Resonant attitude instabilities for a symmetric satellite in a circular orbit

The roll-yaw attitude motion of a spinning symmetric satellite in a circular orbit is investigated with particular emphasis on the behavior near resonance. Resonance in circular orbit occurs if there is a low-order commensurability between the coupled roll-yaw attitude frequencies. For the so-called Delp region where the Hamiltonian describing the linearized attitude oscillations is not positive definite, there can exist, near resonance, a simultaneous growth or decay of the energy of the two normal modes. Two sections of the resonance line ₂=3 ₁ permitting the largest effects are determined and the equations of motion are integrated numerically as a check on the resonance theory. In particular, resonance-induced instabilities are confirmed.     

The motion of a satellite in an axi-symmetric gravitational field

The equations for the variation of the osculating elements of a satellite moving in an axi-symmetric gravitational field are integrated to yield the complete first-order perturbations for the elements of the orbit. The expressions obtained include the effects produced by the second to eighth spherical harmonics. The orbital elements are presented in the most general form of summations by means of Hansen coefficients. Due to their general forms it is a simple matter to estimate the perturbations of any higher harmonic by simply increasing the index of summation. Finally, this paper gives the respective general expressions for the secular perturbations of the orbital elements. The formulae presented should be useful for the reductions of Earth-satellite observations and geopotential studies based on them.List of Symbols semi-major axis - C _jk ⁿ (, ) cosine functions of and - e eccentricity of the orbit - f acceleration vector of perturbing force - f sin²t - i inclination of the orbit - J _n coefficients in the potential expansion - M mean anomaly - n mean motion - p semi-latus rectum of the orbit - R, S, andW components of the perturbing acceleration - r radius-vector of satellite - r magnitude ofr - S _jk ⁿ (, ) sine functions of and - T time of perigee passage - u argument of latitude - U gravitational potential - true anomaly - V perturbing potential - G(M₊+m) (gravitational constant times the sum of the masses of Earth and satellite) - _n,k coefficients ofR component of disturbing acceleration (funtions off) - _n,k coefficients ofS andW components of disturbing acceleration (functions off) - mean anomaly at timet=0 - X ₀ ^n,m zero-order Hansen coefficients - argument of perigee - right ascension of the ascending node     

Non—integrability of Hill's lunar problem

We prove that Hill's lunar problem does not possess a second analytic integral of motion, independent of the Hamiltonian. In order to obtain this result, we avoid the usual normalization in which the angular velocity of the rotating reference frame is put equal to unit. We construct an artificial Hamiltonian that includes an arbitrary parameter b and show that this Hamiltonian does not possess an analytic integral of motion for in an open interval around zero. Then, by selecting suitable values of , b and using the invariance of the Hamiltonian under scaling in the units of length and time, we show that the Hamiltonian of Hill's problem does not possess an integral of motion, analytically continued from the integrable two–body problem in a rotating frame.     

Non-Continuation of Integrals of the Rotating Two-body Problem in Hill's Lunar Problem

We consider Hill's lunar problem as a perturbation of the integrable two-body problem. For this we avoid the usual normalization in which the angular velocity of the rotating frame of reference is put equal to unity and consider as the perturbation parameter. We first express the Hamiltonian H of Hill's lunar problem in the Delaunay variables. More precisely we deduce the expressions of H along the orbits of the two-body problem. Afterwards with the help of the conserved quantities of the planar two-body problem (energy, angular momentum and Laplace–Runge–Lenz vector) we prove that Hill's lunar problem does not possess a second integral of motion, independent of H, in the sense that there exist no analytic continuation of integrals, which are linear functions of in the rotating two-body problem. In connection with the proof of this main result we give a further restrictive statement to the nonintegrability of Hill's lunar problem.     

Concerning isothermal self-similar blast waves

We investigate the one-dimensional self-similar flow behind a blast wave from a plane explosion in a medium whose density varies with distance asx _– with the assumption that the flow is isothermal. If <0 a continuous solution passing through the origin and the shock does not exist. If 1/3>>0 one critical point exists. To be physically acceptable the flow must by-pass this critical point. It is shown that a continuous solution passing through both the origin and through the shock and by-passing the critical point does exist. If 1>>1/3 the first critical point does not exist but a second one appears. To be physically acceptable the flow must again by-pass this new critical point. We show that a continuous solution passing through both the origin and the shock and by-passing the new critical point exists in this case. If >1 no physically acceptable solution exists since the mass behind the shock is infinite.The dependence of the solutions on the parameter is analytic for >0 so that interpolation between neighboring values of is permitted.We investigate the stability of these isothermal blast waves to one-dimensional but non-self-similar perturbations. If 0<<5/7, the solutions are shown to be linearly unstable against short wavelength perturbations near the origin. If the solution crosses the shock with a normalized velocityu>2 the solution is linearly unstable against short wavelength perturbations near the shock for 1>>0. If the solution crosses the shock with normalized velocity 2>u>1 (and it must cross the shock withu>1), the solution is certainly unstable against short wavelength perturbations near the shock for >11/19 and, depending on the crossing velocity, can be unstable there for all .Thus for 1>>0, the solution is always unstable somewhere. Since there is no characteristic time scale in the system all instabilities grow as a power law of time rather than exponentially. The existence of these instabilities implies that initial deviations do not decay and the system does not tend to a self-similar form.     

