An Etymological Dictionary of Astronomy and Astrophysics
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A equation in which the highest → exponent of the → variable is 1. Same as → linear equation.

→ first; → degree; → equation.

A → differential equation containing only the first → derivative. For example, dy/dx = 3x and 2y(dy/dx) + 3x = 5.

→ first; → order; → differential; → equation.

A modified form of → Boltzmann's equation allowing for collision terms in an approximate way. It describes the rate of change of a particle's velocity as a result of small-angle collisional deflections.

After Dutch physicist Adriaan Fokker (1887-1972) and the German physicist Max Planck (1858-1947); → equation.

For an electromagnetic wave incident upon the interface between two media with different indices of refraction, one of a set of equations that give the → reflection coefficient and → transmission coefficient at the optical interface. These coefficients depend on the polarization degree of the incident wave.

→ Fresnel diffraction; → equation.

An equation that expresses energy conservation in an → expanding Universe. It is formally derived from → Einstein's field equations of → general relativity by requiring the Universe to be everywhere → homogeneous and → isotropic. It is expressed by H²(t) = (8πG)/(3c²)ε(t) - (kc²)/R²(t), where H(t) is the → Hubble parameter, G is the → gravitational constant, c is the → speed of light, ε(t) is the → energy density, k is the → curvature of space-time, and R(t) is the → cosmic scale factor. See also → Big Bang, → accelerating Universe. See also → Friedmann-Lemaitre Universe.

Named after the Russian mathematician and physical scientist Aleksandr Aleksandrovich Friedmann (1888-1925), who was the first to formulate an → expanding Universe based on Einstein's theory of → general relativity ; → equation.

An equation that links the pressure and volume of a quantity of gas with the absolute temperature. For a gram-molecule of a perfect gas, PV = RT, where P = pressure, V = volume, T = absolute temperature, and R = the gas constant.

→ gas; → equation.

The main equation of gravitational lens theory that sets a relation between the angular position of the point source and the observable position of its image.

→ gravitational; → lens; → equation.

One of a set of equations that describe the motion of a → dynamical system in terms of the → Hamiltonian function and the → generalized coordinates. For a → holonomic system with n degrees of freedom, Hamilton's equations are expressed by: q^._i = ∂H/∂p_i and p^._i = - ∂H/∂q_i, i = 1, ..., n.

→ Hamiltonian function; → equation.

A → linear differential equation if the right-hand member is zero, Q(x) = 0, on interval I.

→ homogeneous; → linear; → differential; → equation.

Fluid mechanics: A → partial differential equation which describes the motion of an element of fluid subjected to different forces such as pressure, gravity, and frictions.

→ hydrodynamic; → equation.

The equation describing the → hydrostatic equilibrium in a star, expressed as: dP/dr = -GMρ/r², where P and M are the mass and pressure of a spherical shell with thickness dr at some distance r around the center of the star, ρ is the density of the gas, and G the → gravitational constant.

→ hydrostatic; → equation.

In magnetohydrodynamics, an equation that describes the transport of plasma and magnetic field lines over time:
∂B/∂t = ∇ x (v x B) + η∇²B,
where B is the → magnetic induction, v is the plasma velocity, and η = (μσ)^-1 the → magnetic diffusivity. The first term on the right side represents → magnetic advection and the second term → magnetic diffusion. The induction equation can also be expressed as:
∂B/∂t = -(v.∇)B + (B.∇)v - B(∇.v),
where the terms of the right-hand side stand for advection, stretching, and compression, respectively. Among these terms, net increase of the field can be done only through the stretching and compression.

→ induction; → equation.

An equation involving an unknown function that appears as part of an integrand.

→ integral; → equation.

An equation that enables the position of a body in an elliptical orbit to be calculated at any given time from its orbital elements. It relates the → mean anomaly of the body to its → eccentric anomaly.

Keplerian, adj. of → Kepler; → equation.

A set of second order → differential equations for a system of particles which relate the kinetic energy of the system to the → generalized coordinates, the generalized forces, and the time. If the motion of a → holonomic system is described by the generalized coordinates q₁, q₂, ..., q_n and the → generalized velocities  q^.₁, q^.₂, ..., q^._n, the equations of the motion are of the form: d/dt (∂T/∂q^._i) - ∂T/∂q^._i = Q_i (i = 1, 2, ..., n), where T is the kinetic energy of the system and Q_i the generalized force.

→ Lagrangian; → equation.

A second-order nonlinear → differential equation that gives the structure of a → polytrope of index n.

Named after the American astrophysicist Jonathan Homer Lane (1819-1880) and the Swiss astrophysicist Robert Emden (1862-1940); → equation

Equation of motion for a weakly ionized cold plasma.

Paul Langevin (1872-1946), French physicist, who developed the theory of magnetic susceptibility of a paramagnetic gas; → equation.

A → linear differential equation of the second order the solutions of which are important in many fields of science, mainly in electromagnetism, fluid dynamics, and is often used in astronomy. It is expressed by: ∂²V/ ∂x² + ∂²V/ ∂y² + ∂²V/ ∂z² = 0. Laplace's equation can more concisely expressed by: ∇²V = 0. The function V may, for example, be the potential at any point in the electric field where there is no free charge. The general theory of solutions to Laplace's equation is known as potential theory.

→ Laplace; → equation.

The ordinary Newtonian energy conservation equation when expressed in expanding cosmological coordinates. More specifically, it is the relation between the → kinetic energy per unit mass associated with the motion of matter relative to the general → expansion of the Universe and the → gravitational potential energy per unit mass associated with the departure from a homogeneous mass distribution. In other words, it deals with how the energy of the → Universe is partitioned between kinetic and potential energy. Also known as → cosmic energy equation. In its original form, the Layzer-Irvine equation accounts for the evolution of the energy of a system of → non-relativistic particles, interacting only through gravity, until → virial equilibrium is reached. But it has recently been generalized to account for interaction between → dark matter and a homogeneous → dark energy component. Thus, it describes the dynamics of local dark matter perturbations in an otherwise homogeneous and → isotropic Universe (P. P. Avelino and C. F. V. Gomes, 2013, arXiv:1305.6064).

W. M. Irvine, 1961, Ph.D. thesis, Harvard University; D. Layzer, 1963, Astrophys. J. 138, 174; → equation.

The → differential equation of the form: d/dx(1 - x²)dy/dx) + n(n + 1)y = 0. The general solution of the Legendre equation is given by y = c₁P_n(x) + c₂Q_n(x), where P_n(x) are Legendre polynomials and Q_n(x) are called Legendre functions of the second kind.

Named after Adrien-Marie Legendre (1752-1833), a French mathematician who made important contributions to statistics, number theory, abstract algebra, and mathematical analysis; → equation.