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NASA NTRS1966

Numerical integration of periodic orbits in the elliptic restricted three-body problem

The problem of finding periodic orbits in the circular restricted three-body problem has been very extensively studied in celestial mechanics. It is well known that continuous families of periodic orbits exist, for which the period varies in a continuous way. However, all the applications which are found in the solar system correspond to cases with non-zero eccentricities, and the elliptic restricted three-body problem is thus a better approximation than the circular one. For instance, for the motion of a satellite in the Earth-Moon system, as a first approximation, we may assume that the Moon moves around the Earth in circular motion; but as a much better approximation, we can also assume that the Moon moves in an elliptic orbit around the Earth.

NUMERICAL INTEGRATION↗
NASA NTRS1995

Eclipse timings of the low-mass X-ray binary EXO 0748-676: Statistical arguments against orbital period changes

EXO 0748-676, an eclipsing low-mass X-ray binary, is one of only about four or five low-mass X-ray binaries for which orbital period evolution has been reported. We observed a single eclipse egress with ROSAT . The time of this egress is consistent with the apparent increase in P(sub orb) previously reported on the basis of EXOSAT and Ginga observations. Standard analysis, in which O-C (observed minus calculated) timing residuals are examined for deviations from a constant period, implicitly assume that the only uncertainty in each residual is measurement error and that these errors are independent. We argue that the variable eclipse durations and profiles observed in EXO 0748-676 imply that there is an additional source of uncertainty in timing measurements, that this uncertainty is intrinsic to the binary system, and that it is correlated from observation to observation with a variance which increases as a function of the number of binary cycles between observations. This intrinsic variability gives rise to spurious trends in O-C residuals which are misinterpreted as changes in the orbital period. We describe several statistics tests which can be used to test for the presence of intrinsic variability. We apply those statistical tests which are suitable to the EXO 0748-676 observations. The apparent changes in the orbital period of EXO 0748-676 can be completely accounted for by intrinsic variability with an rms variability of approximately 0.35 s per orbital cycle. The variability appears to be correlated from cycle-to-cycle on timescales of less than 1 yr. We suggest that the intrinsic variability is related to slow changes in either the source's X-ray luminosity or the structure of the companion star's atmosphere. We note that several other X-ray binaries and cataclysmic variables have previously reported orbital period changes which may also be due to intrinsic variability rather than orbital period evolution.

Hertz, Paul↗
NASA NTRS1990

The perturbation of ground tracks of periodic orbits

Some geometric characteristics of the ground tracks of periodic orbits are examined. In particular, two theorems are proved concerning points on the ground tracks that are invariant with respect to two periodic orbits. An application of the results to the Topex/Poseidon mission is presented.

Lo, Martin W.↗
NASA NTRS2021

Periodic Orbits As Viable Landing Solutions with an Abort Option at Europa

Moons of the outer planets offer some of the greatest potential scientific treasures in the solar system, yet they are exceedingly hard to reach. A previous work found many low-energy landing trajectories to various moons in the solar system by studying the unstable manifolds of Lyapunov, vertical, and halo periodic orbits. A potential shortcoming of this approach is the lack of abort options for a landing mission. If anything goes wrong prior to landing, there may not be an solution if the spacecraft has already departed the periodic orbit. Similarly, a solution which could return to its initially planned landing site could also perform observation flybys prior to landing to ensure the safety of the site. In the current work, we discuss a new strategy for finding periodic orbits that tangentially intersect the surface of Europa and are themselves either stable or nearly-stable. These qualities may allow certain periodic orbits to act as their own abort solutions or to closely observe their landing sites prior to landing. 18 such periodic orbits that have been identified so far are presented here.

Martin, W. Lo↗
NASA NTRS1985

Periodic orbits of the general three-body problem for the sun-Jupiter-Saturn system

Two families of symmetric periodic orbits of the planar, general, three-body problem are presented. The masses of the three bodies include ratios equal to the sun-Jupiter-Saturn system and the periods of the orbits of Jupiter and Saturn are in a 2:5 resonance. The (linear) stability of the orbits are studied in relation to eccentricity and mass variations. The generation of the two families of periodic orbits follows a systematic approach and employs (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to the elliptic restricted problem and from the circular and elliptic problems to the general problem through bifurcation phenomena relating the three dynamical systems. The approach also provides insight into the evolutionary process of periodic orbits continued from the restricted problems to the general problem.

Kwok, J. H.↗
NASA NTRS1980

Doubly-periodic orbits in the Sun-Earth-Moon system

A series of periodic orbits in the Earth-Moon circular restricted problem of three bodies was found which is ideally suited for exploring the Earth's geomagnetic tail. The mean apsidal motion of the basic highly elliptical Earth orbit was maintained at about one degree per day by a sequence of lunar swingbys, keeping the apogees in the anti-Sun direction. The orbits were periodic in reference frames rotating at both lunar and solar rates. Apogee distances were alternately raised and lowered by the lunar swingby maneuvers. Several categories of these Sun-synchronous double lunar swingby orbits were identified. The strength and flexibility of this trajectory concept was demonstrated with real world simulations.

Farohar, R.↗
NASA NTRS1990

Orbital periods of recurrent novae

The class of recurrent novae (RN) with thermonuclear runaways contains only three systems (T Pyx, U Sco, and V394 CrA), for which no orbital periods are known. This paper presents a series of photometric observations where the orbital periods for all three systems are discovered. T Pyx is found to have sinusoidal modulation with an amplitude of 0.08 mag and a period of 2.3783 h (with a possible alias of 2.6403 h). U Sco is found to be an eclipsing system with an eclipse amplitude of roughly 1.5 mag and an orbital period of 1.2344 days. V394 CrA is found to have sinusoidal modulation with an amplitude of 0.5 mag and a period of 0.7577 days. Thus two out of three RN with thermonuclear runaways (or five out of six for all RN) have evolved companions.

