An Etymological Dictionary of Astronomy and Astrophysics
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A general equation applicable to a gravitationally → bound system of equal mass objects (stars, galaxies, etc.), which is stable against → dynamical disruption. It states that in such a system the average → gravitational potential energy (W_vir) is twice the average → kinetic energy (K_vir) of the system: W_vir = -2K_vir. This general proposition, first derived by Rudolf Clausius (1822-1888), has important applications in a variety of fields ranging from statistical mechanics to astrophysics. See also → virialization, → virial equilibrium, → virialized.

→ virial; → theorem.

The internal structure and all observable characteristics of a star (such as luminosity and temperature) are determined uniquely by its mass, chemical composition, and age. Same as → Russell-Vogt theorem.

Named after the German astronomer Heinrich Vogt (1890-1968) and the American astronomer Henry Norris Russell (1877-1957); → theorem.

A theorem that establishes a relation between the → radiative flux at some → colatitude on the surface of a → rotating star and the local → effective gravity (which is a function of the → angular velocity and colatitude). For a rotating star in which → centrifugal forces are not negligible, the → equipotentials where gravity, centrifugal force, and pressure are balanced will no longer be spheres. The theorem states that the radiative flux is proportional to the local effective gravity at the considered colatitude, F(θ) ∝ g_eff (θ)^α, where α is the → gravity darkening coefficient. As a consequence, the stellar surface will not be uniformly bright, because there is a much larger flux and a higher → effective temperature at the pole than at the equator (T_eff (θ) ∝ g_eff (θ)^β, where β is the → gravity darkening exponent. In → massive stars this latitudinal dependence of the temperature leads to asymmetric → mass loss and also to enhanced average → mass loss rates. Also called → gravity darkening. See also → von Zeipel paradox; → meridional circulation; → baroclinic instability; → Eddington-Sweet time scale.

Named for Edvard Hugo von Zeipel, Swedish astronomer (1873-1959), who published his work in 1924 (MNRAS 84, 665); → theorem.

If a function φ(x) is continuous on a closed interval [a,b], then for every ε > 0 there exists a polynomial P(x) such that |f(x) - P(x)| <ε, for every x in the interval.

After German mathematician Karl Wilhelm Theodor Weierstrass (1815-1897); → approximation; → theorem.

A theorem used in signal processing whereby the → spectral density of a random signal is the → Fourier transform of the corresponding → autocorrelation function. In other words, the autocorrelation function and the spectral density function constitute a → Fourier transform pair. The Wiener-Khinchin theorem allows one to estimate the spectral density function from the Fourier transform of the autocorrelation function, which is easier to handle. The theorem has an important application particularly in radio astronomy. The two following → Fourier integrals are called the Wiener-Khinchin relations: K(τ) = ∫ J(f)e^-iωτdf and J(f) = ∫ K(τ)e^iωτdτ (both summed over -∞ to +∞), where K(τ) is the autocorrelation function and J(f) is the spectral density.

Named after Norbert Wiener (1894-1964), American mathematician, who first published this theorem in 1930, and Aleksandr Khinchin (1894-1959), Russian mathematician, who did so independently in 1934; → theorem.

In → magnetohydrodynamics, in the limit of zero → resistivity, the → magnetic field  B satisfies the → induction equation  ∂B/∂t = ∇ x (v x B), then for a → plasma confined by a perfectly conducting boundary, the → magnetic helicity is conserved. If the normal field is fixed on the boundary, the minimum-energy state is the linear → force-free magnetic field that conserves the total magnetic helicity.

Named after the Dutch astrophysicist Lodewijk Woltjer (1930-2019), who discovered the phenomenon in 1958 while studying the → Crab Nebula; → theorem.