Engineering topics

Malhotra, Renu

Publications and source records attributed to Malhotra, Renu.

16 records

NASA NTRS2000

The nu Andromedae System: Models and Stability

Radial velocity observations of the F8 V star nu Andromedae taken at Lick and at Whipple Observatories have revealed evidence of three periodicities in the line-of-sight velocity of the star. These periodicities have been interpreted as evidence for at least three low-mass companions (LMCs) revolving around nu Andromedae. The mass and orbital parameters inferred for these companions raise questions about the dynamical stability of the system. We report here results from our independent analysis of the published radial velocity data, as well as new unpublished data taken at Lick Observatory. Our results confirm the finding of three periods in the data. Our best fits to the data, on the assumption that these periods arise from the gravitational perturbations of companions in Keplerian orbits, are also generally in agreement but with some differences from the earlier findings. We find that the available data do not constrain well the orbital eccentricity of the middle companion in a three-companion model of the data. We also find that in order for our best-fit model to the Lick data to be dynamically stable over the lifetime of the star (approximately 2 billion years), the system must have a mean inclination to the plane of the sky greater than 13 deg. The corresponding minimum inclination for the best fit to the Whipple data set is 19 deg. These values imply that the maximum mass for the outer companion can be no greater than about 20 Jupiter masses. Our analysis of the stability of the putative systems also places constraints on the relative inclinations of the orbital planes of the companions. We comment on global versus local (i.e., method of steepest descent) means of finding best-fit orbits from radial velocity data sets.

Stepinski, Tomasz F.↗

NASA NTRS1997

Depletion of the Outer Asteroid Belt

During the early history of the solar system, it is likely that the outer planets changed their distance from the sun, and hence, their influence on the asteroid belt evolved with time. The gravitational influence of Jupiter and Saturn on the orbital evolution of asteroids in the outer asteroid belt was calculated. The results show that the sweeping of mean motion resonances associated with planetary migration efficiently destabilizes orbits in the outer asteroid belt on a time scale of 10 million years. This mechanism provides an explanation for the observed depletion of asteroids in that region.

Liou, Jer-Chyi↗

NASA NTRS1996

The Phase Space Structure Near Neptune Resonances in the Kuiper Belt

The Solar system beyond Neptune is believed to house a population of small primordial bodies left over from the planet formation process. The region up to heliocentric distance -50 AU (a.k.a. the Kuiper Belt) may be the source of the observed short-period comets. In this region, the phase space structure near orbital resonances with Neptune is of special interest for the long-term stability of orbits. There is reason to believe that a significant fraction (perhaps most) of the Kuiper Belt objects reside preferentially in these resonance locations. This paper describes the dynamics of small objects near the major orbital resonances with Neptune. Estimates of the widths of stable resonance zones as well as the properties of resonant orbits are obtained from the circular, planar restricted three-body model. Although this model does not contain the full complexity of the long-term orbital dynamics of Kuiper Belt objects subject to the full N-body perturbations of all the planets, it does provide a baseline for the phase space structure and properties of resonant orbits in the trans-Neptunian Solar system.

Malhotra, Renu↗

NASA NTRS1995

The Origin of Pluto's Orbit: Implications for the Solar System Beyond Neptune

The origin of the highly eccentric, inclined, and resonance-locked orbit of Pluto has long been a puzzle. A possible explanation has been proposed recently which suggests that these extraordinary orbital properties may be a natural consequence of the formation and early dynamical evolution of the outer solar system. A resonance capture mechanism is possible during the clearing of the residual planetesimal debris and the formation of the Oort Cloud of comets by planetesimal mass loss from the vicinity of the giant planets. If this mechanism were in operation during the early history of the planetary system, the entire region between the orbit of Neptune and approximately 50 AU would have been swept by first-order mean motion resonances. Thus, resonance capture could occur not only for Pluto, but quite generally for other trans-Neptunian small bodies. Some consequences of this evolution for the present-day dynamical structure of the trans-Neptunian region are (1) most of the objects in the region beyond Neptune and up to approximately 50 AU exist in very narrow zones located at orbital resonances with Neptune (particularly the 3:2 and the 2:1 resonances); and (2) these resonant objects would have significantly large eccentricities. The distribution of objects in the Kuiper Belt as predicted by this theory is presented here.

