Mean Motion Research Articles

TOI-216: Resonant Constraints on Planet Migration

Abstract TOI-216 is a pair of close-in planets with orbits deep in the 2:1 mean motion resonance. The inner Neptune-class planet (TOI-216b) is near 0.12 au (orbital period P b ≃ 17 days) and has a substantial orbital eccentricity (e b ≃ 0.16) and large libration amplitude (A ψ ≃ 60°) in the resonance. The outer planet (TOI-216c) is a gas giant on a nearly circular orbit. We carry out N-body simulations of planet migration in a protoplanetary gas disk to explain the orbital configuration of TOI-216 planets. We find that TOI-216b's migration must have been halted near its current orbital radius to allow for a convergent migration of the two planets into the resonance. For the inferred damping-to-migration timescale ratio τ e /τ a ≃ 0.02, overstable librations in the resonance lead to a limit cycle with A ψ ≃ 80° and e b < 0.1. The system could have remained in this configuration for the greater part of the protoplanetary disk lifetime. If the gas disk was removed from inside out, this would have reduced the libration amplitude to A ψ ≃ 60° and boosted e b via the resonant interaction with TOI-216c. Our results suggest a relatively fast inner-disk removal (∼105 yr). Another means of explaining the large libration amplitude is stochastic stirring from a (turbulent) gas disk. For that to work, overstable librations would need to be suppressed, τ e /τ a ≃ 0.05, and very strong turbulent stirring (or some other source of large stochastic forcing) would need to overcome the damping effects of gas. Hydrodynamical simulations can be performed to test these models.

Влияние эффекта Ярковского на орбитальные резонансы астероидов с малыми перигелийными расстояниями

The paper presents the study results of the Yarkovsky effect influence on the resonant characteristics of asteroids with small perihelion distances. Out of 52 studied objects, 27 asteroids were identified that move in the vicinity of stable and unstable low-order mean motion resonances with major planets. The evolution of the asteroids orbits is constructed with and without the Yarkovsky effect, and the behavior of the resonance characteristics in both cases is analyzed. The effect was taken into account by including the perturbation from the transversal acceleration A2, the values of which for the presented objects were obtained earlier by the authors of this work. It is shown that the Yarkovsky effect influence on stable resonances is insignificant and leads to small changes in the libration amplitude of the resonance characteristics. The stability of the resonance is preserved. With unstable resonant interaction, the number of passages through the exact commensurability changes, and for some asteroids the resonance becomes more stable.

ПРОБЛЕМА ПОХОДЖЕННЯ ЛЮКІВ КІРКВУДА В ПОЯСІ АСТЕРОЇДІВ ТА УМОВИ В РАННІЙ СОНЯЧНІЙ СИСТЕМІ

The orbits of asteroids from the MPC catalogue of January 21, 2022 with absolute magnitudes H < 16 m, in the 3:1, 5:2 and 2:1 mean motion resonances (MMRs) with Jupiter were selected. The number of the orbits in the 2:1 MMR is dozens of times greater than in two other resonances. The are fragments of parent bodies of neighbour asteroid families, in particular the Themis family, in the 2:1 MMR. The sizes of almost all bodies in resonant orbits do not exceed 10 km. It is shown that in the past, there were bodies of much larger sizes in these tree gaps. In particular, more than two billion years ago there were several tens of bodies with sizes of more than 50 km in the 2:1 gap, including those with sizes of more than 100 km. Numerical calculations were performed to evaluate the evolution of the selected resonant orbits over hundreds of thousands years. Perturbations from all eight major planets and the relativistic effects of orbital perihelion displacement were taken into account. For all orbits in the 3:1 and 5:2 MMRs an increase in the orbit eccentricities, which are sufficient for the bodies to approach Mars, was confirmed. It can explain the exit of asteroids from these two gaps. In the 2:1 MMR, a sufficient increase in the orbit eccentricities was not detected. Such increase in orbit eccentricities in this resonance may occur due to the action of non-gravitational effects (NGEs). The action of the Yarkovsky effect may explain exit of an asteroid with a size of 5 km from the 2:1 MMR over a period about 1 billion of years or more. It takes tens of billions years for large bodies that were once there, to exit from the 2:1 gap, under the action of the Yarkovsky effect of today power. To form the gap in the 2:1 resonance at the very beginning, the physical conditions for bodies in the asteroid belt had to be significantly different from the today ones. Most likely, the NGEs were influencing a much stronger on bodies in the asteroid belt in the early Solar system as compared with the today NGEs. In particular, the radiation of the Sun in the early Solar system could be much more intense, in comparison with the today one.

