. The black dashed line shows the linear fit. The pink bands, ranging from dark to light, indicate the 1-σ to 3-σ confidence intervals for this fit. Beige triangles indicate results from Sosa and Fernández⁵³ (corresponding to JPL absolute magnitudes), with the dashed line being the linear fit of their data. b Linear correlations between absolute magnitudes and effective diameters of SPCs. Green circles represent HTCs, and blue circles represent JFCs. All other follow the conventions described for Panel a. The error bars in Panels (a, b) indicate uncertainties of the estimated size as reported in Table 1. c Dependence of the OC/SD population ratio on the slope and intercept of the linear relationship between absolute magnitude and effective diameter of LPCs. The black dashed line denotes the contour derived from the direct simulation result of the Nice model⁹. The purple asterisk denotes the ratio estimated by Brasser and Morbidelli⁹ based on the linear relationship derived by Sosa and Fernández⁵³. The red asterisk denotes the ratio derived in this work. Source data are provided as a Source Data file.
The D–H_T distribution for SPCs shows greater scatter, likely due to their more evolved nuclei (Fig. 5b). To provide a direct and intuitive comparison with D–H_T of LPCs, we also fit their correlation of
The SPC regression line exhibits a shallower slope and lower intercept than that of LPCs, consistent with the trend observed in our statistical comparison of their nucleus sizes (Fig. 4). Interestingly, for comet 81P/Wild, which has a JPL absolute magnitude of 9.8, this correlation yields a diameter of 4.7 km, closely matching the space mission-derived value of 4.17 km⁶⁰.
Assuming similar size-frequency distribution (SFD) slopes for LPCs and SPCs with diameters exceeding 1 km⁶¹, the population ratio between OC and SD can be inferred from the parameters of the D_LPC − H_T relationship (see “Methods”). By applying our newly derived D_LPC − H_T correlation, we estimate a revised OC/SD population ratio of (99{8}_{-794}^{+2045}) (Fig. 5c).
This ratio, approaching three orders of magnitude, confirms that OC is significantly more populated than SD, substantiating the long-standing discrepancy between OC and SD population estimates derived from comet flux observations and formation models^4,9. It naturally results in a more massive OC than previous estimations, as larger nuclei are inferred with the same magnitude. This suggests that mechanisms beyond giant planet migration may be required to account for the populating of the OC, and/or possibly depopulating of the SD.
The influence of the Sun’s birth cluster has been suggested as a mechanism that could significantly alter the populations of both the OC and SD^8,16,62,63. More recently, stellar flyby events have gained attention as a plausible explanation for the orbital distributions of Sedna-like objects, high-inclination trans-Neptunian objects (TNOs), and cold classical Kuiper Belt objects¹⁸. To investigate the potential impact of such stellar encounters on OC and SD formation, we constructed a toy model using similar parameters as the stellar flyby event proposed by Pfalzner et al.¹⁸. By integrating the trajectories of 10⁵ massless test particles initially distributed between 100 and 500 au (see “Methods”), we find that such a flyby could efficiently disperse the initial disk with only approximately 40% of the particles remaining gravitationally bound to the Sun 100 kyr after the encounter. Intriguingly, a substantial fraction (about 30%) of the remaining bound particles subjected to significant gravitational perturbations are excited to large semi-major axes (lower right section of Fig. 6), shown as particles orbiting along large elliptical trajectories in the animation (Supplementary Movie 1). These particles are likely to become part of the OC under the influence of Galactic tides and stochastic stellar perturbations^3,19. Approximately 60% of the particles were scattered into hyperbolic trajectories and escaped the Solar System, shown in the animation as particles flying out of the field of view at high speed. They have, in general, high inclinations due to the inclined flyby trajectory of the star. If such a flyby event happens early when the Sun was still in its birth cluster⁶⁴, it could have created a heterogeneous initial distribution of OC and SD objects, setting the stage for subsequent planet migration to reshape them further. Conversely, a flyby occurring after giant planet migration may have transferred some SD objects into the OC, thereby amplifying the OC/SD ratio.
