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Uniform circular motion of radius $a$ and angular frequency $\omega$ has centripetal acceleration of magnitude $a{\omega }^2$ toward the centre.

A force is an interaction that changes momentum; in Newtonian mechanics it satisfies $F=ma$ for constant mass.

The period is the time required for a periodic motion to complete one full cycle.

Energy is a conserved scalar quantity associated with time-translation symmetry.

For a nonrelativistic point mass of fixed mass $m>0$, measured in an inertial frame, its momentum isMomentum is a vector pointing along the velocity, with International System of Units unit $\mathrm{kg}m/s$. Doubling either mass or velocity doubles momentum. In one dimension, a chosen positive direction makes $p=mv$ a signed quantity: a $2\mathrm{kg}$ body moving right at $3m/s$ has $p=+6\mathrm{kg}m/s$; moving left at the same speed gives $p=-6\mathrm{kg}m/s$. ^[ref]

For a collection, total momentum is the vector sum $\mathbf{P}={\sum }_i{m}_i{\mathbf{v}}_i$. Two equal masses moving with opposite velocities can have zero total momentum while both have positive kinetic energy. Direction is therefore essential when using momentum conservation; adding speeds instead of signed velocities gives the wrong balance.

Newton's second law connects the net force on one fixed-mass point mass to its momentum through ${\mathbf{F}}_{\mathrm{net}}=d\mathbf{p}/dt$. Integrating that change gives impulse. Here $\mathbf{p}$ denotes mechanical momentum; when canonical momentum differs from mass times velocity, the distinction in mechanical momentum below applies. The nonrelativistic formula also has a limited regime: special relativity is needed at speeds comparable with the speed of light.

Position specifies the location of a particle relative to a coordinate system.

A degree of freedom is an independent coordinate needed to specify a mechanical configuration after all mechanical constraints have been imposed.

The Newtonian gravitational field is the gravitational force per unit mass of a test particle. For a point mass $M$ at the origin,The test particle's mass cancels: different small masses have the same gravitational acceleration at the same position. In SI, the Newtonian gravitational field has units $\mathrm{N}\mathrm{k}{\mathrm{g}}^{-1}=\mathrm{m}{\mathrm{s}}^{-2}$. The same exterior expression holds for a spherically symmetric source, by the spherical shell theorem.

Choose a reference radius $r_0>0$, set $g_0=GM/r_0^2$, and use nondimensionalization with $z=r/r_0$. Then $g/g_0=z^{-2}$: at $z=1,2,4$, the magnitudes are $1,1/4,1/16$. This makes the spatial variation visible without specifying a planet. ^[ref]

The order of a unit modulo a product of coprime moduli is the least common multiple of the orders of its components.

An astrophysical black hole is a black hole studied through its formation, accretion, mergers, dynamical environment, and electromagnetic or gravitational-wave signatures.

A galaxy is a gravitationally bound system of stars, interstellar gas and dust, stellar remnants, and dark matter.

An astrophysical disk is a flattened rotating distribution of gas, dust, stars, or other matter whose orbital motion strongly influences its structure and evolution.

Stellar astrophysics studies the structure, formation, energy production, observable properties, and evolution of stars.

Planetary science studies planets, moons, small bodies, planetary systems, and the processes that form and evolve them.

Radiative transfer describes emission, absorption, and scattering of radiation through matter.

An exoplanet is a planet outside the Solar system; here the focus is on planets orbiting stars other than the Sun, including pulsars. Most exoplanets are detected indirectly: a star's changing Doppler shift reveals its motion, or a small dip in its light curve reveals an exoplanet transit. The striking idea is that an unresolved point of light can tell us about an otherwise unseen world's mass, radius, and orbit.

The measurements answer different questions. Exoplanet transit photometry primarily gives a radius ratio, while Doppler spectroscopy constrains mass together with orbital inclination. Combining them with the host star's properties gives a mean mass density, ${\bar{\rho }}_p=3M_p/(4\pi R_p^3)$, useful for exoplanet interior models. Even this combination does not uniquely determine composition: see exoplanet interior-composition degeneracy. Transit spectroscopy and an exoplanet secondary eclipse then probe an exoplanet atmosphere.

