Relative Effect of Inclinations for Moonlets in the Triple Asteroidal Systems

We present the analysis and computational results for the inclination relative effect of moonlets of triple asteroidal systems. Perturbations on moonlets due to the primary’s non-sphericity gravity, the solar gravity, and moonlets’ relative gravity are discussed. The inclination vector for each moonlet follows a periodic elliptical motion; the motion period depends on the moonlet’s semi-major axis and the primary’s J2 perturbations. Perturbation on moonlets from the Solar gravity and moonlet’s relative gravity makes the motion of the x component of the inclination vector of moonlet 1 and the y component of the inclination vector of moonlet 2 to be periodic. The mean motion of x component and the y component of the inclination vector of each moonlet forms an ellipse. However, the instantaneous motion of x component and the y component of the inclination vector may be an elliptical disc due to the coupling effect of perturbation forces. Furthermore, the x component of the inclination vector of moonlet 1 and the y component of the inclination vector of moonlet 2 form a quasi-periodic motion. Numerical calculation of dynamical configurations of two triple asteroidal systems (216) Kleopatra and (153591) 2001 SN263 validates the conclusion.     

Ringlet–satellite interactions

We study the interaction of a satellite and a nearby ringlet on eccentric and inclined orbits. Secular torques originate from mean motion resonances and the secular interaction potential which represents the m  = 1 global modes of the ring. The torques act on the relative eccentricity and inclination. The resonances damp the relative eccentricity. The inclination instability owing to the resonances is turned off by a finite differential eccentricity of the order of 0.27 for nearly coplanar systems. The secular potential torque damps the eccentricity and inclination and does not affect the relative semi-major axis; also, it suppresses the inclination instability that persists at small differential eccentricities. The damping of the relative eccentricity and inclination forces an initially circular and planar small mass ringlet to reach the eccentricity and inclination of the satellite. When the planet is oblate, the interaction of the satellite damps the proper precession of a small mass ringlet so that it precesses at the satellite's rate independently of their relative distance. The oblateness of the primary modifies the long-term eccentricity and inclination magnitudes and introduces a constant shift in the apsidal and nodal lines of the ringlet with respect to those of the satellite. These results are applied to Saturn's F-ring, which orbits between the moons Prometheus and Pandora.     

《大气一号》气球卫星轨道倾角变化分析

引起《大气一号》两颗气球卫星（ＤＱ－１Ａ和ＤＱ－１Ｂ）轨道倾角变化的摄动因素主要是太阳光压摄动、大气旋转和日月引力摄动。太阳光压摄动引起气球卫星轨道倾角增大，平均每天变化约０．００１７，大气旋转引起轨道倾角减小，平均每天变化不到０．０００１，但随着高度下降，变化量亦增大，陨落前达０．００２。本文根据卫星轨道摄动理论，给出气球卫星轨道倾角变化的一种定量分析方法，得到的分析结果为：（１）由太阳光压摄动     

Extension of the critical inclination

The critical inclination is of special interest in artificial satellite theory. The critical inclination can maintain minimal deviations of eccentricity and argument of pericentre from the initial values, and orbits at this inclination have been applied to some space missions. Most previous researches about the critical inclination were made under the assumption that the oblateness term J ₂ is dominant among the harmonic coefficients. This paper investigates the extension of the critical inclination where the concept of the critical inclination is different from that of the traditional sense. First, the study takes the case of Venus for instance, and provides some preliminary results. Then for general cases, given the values of argument of pericentre and eccentricity, the relationship between the multiplicity of the solutions for the critical inclination and the values of J ₂ and J ₄ is analyzed. Besides, when given certain values of J ₂ and J ₄, the relationship between the multiplicity of the solutions for the critical inclination and the values of semimajor axis and eccentricity is studied. The results show that for some cases, the value of the critical inclination is far away from that of the traditional sense or even has multiple solutions. The analysis in this paper could be used as starters of correction methods in the full gravity field of celestial bodies.     

Lunar cratering asymmetries with high lunar orbital obliquity and inclination of the Moon

Accurate estimation of cratering asymmetry on the Moon is crucial for understanding Moon evolution history. Early studies of cratering asymmetry have omitted the contributions of high lunar obliquity and inclination. Here, we include lunar obliquity and inclination as new controlling variables to derive the cratering rate spatial variation as a function of longitude and latitude. With examining the influence of lunar obliquity and inclination on the asteroids population encountered by the Moon, we then have derived general formulas of the cratering rate spatial variation based on the crater scaling law. Our formulas with addition of lunar obliquity and inclination can reproduce the lunar cratering rate asymmetry at the current Earth-Moon distance and predict the apex/ant-apex ratio and the pole/equator ratio of this lunar cratering rate to be 1.36 and 0.87, respectively. The apex/ant-apex ratio is decreasing as the obliquity and inclination increasing. Combining with the evolution of lunar obliquity and inclination, our model shows that the apex/ant-apex ratio does not monotonically decrease with Earth-Moon distance and hence the influences of obliquity and inclination are not negligible on evolution of apex/ant-apex ratio. This model is generalizable to other planets and moons, especially for different spin-orbit resonances.     