Dots represent the distribution of semi-major axes and eccentricities for the massless model particles after 10 kyr of the flyby, with colors indicating their orbital inclinations. Particles with eccentricities larger than one (denoted by the horizontal dashed line) are considered escaped. The ratio between the number of particles escaped and bounded is approximately 3:2. The region on the right side of the vertical dashed line denotes the classical Oort cloud. An animation of the flyby event is displayed in Supplementary Movie 1. Source data are provided as a Source Data file.
Comets are not only key to understanding the primordial conditions of the Solar System but also serve as critical tracers of the dynamical processes that occurred during planetary system formation. However, gaining insights from comets hinges on accurately determining their physical properties, particularly their nucleus sizes. Recent advancements in deep learning, as demonstrated in this study, have enabled an alternative and efficient approach for independently estimating nucleus sizes from cometary water production curves.
Despite this progress, the number of comets with continuous water production rate measurements remains limited, particularly for those with large perihelia. Expanding these observational datasets is essential to reduce selection biases and to derive a statistically more robust relationship between absolute magnitude and effective diameter. To date, spacecraft explorations have exclusively visited SPCs. Observations of LPCs, especially for their activity patterns and direct size measurements, remain a critical knowledge gap. The upcoming ESA Comet Interceptor mission is expected to help address this need by providing the first in situ observations of a dynamically new LPC⁶⁵.
As a close stellar flyby offers a compelling explanation for the high OC/SD population ratio, its actual impact depends on the detailed configuration. Determining the precise timing, geometry, and influence of such an event needs further investigation.
Additionally, recent results from Atacama Large Millimeter Array (ALMA) observations have shown that primordial protoplanetary disks are often far from uniform⁶⁶. Many exhibit irregular density distributions and concentric ring-like substructures. Such complexities in the initial mass distribution of planetesimals can significantly influence their dynamical response to perturbations, including planetary migration and stellar encounters⁶⁷. Thus, adjusting assumptions about the primordial architecture of the Solar System’s protoplanetary disk may also strongly affect the resulting OC/SD population ratio. A more comprehensive understanding of the Solar System’s early evolution requires the synthesis of nuanced dynamical models with refined observational evidence.
We model the cometary water outgassing in the framework of the so-called “dust mantle” model for cometary water activity, where the cometary nucleus is treated as a thin desiccated mantle overlying the dust-ice mixture⁶⁸. Assuming the thickness of the dust mantle is X, the energy balance at the interface between the dust mantle and dust-ice mixture can be expressed as^40,68,69:
The temperature T at the interface is determined by the solar irradiation on the surface of the nucleus, based on the law of energy conservation along with the longitudinal heat conduction^28,68. κ_d, κ_m respectively correspond to the thermal conductivity of the dust mantle and dust-ice mixture. l is the latent heat of water ice, and Z_(X) is the mass flux of sublimation determined by^68,70
where f is called the icy area fraction and represents the microscopic areal abundance of water ice at given depth within the dust-ice mixture. f will enter inside the thermal equilibrium calculation, influencing the temperature at the sublimation front in a nonlinear way. Ψ is a reduction factor determining the permeability of the dust mantle to gas flow and is determined by 1/(1 + pH)^68,71, with a constant coefficient p and H = X/d related to the diameter d of the spherical dust aggregates forming the dust mantle⁷². Z^H−K(T) is the Hertz-Knudsen formula, describing the sublimation flux from the surface of pure solid ice into vacuum^71,73,
where (widehat{m}) is the mass of a water molecule, and k_B is the Boltzmann constant. P_V(T) is the saturation vapor pressure. α(T) is the sublimation coefficient, representing the reduction in sublimation flux due to the reattachment of gas molecules to the ice surface after impacting with each other. They are respectively evaluated as^71,73,74,75
with a, b, c₀, c₁, c₂, c₃ constants⁶⁸.
ThermoONet, the DL-based comet thermophysical modeling tool, was developed to approximate the above thermophysical process⁴⁰. It retrieves an accuracy of T_(X) better than 95% compared to traditional numerical approaches, while reduces simulation time by nearly six orders of magnitude⁴⁰. Each iteration of the global optimization requires a full thermophysical simulation of the water production rate, which involves solving the heat conduction equation for every facet at all orbital phases. A single solve of global temperature takes about 10⁴ s using traditional finite-difference methods, but only 10⁻² s with ThermoONet. As a result, a full SA inversion, which would formerly require months of CPU time, can now be completed in a matter of minutes within our ThermoONet-based framework.
