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, and is analogous to the physical parameters of a speculatively constructible megastructure in self-suspension, such as a Dyson sphere, ringworld, Stanford torus, or Bishop Ring.
Physical Description
At sufficiently large scales, rigid matter such as the typical silicate-ferrous composition of rocky planets behaves fluidly, satisfying the conditions for evaluating the mechanics of toroidal self-gravitating fluid bodies. A rotating mass in the form of a torus allows an effective balance between gravitational attraction and centrifugal acceleration when the angular momentum is adequately large.
Ring-shaped masses without a relatively massive central nucleus in equilibrium have been analyzed by Henri Poincaré (1885), 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 others. In the simple model of parallel sections, beaded instability commences when the aspect ratio of major to minor radius exceeds 3.
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. Prior to this, Chandrasekhar (1965, 1967) and Bardeen (1971) had shown that a Maclaurin spheroid with an eccentricity e ≥ 0.98523 is unstable against displacements leading to toroidal shapes and that this Newtonian instability is excited by the effects of general relativity.
Toroidal planets would experience a tidal force pulling matter in the inner part of the toroid toward the opposite rim, consequently flattening the object across the z-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.
Formation
Since the existence of toroidal planets is strictly hypothetical, no empirical basis for protoplanetary formation has been established. One related concept 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 of the Earth-Moon system that occurred during the early-stage process of formation according to the leading giant-impact hypothesis.
Occurrence
To date, no distinctly torus-shaped planet has ever been observed. Given the improbability of their occurrence, it is extremely unlikely that any will ever be observationally confirmed to exist even within our cosmological horizon; the corresponding search field being approximately 140·(c/H₀)³ Hubble volumes, or ~4.211×10³² cubic light years. However, as some have noted, if the odds of a toroidal planet forming are infinitesimally small yet nonzero, in an infinite universe a donut-shaped planet would almost certainly occur infinitely often (per the second Borel–Cantelli lemma).