Some useful results on initial node locations for near-equatorial circular satellite orbits

Perturbation theory based on Lie transforms is used to obtain a second-order long period solution for inclination and right ascension of ascending node, of near-equatorial circular satellite orbits. The solution includes the average effects of the Earth's oblateness and the luni-solar perturbations. Three algorithms, useful in mission analysis, are then given. The first algorithm finds the initial node location that results in a decrease of inclination to zero and it also finds the corresponding time to arrive at this zero inclination. The second algorithm determines the initial nodal band that maintains the orbital inclination below a specified value for a given time interval. The third algorithm obtains the initial node location that maximizes the time in which the satellite can be maintained within a given inclination tolerance without the use of any active control and it also obtains the corresponding maximum time. The results of the first and the third algorithms are given for 24-h near-equatorial circular satellite orbits and are cast in simple closed forms.     

Derivative of Kaula's inclination function

The derivative of Kaula's inclination function is expressed as a linear combination of two inclination functions themselves.On leave from Institute of Applied Astronomy, 197042 St.-Petersburg, Russia.     

Stability VS inclination of motions inE 3

Five families of three-dimensional doubly symmetric motions are computed after establishing their existence by means of a grid-search technique. It is confirmed that within the same family orbits of lower inclination with respect to the plane of motion of the primaries are stable while the critical inclination at which instability occurs varies between families. The maximum inclination at which stable motions of the type presented here were found is about 52°.     

Critical inclination of Trojan asteroids

The author's earlier solution for Trojan asteroids is developed further. It is shown that depending on the amplitude of libration around the Lagrangian point L₄, there is a critical inclination which determines the sign of the variation of the ascending node. If the orbital inclination of a Trojan is smaller than the critical one, then the ascending node decreases and otherwise it increases. The variation of the eccentricity and of the longitude of the perihelion has also a dependence on the critical inclination.     

倾角函数展开及其在分析法轨道预报中的应用

田谐项摄动是分析法轨道预报中的重要部分,其中包含大量倾角函数及其偏导数的计算.由于具有精度更高、速度更快的优点,倾角函数一般通过递推方法计算.以文献中提出的改进Gooding方法为基础,将其给出的程序稍加改进,在计算2–50阶倾角函数时缩短了约24%的计算时间.考虑到分析法预报过程中轨道平倾角变化很小,以泰勒展开式计算倾角函数,可极大提高计算速度,较大程度地减小分析法预报耗时,且引力场阶次越高,减小幅度越大,取50阶时预报耗时缩短了48%.另一方面,以2阶展开式计算倾角函数时,与改进Gooding法相比,分析法预报星历偏差很小.对于500 km高度的低轨卫星,分别以改进Gooding法和2阶泰勒展开式计算倾角函数,预报3天,当地球引力场阶次不高于50时,二者预报星历偏差RMS (Root Mean Square)低于1 mm,且随着轨道高度的增加,预报星历偏差RMS逐渐减小.     

Further investigation into a recent model for toroidal rings of Saturn

A previous paper described a toroidal ringlet consisting of a large number of charged particles distributed uniformly along slightly inclined, slightly eccentric, orbits around Saturn. The orbits have identical semimajor-axis, identical eccentricity and identical inclination, but their ascending nodes are distributed uniformly around Saturn. Assuming an inclination (in radians) much smaller than the eccentricity, it was shown that the toroidal cross-section could rock slowly back and forth as the inclination fluctuates. In this paper a similar result is obtained for the more general case when inclination is comparable with eccentricity and both fluctuate.     

Balanced earth satellite orbits

An analytic model for third-body perturbations and for the second zonal harmonic of the central body's gravitational field is presented. A simplified version of this model applied to the Earth-Moon-Sun system indicates the existence of high-altitude and highly-inclined orbits with their apsides in the equator plane, for which the apsidal as well as the nodal motion ceases. For special positions of the node, secular changes of eccentricity and inclination disappear too (balanced orbits). For an ascending node at vernal equinox, the inclination of balanced orbits is 94.56°, for a node at autumnal equinox 85.44°, independent of the eccentricity of the orbit. For a node perpendicular to the equinox, there exist circular balanced orbits at 90° inclination. By slightly adjusting the initial inclination as suggested by the simplified model, orbits can be found — calculated by the full model or by different methods — that show only minor variations in eccentricity, inclination, argument of perigee, and longitude of the ascending node for 10⁵ revolutions and more. Orbits near the unstable equilibria at 94.56° and 85.44° inclination show very long periodic librations and oscillations between retrogade and prograde motion.Retired from IBM Vienna Software Development Laboratory.     