Eviction-like resonances for satellite orbits

The motion of a satellite can experience secular resonances between the precession frequencies of its orbit and the mean motion of the host planet around the star. Some of these resonances can significantly modify the eccentricity (evection resonance) and the inclination (eviction resonance) of the satellite. In this paper, we study in detail the secular resonances that can disturb the orbit of a satellite, in particular the eviction-like ones. Although the inclination is always disturbed while crossing one eviction-like resonance, capture can only occur when the semi-major axis is decreasing. This is, for instance, the case of Phobos, the largest satellite of Mars, that will cross some of these resonances in the future because its orbit is shrinking owing to tidal effects. We estimate the impact of resonance crossing in the orbit of the satellite, including the capture probabilities, as a function of several parameters, such as the eccentricity and the inclination of the satellite, and the obliquity of the planet. Finally, we use the method of the frequency map analysis to study the resonant dynamics based on stability maps, and we show that some of the secular resonances may overlap, which leads to chaotic motion for the inclination of the satellite.

Neptune’s ring arcs from VLT/SPHERE-IRDIS near-infrared observations

Context. Neptune’s incomplete ring arcs have been stable since their discovery in 1984 by stellar occultation. Although these structures should be destroyed within a few months through differential Keplerian motion, imaging data over the past couple of decades have shown that these structures remain stable. Aims. We present the first SPHERE near-infrared observations of Neptune’s ring arcs taken at 2.2 μm (broadband Ks) with the IRDIS camera at the Very Large Telescope (VLT) in August 2016. Methods. The images were aligned using the ephemerides of the satellite Proteus and were suitably co-added to enhance ring and satellite signals. Results. We analyse high-angular-resolution near-infrared images of Neptune’s ring arcs obtained in 2016 at the ESO VLT-UT3 with the adaptive-optics-fed camera SPHERE-IRDIS. We derive accurate mean motion values for the arcs and the nearby satellite Galatea. The trailing arcs Fraternité and Égalité have been stable since they were last observed in 2007. Furthermore, we confirm the fading away of the leading arcs Courage and Liberté. Finally, we confirm the mismatch between the arcs’ position and the 42:43 inclined and eccentric corotation resonances with Galatea, thus demonstrating that no 42:43 corotation model works to explain the azimuthal confinement of the arcs’ materiel.

Survey of secular resonances in the asteroid belt

Using a recently introduced synthetic method to compute the asteroid secular frequencies (Knezevic and Milani 2019), in this paper we survey the locations of secular resonances in the 9 dynamically distinct zones of the asteroid belt. Positions of all resonances up to order four, of a significant fraction of the order six resonances, and of a several order eight ones were determined, plotted in the space of proper elements, and discussed in relation to the local dynamics and to the structure and shape of the nearby asteroid collisional families. Only the resonant combinations with fundamental frequencies of Jupiter and Saturn were considered, with a few special cases involving other planets and largest asteroids. Accuracy of the polynomial fit to determine the frequencies was found to be satisfactory for the purpose of determination of secular resonance positions. This enabled a precise identification of dynamical mechanisms affecting the computation of frequencies (close vicinity of the mean motion resonances and libration in secular resonances), and of the ?cycle slips? as a primary technical drawback causing deterioration of the results. For each zone we also presented and discussed a fairly complete sample of recent works dealing with interaction of the secular resonances with asteroid families present in that zone. Finally, a few words were devoted to possibilities for future work.

Mathematical Model of Satellite Rotation near Spin-Orbit Resonance 3:2

This paper considers the rotational motion of a satellite equipped with flexible viscoelastic rods in an elliptic orbit. The satellite is modeled as a symmetric rigid body with a pair of flexible viscoelastic rods rigidly attached to it along the axis of symmetry. A planar case is studied, i. e., it is assumed that the satellite’s center of mass moves in a Keplerian elliptic orbit lying in a stationary plane and the satellite’s axis of rotation is orthogonal to this plane. When the rods are not deformed, the satellite’s principal central moments of inertia are equal to each other. The linear bending theory for thin inextensible rods is used to describe the deformations. The functionals of elastic and dissipative forces are introduced according to this model. The asymptotic method of motions separation is used to derive the equations of rotational motion reflecting the influence of the fluctuations, caused by the deformations of the rods. The method of motion separation is based on the assumption that the period of the autonomous oscillations of a point belonging to the rod is much smaller than the characteristic time of these oscillations’ decay, which, in its turn, is much smaller than the characteristic time of the system’s motion as a whole. That is why only the oscillations induced by the external and inertial forces are taken into account when deriving the equations of the rotational motion. The perturbed equations are described by a third-order system of ordinary differential equations in the dimensionless variable equal to the ratio of the satellite’s absolute value of angular velocity to the mean motion of the satellite’s center of mass, the angle between the satellite’s axis of symmetry and a fixed axis and the mean anomaly. The right-hand sides of the equation depend on the mean anomaly implicitly through the true anomaly. A new slow angular variable is introduced in order to perform the averaging for the perturbed system near the 3:2 resonance, and the averaging is