The distribution law of stellarp-modes

We show that the overall densityg() of asymptotic acoustic frequencies of a star obeys a Weyl lawg() ^D–1, whereD is the dimensionality of the oscillating stellar configuration. For realistic stars with a finite non-zero surface sound speed,D is equal to the actual dimensionality of the star,D=3. For formal models with a vanishing sound velocity at the surface, heuristic arguments lead to a dimensionality parameterD=4.5. The empirical frequencies of Eddington's standard model are found to be consistent with the latter distribution, with reasonable agreement already occurring in the low-frequency range > _i 2× fundamental radial mode. We argue that real stars obey this 3.5-power law in some finite frequency interval _i << _f , _f being a very high frequency critically depending on the surface sound velocity, while the full asymptotic law, withD=3, holds for > _f .     

Pulsar radio emission from expanding charge sheets

We semi-quantitatively calculate the distribution of energy in frequency and angle emitted from a sheet of charges that are moving out relativistically along dipolar magnetic field lines originating near the magnetic polar caps of a rotating neutron star. The angular distribution is conical with the angle of maximum intensity varying with frequency as ^–1/4 for _c 2 _c /(R _M ²), whereR_M is the initial angular radius of the charge sheet at the surface of the star of radiusR. At higher frequencies the width of the angular cone remains constant. The radiation is linearly polarized with the polarization vector in the plane of the line of sight and the magnetic axis. A sheet of uniform charge density and finite thickness has a frequency spectrum that varies from ^–3/2 to ^–4 for _c and _c , respectively. These features are in good general agreement with the observed characteristics of the intensity, pulse shape, and frequency spectrum of the radio pulses from pulsars.Operated by Associated Universities, Inc., under contract with the National Science Foundation.     

The numerical evaluation of theH-functions of radiative transfer in active amplifying media

The present paper contains extensive tables of the values of theH-functionH(z,<0) and of the moments ofQ(x) (in terms of which the moments ofH(z, ) can be determined) appropriate for transfer of radiation in active amplifying media in which<0. These values have been computed correct to the 7th decimal place for values of in the range (–10^–12)-(–10³⁰) and for values of z[0,1] with the aid of a 48-point gaussian quadrature formula.     

Stability and motion in two degree-of-freedom hamiltonian systems for two-to-one commensurability

The two time variable method is used to investigate the stability of and the motion about the equilibrium point of an autonomous Hamiltonian system of two degrees of freedom when the HamiltonianH is indefinite and the relation between the frequencies ₁ and ₂ of the linearized system is ₁ 2₂. Also, the conditions for periodic orbits and the stability of these orbits are obtained.     

Expansion of the perturbing function of a satellite restricted four-body problem

A satellite four-body problem is the problem of motion of an artificial satellite of a planet in a region of the space where perturbations due to the gravitational field of the planet are of the same order as perturbations due to influences of two perturbing bodies. In this paper an expansion of the perturbing function into a Fourier series in terms of angular Keplerian elements ( _j , _j ,M _j :j=0,1,2) (designations are standard) is obtained taking into account a sharp commensurability of the typen/ ₀=(p+q)/p (n is the mean motion of the artificial satellite and ₀ is the angular velocity of rotation of the planet,p andq are integers).The coefficients of the Fourier series are the functions of the positional Keplerian elements (a _j ,e _j ,i _j ;j=0, 1, 2) (designations are standard) and, in particular, are series in terms ofe _j that, generally speaking, can be written out to an accuracy ofe _j ¹⁹ .The expansion obtained can be used for the construction of a semianalytical theory of motion of resonant satellites on the basis of conditionally periodic solutions of the restricted four-body problem.     

Stability of the triangular libration points of the circular restricted problem in the presence of resonances

The stability of triangular libration points, when the bigger primary is a source of radiation and the smaller primary is an oblate spheroid. has been investigated in the resonance cases ₁ = 2₂ and ₁ = 3₂. The motion is unstable for all the values of parameters q and A when ₁ = 2₂ and the motion is unstable and stable depending upon the values of the parameters q and A when ₁ = 3₂. Here q is the radiation parameter and A is the oblateness parameter.     

