Orbital Resonance Research Articles

Radii, masses, and transit-timing variations of the three-planet system orbiting the naked-eye star TOI-396

TOI-396 is an F6\,V bright naked-eye star ($V$\,approx \,6.4) orbited by three small ($R_p$\,approx \,2 $R_ transiting planets discovered thanks to space-based photometry from two sectors. The orbital periods of the two innermost planets, namely TOI-396\,b and c, are close to the 5:3 commensurability ($P_b$\,sim \,3.6\,d and $P_c$\,sim \,6.0\,d), suggesting that the planets might be trapped in a mean motion resonance (MMR). To measure the masses of the three planets, refine their radii, and investigate whether planets b and c are in MMR, we carried out radial velocity (RV) observations of TOI-396 and retrieved archival high-precision transit photometry from four sectors. We extracted the RVs via a skew-normal fit onto the cross-correlation functions and performed a Markov chain Monte Carlo joint analysis of the Doppler measurements and transit photometry, while employing the breakpoint method to remove stellar activity from the RV time series. We also performed a transit timing variation (TTV) dynamical analysis of the system and simulated the temporal evolution of the TTV amplitudes of the three planets following an N-body numerical integration. Our analysis confirms that the three planets have similar sizes ($R_b=2.004_ oplus $; $R_c=1.979_ oplus $; $R_d=2.001_ oplus $) and is thus in agreement with previous findings. However, our measurements are sim \,1.4 times more precise thanks to the use of two additional sectors. For the first time, we have determined the RV masses for TOI-396\,b and d, finding them to be oplus $ and $M_d=7.1 oplus $, which implies bulk densities of $ $ and $ $, respectively. Our results suggest a quite unusual system architecture, with the outermost planet being the densest. Based on a frequency analysis of the activity indicators and light curves, we find the rotation period of the star to be $ d, in agreement with the value predicted from $ R'_ HK $-based empirical relations. The Doppler reflex motion induced by TOI-396\,c remains undetected in our RV time series, likely due to the proximity of the planet's orbital period to the star's rotation period. We also discovered that TOI-396 b and c display significant TTVs. While the TTV dynamical analysis returns a formally precise mass for TOI-396\,c of c dyn oplus $, the result might not be accurate, owing to the poor sampling of the TTV phase. We also conclude that TOI-396\,b and c are close to but out of the 5:3 MMR. A TTV dynamical analysis of additional transit photometry evenly covering the TTV phase and super-period is likely the most effective approach for precisely and accurately determining the mass of TOI-396\,c. Our numerical simulation suggests TTV semi-amplitudes of up to five hours over a temporal baseline of sim \,5.2 years, which should be duly taken into account when scheduling future observations of TOI-396.

Long-term Dynamical Stability in the Outer Solar System. II. Detailed Secular Evolution of Four Large Regular and Resonant Trans-Neptunian Objects

The long-term evolution of the outer solar system is subject to the influence of the giant planets, however, perturbations from other massive bodies located in the region imprint secular signatures, which are discernible in long-term simulations. In this work, we performed an in-depth analysis of the evolution of massive objects Eris, 2015 KH162, Pluto, and 2010 EK139 (aka, Dziewanna), subject to perturbations from the giant planets and the 34 largest trans-Neptunian objects. We do this by analyzing 200, 1 Gyr long simulations with identical initial conditions, but requiring the numerical integrator to take different time steps for each realization. Despite the integrator’s robustness, each run’s results are surprisingly different, showing the limitations of individual realizations when studying the trans-Neptunian region due to its intrinsic chaotic nature. For each object, we find orbital variables with well-defined oscillations and limits, and others with surprisingly large variances and seemingly erratic behaviors. We found that 2015 KH162 is a nonresonant and very stable object that experiences only limited orbital excursions. Pluto is even more stable and we found a new underlying constraining mechanism for its orbit; 2010 EK139 is not well trapped in the 7:2 mean motion resonance in the long term and cannot be trapped simultaneously in von Zeipel–Lidov–Kozai resonance; and finally, we found that at present Eris’s longitude of perihelion is stationary, tightly librating around 190°, but unexpectedly loses its confinement, drifting away after 150 Myr, suggesting a missing element in our model.