V1298 Tau with TESS: Updated Ephemerides, Radii, and Period Constraints from a Second Transit of V1298 Tau e

Abstract V1298 Tau is a young (20–30 Myr) solar-mass K star hosting four transiting exoplanets with sizes between 0.5 and 0.9 R J. Given the system’s youth, it provides a unique opportunity to understand the evolution of planetary radii at different separations in the same stellar environment. V1298 Tau was originally observed 6 yr ago during K2 Campaign 4. Now, V1298 Tau has been re-observed during the extended mission of NASA’s Transiting Exoplanet Survey Satellite (TESS). Here, we present new photometric observations of V1298 Tau from TESS. We use the TESS data to update the ephemerides for V1298 Tau bcde as well as compare newly observed radii to those measured from the K2 light curve, finding shallower transits for V1298 Tau bcd in the redder TESS bandpass at the 1–2σ level. We suspect the difference in radii is due to starspot crossing events or contamination from nearby faint stars on the same pixels as V1298 Tau. Additionally, we catch a second transit of V1298 Tau e and present a new method for deriving the marginalized posterior probability of a planet’s period from two transits observed years apart. We find the highest probability period for V1298 Tau e to be in a near 2:1 mean motion resonance with V1298 Tau b which, if confirmed, could make V1298 Tau bcde a four-planet resonant chain. V1298 Tau is the target of several ongoing and future observations. These updated ephemerides will be crucial for scheduling future transit observations and interpreting future Doppler tomographic or transmission spectroscopy signals.

A survey on Hungaria asteroids involved in mean motion resonances with Mars

Context. A region at the inner edge of the main asteroid belt is populated by the Hungaria asteroids. Among these objects, the Hungaria family formed as the result of a catastrophic disruption of the (434) Hungaria asteroid a few hundred million years ago. Due to the Yarkovsky effect, the fragments depending on their direction of rotation are slowly drifting inwards or outwards from the actual place of collision. Due to this slow drift, these bodies could approach the locations of the various outer-type mean-motion resonances (MMRs) with Mars. Aims. We aim to study the actual dynamical structure of Hungaria asteroids, which is primarily shaped by various outer-type MMRs with Mars. Moreover, we also seek connections between the orbital characteristics of Hungaria asteroids and their absolute magnitude. Methods. To map the resonant structure and dynamics of asteroids belonging to the Hungaria group, we used the FAst Identification of mean motion Resonances method, which can detect MMRs without the a priori knowledge of the critical argument. We also compiled stability maps of the regions around the MMRs by using the maximal variations in the asteroids’ eccentricities, semi-major axes, and inclinations. We numerically integrated the orbits of all asteroids belonging to the Hungaria group available in the JPL Horizon database together with the Solar System planets for one and ten million years. Results. Having studied the resonant structure of the Hungaria group, we find that several asteroids are involved in various MMRs with Mars. We identify both short- and long-term MMRs. Besides this, we also find a relationship between the absolute magnitude of asteroids and the MMR in which they are involved.

Black Sea Mnemiopsis leidyi (Ctenophora) adult locomotion and light-induced behavior in laboratory experiments

The capacity of Ctenophores to sense light is indicated by comparative morphology and genomics data, but the relevant behavioral responses of the animals to light stimulus have not been studied yet. We developed the design of a laboratory experiment to reveal the light-induced behavior of adult Mnemiopsis leidyi from the Black Sea and identified criteria for its evaluation. Average swimming speed up to ~6–7 mm s−1 can be considered specific for the M. leidyi adults under rest or active feeding. Ctenophore reacts to a sharp light rise as an external stimulus, briefly increasing the rate several times (up to 40 mm sec−1). Analysis of motion vectors revealed no directional movement from the light source or towards it. Maximum and average rates decrease significantly for two minutes after the light is turned on, and speeds return to the original values within an hour. Small animals are more active than large ones. The mean motion rates of small comb jellies (<60 mm) were 0.7 ± 0.84 mm s−1 and 0.18 ± 0.19 mm s−1 - for large animals (>80 mm) when averaging the data for the entire observation period (from February to June).

