Brown dwarfBrown dwarfs (also called failed stars) are substellar objects that are not massive enough to sustain nuclear fusion of ordinary hydrogen (1H) into helium in their cores, unlike a main-sequence star. Instead, they have a mass between the most massive gas giant planets and the least massive stars, approximately 13 to 80 times that of Jupiter (). However, they can fuse deuterium (2H) and the most massive ones (> ) can fuse lithium (7Li).
Differential equationIn mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.
Integral equationIn mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: where is an integral operator acting on u. Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals.
Ordinary differential equationIn mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equations which may be with respect to one independent variable. A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a_0(x), .
Linear differential equationIn mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where a0(x), ..., an(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y′, ..., y(n) are the successive derivatives of an unknown function y of the variable x. Such an equation is an ordinary differential equation (ODE).
Partial differential equationIn mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similar to how x is thought of as an unknown number to be solved for in an algebraic equation like x2 − 3x + 2 = 0. However, it is usually impossible to write down explicit formulas for solutions of partial differential equations.
Red dwarfA red dwarf is the smallest and coolest kind of star on the main sequence. Red dwarfs are by far the most common type of star in the Milky Way, at least in the neighborhood of the Sun. However, as a result of their low luminosity, individual red dwarfs cannot be easily observed. From Earth, not one star that fits the stricter definitions of a red dwarf is visible to the naked eye. Proxima Centauri, the nearest star to the Sun, is a red dwarf, as are fifty of the sixty nearest stars.
MetallicityIn astronomy, metallicity is the abundance of elements present in an object that are heavier than hydrogen and helium. Most of the normal currently detectable (i.e. non-dark) matter in the universe is either hydrogen or helium, and astronomers use the word "metals" as a convenient short term for "all elements except hydrogen and helium". This word-use is distinct from the conventional chemical or physical definition of a metal as an electrically conducting solid.
G-type main-sequence starA G-type main-sequence star (spectral type: G-V), also often, and imprecisely called a yellow dwarf, or G star, is a main-sequence star (luminosity class V) of spectral type G. Such a star has about 0.9 to 1.1 solar masses and an effective temperature between about 5,300 and 6,000 K. Like other main-sequence stars, a G-type main-sequence star converts the element hydrogen to helium in its core by means of nuclear fusion, but can also fuse helium when hydrogen runs out.
Integro-differential equationIn mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function. The general first-order, linear (only with respect to the term involving derivative) integro-differential equation is of the form As is typical with differential equations, obtaining a closed-form solution can often be difficult. In the relatively few cases where a solution can be found, it is often by some kind of integral transform, where the problem is first transformed into an algebraic setting.
Stochastic differential equationA stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs have many applications throughout pure mathematics and are used to model various behaviours of stochastic models such as stock prices, random growth models or physical systems that are subjected to thermal fluctuations. SDEs have a random differential that is in the most basic case random white noise calculated as the derivative of a Brownian motion or more generally a semimartingale.
White dwarfA white dwarf is a stellar core remnant composed mostly of electron-degenerate matter. A white dwarf is very dense: its mass is comparable to the Sun's, while its volume is comparable to the Earth's. A white dwarf's low luminosity comes from the emission of residual thermal energy; no fusion takes place in a white dwarf. The nearest known white dwarf is Sirius B, at 8.6 light years, the smaller component of the Sirius binary star. There are currently thought to be eight white dwarfs among the hundred star systems nearest the Sun.
Error functionIn mathematics, the error function (also called the Gauss error function), often denoted by erf, is a complex function of a complex variable defined as: Some authors define without the factor of . This nonelementary integral is a sigmoid function that occurs often in probability, statistics, and partial differential equations. In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real.
Function (mathematics)In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity).
First-order partial differential equationIn mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of n variables. The equation takes the form Such equations arise in the construction of characteristic surfaces for hyperbolic partial differential equations, in the calculus of variations, in some geometrical problems, and in simple models for gas dynamics whose solution involves the method of characteristics.
Gamma functionIn mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n, Derived by Daniel Bernoulli, for complex numbers with a positive real part, the gamma function is defined via a convergent improper integral: The gamma function then is defined as the analytic continuation of this integral function to a meromorphic function that is holomorphic in the whole complex plane except zero and the negative integers, where the function has simple poles.
Stellar massStellar mass is a phrase that is used by astronomers to describe the mass of a star. It is usually enumerated in terms of the Sun's mass as a proportion of a solar mass (). Hence, the bright star Sirius has around . A star's mass will vary over its lifetime as mass is lost with the stellar wind or ejected via pulsational behavior, or if additional mass is accreted, such as from a companion star. Stars are sometimes grouped by mass based upon their evolutionary behavior as they approach the end of their nuclear fusion lifetimes.
Planetary systemA planetary system is a set of gravitationally bound non-stellar objects in or out of orbit around a star or star system. Generally speaking, systems with one or more planets constitute a planetary system, although such systems may also consist of bodies such as dwarf planets, asteroids, natural satellites, meteoroids, comets, planetesimals and circumstellar disks. The Sun together with the planetary system revolving around it, including Earth, forms the Solar System.
F-type main-sequence starAn F-type main-sequence star (F V) is a main-sequence, hydrogen-fusing star of spectral type F and luminosity class V. These stars have from 1.0 to 1.4 times the mass of the Sun and surface temperatures between 6,000 and 7,600 K.Tables VII and VIII. This temperature range gives the F-type stars a whitish hue when observed by the atmosphere. Because a main-sequence star is referred to as a dwarf star, this class of star may also be termed a yellow-white dwarf (not to be confused with white dwarfs, remnant stars that are a possible final stage of stellar evolution).
Stellar kinematicsIn astronomy, stellar kinematics is the observational study or measurement of the kinematics or motions of stars through space. Stellar kinematics encompasses the measurement of stellar velocities in the Milky Way and its satellites as well as the internal kinematics of more distant galaxies. Measurement of the kinematics of stars in different subcomponents of the Milky Way including the thin disk, the thick disk, the bulge, and the stellar halo provides important information about the formation and evolutionary history of our Galaxy.
