Semi-analytic solution of a two-body problem with drag

An approximate semi-analytic solution of a two-body problem with drag is presented. The solution describesnon-lifting orbital motion in a central, inverse-square gravitational field. Drag deceleration is a non-linear function of velocity relative to a rotating atmosphere due to dynamic pressure and velocity-dependent drag coefficient. Neglected are aerodynamic lift, gravitational perturbations of the inverse-square field, and kinematic accelerations due to coordinate frame rotation at earth angular rate. With these simplifications, it is shown that (i) orbital motion occurs in an earth-fixed invariable plane defined by the radius and relative velocity vectors, and (ii) the simplified equations of motion are autonomous and independent of central angle measured in the invariable plane. Consequently, reduction of the differential equations from sixth to second-order is possible. Solutions for the radial and circumferential components of relative velocity are reduced to quadratures with respect to radial distance. Since the independent variable is radial distance, the solutions are singular at zero radial velocity (e. g., for circular orbits). General atmospheric density and drag coefficient models may be used to evaluate the velocity quadratures. The central angle and time variables are recovered from two additional quadratures involving the velocity quadratures. Theoretical results are compared with numerical simulation results.Presently affiliated with AVCO Systems Division, Wilmington, MA 01887, U.S.A.     

An alternative deduction of the hill-type surfaces of the spatial 3-body problem

In the present paper, inequalities stronger than Sundman's and the best possible zero velocity surfaces of the spatial 3-body problem first obtained by Saari (1987) are deduced using a modified version of the transformation developed by Zare (1976). The notion of inertia ellipsoid is used to show the equivalence of the present authors' result to that of Saari's.     

Roche limit for homogeneous incompressible masses

The aim of the present investigation has been to establish the minimum distance (commonly referred to as the ‘Roche limit’), to which a small satellite can approach its central star without the loss of its stability. In order to do so, we shall depart from hydrodynamical equations governing small oscillations of stellar structures, and set out to establish the limit at which their distorted form of equilibrium can no longer vibrate periodically in response to arbitrary perturbations. To this end, such equations will be rewritten in terms of curvilinear Clairaut coordinates (Kopal, 1980) in which the gravitational potential defining equilibrium surfaces plays the role of the radial coordinate; and their solution constructed for the classical Roche problem in which the oscillating satellite of infinitesimal mass consists of material which is homogeneous and incompressible, while its primary component acts gravitationally as a mass-point. The outcome of such a solution agrees satisfactorily with that previously established by Chandrasekhar (1963) on the basis of the virial theorem; but the method employed by us lends itself more readily to a generalization of the Roche limit to systems of finite mass ratios and consisting of the components of finite size.     

Position and velocity perturbations in the orbital frame in terms of classical element perturbations

The transformation of classical orbit element perturbations to perturbations in position and velocity in the radial, transverse and normal directions of the orbital frame is developed. The formulation is given for the case of mean anomaly perturbations as well as for eccentric and true anomaly perturbations. Approximate formulas are also developed for the case of nearly circular orbits and compared with those found in the literature.     

Zero velocity hypersurfaces for the general three-dimensional three-body problem

The equation of zero velocity surfaces for the general three-body problem can be derived from Sundman's inequality. The geometry of those surfaces was studied by Bozis in the planar case and by Marchal and Saari in the three-dimensional case. More recently, Saari, using a geometrical approach, has found an inequality stronger than Sundman's. Using Bozis' algebraic method, and a rotating frame which does not take into account entirely the rotation of the three-body system, we also find an inequality stronger than Sundman's. The comparison with Saari's inequality is more difficult. However, our result can be expressed in four-dimensional space and the regions where motion is allowed can be seen (numerically) to lie inside those obtained by the use of Sundman's inequality.Agrégé de Faculté.     