Long-term dynamical evolution of Pallene (Saturn XXXIII) and its diffuse, dusty ring

ABSTRACT The distinctive set of Saturnian small satellites, Aegaeon, Methone, Anthe, and Pallene, constitutes an excellent laboratory to understand the evolution of systems immersed in co-orbital dusty rings/arcs, subjected to perturbations from larger satellites and non-gravitational forces. In this work, we carried out a comprehensive numerical exploration of the long-term evolution of Pallene and its ring. Through frequency map analysis, we characterized the current dynamical state around Pallene. A simple tidal evolution model serves to set a time frame for the current orbital configuration of the system. With detailed short- and long-term N-body simulations we determine whether Pallene is currently in resonance with one or more of six of Saturn’s major moons. We analysed a myriad of resonant arguments extracted from the direct and indirect parts of the disturbing function, finding that Pallene is not in mean motion resonance from the present up to 5 Myr into the future; none the less, some resonant arguments exhibit intervals of libration and circulation at different time-scales and moon pairings. We studied the dynamical evolution of micrometric particles forming the ring, considering gravitational and non-gravitational forces. Non-gravitational forces are responsible for particles vertical excursions and outward migration. By estimating the satellite’s mass production rate, we find that Pallene could be responsible for keeping its ring in steady-state only if it is mainly composed of large micrometre-sized particles. If mainly composed of particles with a few micrometres for which Pallene is the only source, the ring will spread out, both radially and vertically, until it finally disappears.

Study on Motion Management of Pancreatic Cancer Treated by CyberKnife.

PurposeWe investigated the movement characteristics of pancreas and the clinical accuracy of tracking pancreas with the Synchrony Respiratory Tracking System (SRTS) during the CyberKnife treatment. These data provide a clinical data basis for the expansion margins of pancreatic tumor target.Methods and MaterialsForty-two patients with pancreatic cancer treated by CyberKnife were retrospectively studied. The pancreatic displacement calculated from the x-ray images collected during the time interval between two consecutive movements constituted a data set.ResultsThe total mean motion amplitudes and standard deviations of pancreatic tumors in SI, LR, AP, and radial directions were 3.66 ± 1.71 mm, 0.97 ± 0.62 mm, 1.52 ± 1.02 mm, and 1.36 ± 0.49 mm, respectively. The overall mean correlation errors and standard deviations were 0.82 ± 0.46 mm, 0.47 ± 0.33 mm, 0.41 ± 0.24 mm, and 0.98 ± 0.37 mm, respectively. The overall mean prediction errors and standard deviations were 0.57 ± 0.14 mm, 0.62 ± 0.28 mm, 0.39 ± 0.17 mm, and 1.58 ± 0.36 mm, respectively. The correlation errors and prediction errors of pancreatic tumors at different anatomical positions in SI, LR, and AP directions were statistically significant (p < 0.05).ConclusionsThe tumor motion amplitude, the tumor location, and the treatment time are the main factors affecting the tracking accuracy. The pancreatic tumors at different anatomical locations should be treated differently to ensure sufficient dose coverage of the pancreatic target area.

A Pair of Warm Giant Planets near the 2:1 Mean Motion Resonance around the K-dwarf Star TOI-2202*

Abstract TOI-2202 b is a transiting warm Jovian-mass planet with an orbital period of P = 11.91 days identified from the Full Frame Images data of five different sectors of the TESS mission. Ten TESS transits of TOI-2202 b combined with three follow-up light curves obtained with the CHAT robotic telescope show strong transit timing variations (TTVs) with an amplitude of about 1.2 hr. Radial velocity follow-up with FEROS, HARPS, and PFS confirms the planetary nature of the transiting candidate (a b = 0.096 ± 0.001 au, m b = 0.98 ± 0.06 M Jup), and a dynamical analysis of RVs, transit data, and TTVs points to an outer Saturn-mass companion (a c = 0.155 ± 0.002 au, m c = 0.37 ± 0.10 M Jup) near the 2:1 mean motion resonance. Our stellar modeling indicates that TOI-2202 is an early K-type star with a mass of 0.82 M ⊙, a radius of 0.79 R ⊙, and solar-like metallicity. The TOI-2202 system is very interesting because of the two warm Jovian-mass planets near the 2:1 mean motion resonance, which is a rare configuration, and their formation and dynamical evolution are still not well understood.

