A toroidal planet is a hypothetical type of telluric exoplanet with a toroidal or doughnut shape. While no firm theoretical understanding as to how toroidal planets could form naturally is necessarily known, the shape itself is potentially quasistable, [1] and is analogous to the physical parameters of a speculatively constructible megastructure in self-suspension, such as a Dyson Ring, ringworld, Stanford torus or Bishop Ring.
At sufficiently large enough scales, rigid matter such as the typical silicate-ferrous composition of rocky planets behaves fluidly, and satisfies the condition for evaluating the mechanics of toroidal self-gravitating fluid bodies in context. [2] A rotating mass in the form of a torus allows an effective balance between the gravitational attraction and the force due to centrifugal acceleration, when the angular momentum is adequately large. Ring-shaped masses without a relatively massive central nuclei in equilibrium have been analyzed in the past by Henri Poincaré (1885), [3] Frank W. Dyson (1892), and Sophie Kowalewsky (1885), wherein a condition is allowable for a toroidal rotating mass to be stable with respect to a displacement leading to another toroid. Dyson (1893) investigated other types of distortions and found that the rotating toroidal mass is secularly stable against "fluted" and "twisted" displacements but can become unstable against beaded displacements in which the torus is thicker in some meridians but thinner in some others. In the simple model of parallel sections, beaded instability commences when the aspect ratio of major to minor radius exceeds 3. [4] [5]
Wong (1974) found that toroidal fluid bodies are stable against axisymmetric perturbations for which the corresponding Maclaurin sequence is unstable, yet in the case of non-axisymmetric perturbation at any point on the sequence is unstable. [6] Prior to this, Chandrasekhar (1965, 1967), and Bardeen (1971), [7] had shown that a Maclaurin spheroid with an eccentricity is unstable against displacements leading to toroidal shapes and that this Newtonian instability is excited by the effects of general relativity. Eriguchi and Sugimoto (1981) improved on this result, and Ansorg, Kleinwachter & Meinel (2003) achieved near-machine accuracy, which allowed them to study bifurcation sequences in detail and correct erroneous results. [8]
While an integral expression for gravitational potential of an idealized homogeneous circular torus composed of infinitely thin rings is available, [9] more precise equations are required to describe the expected inhomogeneities in the mass-distribution per the differentiated composition of a toroidal planet. The rotational energy of a toroidal planet in uniform rotation is where is the angular momentum and the rigid-body moment of inertia about the central symmetry axis. Toroidal planets would experience a tidal force pulling matter in the inner part of toroid toward the opposite rim, consequently flattening the object across the -axis. Tectonic plates drifting hubward would undergo significant contraction, resulting in mountainous convolutions inside the planet's inner region, whereby the elevation of such mountains would be amplified via isostasy due to the reduced gravitational effect in that region.
Since the existence of toroidal planets is strictly hypothetical, no empirical basis for protoplanetary formation has been established. One homolog is a synestia, a loosely connected doughnut-shaped mass of vaporized rock, proposed by Simon J. Lock and Sarah T. Stewart-Mukhopadhyay to have been responsible for the isotopic similarity in composition, particularly the difference in volatiles, of the Earth-Moon system that occurred during the early-stage process of formation, according to the leading giant-impact hypothesis. [10] The computer modelling incorporated a smoothed particle hydrodynamics code for a series of overlapping constant-density spheroids to obtain the result of a transitional region with a corotating inner region connected to a disk-like outer region.
To date, no distinctly torus-shaped planet has ever been observed. Given how improbable their occurrence, it is extremely unlikely any will ever be observationally confirmed to exist even within our cosmological horizon; the corresponding search field being approximately Hubble volumes, or cubic light years. [11]
The following is a timeline of gravitational physics and general relativity.
An equatorial bulge is a difference between the equatorial and polar diameters of a planet, due to the centrifugal force exerted by the rotation about the body's axis. A rotating body tends to form an oblate spheroid rather than a sphere.
Sergei Kopeikin is a USSR-born theoretical physicist and astronomer presently living and working in the United States, where he holds the position of Professor of Physics at the University of Missouri in Columbia, Missouri. He specializes in the theoretical and experimental study of gravity and general relativity. He is also an expert in the field of the astronomical reference frames and time metrology. His general relativistic theory of the Post-Newtonian reference frames which he had worked out along with Victor A. Brumberg, was adopted in 2000 by the resolutions of the International Astronomical Union as a standard for reduction of ground-based astronomical observation. A computer program Tempo2 used to analyze radio observations of pulsars, includes several effects predicted by S. Kopeikin that are important for measuring parameters of the binary pulsars, for testing general relativity, and for detection of gravitational waves of ultra-low frequency. Sergei Kopeikin has worked out a complete post-Newtonian theory of equations of motion of N extended bodies in scalar-tensor theory of gravity with all mass and spin multipole moments of arbitrary order and derived the Lagrangian of the relativistic N-body problem.
The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which makes exact solutions very difficult to find.
A bubble ring, or toroidal bubble, is an underwater vortex ring where an air bubble occupies the core of the vortex, forming a ring shape. The ring of air as well as the nearby water spins poloidally as it travels through the water, much like a flexible bracelet might spin when it is rolled on to a person's arm. The faster the bubble ring spins, the more stable it becomes. The physics of vortex rings are still under active study in fluid dynamics. Devices have been invented which generate bubble vortex rings.
An asymptotically flat spacetime is a Lorentzian manifold in which, roughly speaking, the curvature vanishes at large distances from some region, so that at large distances, the geometry becomes indistinguishable from that of Minkowski spacetime.
