Address to the International Congress of Mathematicians at Cambridge in 1912

Darwin, George Howard; Stratton, F. J. M.; Jackson, John Brinckerhoff

doi:10.1017/cbo9780511703515.008

Cited by 7 publications

(7 citation statements)

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“…where R p is the radius of the planet and q p and q s are the mean densities of the planet and moon respectively; while M P is the mass of the planet and f is a numerical factor which for a rigid spherical satellite is f ¼ 2 1 3 ¼ 1:26 (Scharf 2006); while for a fluid-like body it is given by f = 2.44, (Darwin 1908). A moon moving inside this limit would either collide with the planet or be torn apart very rapidly by the tidal forces exerted by the planet (see Donnison and Williams 1975 for an estimate of the likely dispersal timescale).…”

Section: Stability Limitsmentioning

confidence: 99%

Limits on the Orbits of Possible Eccentric and Inclined Moons of Extrasolar Planets Orbiting Single Stars

Donnison

2014

Earth Moon Planets

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Limits are placed on the range of orbits and masses of possible moons orbiting extrasolar planets which orbit single central stars. The Roche limiting radius determines how close the moon can approach the planet before tidal disruption occurs; while the Hill stability of the star-planet-moon system determines stable orbits of the moon around the planet. Here the full three-body Hill stability is derived for a system with the binary composed of the planet and moon moving on an inclined, elliptical orbit relative the central star. The approximation derived here in Eq. (17) assumes the binary mass is very small compared with the mass of the star and has not previously been applied to this problem and gives the criterion against disruption and component exchange in a closed form. This criterion was applied to transiting extrasolar planetary systems discovered since the last estimation of the critical separations (Donnison in Mon Not R Astron Soc 406:1918, 2010a) for a variety of planet/ moon ratios including binary planets, with the moon moving on a circular orbit. The effects of eccentricity and inclination of the binary on the stability of the orbit of a moon is discussed and applied to the transiting extrasolar planets, assuming the same planet/moon ratios but with the moon moving with a variety of eccentricities and inclinations. For the non-zero values of the eccentricity of the moon, the critical separation distance decreased as the eccentricity increased in value. Similarly the critical separation decreased as the inclination increased. In both cases the changes though very small were significant.

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Section: Stability Limitsmentioning

confidence: 99%

Limits on the Orbits of Possible Eccentric and Inclined Moons of Extrasolar Planets Orbiting Single Stars

Donnison

2014

Earth Moon Planets

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“…A rapid phase of mass loss from the more evolved star may occur, leading to the engulfment of its companion provided that the less-evolved companion rapidly expands to fill its own Roche lobe (see Kippenhahn & Meyer-Hofmeister 1977) in response to mass accretion. An alternative evolutionary channel to the common envelope phase can take place as a result of a tidal instability (Darwin 1908) if there is insufficient orbital angular momentum in the system for the evolved star to be in a state of synchronous rotation with its companion. This occurs when…”

Section: Primary Main-sequence Evolutionmentioning

confidence: 99%

A population synthesis study of high-mass X-ray binaries

Terman¹,

Taam²,

Savage³

1998

Monthly Notices RAS

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A B S T R A C TThe formation of high-mass X-ray binary systems is followed using a statistical Monte Carlo approach. The evolution of an initial distribution of binary systems is followed through mass transfer and mass loss or common envelope evolution to the supernova phase. The population synthesis is explored as a function of the form and magnitude of the kick velocity associated with asymmetric supernova explosions, the specific angular momentum of the matter lost from the system, the mass ratio above which the system enters into the common envelope phase, and the common envelope efficiency. To account for the observed systems with B-type companions in the orbital period-eccentricity plane, the results of the study strongly suggest that asymmetric explosions are required. Furthermore, the existence of systems with Betype companions, in conjunction with the above, indicates that the specific angular momentum of matter lost (relative to that of the binary) during the mass transfer/ loss phase is high (#5).

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“…The researches of George Darwin were the most detailed in the evaluation of specific results. 6 He introduced a principal-axis frame and derived equations of motion for this frame. He was able to show that a movement of the poles of 8 deg would require one half of the Earth's surface to be deflected by nearly 2 miles.…”

Section: Floating Frames For Deformable Bodiesmentioning

confidence: 99%

Floating Reference Frames for Flexible Spacecraft

Canavin

Likins

1977

Journal of Spacecraft and Rockets

125

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Floating reference frames which move with the flexible body under dynamic analysis offer the advantages of a linear vibration analysis in the presence of large sytem rotations. When the deformations of an elastic continuum are expanded in terms of the free-free modes on an unconstrained system, the rigid-body modes are found to be fixed in a reference frame called the Tisserand frame, with respect to which the relative momentum is zero. This result also guarantees the independence of small variations of frame motions and coordinates for all modes with nonzero natural frequencies, a condition which can greatly simplify the formulation of equations of motion. A modified Tisserand constraint is introduced in order to define a floating reference frame with similar properties for an elastic body which contains spinning rotors.

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Address to the International Congress of Mathematicians at Cambridge in 1912

Cited by 7 publications

References 0 publications

Limits on the Orbits of Possible Eccentric and Inclined Moons of Extrasolar Planets Orbiting Single Stars

Limits on the Orbits of Possible Eccentric and Inclined Moons of Extrasolar Planets Orbiting Single Stars

A population synthesis study of high-mass X-ray binaries

Floating Reference Frames for Flexible Spacecraft

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