On slowly-rotating cosmological viscous-fluid model universes in general relativity

The dynamics of a slowly-rotating cosmological viscous-fluid universe is investigated and the rotational perturbations of such models are studied in order to substantiate the possibility that the Universe is endowed with slow rotation, in the course of presentation of some new analytic solutions. Three different cases are taken up in which the nature and role of the metric rotation (r, t) as well as that of the matter rotation (r, t) are discussed. The periods of physical validity of some of the models and the effect of viscosity on the rotational motion are also found out. Rotating models which are expanding as well are obtained, where in all the cases the rotational velocities are found to decay with the time; and these models may be taken as good examples of real astrophysical situations.     

The motion of a satellite in a ring potential

This investigation presents the orbital elements of a satellite moving in a circular ring potential. The ring is considered to be of infinitesimal thickness and of unit radius. The components of the perturbing accelerations due to the ring potential have been substituded into the Gauss form of Lagrange's planetary equations to yield the first-order approximations. The elements of the orbit have been expressed by means of Hansen coefficients. The results include the effects produced by the 2nd, 4th, 6th, and 8th spherical harmonics. Due to their importance we present separately the secular terms from the periodic ones. The general expressions for the orbital elements can be easily extended to include the effects produced by any other higher harmonic.List of Symbols semi-major axis - C _jK ⁿ (u, ) cosine functions ofu and - e eccentricity of the orbit - f sin² - inclination of the orbit - M mean anomaly - n mean motion - p semi-latus rectum of the orbit - R, S, andW components of the perturbing acceleration - r magnitude of position vector - S _jK ⁿ (u, ) sine functions ofu and - T time of periapse passage - u argument of latitude - U gravitational potential - V perturbing potential - G(M _r +m) (gravitational constant times the sum of the masses of ring and satellite) - _{n, k} coefficients ofR component of disturbing acceleration (functions off) - _{n, k} coefficients ofS andW components of disturbing acceleration (functions off) - mean anomaly at timet=0 - X ₀ ^{n, m} zero-order Hansen coefficients - argument of periapse - longitude of the ascending node     

Rotational perturbations of magneto-viscous fluid universes coupled with zero-mass scalar field

The dynamics of slowly rotating magneto-viscous fluid universe coupled with zero-mass scalar field is investigated, and the rotational perturbations of such models are studied in order to substantiate the possibility that the Universe is endowed with slow rotation, in the course of presentation of several new analytic solutions. Four different cases are taken up in which the nature and role of the metric rotation (r, t) as well as that of the matter rotation(r,t) are discussed. Except for the case of perfect drag, the scalar field is found to have a damping effect on the rotational motion. This damping effect is seen to be roughly analogous to the viscosity. The periods of physical validity of some of the models are also found out. Most of the rotating models obtained here come out to be expanding ones as well which may be taken as good examples of real astrophysical situations.     

Models of pulsating low-massive yellow supergiants

The nonlinear self-excited oscillations of the envelopes of low-massive highly luminous stars are described. The parameters for these models wereM=0.8M ,M _bol=–5.5, –5.84 mag,T _eff=4500, 5000, 5500 K. The oscillations have been found to consist of the standing wave pulsation near the envelope bottom and running waves in outer layers. The ratio of the standing wave frequency _s to the average frequency of the running waves _r increases with the stellar luminosity: _s / _r =1.7 whenM _bol=–5.5 mag and _s / _r =2.4 whenM _bol=–5.84 mag. The frequency of oscillations near the photosphere is found to be in close agreement with the critical frequency for running waves. Mass loss from these stars is caused by shocks. It has been shown that agreement between FG Sge's period change observed during the last decade and the period-luminosity relation for double shell stars takes place when FG Sge's luminosity isM _bol=–5.96 mag.     