Malhotra, Renu↗

NASA NTRS1994

A mapping method for the gravitational few-body problem with dissipation

Recently a new class of numerical integration methods - 'mixed variable symplectic integrators' - has been introduced for studying long-term evolution in the conservative gravitational few-body problem. These integrators are an order of magnitude faster than conventional ordinary differential equations (ODE) integration methods. Here we present a simple modification of this method to include small non-gravitational forces. The new scheme provides a similar advantage of computational speed for a larger class of problems in Solar System dynamics.

Malhotra, Renu↗

NASA NTRS1994

Nonlinear Resonances in the Solar System

Orbital resonances are ubiquitous in the Solar system. They play a decisive role in the long term dynamics, and in some cases the physical evolution, of the planets and of their natural satellites, as well as the evolution of small bodies (including dust) in the planetary system. The few-body gravitational problem of hierarchical planetary-type systems allows for a complex range of dynamical timescales, from the fast orbital periods to the very slow orbit precession rates. The interaction of fast and slow degrees of freedom produces a rich diversity of resonance phenomena. Weak dissipative effects - such as tides or radiation drag forces - also produce unexpectedly rich dynamical behaviors. This paper provides a mostly qualitative discussion of simple dynamical models for the commonly encountered orbital resonance phenomena in the Solar system.

Malhotra, Renu↗

NASA NTRS1994

Pluto's Heliocentric Orbit

We review the current state of knowledge regarding Pluto's orbital motion.

Pluto planetary orbit orbit charon↗

NASA NTRS1993

Orbital Resonances in the Solar Nebula: Strengths and Weaknesses

A planetesimal moving in the Solar Nebula experiences an aerodynamic drag which causes its orbit to circularize and shrink. However, resonant perturbations from a protoplanet interior to the planetesimal's orbit can counteract both the orbital decay and the damping of the eccentricity: the planetesimal can be captured into an orbital resonance and its eccentricity pumped up to a modestly high equilibrium value. Thus, orbital resonances constitute (partial) barriers to the delivery of planetesimals into the feeding zone of the protoplanet. We have established the characteristics of the phenomenon of resonance capture by gas drag in the circular restricted three-body approximation. We have determined the strengths of the equilibrium resonant orbits with respect to impulsive velocity perturbations. We conclude that planetesimals captured in orbital resonances are quite vulnerable to being dislocated from these orbits by mutual planetesimal interactions, but that the resonances are effective in slowing down the rate of orbital decay of planetesimals. Only very small bodies, less than or equal to 100 m, are able to reach a approx. 1 Earth mass protoplanet without being slowed down by resonances.

Malhotra, Renu↗

NASA NTRS1993

Three-body effects in the PSR 1257+12 planetary system

A detailed theoretical analysis of the three-body effects in the putative planetary system of PSR 1257+12 is presented. Ways in which these effects are manifested in the pattern of pulse arrival times are discussed; the dominant perturbation is described as a modulation of the phases of the near-sinusoidal signals of the two planetary companions. Explicit formulas for the time dependence of the oscillating orbital elements that are needed for an improved timing model for this system are provided. If a timing model with fixed, independent Keplerian orbits continues to be used for the timing analysis, and if two planets are indeed orbiting this pulsar, then the three-body effects should become detectable by means of a growth in the postfit residuals as more observations are accumulated. If the typical error in the pulse arrival time measurements is about 10 microsec, the amplitude of the postfit residuals will increase beyond this level with three to five years of timing observations.

Malhotra, Renu↗

NASA NTRS1993

Orbital dynamics of PSR1257+12 and its two planetary companions

I present a review of the orbital dynamics of the three body problem of PSR1257+12 and its two putative planetary companions. The orbital dynamics provides two 'reality checks' on the planetary interpretation of the pulse arrival time residuals observed by Wolszczan and Frail (1992). First, by examining the dynamical stability of the system for the deduced orbital periods, one can place upper limits on the companion masses of about two times the mass of Jupiter. Since the lower limits of the companion masses deduced from the observations are well below this upper limit, the planetary interpretation passes this first 'reality check'. Second, the 3:2 near-commensurability of the orbital periods effects periodic variations of the Keplerian orbital parameters. The amplitude of these variations provides a direct measure of the masses of the companions relative to the mass of the pulsar. The signature of these variations in pulse time-of-arrival data is not straightforward, but, if the planetary interpretation is correct, it should become detectable as an additional quasi-periodic phase residual over a period of a few years.