Radial,transverse and normal satellite position perturbations due to the geopotential

Perturbations in the position of a satellite due to the Earth's gravitational effects are presented. The perturbations are given in the radial, transverse (or alongtrack) and normal (or cross-track) components. The solution is obtained by projecting the Kepler element perturbations obtained by Kaula [Kaula, 1966] into each of the three components. The resulting perturbations are presented in a form analogous to the form of Kaula's solution which facilitates implementation and interpretation.     

Families of periodic orbits in the three-body problem

We show by a general argument that periodic solutions of the planar problem of three bodies (with given masses) form one-parameter families. This result is confirmed by numerical investigations: two orbits found earlier by Standish and Szebehely are shown to belong to continuous one-parameter families of periodic orbits. In general these orbits have a non-zero angular momentum, and the configuration after one period is rotated with respect to the initial configuration. Similar general arguments whow that in the three-dimensional problem, periodic orbits form also one-parameter families; in the one-dimensional problem, periodic orbits are isolated.     

The mapping of Kaula's solution into the orbital reference frame

Kaula's celebrated solution to the problem of satellite motion in the gravitational field of a rigid body is transformed to give the perturbation spectra in both position and velocity in the radial, transverse and normal directions of the orbital reference frame. This work is an extension and a refinement of the theory of orbital perturbations due to the geopotential previously published by Rosborough and Tapley (1987).     

Unified analytical solutions to two-body problems with drag

The two-body problem with a generalized Stokes drag is discussed. The drag force is proportional to the product of the velocity vector and the inverse square of the distance. The generalization consists of allowing two different proportionality constants for the radial and the transverse components of the force. Under the 'generalized Robertson transformation', the equation of the orbit takes the form of the Lommel equation and admits solutions in terms of Bessel and Lommel functions. The exact, analytical solutions for this type of drag reveal a paradoxical effect of increasing eccentricity for all trajectories. The Poynting–Robertson drag and Poynting–Plummer–Danby problems are discussed as particular cases of the general solution.     

On the propagation of magnetic disturbances in the solar wind

Under the geometrical optics approximation we discuss the propagation of a polarized magnetic profile, made up of Alfvén waves, in the solar wind. We show that (i) the profile propagates at an angle to the radial direction (the direction of the solar wind flow), (ii) the radial half-width of the profile stays essentially constant, or even diminishes a little, with distance from the Sun, (iii) the half-width in a direction transverse to the radial direction increases without limit as the magnetic profile moves outward from the Sun. Thus the profile stretches out into a ‘ribbon’ which could, of course, be experimentally identified as a discontinuity. We also give equations for the variation of polarization of the profile, and illustrate the behavior of polarization in a simple case. We have done these calculations to show that the production of ‘discontinuities’ in the solar wind can arise from propagation effects on irregularly shaped ‘blobs’ of magnetic field, as well as from other causes.     

A new method for computing the spectrum of the gravitational perturbations on satellite orbits

Some aspects for efficient computation of the tidal perturbation due to the ellipticity effects of the Earth, the luni-solar potential on an Earth-orbiting satellite and the perturbations of the satellite's radial, transverse and normal position components due to the effects of the Earth's gravitational and ocean tide fields are presented. A straightforward method for computing the spectrum of the geopotential and the tidal-induced perturbations of the orbit elements and the radial, transverse and normal components is described.     