The discovery history has two distinct milestones. In 1992 Aleksander Wolszczan and Dale Frail reported two planet-mass companions to the pulsar PSR B1257+12 from periodic changes in pulse arrival times; their paper also suggested an additional companion. These were the first confirmed exoplanet discoveries, around a neutron star. ^[ref]. Wolszczan's contemporary conference account, archived by NASA, likewise distinguishes the two reported companions from a possible third. ^[ref].

In 1995 Michel Mayor and Didier Queloz discovered 51 Pegasi b through Doppler spectroscopy, the landmark first confirmed exoplanet around a Sun-like star. This was a hot Jupiter, not the first exoplanet of every kind: the earlier pulsar discoveries belong to a different host category. ^[ref]. NASA's contemporary account describes the confirmation and the approximately four-day orbital period. ^[ref].

Non-Archimedean analysis studies fields whose distance is induced by an ultrametric absolute value, especially local fields and valued function fields.

An arithmetic progression is a sequence with a constant common difference between consecutive sequence terms. It has the form $a,a+r,\dots ,a+(m-1)r$ and is nonconstant when $r\ne 0$. Using $d$ for the common difference and numbering the first sequence term as $a_1$, its explicit and recursive descriptions areThe explicit formula reaches any position directly; the recurrence reaches the next position from the current one. The factor $k-1$ counts the steps after the first entry. For real $a_1,d$, a positive common difference makes the sequence increase, a negative one makes it decrease, and zero makes it constant.^[ref]

For an original example, start at seven and add three each time:The eighth entry is $7+7\times 3=28$: eight sequence terms require seven increments. This counting principle makes arithmetic progressions useful for regularly spaced positions, repeated equal additions, and iteration over evenly spaced integers. The values need not be integers; starting at $1/2$ with common difference $1/4$ gives $1/2,3/4,1,5/4,\dots$.

A geometric progression instead holds a ratio constant. In the displayed arithmetic progression, $10/7$ and $13/10$ differ even though $10-7=13-10=3$. Multiplying each entry by two would give a different kind of repeated update from adding three. Distinguishing the two rules matters before predicting later entries or adding the sequence terms.

Equal differences also appear in Nicomachus's treatment of arithmetic proportion. He describes successions of at least three numbers whose neighboring differences agree, then explains how regularly omitting counting numbers changes the common gap. This connects the progression to the arithmetic mean: a middle entry is equally far from its two neighbors.^[ref]

Multiplication combines two factors into their product of numbers. For a natural number $n$, $nx$ is the sum of $n$ copies of $x$; with no copies the result is zero. This repeated-addition interpretation directly explains integer grouping, while multiplication of arbitrary real numbers also describes scaling by noninteger factors.

A rectangle of side lengths $3$ and $6$ has area $18$. Split the second side into lengths $2$ and $4$. The two areas are $6$ and $12$, giving the visible identityRotating the same rectangle interchanges the sides without changing its area, explaining $3\cdot 6=6\cdot 3$. These pictures illustrate distributivity and commutativity for ordinary positive numbers; matrix multiplication and other operations require their own rules.

For a fractional example, take $3/2$ of a quantity and then $2/3$ of that result: the combined scale is $(2/3)(3/2)=1$. The two fractions are multiplicative inverses. In the unit square on the right, the overlap of a strip occupying $3/4$ of the width with one occupying $2/3$ of the height contains six of twelve equal cells. Its area is $(3/4)(2/3)=1/2$. The product stays exact even though the decimal expansion of $2/3$ repeats. ^[ref]

Subtraction adds the additive inverse: $a-b=a+(-b)$. On the number line, $2-3=-1$ is a shift three units left from $2$. Reversing the two inputs changes the sign: $3-2=1$, so subtraction is generally not commutative. Solving $x+3=2$ gives the same answer $x=-1$ by adding $-3$ to both sides; the operation preserves equality while undoing the added amount.