How critical is the critical inclination?

It has been claimed that the representation of satellite motion in the vicinity of the critical inclination is a matter of practical, as well as theoretical interest, since the perturbations in the coordinates are of the order of 25 times greater near the critical inclination than away from it (Messageet al., 1962). In this paper we show, using Encke's method of numerical integration for satellites which are at, near, and away from the critical inclination, that there are no discernible features in the coordinate perturbations which distinguish the critical inclination from any other.     

倾角函数及其导数的定积分计算方法

给出了一种倾角函数及其导数的定积分计算方法,表达式十分简单,其计算精度:倾角函数可达10^-15,导数可达10^-13,可与Gooding方法相媲美.该方法的稳定性和适用倾角范围均较好,可供倾角函数的最高阶数L_max≤50时使用.     

The inclination function in terms of nonsingular elements

Different formulas for computation of the inclination function in terms of nonsingular elements have been presented and compared. Among the ways to compute the inclination function presented below, the best one is based on the recurrence relations (eqs. 16) derived in this paper.     

High-inclination planets and asteroids in multistellar systems

The Kozai mechanism often destabilizes high-inclination orbits. It couples changes in the eccentricity and inclination, and drives high inclination, circular orbits to low inclination, eccentric orbits. In a recent study of the dynamics of planetesimals in the quadruple star system HD 98800, there were significant numbers of stable particles in circumbinary polar orbits about the inner binary pair which are apparently able to evade the Kozai instability.
Here, we isolate this feature and investigate the dynamics through numerical and analytical models. The results show that the Kozai mechanism of the outer star is disrupted by a nodal libration induced by the inner binary pair on a shorter time-scale. By empirically modelling the period of the libration, a criteria for determining the high-inclination stability limits in general triple systems is derived. The nodal libration feature is interesting and, although affecting inclination and node only, shows many parallels to the Kozai mechanism. This raises the possibility that high-inclination planets and asteroids may be able to survive in multistellar systems.     

Spectral analysis of archaeomagnetic inclinations for the last 2000 years

The analytical representation and the application of maximum entropy spectral analysis to inclination data of the last 2000 years revealed the possible existence of periodic terms of approximately 1000, 500 and 260 years. The best fitting of the inclination data was achieved by a network of cosinusoidals as well as cubic splines.     

Influence of periodic orbits on the formation of giant planetary systems

The late-stage formation of giant planetary systems is rich in interesting dynamical mechanisms. Previous simulations of three giant planets initially on quasi-circular and quasi-coplanar orbits in the gas disc have shown that highly mutually inclined configurations can be formed, despite the strong eccentricity and inclination damping exerted by the disc. Much attention has been directed to inclination-type resonance, asking for large eccentricities to be acquired during the migration of the planets. Here we show that inclination excitation is also present at small to moderate eccentricities in two-planet systems that have previously experienced an ejection or a merging and are close to resonant commensurabilities at the end of the gas phase. We perform a dynamical analysis of these planetary systems, guided by the computation of planar families of periodic orbits and the bifurcation of families of spatial periodic orbits. We show that inclination excitation at small to moderate eccentricities can be produced by (temporary) capture in inclination-type resonance and the possible proximity of the non-coplanar systems to spatial periodic orbits contributes to maintaining their mutual inclination over long periods of time.     

Determining the spin of two stellar-mass black holes from disc reflection signatures

We present measurements of the dimensionless spin parameters and inner-disc inclination of two stellar-mass black holes. The spin parameter of SWIFT J1753.5−0127 and GRO J1655−40 is estimated by modelling the strong reflection signatures present in their XMM–Newton observations. Using a newly developed, self-consistent reflection model which includes the blackbody radiation of the disc as well as the effect of Comptonization, blurred with a relativistic line function, we infer the spin parameter of SWIFT J1753.5−0127 to be  0.76^+0.11_−0.15  . The inclination of this system is estimated at  55°⁺²₋₇  . For GRO J1655−40, we find that the disc is significantly misaligned to the orbital plane, with an innermost inclination of  30°⁺⁵₋₁₀  . Allowing the inclination to be a free parameter, we find a lower limit for the spin of 0.90, this value increases to that of a maximal rotating black hole when the inclination is set to that of the orbital plane of J1655−40. Our technique is independent of the black hole mass and distance, uncertainties in which are among the main contributors to the spin uncertainty in previous works.     

A Definite Integration Method for Calculating Inclination Function and Its Derivative

This paper gives a de?nite integration method for calculating the inclination function and its derivative, which has a very simple expression, and the accuracies as high as 10^-15 for the inclination function, and 10^-13 for its derivative, comparable with the accuracy of Gooding's method. By through a lot of numerical simulations, it is proved that this method has a good stability and an wide applicable range of inclinations, hence it can be used to calculate the inclination function to the maximum order of L_max ≤ 50.