An Etymological Dictionary of Astronomy and Astrophysics
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An equation expressing a relationship between an → independent variable, x, an unknown → function, y = f(x), and its → derivatives. The general form of a differential equation is: F(x, y, y', y'', ..., y⁽ⁿ⁾) = 0, or F(x,y, dy/dx, d²y/dx², ..., dⁿy/dxⁿ) = 0. See also: → ordinary differential equation; → partial differential equation; → linear differential equation; → exact differential equation; → first-order differential equation; → homogeneous linear differential equation; → nonhomogeneous linear differential equation; → differential equation with separated variables; → differential equation with separable variables.

→ differential; → equation.

A → differential equation of the form: M₁(x) N₁(y) dx + M₂(x) N₂(y) dy = 0, which can be reduced to a → differential equation with separated variables.

→ differential; → equation; → separate; → variable.

A → differentail equation that can be transformed into the form: M(x)dx + N(x)dy = 0.

→ differential; → equation; → separate; → variable.

An equation that expresses the time rate of change of a quantity in terms of the product of the diffusion coefficient and the → Laplacian operating on the quantity. For example the diffusion equation for temperature is: ∂T/∂t = D∇²T.

→ diffusion; → equation.

The equation that describes the behavior of an → electron in a way that combines the requirements of → quantum mechanics with the requirements of → special relativity. The Dirac equation predicted the existence of antimatter

→ Dirac; → equation.

An equation representing the variation of → refractive index as a function of → wavelength; for example → Cauchy's equation and → Sellmeier's equation.

→ dispersion; → equation.

A probabilistic argument used to estimate the number of → intelligent, communicating → extraterrestrial civilizations in the → Milky Way galaxy. The Drake equation is:
N = R_* . f_p . n_e . f_l . f_i . f_c . L, where:
N = the number of → civilizations in our Galaxy with which → communication might be possible,
R_* = the average rate of → star formation in our Galaxy,
f_p = the fraction of those stars that have → planets,
n_e = the average number of planets that can potentially support → life per star that has planets,
f_l = the fraction of planets that could develop life at some point,
f_i = the fraction of planets with life that actually go on to develop intelligent life,
f_c = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space,
L = the length of time for which such civilizations release detectable signals into space.
The first three terms of the equation have been successfully investigated by astronomers and are to some extent known. In contrast, values for the last four are very speculative. Drake himself estimates that N might be as high as 10,000. Carl Sagan was more optimistic, and came up with the value of a million or more for N. These estimates may be too optimistic. A pessimistic choice of parameters leads to N smaller than 1, which means that we might be the only technically sophisticated civilization in the Galaxy.

Frank Donald Drake (1930-); → equation.

A system of ten non-linear → partial differential equations in the theory of → general relativity which relate the curvature of → space-time with the distribution of matter-energy. They have the form: G_μν = -κ T_μν, where G_μν is the → Einstein tensor (a function of the → metric tensor), κ is a coupling constant called the → Einstein gravitational constant, and T_μν is the → energy-momentum tensor. The field equations mean that the curvature of space-time is due to the distribution of mass-energy in space. A more general form of the field equations proposed by Einstein is: G_μν + Λg_μν = - κT_μν, where Λ is the → cosmological constant.

Named after Albert Einstein (1879-1955); → field; → equation.

An equation with surprising simplicity that expresses a fundamental result relating several apparently unassociable elements. For example, → Euler's formula for the particular case of θ = π, and the → mass-energy relation.

→ elegant; → equation.

Of a stellar → nuclear fusion, the equation describing the → energy generation rate as a function of → density and → temperature.

→ energy; → generation; → equation.

A statement asserting the equality of two numbers or two expressions. It consists of two parts, called sides or members of the equation, separated by the Same as → equality sign.

From L. æquation- "an equalizing," noun of → equate.

Verbal noun of hamugidan, → equate.

1) Any equation that describes the motion of objects, i.e., variation of velocity, distance covered, acceleration, etc., as a function of time; e.g., V = V₀ + at, S = Vt + (1/2)at².
2) For a fluid, a relation, in its most fundamental form, equating the rate of change of momentum of a selected portion of fluid and the sum of all forces acting on that portion of fluid.
3) In quantum mechanics, an equation that governs the time variation of the → state of the system. → Schrodinger equation. However, in the Heisenberg formulation of quantum mechanics the equation of motion does not involve the states, which in this case is time independent, but rather the → observables of the system.

→ equation; → motion.

In physics and thermodynamics, the equation that describes the relationship between pressure, density, and temperature, e.g. → ideal gas law, → van der Waals equation, → polytropic process, → virial equation of state.

→ equation; → state.

In cosmology, a → dimensionless parameter introduced by the → equation of state representing the ratio of the pressure to the energy density of a fluid, such as the → dark energy: w = p/ρ. The → deceleration or → acceleration of an → expanding Universe depends on this parameter (→ accelerating Universe). A number of numerical values of this parameter are as follows: for the → cosmological constant: w = -1, for → non-relativistic matter (present-day → baryons): w = 0, and for → relativistic matter (photons, neutrinos): w = +1/3. Together with Ω(dark energy) and Ω(matter), w provides a three-parameter description of the dark energy. The simplest parametrization of the dark energy is w = constant, although w might depend on → redshift.

→ equation; → state; → parameter.

The difference between → apparent sidereal time and → mean sidereal time. It is due to the nutation of the Earth's polar axis of rotation about its precessional motion. It ranges from +0.8 to +1.2 seconds. Also known as → nutation in right ascension.

→ equation; → equinox.

The difference, due to Earth's elliptical orbit and variable orbital velocity, between apparent solar time and mean solar time. It varies throughout the year, and slightly from year to year. At present, it reaches extremes of about -14 minutes in February, and about +16 minutes in November. The equation of time is visually illustrated by an → analemma.

→ equation; → time.

In → fluid mechanics, one of a set of → differential equations that govern the motion of a → compressible, → inviscid fluid. Euler equations correspond to the → Navier-Stokes equations with zero → viscosity.

→ Euler; → equation.

A → differential equation composed of → continuous  → differentiable functions for which certain conditions are fulfilled. The equation M(x,y)dx + N(x,y)dy = 0 is called exact if M(x,y) and N(x,y) are continuous differentiable functions for which the following relationship is fulfilled: ∂M/∂y = ∂N/∂x, and ∂M/∂y and ∂N/∂x are continuous in some region.

→ exact; → differential; → equation.

An equation in which unknowns appear as exponents. Examples: 2^{3x + 1} = 32.

→ exponential; → equation.

In a physical theory, an equation that describe how a fundamental force interacts with matter. Einstein's equations of → general relativity are called field equations since they describe the → gravitational field. Similarly, → Maxwell's equations describe the electromagnetic field.

→ field; → equation.