The surface gravity, g, of an astronomical object is the gravitational acceleration experienced at its surface at the equator, including the effects of rotation. The surface gravity may be thought of as the acceleration due to gravity experienced by a hypothetical test particle which is very close to the object's surface and which, in order not to disturb the system, has negligible mass. For objects where the surface is deep in the atmosphere and the radius not known, the surface gravity is given at the 1 bar pressure level in the atmosphere.
Tests of general relativity serve to establish observational evidence for the theory of general relativity. The first three tests, proposed by Albert Einstein in 1915, concerned the "anomalous" precession of the perihelion of Mercury, the bending of light in gravitational fields, and the gravitational redshift. The precession of Mercury was already known; experiments showing light bending in accordance with the predictions of general relativity were performed in 1919, with increasingly precise measurements made in subsequent tests; and scientists claimed to have measured the gravitational redshift in 1925, although measurements sensitive enough to actually confirm the theory were not made until 1954. A more accurate program starting in 1959 tested general relativity in the weak gravitational field limit, severely limiting possible deviations from the theory.
Stokes flow, also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. . This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature, this type of flow occurs in the swimming of microorganisms and sperm. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.
A Maclaurin spheroid is an oblate spheroid which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. This spheroid is named after the Scottish mathematician Colin Maclaurin, who formulated it for the shape of Earth in 1742. In fact the figure of the Earth is far less oblate than Maclaurin's formula suggests, since the Earth is not homogeneous, but has a dense iron core. The Maclaurin spheroid is considered to be the simplest model of rotating ellipsoidal figures in hydrostatic equilibrium since it assumes uniform density.
Scalar–tensor–vector gravity (STVG) is a modified theory of gravity developed by John Moffat, a researcher at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario. The theory is also often referred to by the acronym MOG.
In general relativity, post-Newtonian expansions are used for finding an approximate solution of Einstein field equations for the metric tensor. The approximations are expanded in small parameters that express orders of deviations from Newton's law of universal gravitation. This allows approximations to Einstein's equations to be made in the case of weak fields. Higher-order terms can be added to increase accuracy, but for strong fields sometimes it is preferable to solve the complete equations numerically. This method is a common mark of effective field theories. In the limit, when the small parameters are equal to 0, the post-Newtonian expansion reduces to Newton's law of gravity.
The magnetorotational instability (MRI) is a fluid instability that causes an accretion disk orbiting a massive central object to become turbulent. It arises when the angular velocity of a conducting fluid in a magnetic field decreases as the distance from the rotation center increases. It is also known as the Velikhov–Chandrasekhar instability or Balbus–Hawley instability in the literature, not to be confused with the electrothermal Velikhov instability. The MRI is of particular relevance in astrophysics where it is an important part of the dynamics in accretion disks.
Modified Newtonian dynamics (MOND) is a hypothesis that proposes a modification of Newton's law of universal gravitation to account for observed properties of galaxies. It is an alternative to the hypothesis of dark matter in terms of explaining why galaxies do not appear to obey the currently understood laws of physics.
The firehose instability is a dynamical instability of thin or elongated galaxies. The instability causes the galaxy to buckle or bend in a direction perpendicular to its long axis. After the instability has run its course, the galaxy is less elongated than before. Any sufficiently thin stellar system, in which some component of the internal velocity is in the form of random or counter-streaming motions, is subject to the instability.
Gravitoelectromagnetism, abbreviated GEM, refers to a set of formal analogies between the equations for electromagnetism and relativistic gravitation; specifically: between Maxwell's field equations and an approximation, valid under certain conditions, to the Einstein field equations for general relativity. Gravitomagnetism is a widely used term referring specifically to the kinetic effects of gravity, in analogy to the magnetic effects of moving electric charge. The most common version of GEM is valid only far from isolated sources, and for slowly moving test particles.
A Jacobi ellipsoid is a triaxial ellipsoid under hydrostatic equilibrium which arises when a self-gravitating, fluid body of uniform density rotates with a constant angular velocity. It is named after the German mathematician Carl Gustav Jacob Jacobi.
Rossby Wave Instability (RWI) is a concept related to astrophysical accretion discs. In non-self-gravitating discs, for example around newly forming stars, the instability can be triggered by an axisymmetric bump, at some radius , in the disc surface mass-density. It gives rise to exponentially growing non-axisymmetric perturbation in the vicinity of consisting of anticyclonic vortices. These vortices are regions of high pressure and consequently act to trap dust particles which in turn can facilitate planetesimal growth in proto-planetary discs. The Rossby vortices in the discs around stars and black holes may cause the observed quasi-periodic modulations of the disc's thermal emission.
An accretion disk is a structure formed by diffuse material in orbital motion around a massive central body. The central body is most frequently a star. Friction, uneven irradiance, magnetohydrodynamic effects, and other forces induce instabilities causing orbiting material in the disk to spiral inward toward the central body. Gravitational and frictional forces compress and raise the temperature of the material, causing the emission of electromagnetic radiation. The frequency range of that radiation depends on the central object's mass. Accretion disks of young stars and protostars radiate in the infrared; those around neutron stars and black holes in the X-ray part of the spectrum. The study of oscillation modes in accretion disks is referred to as diskoseismology.
A synestia is a hypothesized rapidly spinning doughnut-shaped mass of vaporized rock. It was named by Sarah T. Stewart-Mukhopadhyay, taken from Hestia, goddess of the hearth, combined with syn- meaning together. In computer simulations of giant impacts of rotating objects, a synestia can form if the total angular momentum is greater than the co-rotational limit. Beyond the co-rotational limit, the velocity at the equator of a body would exceed the orbital velocity.
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