A Fourth Planet in the Kepler-51 System Revealed by Transit Timing Variations

Kepler-51 is a ≲1 Gyr old Sun-like star hosting three transiting planets with radii ≈6–9 R ⊕ and orbital periods ≈45–130 days. Transit timing variations (TTVs) measured with past Kepler and Hubble Space Telescope (HST) observations have been successfully modeled by considering gravitational interactions between the three transiting planets, yielding low masses and low mean densities (≲0.1 g cm−3) for all three planets. However, the transit time of the outermost transiting planet Kepler-51d recently measured by the James Webb Space Telescope 10 yr after the Kepler observations is significantly discrepant from the prediction made by the three-planet TTV model, which we confirmed with ground-based and follow-up HST observations. We show that the departure from the three-planet model is explained by including a fourth outer planet, Kepler-51e, in the TTV model. A wide range of masses (≲M Jup) and orbital periods (≲10 yr) are possible for Kepler-51e. Nevertheless, all the coplanar solutions found from our brute-force search imply masses ≲10 M ⊕ for the inner transiting planets. Thus, their densities remain low, though with larger uncertainties than previously estimated. Unlike other possible solutions, the one in which Kepler-51e is around the 2:1 mean motion resonance with Kepler-51d implies low orbital eccentricities (≲0.05) and comparable masses (∼5 M ⊕) for all four planets, as is seen in other compact multiplanet systems. This work demonstrates the importance of long-term follow-up of TTV systems for probing longer-period planets in a system.

Possible origin of Mars-crossing asteroids and related dynamical properties of inner main-belt asteroids

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.

A dynamical study of Hilda asteroids in the Circular and Elliptic RTBP.

The Hilda group is a set of asteroids whose mean motion is in a 3:2 orbital resonance with Jupiter. In this paper, we use the planar Circular Restricted Three-Body Problem (CRTBP) as a dynamical model and we show that there exists a family of stable periodic orbits that are surrounded by islands of quasi-periodic motions. We have computed the frequencies of these quasi-periodic motions and we have shown how the Hilda family fits inside these islands. We have compared these results with the ones obtained using the Elliptic Restricted Three-Body Problem and they are similar, showing the suitability of the CRTBP model. It turns out that, to decide if a given asteroid belongs to the Hilda class, it is much better to look at its frequencies in the planar CRTBP rather than to use two-body orbital elements as it is commonly done today.

Evaluation of the Possibility of Using Resonant Orbits for Accompanying Relay Satellites

Evaluation of the Possibility of Using Resonant Orbits for Accompanying Relay Satellites

Quantitative Risk Assessment for Autonomous Vehicles: Integrating Functional Resonance Analysis Method and Bayesian Network

ABSTRACTIn autonomous vehicles (AVs), intricate functional‐level couplings exist among the components. Accidents can occur even when all functions are operating normally, as subtle performance variabilities in these functions can aggregate through these couplings, leading to functional resonance. The aim of this study is to identify, analyze and quantitatively assess the safety issues caused by these complex interactions in AVs and to propose appropriate risk management strategies to improve vehicle safety. Commonly used modern methods of risk assessment, such as system‐theoretical process analysis and accident mapping, struggle to capture this resonance in AVs and lack quantitative analysis. To this end, this paper proposes a quantitative risk assessment method that integrates functional resonance analysis method (FRAM) with Bayesian network (BN) to reveal the complex interactions and quantify risks within AVs. Initially, a FRAM model is constructed to characterize the function couplings of a system, which are subsequently aggregated into functional resonance chains to identify potential hazards. Then, these functional resonance chains are used to develop a BN model for quantitative assessment of system risk. A case study of an automatic emergency braking (AEB) system on an open‐source vehicle is conducted to verify its effectiveness. The results demonstrate that the proposed approach not only identifies functional resonance but also effectively quantifies risks in the AEB system.