A new expansion of planetary disturbing function and applications to interior, co-orbital and exterior resonances with planets

In this study, a new expansion of planetary disturbing function is developed for describing the resonant dynamics of minor bodies with arbitrary inclinations and semimajor axis ratios. In practice, the disturbing function is expanded around circular orbits in the first step and then, in the second step, the resulting mutual interaction between circular orbits is expanded around a reference point. As usual, the resulting expansion is presented in the Fourier series form, where the force amplitudes are dependent on the semimajor axis, eccentricity and inclination, and the harmonic arguments are linear combinations of the mean longitude, longitude of pericenter and longitude of ascending node of each mass. The resulting new expansion is valid for arbitrary inclinations and semimajor axis ratios. In the case of mean motion resonant configuration, the disturbing function can be easily averaged to produce the analytical expansion of resonant disturbing function. Based on the analytical expansion, the Hamiltonian model of mean motion resonances is formulated, and the resulting analytical developments are applied to Jupiter’s inner and co-orbital resonances and Neptune’s exterior resonances. Analytical expansion is validated by comparing the analytical results with the associated numerical outcomes.

A Sublime Opportunity: The Dynamics of Transitioning Cometary Bodies and the Feasibility of In Situ Observations of the Evolution of Their Activity

Abstract The compositional and morphological evolution of minor bodies in the solar system is primarily driven by the evolution of their heliocentric distances, as the level of incident solar radiation regulates cometary activity. We investigate the dynamical transfer of Centaurs into the inner solar system, facilitated by mean motion resonances with Jupiter and Saturn. The recently discovered object P/2019 LD2 will transition from the Centaur region to the inner solar system in 2063. In order to contextualize LD2, we perform N-body simulations of a population of Centaurs and Jupiter-family comets. Objects between Jupiter and Saturn with Tisserand parameter T J ∼ 3 are transferred onto orbits with perihelia q &lt; 4 au within the next 1000 yr with notably high efficiency. Our simulations show that there may be additional LD2-like objects transitioning into the inner solar system in the near future, all of which have low ΔV with respect to Jupiter. We calculate the distribution of orbital elements resulting from a single Jovian encounter and show that objects with initial perihelia close to Jupiter are efficiently scattered to q &lt; 4 au. Moreover, approximately 55% of the transitioning objects in our simulated population experience at least one Jovian encounter prior to reaching q &lt; 4 au. We demonstrate that a spacecraft stationed near Jupiter would be well positioned to rendezvous, orbit-match, and accompany LD2 into the inner solar system, providing an opportunity to observe the onset of intense activity in a pristine comet in situ. Finally, we discuss the prospect of identifying additional targets for similar measurements with forthcoming observational facilities.

On mean motion resonances in the Geminid meteoroid stream

Analysing numerical models of the Geminid meteoroid stream for particle masses 0.00003–0.3 ​g we found evidence of several mean motion resonances: 1:2, 2:5, 3:7, 3:8 and 4:9 with Venus, 2:3 and 5:7 with the Earth and 7:1 with Jupiter. The resonant particles form dust trails located far from the Earth's orbit. Indeed, there is no observational support for resonant effects in the observed Geminid meteor shower. The trails are stable and compact, and consist of large (0.003–0.3 ​g) meteoroids.

The ‘breaking the chains’ migration model for super-Earth formation: the effect of collisional fragmentation

ABSTRACT Planets between 1 and 4 R⊕ (Earth radius) with orbital periods &amp;lt;100 d are strikingly common. The migration model proposes that super-Earths migrate inwards and pile up at the disc inner edge in chains of mean motion resonances. After gas disc dispersal, simulations show that super-Earth’s gravitational interactions can naturally break their resonant configuration leading to a late phase of giant impacts. The instability phase is key to matching the orbital spacing of observed systems. Yet, most previous simulations have modelled collisions as perfect accretion events, ignoring fragmentation. In this work, we investigate the impact of imperfect accretion on the ‘breaking the chains’ scenario. We performed N-body simulations starting from distributions of planetary embryos and modelling the effects of pebble accretion and migration in the gas disc. Our simulations also follow the long-term dynamical evolution of super-Earths after the gas disc dissipation. We compared the results of simulations where collisions are treated as perfect merging events with those where imperfect accretion and fragmentation are allowed. We concluded that the perfect accretion is a suitable approximation in this regime, from a dynamical point of view. Although fragmentation events are common, only ∼10 per cent of the system mass is fragmented during a typical ‘late instability phase’, with fragments being mostly reacreted by surviving planets. This limited total mass in fragments proved to be insufficient to alter qualitatively the final system dynamical configuration – e.g. promote strong dynamical friction or residual migration – compared to simulations where fragmentation is neglected.