The leap in computation efficiency enables ThermoONet to reproduce the water curve of a comet with an irregular shape, as well as the exploration of a large and high-dimensional parameter space.
For a nucleus represented as a polyhedron with k facets, its water outgassing rate n_Z(t₀) at time t₀ could be expressed as the total global outgassing rate averaged over one spin period P, starting from t₀:
where Z_k, S_k indicate the sublimation flux and the area of the k-th facet.
In this sense, the size of the nucleus can be treated as an independent scaling factor for a normalized water curve. However, it is essential to first assess the influence of other model parameters to avoid the potential degeneracy. To this end, we performed benchmark analysis with comet C/2001 Q4 (Neat) and C/2012 F6 (Lemmon), focusing on the influence of the key parameters shaping the water curve, including the icy area fraction f, the thickness of dust mantle X, thermal conductivity κ, a composite parameter p_s (defined as the square root of the product of density ρ, specific heat capacity C, and rotational angular velocity ω), the obliquity of spin axis β, and the nucleus effective diameter d_n. To isolate the effects of each parameter, we apply the control variate method by oscillating a certain parameter while fixing the others. The results demonstrate that the size of the nucleus only controls the overall intensity of the water emission and has no effect on the shape of the water curve. On the contrary, other parameters do not contribute significantly to the level of production rate compared to the size, but mainly alter its trend (Fig. 7). Therefore, as long as the highly nonlinear shape of the water curve is fitted by optimizing these parameters, we could retrieve the size of the nucleus simply as a scaling factor.
Column (a) Results based on the orbital parameters of C/2001 Q4. Column (b) Results based on the orbital parameters of C/2012 F6. The first row shows the influence of the icy area fraction of 1% (black), 5% (red), and 10% (blue). The second row shows the influence of dust mantle thickness of 5 mm (black), 8 mm (red), 10 mm (blue). The third row shows the influence of thermal conductivity (in W m⁻¹ K) corresponding to 0.002 (black), 0.004 (red), and 0.007 (blue). The fourth row shows the influence of a combined parameter p_s (defined in Zhao et al.⁴⁰) of 5 (black), 10 (red), 15 (blue). The fifth row shows the influence by the obliquity of the spin axis of 20^∘ (black), 50^∘ (red), and 80^∘ (blue). And the last row shows the influence of the effective diameter, corresponding to 1.52 km (black), 7.63 (red), and 15.26 km (blue). The vertical dashed lines indicate times of perihelia. Source data are provided as a Source Data file.
The influence of icy area fraction f on the slope of the water curve is a noteworthy phenomenon, for which we provide a concise explanation. A lower f reduces the energy consumed by sublimation, resulting in a higher equilibrium temperature at the ice front under the same insolation. Since the Hertz-Knudsen sublimation rate depends exponentially on temperature, a given temperature increase (e.g., 20 K) causes a more substantial rise in the sublimation rate at higher temperature ranges (e.g., 200 K to 220 K) than at lower ones (e.g., 160 K to 180 K). Consequently, a lower f leads to a steeper increase in water production as the comet approaches the Sun, thereby altering the slope of the curve, the behavior clearly illustrated in Fig. 7(first row).
We initialize our inversion by constructing the nucleus as a normalized tri-axial ellipsoid with semi-axes (a_n, b_n, c_n) where a_n = 1 km. The actual axial ratios are applied for those comets with measured shape. Statistical investigations of light curves have demonstrated that the axial ratios of most comets distribute between 1 and 2, with a median value of 1.5^21,23,76. In our previous work on comet 21P⁴⁰, we showed that the axial ratio has a negligible impact on insolation received and the resultant water production rate per unit area, unless it arrives at an extremely large value (i.e.,  > 4). The axial ratio primarily affects the results through its determination of the surface area and volume of the ellipsoid. We quantify the effect on diameter estimation by comparing results for different axial ratios while holding the semi-major axis constant at 1 km and the observed-to-modeled water production ratio at unity (Eq. (9)). The derived diameters are respectively 0.55 km, 0.56 km, and 0.54 km for axial ratios of 1.5, 1 (sphere), and 2. This narrow range of values demonstrates that the assumed shape has only a minor effect on the estimated diameter for typical cometary shapes. Therefore, for comets with unknown axial ratios, we model their nuclei as a spheroid with an axial ratio of 1.5.