Constraining the Presence of Companion Planets in Hot Jupiter Planetary Systems Using Transit-timing Variation Observations from TESS

The presence of another planetary companion in a transiting exoplanet system can impact its transit light curve, leading to sinusoidal transit-timing variations (TTV). By utilizing both χ 2 and rms analysis, we have combined the TESS observation data with an N-body simulation to investigate the existence of an additional planet in the system and put a limit on its mass. We have developed CMAT, an efficient and user-friendly tool for fitting transit light curves and calculating TTV with a theoretical period, based on which we can give a limit on its hidden companion’s mass. We use 260 hot Jupiter systems from the complete TESS data set to demonstrate the use of CMAT. Our findings indicate that, for most systems, the upper mass limit of a companion planet can be restricted to several Jupiter masses. This constraint becomes stronger near resonance orbits, such as the 1:2, 2:1, 3:1, and 4:1 mean-motion resonance, where the limit is reduced to several Earth masses. These findings align with previous studies suggesting that a lack of companion planets with resonance in hot Jupiter systems could potentially support the high-eccentricity migration theory. Additionally, we observed that the choice between χ 2 or rms method does not significantly affect the upper limit on companion mass; however, χ 2 analysis may result in weaker restrictions but is statistically more robust compared to rms analysis in most cases.

Hydrogen fuel cell energy regulators based on Boost DC-DC converters with switched capacitors

A new principle for constructing small-sized capacitor DC-DC regulators that provide energy-efficient conversion and multi-zone energy regulation of hydrogen fuel cells is proposed. It consists in converting the energy of their current into the energy of a constant voltage source with further alternating changes in its low-voltage regulation subranges. It is carried out by discretely changing the number of resonant chains of a step-up capacitor DC-DC converter in combination with a smooth change in the charge level of its first chain using a low-power classic PWM constant voltage regulator. Improving the energy properties of the regulator is achieved by multi-zone regulation and soft switching of transistors in the power circuit of the capacitor DC-DC converter. The energy properties of the proposed regulator have been studied. It is shown that with an increase in the number of resonant chains in the capacitor DC-DC converter, there is a tendency for the controller efficiency to increase. For the various regulator circuits considered in the work, the regulating characteristics were obtained and a comparative assessment of their efficiency was carried out. A comparative analysis of energy and design properties proves the significant advantage of a circuit using a Boost PWM constant voltage regulator.

Hic Sunt Dracones: Uncovering Dynamical Perturbers within the Habitable Zone

The continuing exploration of neighboring planetary systems is providing deeper insights into the relative prevalence of various system architectures, particularly with respect to the solar system. However, a full assessment of the dynamical feasibility of possible terrestrial planets within the habitable zones (HZs) of nearby stars requires detailed knowledge of the masses and orbital solutions of any known planets within these systems. Moreover, the presence of as-yet undetected planets in or near the HZ will be crucial for providing a robust target list for future direct imaging surveys. In this work, we quantify the distribution of uncertainties on planetary masses and semimajor axes for 1062 confirmed planets, finding median uncertainties of 11.1% and 2.2%, respectively. We show the dependence of these uncertainties on stellar mass and orbital period and discuss the effects of these uncertainties on dynamical analyses and the locations of mean motion resonance. We also calculate the expected radial velocity (RV) semiamplitude for a Neptune-mass planet in the middle of the HZ for each of the proposed Habitable Worlds Observatory target stars. We find that for more than half of these stars, the RV semiamplitude is less than 1.5 m s−1 rendering them unlikely to be detected in archival RV data sets and highlighting the need for further observations to understand the dynamical viability of the HZ for these systems. We provide specific recommendations regarding stellar characterization and RV survey strategies that work toward the detection of presently unseen perturbers within the HZ.

Leaning Sideways: VHS 1256−1257 b is a Super-Jupiter with a Uranus-like Obliquity

We constrain the angular momentum architecture of VHS J125601.92-125723.9, a 140 ± 20 Myr old hierarchical triple system composed of a low-mass binary and a widely separated planetary-mass companion, VHS 1256 b. VHS 1256 b has been a prime target for multiple characterization efforts, revealing the highest measured substellar photometric variability to date and the presence of silicate clouds and disequilibrium chemistry. Here we add a key piece to the characterization of this super-Jupiter on a Tatooine-like orbit: we measure its spin-axis tilt relative to its orbit, i.e., the obliquity of VHS 1256 b. We accomplish this by combining three measurements. We find a projected rotation rate vsinip=8.7±0.1kms−1 for VHS 1256 b using near-infrared high-resolution spectra from Gemini/IGRINS. Combining this with a published photometric rotation period indicates that the companion is viewed edge on, with a line-of-sight spin axis inclination of i p = 90° ± 18°. We refit available astrometry measurements to confirm an orbital inclination of io=23−13+10 degrees. Taken together, VHS 1256 b has a large planetary obliquity of ψ = 90° ± 25°. In total, we have three measured angular momentum vectors for the system: the binary orbit normal, companion orbit normal, and companion spin axis. All three are misaligned with respect to each other. Although VHS 1256 b is tilted like Uranus, their origins are distinct. We rule out planet-like scenarios including collisions and spin–orbit resonances, and suggest that top-down formation via core/filament fragmentation is promising.