Three-dimensional retrograde periodic orbits of asteroids moving in mean motion resonances with Jupiter

We study the dynamics of interior mean motion resonances of first, second and third order with Jupiter using the model of the restricted three-body problem with the Sun and Jupiter as primaries and we focus on asteroids which are in retrograde motion with the perturbing planet. The basis of our study is the computation of three-dimensional symmetric periodic orbits and their linear stability. The position and stability of periodic orbits determine critically the phase space topology in the resonant regimes. The periodic orbits are continued analytically forming families which show particular structures. They may continue from the planar model to the three-dimensional one forming “direct” families, which start from inclination i ​= ​0°, or “retrograde” families, which start from i ​= ​180°. Also there exist families that start from inclined circular orbits and do not contain any planar orbits. Families that contain a polar orbit (i ​= ​90°) consist of both direct and retrograde parts and are defined in the whole inclination domain. Instead, pure direct or retrograde families tend to high eccentricity values, i.e., they terminate to a collision with Sun. The resonance 3/1 is studied further in order to understand the dynamics of the asteroid (343158) Marsyas, which is in retrograde motion.

Taurid Stream #628: A Reservoir of Large Cometary Impactors

Abstract The Desert Fireball Network observed a significant outburst of fireballs belonging to the Southern Taurid Complex of meteor showers between 2015 October 27 and November 17. At the same time, the Cameras for Allsky Meteor Surveillance project detected a distinct population of smaller meteors belonging to the irregular IAU shower #628, the s-Taurids. While this returning outburst was predicted and observed in previous work, the reason for this stream is not yet understood. 2015 was the first year that the stream was precisely observed, providing an opportunity to better understand its nature. We analyze the orbital elements of stream members and establish a size–frequency distribution from millimeter to meter size range. The stream is highly stratified with a large change of entry speed along Earth’s orbit. We confirm that the meteoroids have orbital periods near the 7:2 mean motion resonance with Jupiter. The mass distribution of this population is dominated by larger meteoroids, unlike that for the regular Southern Taurid shower. The distribution index is consistent with a gentle collisional fragmentation of weak material. A population of meter-sized objects is identified from satellite observations at a rate consistent with a continuation of the size–frequency distribution established at centimeter size. The observed change of longitude of perihelion among the s-Taurids points to recent (a few centuries ago) activity from fragmentation involving surviving asteroid 2015 TX24. This supports a model for the Taurid Complex showers that involves an ongoing fragmentation cascade of comet 2P/Encke siblings following a breakup some 20,000 yr ago.

Development of a Monsoon Depression and Its Interaction with the Large-Scale Background: A Case Study

Abstract Structure and time evolution of the large-scale background and an embedded synoptic-scale monsoon depression and their interactions are studied. The depression formation is preceded by a cyclonic circulation around 400 hPa. The Fourier-based scale separation technique is used to isolate large (wavenumbers 0–8) and synoptic-scale (wavenumbers 12–60). The wavelength and depression center is determined objectively. The synoptic-scale depression has an average longitudinal wavelength of around 1900 km and a north–south size of 1100 km; it is most intense with a vorticity of 20.5 × 10 −5 s −1 at 900 hPa. The strongest cold core of −3.0°C below 850 hPa and the above warm core of around 2.0°C are evident. The depression is tilted southwestward in the midtroposphere with no significant vertical tilt in the lower troposphere. The mean maximum intensity and upward motion over the life cycle of depression are in close agreement with the composite values. A strong cyclonic shear zone is developed in the midtroposphere preceding the depression. The necessary condition for barotropic (baroclinic) instability is satisfied in the midtroposphere (boundary layer). Strong northward transport of momentum by the depression against the southward shear is found. The strong growth of the MD in the lower troposphere is due to downward transfer of excess energy gained in the midtroposphere from the barotropic energy conversion and east–west direct thermal circulation as the vertical energy flux. The baroclinic interaction contributes to the maintenance of the cold core in the lower troposphere. The diabatic heating rate is computed and its role in the genesis and growth of MD is investigated.