Malhotra, Renu↗

NASA NTRS1993

On the delivery of planetesimals to a protoplanet in the solar nebula

A planetesimal moving in the solar nebula experiences an aerodynamic drag which causes its orbit to circularize and shrink. However resonant perturbations from a protoplanet interior to the planetesimal's orbit can counteract both the orbital decay and the damping of the eccentricity: the planetesimal can be captured into an orbital resonance and its eccentricity pumped up to an equilibrium value. The resonance trapping phenomenon is sensitive to several factors and we elaborate on these below. Orbital resonances form (partial) barriers to the transport of planetesimals into the feeding zone of the protoplanet. Here we report on a systematic study of the idealized system of a single protoplanet of mass much larger than the masses of the planetesimals, comparing relevant analytic estimates with numerical simulations. Our principal conclusion is that in a gas-rich environment, the bulk of the solid material delivered to the feeding zone of the protoplanet is in the form of bodies of small radius.

Malhotra, Renu↗

NASA NTRS1993

Orbital Resonances in the Solar Nebula: Strengths and Weaknesses

A planetesimal moving in the Solar Nebula experiences an aero- dynamic drag which causes its orbit to circularize and shrink. However, resonant perturbations from a protoplanet interior to the planetesimal's orbit ran counteract both the orbital decay and the damping of the eccentricity: the planetesimal can be captured into an orbital resonance and its eccentricity pumped up to a modestly high equilibrium value. Thus, orbital resonances constitute (partial) barriers to the delivery of planetesimals into the feeding zone of the protoplanet. We have established the characteristics of the phenomenon of resonance capture by gas drag in the circular restricted three-body approximation. We have determined the strengths of the equilibrium resonant orbits with respect to impulsive velocity perturbations. We conclude that planetesimals captured in orbital resonances are quite vulnerable to being dislocated from these orbits by mutual planetesimal interactions, but that the resonances are effective in slowing down the rate of orbital decay of planetesimals. Only very small bodies, less or approx. equal to 100 m, are able to reach a approx. 1 mass of the earth protoplanet without being slowed down by resonances.

Malhotra, Renu↗

NASA NTRS1991

Tidal origin of the Laplace resonance and the resurfacing of Ganymede

In the present scenario for the Laplace resonance's tidal origin, the Laplace relation for the three inner Galilean satellites Io, Europa, and Ganymede may have been established within a previous three-body resonance which excited Ganymede's orbital eccentricity to a high value. The tidal heating enhancement associated with that high-eccentricity period furnishes a plausible explanation for the Ganymede surface, which appears to be geologically younger than that of Callisto.

Malhotra, Renu↗

NASA NTRS1990

Capture probabilities for secondary resonances

A perturbed pendulum model is used to analyze secondary resonances, and it is shown that a self-similarity between secondary and primary resonances exists. Henrard's (1982) theory is used to obtain formulas for the capture probability into secondary resonances. The tidal evolution of Miranda and Umbriel is considered as an example, and significant probabilities of capture into secondary resonances are found.

Malhotra, Renu↗

NASA NTRS1990

The role of secondary resonances in the orbital history of Miranda

A simple 'perturbed pendulum' model is developed in order to ascertain the origin and dynamics of the secondary resonances in the tidal evolution of Miranda and Umbriel through the 1:3 mean motion commensurability. Long-term evolution is addressed on the basis of the computation-accelerating method of algebraic mappings. It is found that the present 4.34-deg value of Miranda's orbital inclination is probably a result of capture into the primary resonance, as well as subsequent capture into the 3/1 secondary resonance. The damping of an eccentricity as high as 0.035 by tidal dissipation in the satellite may account for the surface features of Miranda.

Malhotra, Renu↗

NASA NTRS1988

Dynamics of the Uranian and Saturnian satellite systems - A chaotic route to melting Miranda?

Miranda's anomalously large inclination, in conjunction with the postaccretional resurfacing of both Miranda and Ariel and anomalously large eccentricities characterizing the inner Uranian satellites, are presently held to suggest the disruption of resonant configurations that once existed in this satellite system. Classical analytical methods for the dynamics of resonance are here used to demonstrate how temporary capture into a second- or higher-order resonance can generate large increases in eccentricity and inclination on comparatively short time-scales. Such capture into resonance may result in chaotic motion.

Dermott, Stanley F.↗