Schaefer, Bradley E.↗
NASA NTRS1992

Orbital period change of the low-mass X-ray binary EXO 0748-676

The transient low-mass X-ray binary, EXO 0748-676, discovered with EXOSAT, is known to exhibit eclipses of a 492-s duration with a 3.82-hr period, intensity dips at pre-eclipse phases and type-I X-ray bursts. We observed this source with Ginga in 1989 March, 1990 December, 1991 January, and 1991 August and determined nine eclipse center times. Combining these eclipse center times with the previous result of the EXOSAT observations, we find that the orbital period of this source is not decaying monotonically, contrary to the previously reported suggestion. Instead, it shows a more complex behavior. A quadratic fit to the eclipse data yields a positive rate of change in orbital period with an approximate rate of 0.9 x 10 exp 7/yr, although the EXOSAT observations made in 1985 do not fit this trend. A sinusoidal function gives a better fit to the observed orbital period changes with a period of about 12 yr and an amplitude of about 44 lt-s, although the period is much longer than the observation interval of about 6.5 yr. Possible mechanisms for the orbital period change are discussed.

Asai, Kazumi↗
NASA NTRS1997

On the Existence of Low-Luminosity Cataclysmic Variables Beyond the Orbital Period Minimum

Models of the present-day intrinsic population of cataclysmic variables predict that 99 per cent of these systems should be of short orbital period. The Galaxy is old enough that approx. 70 per cent of these stars will have already reached their orbital period minimum (approx. 80 min), and should be evolving back toward longer periods. Mass-transfer rates in these highly evolved binaries are predicted to be less or equal to 10(exp -11), leading to M(sub V) of approx. 10 or fainter, and the secondaries would be degenerate, brown dwarf-like stars. Recent observations of a group of low-luminosity dwarf novae (TOADS) provide observational evidence for systems with very low intrinsic M,. and possibly low-mass secondaries. We carry out population synthesis and evolution calculations for a range of assumed ages of the Galaxy in order to study P(sub orb) and M distributions for comparison with the TOAD observations. We speculate that at least some of the TOADs are the predicted very low- luminosity, post-period-minimum cataclysmic variables containing degenerate (brown dwarf-like) secondaries having masses between 0.02 and 0.06 M, and radii near 0.1 R., We show that these low-luminosity systems are additionally interesting in that they can be used to set a lower limit on the age of the Galaxy. The TOAD with the longest orbital period currently known (123 min), corresponds to a Galaxy age of at least 8.6 x 10(exp 9) yr.

Howell, Steve B.↗
NASA NTRS1985

Evolution of rotationally and tidally distorted low-mass, close binary systems - Implications for the minimum orbital period of cataclysmic variables

A (1 + 0.4) solar mass close binary system consisting of a compact primary and a red dwarf secondary has been evolved numerically. Such a binary system should effectively model cataclysmic variables for which a minimum orbital period cutoff of about 81 minutes has been observed. The influence of gravitational radiation losses which drive Roche lobe overflow has been studied, and the effects of rotational and tidal distortion have also been incorporated in the calculation. The evolution of the He-3 abundance, which has been suggested as a possible explanation for the upper limit of the apparent orbital period gap exhibited by cataclysmic variables, is also considered. It is found that both the distortional effects and the He-3 chemical profile can play an important role in determining the subsequent evolution of these systems. Specifically, when distortion is included, the theoretical minimum orbital period is increased by about 10 percent, yielding better agreement with observations.

Nelson, L. A.↗
NASA NTRS1986

Orbital period changes in massive X-ray binaries

Data from long-term pulse-timing observations of Cen X-3 are compiled and analyzed to determine the evolution of the orbital period, and the implications of the results for other massive X-ray binaries are discussed. Third-body effects, tidal effects, and mass-loss/transfer effects are considered as possible causes for the orbital-period decay observed in Cen X-3, and it is concluded that strong tidal coupling between the companion star and the orbital causes the period decay via orbital instability and/or mass loss. Applying these results to the Roche-lobe-overflow models of Savonije (1983), the evolutionary expansion rate of the Cen X-3 companion and the Roche-lobe shrinking rate are estimated as about 2 x 10 to the -7th/yr and about 1 x 10 to the -6th/yr, respectively, implying an X-ray lifetime of about 5000 yr. Estimates of the companion mass-change rate and period decay for Cen X-3 and four other objects are presented in a table.

Kelley, Richard L.↗
NASA NTRS1981

A four-hour orbital period of the X-ray burster 4U/MXB1636-53

The characterization of X-ray burst sources and other, non-bursting galactic bulge X-ray sources as low-mass close binary systems is generally accepted. The companion stars of the transient burst sources Aql X-1 and Cen X-4 are of spectral types G7-K3V and K3-7V, respectively. If these stars fill their Roche lobe, their orbital periods are in the range of 5-8 hr. X-ray observations offer evidence for a Cen X-4 orbital period of about 8 hr. An analysis of average optical properties yields values for the typical companion star masses and orbital periods of this class of low-mass X-ray binaries of about 0.6 solar masses and 6 hr, respectively.

Pedersen, H.↗