We adopt the Simulated Annealing (SA) global optimization scheme⁷⁷ to find the parameters that reproduce the overall normalized water production curves. The algorithm proceeds as follows: It starts with an initial set of parameters. At each step, it proposes a new set of parameters by randomly perturbing the current set. If the new parameters yield a better fit, they are always accepted. Critically, if they yield a worse fit, they can still be accepted with a probability that depends on the current fitting quality and the level of increase in the misfit. This controlled acceptance of worse solutions allows the algorithm to escape from local minima in the parameter space. The misfit is gradually decreased according to a predefined schedule, and the process converges towards a set of parameters corresponding to a global optimum.
Twelve variables are fitted through the procedure, including f, X(x₁, x₂, x₃, x₄, x₅), κ_d, κ_m, p_s, the spin axis orientation (RA, Dec) and the effective diameter d_n. For comets with measured spin axis orientation, RA and Dec are excluded from the fitting. Investigations of 67P by Rosetta have revealed that comets possess extremely low thermal conductivity⁷⁸. Low thermal conductivity of comets is also predicted by recent works on comet formation and evolution via pebble accretion^24,79,80. Therefore, we set an upper limit of κ as 0.01 W m⁻¹K⁻¹.
Based on the aforementioned sensitivity analysis for parameters, we decouple d_n from the rest of the variables, reserving it as a scaling factor of the water production rate, while optimizing the other five or seven parameters to reproduce the trend of the curve.
Rosetta observations of comet 67P have demonstrated a varying dust mantle thickness through its perihelion passage, thinning on the inbound and thickening on the outbound^81,82. We include this dynamic behavior into our SA optimization by formulating X as a heliocentric distance r-dependent variable with monotonic reduction before peaking of the water activity, followed by a gradual recovery. Based on the spectral slope analysis by Filacchione et al.⁸², we model the thickness of dust mantle X as a piece-wise function:
where ∣*∣₀₋₁ denotes normalization to the [0, 1] interval. Parameters x₃, x₄, x₅ respectively represent the dust mantle thicknesses corresponding to three critical phases of the observed water production rate curve: initiation (t₀), termination (t_end), and peak (t_p). The functional dependence on x₁ and x₂ exhibits a positive correlation with system nonlinearity, where increasing parameter values correlate with enhanced nonlinear behavior. The five-dimensional parameter set (x₁, x₂, x₃, x₄, x₅) is explicitly incorporated into the SA framework as the mathematical parameterization of the crucial variations on the surface dust.
Optimal fitting is achieved by minimizing the mean absolute percentage error (MAPE) between the observed and reconstructed curves, both normalized: ({{{boldsymbol{W}}}}_{o}/{W}_{o,max }) and ({{{boldsymbol{W}}}}_{f}/{W}_{f,max }). Here W_o denotes the observed curve with its maximum value of ({W}_{o,max }), and W_f is the corresponding fitted curve derived from SA and ThermoONet with its maximum of W_f,max. The effective nucleus diameter D is subsequently calculated as
To address potential over-fitting issues arising from the high-dimensional parameter space and the broad distribution of observational data, we implement a post-annealing perturbation strategy. Following the completion of SA, we generate normally distributed random sampling points centered on the optimized parameter set while maintaining a relatively large parameter space. Water curves are produced with these perturbed parameter combinations. We then select all curves exhibiting MAPE below approximately 30% relative to the observed curve, perform size inversion and retrieve a distribution of estimated sizes. For each comet, we analyse approximately 400 qualified parameter sets to establish size distribution statistics. The resulting distributions followed normal distributions, from which the mean values combined with 1-σ confidence intervals (encompassing 68.27% of the data) are adopted as the final size estimates with corresponding uncertainties (Fig. 2).