Azaindole: A Candidate Anchor for Regulating Charge Polarity and Inducing Resonance Transmission at the Fermi Level via Dehydrogenation.

Tuning the polarity of charge carriers is essential for designing molecular logic devices in molecular electronics. In this study, the electrical transport properties of a family of azaindole-anchored single-molecule junctions have been investigated using density functional theory combined with the nonequilibrium Green's function method. The obtained results reveal that dehydrogenation is an effective method for reversing the polarity of charge carriers. The molecular junctions based on the entire azaindole unit are n-type and contain electrons as the principal charge carriers, whereas the dehydrogenated junctions are p-type and contain holes as the main carriers. Furthermore, the azaindole anchors undergo a transition from an electron-rich to an electron-deficient state due to dehydrogenation, which is the original cause of the charge carrier polarity conversion. Dehydrogenated molecular junctions also exhibit the Fermi pinning effect and a sharp highest occupied molecular orbital (HOMO) resonance peak at the Fermi level. In addition, using Pt electrodes instead of Au electrodes is a means of producing a HOMO resonance peak a for azaindole-based molecular junctions. This work demonstrates the enormous potential of utilizing azaindole-anchored molecular junctions for the implementation of molecular logic and multifunctional molecular devices.

New Moons of Uranus and Neptune from Ultradeep Pencil-beam Surveys

We have conducted extremely ultradeep pencil-beam observations for new satellites around both Uranus and Neptune. Tens of images on several different nights in 2021, 2022, and 2023 were obtained, shifted, and added together to reach as faint as 26.9 and 27.2 mag in the r band around Uranus and Neptune, respectively. One new moon of Uranus, S/2023 U1, and two new moons of Neptune, S/2021 N1 and S/2002 N5, were found. S/2023 U1 was 26.6 mag, is about 7 km in diameter, and has a distant, eccentric, and inclined retrograde orbit similar to Caliban and Stephano, implying these satellites are fragments from a once larger parent satellite. S/2021 N1 was 26.9 mag, about 14 km in size, and has a retrograde orbit similar to Neso and Psamathe, indicating they are a dynamical family. We find S/2021 N1 is in Kozai–Lidov orbital resonance. S/2002 N5 was 25.9 mag, is about 23 km in size, and it makes a family of distant prograde satellites with Sao and Laomedeia. This survey mostly completes the outer satellites of Uranus to about 8 km and Neptune to about 14 km in diameter. The size distributions of satellite dynamical families around the giant planets shows a strong steepening in the power-law size distribution smaller than 5 km in diameter. The satellites of a family become much more common at diameters smaller than 5 km and their size distribution is consistent with a collisional breakup of a once larger parent satellite.

Analysing the dynamics of the Kepler-90 planetary system

Abstract Kepler-90 system has a set of eight planets in a hierarchical structure. In this work, we used frequency analysis to study several Kepler-90 analogues to analyse in detail how the values of the eccentricity of each planet can alter the stability of the system. The system was formed by the star and eight planets for three different intervals of eccentricity (e). Our results show that for the first and second intervals all the systems are stable. However, no set of Kepler-90 system with large values of e survived up to 105 orbits of Kepler-90h. In the low eccentricity interval, planets b and c were found to be in a 5:4 mean motion resonance, and the pair g and h in a 3:2 mean motion resonance, although these two pairs are in resonance in different Kepler-90 analogues. In the medium eccentricity interval, only the pair g and h is in resonance. Kepler-90 b and c are in quasi-resonance 5:4; the critical angle circulates and librates intermittently in different periods of time. The results statistically indicate that these resonances directly affect most of them. The influence covers a wide range of possibilities: (i) a strong resonant mean motion coupling added to a coupling of the longitude of the pericentres; (ii) just a single resonant argument librating; (iii) intermittent behaviour, interleaving libration and circulation of the resonant arguments; and (iv) the quasi-resonant influence due to the near commensurability.