Conservation laws and Liapounov stability of the free rotation of a rigid body

The paper derives the well known stabilities of free rotation of a rigid body about its principal axes of least and greatest moments of inertia directly from the constancy of the kinetic energy and of the square of the angular momentum. The resulting proof of Liapounov stability yields new quantitative measures of this stability. Involving only simple algebra, it depends on satisfying a weak sufficient condition that insures an unchanging sign of the main component of the angular velocity . The method cannot be used, however, to prove the well known instability of rotation about the intermediate axis.The quantitative results for the radii of the spheres in -space that occur in the Liapounov proof lead to a physical result that may be of interest. If the earth were truly a rigid body, rotating freely, the angular deviation of its instantaneous polar axis from the nearest principal axis could not increase from a given initial value by more than the factor 2.These same quantitative results for the radii of the Liapounov spheres in -space also lead to suflicient conditions for the rotational stability of almost spherical bodies of various shapes, prolate or oblate. They may be pertinent in designing spheres to be used in currently planned experiments to test general relativity by observing the rate of precession of a rotating sphere in orbit about the earth.The above results follow from restricted Liapounov stability alone. The last section contains the proof of general Liapounov stability.This paper was prepared under the sponsorship of the Electronics Research Center of the National Aeronautics and Space Administration through NASA Grant NGR 22-009-262.     

A nonlinear analysis of the moon's physical libration in longitude

The Euler equations for the forced physical librations of the Moon have already been solved by using a digital computer to perform the semi-literal mathematical manipulations. Very near resonance, the computer solution for the physical libration in longitude is complemented by the solution of the appropriate Duffing equation with a dissipation term. Because of its apparent proximity to a resonant frequency, the term whose argument is 2 - twice the mean angular distance of the Moon's perigee from the ascending node of its orbit - is especially important. Its phase, which soon should be measurable, is related to the Moon's anelasticity. The term's frequency, in units of the sidereal month, increases as the semi-major axis of the Moon's orbit about the Earth increases. Using the Moon's mechanical ellipticity of Koziel and the rate of increase of the semi-major axis of MacDonald, it is estimated that the 2 term will cross the resonant frequency in 130 million years and, if the rate of energy dissipation is sufficiently low, a transient libration will be induced.     

Motion near the 3/1 resonance of the planar elliptic restricted three body problem

The global semi-numerical perturbation method proposed by Henrard and Lemaître (1986) for the 2/1 resonance of the planar elliptic restricted three body problem is applied to the 3/1 resonance and is compared with Wisdom's perturbative treatment (1985) of the same problem. It appears that the two methods are comparable in their ability to reproduce the results of numerical integration especially in what concerns the shape and area of chaotic domains. As the global semi-numerical perturbation method is easily adapted to more general types of perturbations, it is hoped that it can serve as the basis for the analysis of more refined models of asteroidal motion. We point out in our analysis that Wisdom's uncertainty zone mechanism for generating chaotic domains (also analysed by Escande 1985 under the name of slow Hamiltonian chaotic layer) is not the only one at work in this problem. The secondary resonance _p = 0 plays also its role which is qualitatively (if not quantitatively) important as it is closely associated with the random jumps between a high eccentricity mode and a low eccentricity mode.     

Equilibrium and stability of rotating masses with a toroidal magnetic field

We investigate the equilibrium, oscillations, and stability of uniformly rotating masses with a toroidal magnetic field, proportional with the distance to te axis of rotation. The equilibrium is an oblate or prolate spheroid according as the rotational energy is greater or smaller than the magnetic energy. The sequence of equilibrium figures exhibits a maximum value for the angular velocity in the oblate case and a maximum for the angular momentum in the prolate case. The dispersion relation is derived using Bryan's modified spheroidal coordinates. One obtains 2(n–m)+4 solutions for the oscillation frequency ifm0 and 1/2n or 1/2(n+1) solutions for ² according asn is even or odd ifm=0. The point where the Jacobi ellipsoids bifurcate from the MacLaurin sequence is unaffected by the magnetic field. However, the points of the onset of dynamical instability corresponding to the second and third harmonics and the point where a pear-shaped sequence bifurcate, depend upon the magnetic field. They are shifted to higher values for the eccentricity and can be suppressed by a sufficiently large magnetic field.     

Wave and stability properties of black holes

In this paper we transform the wave equation governing gravitational perturbations of a Schwarzschild black hole from its standard Schrödinger or Regge-Wheeler form to a Klein-Gordon type wave equation. This latter form reveals immediately that incoming waves with frequencies () _cml , a critical frequency, are completely reflected (transmitted). This process is entirely due to the radial variation of the cut-off frequency inherent in the dispersive nature of the wave propagation properties of gravitational perturbations of the Schwarzschild metric. Moreover, those high-frequency waves ( _cml) which penetrate through the region near the Schwarzschild radiusr _sare, on crossing this event horizon, attenuated by a factor exp (–r _s/c), thereby dumping most of their energy and momentum into the black hole. It is shown that in the vicinity ofr _sthe metric is locally unstable. This feature and the wave absorption process indicate that the neighbourhood aroundr _sis dynamically active, and, as well as acting like a Hawking-type particle creator, will behave as a wave emitter in order to relax the stresses on the metric.