Leon Blitzer

The Gravitational Potential Due to Uniform Disks and Rings

The gravitational potential due to thin uniform disks and rings is obtained in closed form in terms of complete elliptic integrals.

Perturbations of a Satellite's Orbit Due to the Earth's Oblateness

The free‐flight motion of a satellite around a spherical earth is investigated neglecting atmospheric deceleration. This treatment is then extended by recognizing the earths oblateness, which is represented by an additional quadrupole field. For nearly circular orbits, one finds that perturbations of the usual planar motion may be described analytically. These are characterized by a precession of the orbit plane around the equator, that is, a regression of the nodes, amounting to as much as forty miles per revolution. The large magnitude of these effects indicates that one may exploit a satellites motion, as measured by conventional radio techniques, to determine the earths oblateness to a new precision.

Lunar-Solar Perturbations of an Earth Satellite

The influence of the sun and moon on the orbit of a near satellite of the earth is investigated, and it is shown that the principal effect is a precession of the orbit plane about the pole of the ecliptic, analogous to the precession of the equinoxes. The precessional rate increases with orbit size and eccentricity and decreases with orbit inclination to the equator. For orbits close to the earth the lunar-solar precessional motion is only about 10−4 that due to the earths oblateness. Radial perturbations resulting from the attractions of the sun and moon are similarly extremely small, being of the order of one meter. The induced radial oscillations exhibit twice the frequency of the satellites orbital motion around the earth, analogous to the twice daily motion of the tides. To second-order terms in the orbit eccentricity the expressions derived herein are in exact agreement with the astronomical treatments for the special case of the perturbations of the moons orbit due to the sun.

Excitation Temperature in Time-Resolved Spectra of Single Condensed Spark Discharges*

From relative intensities of Fe I lines, a study has been made of the excitation temperatures in short-time high current condensed spark discharges, both in the integrated light from single sparks and from time-resolved spectrograms. In all cases a Boltzmann distribution of the atoms among the various excited energy-states has been found to exist, with the temperature as the distribution parameter. A statistical equilibrium between the electron gas and the excited atoms is presumed to exist, in which case the excitation temperature is the temperature of the electron gas. The spectrum of the initial phase of the spark is characterized by short-lived broadened lines of N II and O II. During the latter phase of the discharge (4–14 microseconds), in spite of extreme variations in current and radiation intensity from the source, the temperature was found to be approximately constant (6000–7000°K) both for damped and oscillatory discharges.

Spin-orbit coupling: A unified theory of orbital and rotational resonances

The dynamics of the spin-orbit interaction of a sphereM8 and a rotating asymmetrical rigid bodyMa are examined. No restrictions are imposed on the masses, on the orientation of the rotation axis to the orbit plane, or on the orbit eccentricity. The zonal potential harmonics ofMa induce a precession of the spin axis as well as a precession of the orbit plane, the net effect being a uniform precession of the node on an invariant plane normal to the constant total angular momentum of the system. In general, the effect of the tesseral harmonics is to induce short-period perturbations of small amplitude in both the orbital and spin motions. Resonances are shown to exist whenever the orbital and rotational periods are commensurable. In any resonant state a single coordinate is found to represent both orbital and spin perturbations; and the system may be described as trapped in a localized potential well. The resultant spin and orbit librations are in phase with a common period. The relative amplitudes of the spin/orbit modes are determined by the characteristic parameter α=MaMsa2/3(Ma+Ms)C, wherea is the semimajor axis of the orbit, andC is the moment of inertia ofMa about the rotation axis. When ga≪1, the solutions reduce to those for pureorbital resonance, in whichMs librates in an appropriate reference frame while the rotation rate of the asymmetrical body remains constant. In the opposite extreme of α≫1, the solutions are appropriate to purerotational resonance, in which the orbital motion is unperturbed but the spin ofMa librates. In each of these special cases the equations developed herein on the basis of a single theory are in agreement with those previously determined from separate theories of spin and orbital resonances.

Maximum Range of a Projectile in Vacuum on a Spherical Earth

Expressions are derived for the maximum range of a projectile in vacuum on a spherical earth in terms of the ratio of kinetic to potential energy at launch. These results are combined with those for the corresponding launch angle and presented graphically. Limiting conditions are discussed, and the effect of the earths rotation is indicated.

Theory of satellite orbit-orbit resonance

On the basis of the strong mathematical and physical parallels between orbit-orbit and spin-orbit resonances, the dynamics of mutual orbit perturbations between two satellites about a massive planet are examined, exploiting an approach previously adopted in the study of spin-orbit coupling. The satellites are assumed to have arbitrary mass ratio and to move in non-intersecting orbits of arbitrary size and eccentricity. Resonances are found to exist when the mean orbital periods are commensurable with respect to some rotating axis, which condition also involves the apsidal and nodal motions of both satellites. In any resonant state the satellites are effectively trapped in separate potential wells, and a single variable is found to describe the simultaneous librations of both satellites. The librations in longitude are 180° out-of-phase, with fixed amplitude ratio that depends only on their relative masses and semimajor axes. At the same time the stroboscopic longitude of conjunction also librates about the commensurate axis with the same period. The theory is applicable to Saturns resonant pairs Titan-Hyperion and Mimas-Tethys, and in these cases our calculated libration periods are in reasonably good agreement with the observed periods.

Satellite Orbit Paradox: A General View

The satellite orbit paradox is examined from the viewpoint of energy considerations, with no restrictions as to the physical character or direction of the perturbing force. Application is made to three diverse phenomena, in which the paradox is the common link: (i) the increase in speed of a satellite subject to atmospheric drag, (ii) the librational motion of a 24-h equatorial satellite, and (iii) the secular increase in the length of the month.

Equilibrium and Stability of a Pendulum in an Orbiting Spaceship.

A simple pendulum attached to a fixed point inside a spaceship moving in a circular orbit about a planet can be in equilibrium in either 2, 3, 4, or 6 positions, of which 1, 1, 2, and 2, respectively, are stable. The number, orientation, and stability of the equilibria depend on the length of the pendulum and the location of the point of attachment relative to the center of gravity of the spacecraft.

Application of spin-orbit theory to a class of orbit-orbit resonances: The case of Titan-Hyperion

The orbit-orbit interaction of two satellites of greatly unequal mass is studied under the condition that the more massive satellite moves in a circular (unperturbed) orbit that lies entirely inside the orbit of the smaller (perturbed) body. It is shown that this system is equivalent in every respect to a special case ofspin-orbit coupling. On this basis, conditions for resonance are derived, as well as libration periods and bandwidths. Application is made to Saturns resonant pair of satellites, Titan and Hyperion, which approximate the conditions of this problem. The calculated libration period of 646 days is in reasonably good agreement with the observed 640-day period.