The Prevalence of Resonance Among Young, Close-in Planets

Multiple planets undergoing disk migration may be captured into a chain of mean-motion resonances with the innermost planet parked near the disk’s inner edge. Subsequent dynamical evolution may disrupt these resonances, leading to the nonresonant configurations typically observed among Kepler planets that are Gyr old. In this scenario, resonant configurations are expected to be more common in younger systems. This prediction can now be tested, thanks to recent discoveries of young planets, in particular those in stellar clusters, by NASA’s TESS mission. We divided the known planetary systems into three age groups: young (<100 Myr old), adolescent (0.1–1 Gyr old), and mature (>1 Gyr old). The fraction of neighboring planet pairs having period ratios within a few percent of a first-order commensurability (e.g., 4:3, 3:2, or 2:1) is 70% ± 15% for young pairs, 24% ± 8% for adolescent pairs, and 15% ± 2% for mature pairs. The fraction of systems with at least one nearly commensurable pair (either first- or second-order) is 86% ± 13% among young systems, 38% ± 12% for adolescent systems, and 23% ± 3% for mature systems. First-order commensurabilities prevail across all age groups, with an admixture of second-order commensurabilities. Commensurabilities are more common in systems with high planet multiplicity and low mutual inclinations. Observed period ratios often deviate from perfect commensurability by ∼1% even among young planets, too large to be explained by resonant repulsion with equilibrium eccentricity tides. We also find that super-Earths in the radius gap (1.5–1.9R ⊕) are less likely to be near-resonant (11.9% ± 2.0%) compared to Earth-sized planets (R p < 1R ⊕; 25.3% ± 4.4%) or mini-Neptunes (1.9R ⊕ ≤ R p < 2.5R ⊕; 14.4% ± 1.8%).

Dynamical stability of the Laplace resonance

Dynamical stability of the Laplace resonance

Main-belt comets as contributors to the near-Earth objects population

Aims. Most studies of the dynamics of main-belt objects describing the evolution of the bodies in inner Solar System have been carried with models that include weak nongravitational forces, such as the Yarkovsky and YORP effects. Only about 19 objects exhibit cometary-type activity, with sublimation being the principal mechanism. This paper presents a study of the influence of cometary- type nongravitational forces on the dynamics of main-belt comets, and the possible paths and timescales of evolution into other Solar System regions. Methods. We used the standard Marsden model for cometary-type activity. This model was designed for elongated orbits and the continuous ejection of mass, while the main-belt comets exhibit a different mode of activity. For this reason, we propose a simple model of nongravitational force activation that is consistent with observations. The dynamical evolution of objects was studied using a fifteenth-order RADAU integrator implemented in the REBOUND package. Results. The paper presents the dynamical routes of main-belt comets in the inner Solar System when cometary-type nongravitational forces are included in calculations. The forces significantly shorten the time of transition to other regions compared to the Yarkovsky effect, shortening this time to as little as a few thousand years depending on the frequency of the activity and the A1, A2, and A3 Marsden constants. There is a large probability of a transition to near-Earth object(NEO)-type orbits for bodies with A2 &lt; 0, which means that cometary-type nongravitational forces can be a non-negligible mechanism increasing the number of bodies in that population. The forces can also deliver active main-belt objects into mean motion resonances but can equally eject bodies into outer planetary regions on far shorter timescales than the Yarkovsky and YORP effects. Cometary-type nongravitational effects should be included in dynamical studies of individual sublimating active asteroids.