SPCs experience mass loss during repeated perihelion passages. To assess the extent of erosion they underwent and its impact on the estimate of the OC/SD ratio, we introduce an indicator of erosion probability factor. The precise timing of an SPC into its current orbit remains largely unknown, and constructing accurate physical models for cometary mass loss presents significant challenges, making it difficult to precisely quantify mass loss for individual comets. However, we can perform a statistical analysis at a probabilistic level based on their current orbit configurations. Specifically, we combine the Hertz-Knudsen formula (Eq. (5)) with the characteristic temperature of the comet at different orbital phases to qualitatively assess the total amount of erosion over one complete orbital period. The erosion probability factor of the comet is defined as the average erosion over the orbit, obtained by dividing the total erosion by the comet’s orbital period. Statistical analysis between this erosion probability factor and cometary magnitude reveals a general trend (Supplementary Fig. 2). Comets with smaller magnitudes tend to have lower erosion probability factors, implying a lower probability for them to have undergone severe erosion. Based on the relationship between the erosion probability factor and magnitude, we can infer that currently observed JFCs larger than 10 km, those we use for inferring the OC/SD ratio, have statistically experienced relatively limited erosion. Therefore, from a statistical perspective, we propose that the current sizes of these objects can reasonably represent their primordial sizes within the SD.
By integrating comet observations^56,83, the flux of new LPCs with absolute magnitudes  < 6.5 is estimated to be (0.7{6}_{-0.33}^{+0.33}) per year⁹. Combined with the absolute magnitude-size linear relationship proposed by Sosa and Fernández⁵³ and the average fraction of OC reported by Kaib and Quinn¹³, it was predicted that the number of OC objects with nuclei  > 2.3 km was (7.{6}_{-3.3}^{+3.3}times 1{0}^{10})⁹. To revise this estimation, we need to first reassess the linear regression between nucleus sizes from Sosa and Fernández⁵³ and corresponding absolute magnitudes from JPL (the beige dashed line in Fig. 4a). The updated correlation reveals that, according to Sosa and Fernández⁵³, a nucleus size of 2.3 km corresponds to a JPL magnitude of about 10. This critical magnitude translates to a nucleus size of about 10 km according to the updated D_LPC − H_T found in our study. Therefore, based on the estimation by Brasser et al.⁹, there should be (7.{6}_{-3.3}^{+3.3}times 1{0}^{10}) OC objects exceeding 10 km in diameter. The possible higher new LPC influx found by Fouchard et al.¹¹ as well as inactive OC objects detected by Meech et al.⁸⁴ indicate a potentially larger number.
Based on a sample of (11{7}_{-50}^{+50}) visible JFCs with nuclei  > 2 km⁸⁵, and a cumulative JFC size-frequency distribution N(N > D) ∝ D^{−2 86,87,88}, a model connecting the population of SD with JFCs was established⁹, indicating that there are (1.{7}_{-0.9}^{+3}times 1{0}^{9}) SD objects with diameters  > 2.3 km. Above calculations of OC population have revised the nucleus size constraint for LPCs, the minimum nuclear diameter is 10 km, and the size cutoff for JFCs should also be updated. Utilizing the same size distribution of JFC and starting from the count of SD objects with diameters greater than 2.3 km, we estimate that the population of SD objects with nucleus diameter  > 10 km through standard scaling is (7.{6}_{-4}^{+13}times 1{0}^{7}). Given the similar slope of the size-frequency distribution for LPCs and SPCs with diameters exceeding 1 km⁶¹, the OC/SD population ratio is larger than (99{8}_{-794}^{+2045}).
Considering the uncertainties inherent in the determination of cometary absolute magnitudes due to their model-dependent nature, and to underscore the importance of employing a self-consistent photometric model, we have supplemented our analysis with an independent assessment of the comet size-magnitude relationship and the derived OC/SD ratio using the MPC (Minor Planet Center) database. This procedure was conducted in a manner consistent with that used for the JPL database. Our results are presented in the Supplementary Fig. 3. Based on the cometary nuclear size from⁵³, a nucleus of 2.3 km corresponds to an MPC magnitude of approximately 8. Applying this magnitude to our revised relationship yields an estimated LPC size of about 9 km, which is comparable to about 10 km derived from the JPL data. Furthermore, we calculated an OC/SD population ratio of 715 under the MPC photometric model, a value that differences less than 30% with the result of 998 obtained using the JPL model.
