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 game My Singing Monsters takes place in a toroidal planet named simply The Monster World.
In general relativity, a naked singularity is a hypothetical gravitational singularity without an event horizon.
In fluid mechanics, hydrostatic equilibrium is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary physics of Earth, the pressure-gradient force prevents gravity from collapsing the planetary atmosphere into a thin, dense shell, whereas gravity prevents the pressure-gradient force from diffusing the atmosphere into outer space. In general, it is what causes objects in space to be spherical.
Tidal locking between a pair of co-orbiting astronomical bodies occurs when one of the objects reaches a state where there is no longer any net change in its rotation rate over the course of a complete orbit. In the case where a tidally locked body possesses synchronous rotation, the object takes just as long to rotate around its own axis as it does to revolve around its partner. For example, the same side of the Moon always faces Earth, although there is some variability because the Moon's orbit is not perfectly circular. Usually, only the satellite is tidally locked to the larger body. However, if both the difference in mass between the two bodies and the distance between them are relatively small, each may be tidally locked to the other; this is the case for Pluto and Charon, as well as for Eris and Dysnomia. Alternative names for the tidal locking process are gravitational locking, captured rotation, and spin–orbit locking.
The rotation curve of a disc galaxy is a plot of the orbital speeds of visible stars or gas in that galaxy versus their radial distance from that galaxy's centre. It is typically rendered graphically as a plot, and the data observed from each side of a spiral galaxy are generally asymmetric, so that data from each side are averaged to create the curve. A significant discrepancy exists between the experimental curves observed, and a curve derived by applying gravity theory to the matter observed in a galaxy. Theories involving dark matter are the main postulated solutions to account for the variance.
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.
In physics, specifically classical mechanics, the three-body problem involves taking the initial positions and velocities of three point masses that orbit each other in space and calculating their subsequent trajectories using Newton's laws of motion and Newton's law of universal gravitation.
Gravitational microlensing is an astronomical phenomenon caused by the gravitational lens effect. It can be used to detect objects that range from the mass of a planet to the mass of a star, regardless of the light they emit. Typically, astronomers can only detect bright objects that emit much light (stars) or large objects that block background light. These objects make up only a minor portion of the mass of a galaxy. Microlensing allows the study of objects that emit little or no light.
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.
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.
While the presence of any mass bends the path of light passing near it, this effect rarely produces the giant arcs and multiple images associated with strong gravitational lensing. Most lines of sight in the universe are thoroughly in the weak lensing regime, in which the deflection is impossible to detect in a single background source. However, even in these cases, the presence of the foreground mass can be detected, by way of a systematic alignment of background sources around the lensing mass. Weak gravitational lensing is thus an intrinsically statistical measurement, but it provides a way to measure the masses of astronomical objects without requiring assumptions about their composition or dynamical state.
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.
The Love numbers are dimensionless parameters that measure the rigidity of a planetary body or other gravitating object, and the susceptibility of its shape to change in response to an external tidal potential.
WASP-33b is an extrasolar planet orbiting the star HD 15082. It was the first planet discovered to orbit a Delta Scuti variable star. With a semimajor axis of 0.026 AU and a mass likely greater than Jupiter's, it belongs to the hot Jupiter class of planets.
A binary black hole (BBH), or black hole binary, is a system consisting of two black holes in close orbit around each other. Like black holes themselves, binary black holes are often divided into stellar binary black holes, formed either as remnants of high-mass binary star systems or by dynamic processes and mutual capture; and binary supermassive black holes, believed to be a result of galactic mergers.
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.
A circumstellar disc is a torus, pancake or ring-shaped accretion disk of matter composed of gas, dust, planetesimals, asteroids, or collision fragments in orbit around a star. Around the youngest stars, they are the reservoirs of material out of which planets may form. Around mature stars, they indicate that planetesimal formation has taken place, and around white dwarfs, they indicate that planetary material survived the whole of stellar evolution. Such a disc can manifest itself in various ways.
Chandrasekhar–Friedman–Schutz instability or shortly CFS instability refers to an instability that can occur in rapidly rotating stars with which the instability arises for cases where the gravitational radiation reaction is unable to cope with the change in angular momentum associated with the perturbations. The instability was discovered by Subrahmanyan Chandrasekhar in 1970 and later a simple intuitive explanation for the instability was provided by John L. Friedman and Bernard F. Schutz. Specifically, the instability arises when a non-axisymmetric perturbation mode that appears co-rotating in the inertial frame, is in fact is counter-rotating with respect to the rotating star.
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