CZ Aqr: An Oscillating Eclipsing Algol-type System Composed of a δ Sct Primary Star and a Subgiant Star in a Quadruple System

Eclipsing Algol-type systems containing a δ Scuti (hereafter δ Sct) star enable precise determination of physical parameters and the investigation of stellar internal structure and evolution. We present the absolute parameters of CZ Aquarius (hereafter CZ Aqr) based on TESS data. CZ Aqr has an orbital period of 0.86275209 day, a mass ratio of 0.489 (6), and the secondary component nearly fills its Roche lobe. O − C analysis reveals a downward parabolic trend and a cyclical variation with a period of 88.2 yr. The downward parabola suggests a long-term decrease in the orbital period with Ṗ = −3.09 × 10−8 days yr−1. The mass loss rate is estimated to be 4.54 × 10−9 M ⊙ yr−1, which is possibly due to magnetic stellar wind or a hot spot. The cyclical variation might be caused by the light travel time effect via the presence of a third body with a minimum mass of M3min = 0.312 (21) M ⊙. Additionally, there are two possible celestial bodies in a 2:7 resonance orbit around CZ Aqr. The asymmetric light curve is explained by adding a hot spot on the surface of the primary star. After removing the binary model, 26 frequencies were extracted from TESS data. Two radial modes were newly identified among three possible independent frequencies. Our results show that the eclipsing Algol-type system is composed of a δ Sct primary star and a subgiant star in a quadruple system.

High-resolution ALMA Observations of Richly Structured Protoplanetary Disks in σ Orionis

The Atacama Large Millimeter/submillimeter Array (ALMA) has detected substructures in numerous protoplanetary disks at radii from a few to over 100 au. These substructures are commonly thought to be associated with planet formation, either by serving as sites fostering planetesimal formation or by arising as a consequence of planet–disk interactions. Our current understanding of substructures, though, is primarily based on observations of nearby star-forming regions with mild UV environments, whereas stars are typically born in much harsher UV environments, which may inhibit planet formation in the outer disk through external photoevaporation. We present high-resolution (∼8 au) ALMA 1.3 mm continuum images of eight disks in σ Orionis, a cluster irradiated by an O9.5 star. Gaps and rings are resolved in the images of five disks. The most striking of these is SO 1274, which features five gaps that appear to be arranged nearly in a resonant chain. In addition, we infer the presence of gap or shoulder-like structures in the other three disks through visibility modeling. These observations indicate that substructures robustly form and survive at semimajor axes of several tens of au or less in disks exposed to intermediate levels of external UV radiation as well as in compact disks. However, our observations also suggest that disks in σ Orionis are mostly small, and thus millimeter continuum gaps beyond a disk radius of 50 au are rare in this region, possibly due to either external photoevaporation or age effects.

A new multiple-arc model of the resonant Kuiper belt objects – Plutinos

ABSTRACT To incorporate the gravitational influence of Kuiper belt objects (KBOs) in planetary ephemerides, uniform-ring models are commonly employed. In this paper, for representing the KBO population residing in Neptune’s 2:3 mean motion resonance (MMR), known as the Plutinos, we introduce a three-arc model by considering their resonant characteristics. Each ‘arc’ refers to a segment of the uniform ring and comprises an appropriate number of point masses. Then the total perturbation of Plutinos is numerically measured by the change in the Sun–Neptune distance ($\Delta d_{\mathrm {SN}}$). We conduct a comprehensive investigation to take into account various azimuthal and radial distributions associated with the resonant amplitudes (A) and eccentricities (e) of Plutinos, respectively. The results show that over a 100-yr period: (1) at the smallest $e=0.05$, the Sun–Neptune distance change $\Delta d_{\mathrm {SN}}$ caused by Plutinos decreases significantly as A reduces. It can deviate from the value of $\Delta d_{\mathrm {SN}}$ obtained in the ring model by approximately 100 km; (2) as e increases in the medium range of 0.1–0.2, the difference in $\Delta d_{\mathrm {SN}}$ between the arc and ring models becomes increasingly significant; (3) at the largest $e\gtrsim 0.25$, $\Delta d_{\mathrm {SN}}$ can approach zero regardless of A, and the arc and ring models exhibit a substantial difference in $\Delta d_{\mathrm {SN}}$, reaching up to 170 km. Then the applicability of our three-arc model is further verified by comparing it to the perturbations induced by observed Plutinos on the positions of both Neptune and Saturn. Moreover, the concept of the multiple-arc model, designed for Plutinos, can be easily extended to other MMRs densely populated by small bodies.