One recent work demonstrated that a stellar flyby involving a (0.{8}_{-0.1}^{+0.1}) solar mass perturber with closest encounter distance at (11{0}_{-10}^{+10}) au and an inclination of ({7{0}^{circ }}_{-10}^{+5}) can adequately explain the dynamical evolution of Sedna-like TNOs, high-inclination TNOs, and cold Kuiper Belt objects¹⁸. We adopt a similar scenario to investigate the simultaneous effects of this stellar flyby on the more distant disk objects. We initiate the outer debris disk as a nearly circular thin annulus, regardless of whether it originated from nebular evolution or emerged through the giant planet migration process. Based on the typical sizes of protoplanetary and debris disks of 100–500 au^18,89,90, and the size of the scattering disk from giant planet migration^3,4,19, we set the initial annulus to have a size of 100–500 au.
The initial disk consists 10⁵ massless particles with semi-major axes sampled between 100–500 au, following a radially decreasing density, and eccentricities uniformly sampled between 0–0.1, inclinations uniformly sampled between 0–5^∘, and other angular elements (longitude of ascending node, argument of perihelion, eccentric anomaly) uniformly sampled across their full value ranges. Each particle is propagated in the framework of a three-body problem during the stellar encounter with the Runge-Kutta-Fehlberg 7(8) (RKF78) adaptive integration scheme, neglecting the interparticle interactions. To capture both pre-encounter orbital configurations and post-encounter dynamical evolution, the simulation time extends from 10 kyr before to 100 kyr after the stellar closest encounter. Following the simulation, particles escaping the solar gravitational influence are excluded based on their orbital elements. After 100 kyr, the flyby star has receded to a considerable distance, leaving the remaining particles in Keplerian orbits around the Sun. The current simulation omits the subsequent long-term dynamical evolution, since the main goal is to get a qualitative idea of the OC populating efficiency by a stellar flyby. It is foreseeable that during subsequent evolution, perturbations from distant stars would replenish some inner Oort cloud objects into the outer Oort cloud region^91,92. Combined with Galactic tidal effects, the spatial distribution would become isotropic^3,19.
All data supporting the findings of this study are publicly available. The SOHO/SWAN comet water production rate data that support the findings of this study are available in the NASA PDS small body node with the identifier: https://doi.org/10.26007/8716-yz09⁴⁴. The Rosetta-derived water production rate data of comet 67P are available in Läuter et al. 2020⁴⁸. The water production rate data of comet Halley are available in Combi et al. 1993⁴⁷. Source data are provided with this paper.
The AI-empowered small body thermophysical model is available at: https://github.com/zsjnb7/ThermoONet-Comet.
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The authors gratefully acknowledge Dr. Michael R. Combi and his team for their dedicated efforts in deriving and curating the invaluable SOHO/SWAN cometary water production rates collection. S.J.Z thanks Dr. Liu Tao for helpful discussions about the Oort Cloud. This work is financially supported by the National Natural Science Foundation of China (No. 12233003, 12573063).
School of Astronomy and Space Science, Nanjing University, Nanjing, China
Shunjing Zhao & Hanlun Lei
Key Laboratory of Modern Astronomy and Astrophysics in Ministry of Education, Nanjing University, Nanjing, China
Shunjing Zhao & Hanlun Lei
Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai, China
Shunjing Zhao, Xian Shi, Man-To Hui & Jianchun Shi
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X.S. conceived and designed the study. S.J.Z. performed thermophysical modeling, size estimation, and dynamical simulations. S.J.Z. and X.S. drafted the manuscript. H.L.L., M.T.H., and J.C.S. contributed to the interpretation of the results and revising the manuscript.
Correspondence to Xian Shi or Hanlun Lei.
The authors declare no competing interests.
Nature Communications thanks Hai Jiang, Nalin H. Samarasinha and the other anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.
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Zhao, S., Shi, X., Lei, H. et al. Deep learning-enabled size estimation of comets indicates a more dynamic early solar system. Nat Commun 17, 6070 (2026). https://doi.org/10.1038/s41467-026-72646-8
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Received: 05 September 2025
Accepted: 20 April 2026
Published: 06 May 2026
Version of record: 10 July 2026
DOI: https://doi.org/10.1038/s41467-026-72646-8
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