Observations of longperiod mass velocity oscillations in the Sun's chromosphere

Observations made by the differential method in the H line have revealed longperiod (on a timescale of 40 to 80 min) line-of-sight velocity oscillations which increase in amplitude with distance from the centre to the solar limb and, as we believe, give rise to prominence oscillations. As a test, we present some results of simultaneous observations at the photospheric level where such periods are absent.Oscillatory processes in the solar chromosphere have been studied by many authors. Previous efforts in this vein led to the detection of shortperiod oscillations in both the mass velocities and radiation intensity (Deubner, 1981). The oscillation periods obtained do not, normally, exceed 10–20 min (Dubov, 1978). More recently, Merkulenko and Mishina (1985), using filter observations in the H line, found intensity fluctuations with periods not exceeding 78 min. However, the observing technique they used does not exclude the possibility that those fluctuations were due to the influence of the Earth's atmosphere. It is also interesting to note that in spectra obtained by Merkulenko and Mishina (1985), the amplitude of the 3 min oscillations is anomalously small and the 5 min period is altogether absent, while the majority of other papers treating the brightness oscillations in the chromosphere, do not report such periods in the first place. So far, we are not aware of any other evidence concerning the longperiod velocity oscillations in the chromosphere on a timescale of 40–80 min.Longperiod oscillations in prominences (filaments) in the range from 40 to 80 min, as found by Bashkirtsev et al. (1983) and Bashkirtsev and Mashnich (1984, 1985), indicate that such oscillations can exist in both the chromosphere and the corona (Hollweg et al., 1982).In this note we report on experimental evidence for the existence of longperiod oscillations of mass velocity in the solar chromosphere.     

Exosat medium energy observations of Cyg X-2

We report on a series of observations of Cyg X-2 obtained with EXOSAT in September 1983 at five phases in a single orbital cycle (P=9.8 days, see Cowleyet al. 1979). Here we present spectral data obtained with the Argon counters of the Medium Energy experiment (ME) (see Turneret al. 1981), together with the search of Quasi Periodic Oscillations (QPOs) in high time resolution data.Paper presented at the IAU Colloquium No. 93 on Cataclysmic Variables. Recent Multi-Frequency Observations and Theoretical Developments, held at Dr. Remeis-Sternwarte Bamberg, F.R.G., 16–19 June, 1986.     

On Szebehely's equation for the potential of a prescribed family of orbits

In the present note we first give a simple proof of the Dainelli formulas for the force field generating a given family of orbits. We also show that the Szebehely partial differential equation for the potential can be derived from the Dainelli formulas if the energy integral is assumed. The Szebehely equation can be solved directly with the method of characteristics or indirectly with the Joukovsky formulas. Several examples are briefly described in the article. In particular we find some rather general potential functions corresponding to circular motion.     

Certain Comments on V. Szebehely's Paper 'Open Problems on the Eve of the Next Millennium'

An equation for inverse problem considerations, offered by V. Szebehely in the above paper, is amended and its applicability is discussed.     

Properties of the moment of inertia in the problem of three bodies

Sundman's and Birkhoff's results are combined with a recently developed inequality and new qualitative results are given for the problem of three bodies.     

Exponential instability of collision orbit in the anisotropic Kepler problem

The straight-line collision solution in the anisotropic Kepler problem is extended to a periodic solution by means of Sundman's analytic continuation. It is shown that this collision periodic solution is always exponentially unstable.     

A new set of integrals of motion to propagate the perturbed two-body problem

A formulation of the perturbed two-body problem that relies on a new set of orbital elements is presented. The proposed method represents a generalization of the special perturbation method published by Peláez et al. (Celest Mech Dyn Astron 97(2):131–150, 2007) for the case of a perturbing force that is partially or totally derivable from a potential. We accomplish this result by employing a generalized Sundman time transformation in the framework of the projective decomposition, which is a known approach for transforming the two-body problem into a set of linear and regular differential equations of motion. Numerical tests, carried out with examples extensively used in the literature, show the remarkable improvement of the performance of the new method for different kinds of perturbations and eccentricities. In particular, one notable result is that the quadratic dependence of the position error on the time-like argument exhibited by Peláez’s method for near-circular motion under the $J_{2}$ perturbation is transformed into linear. Moreover, the method reveals to be competitive with two very popular element methods derived from the Kustaanheimo-Stiefel and Sperling-Burdet regularizations.     

Three theorems on relative motion

The three theorems treat the analytical aspects of the relative motion of non-interacting particles influenced by arbitrary force fields. As special cases motion in gravitational and central force fields are discussed. The theorems generalize Encke's method, Euler's transformation and Sundman's regularization.