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For $m>0$, the three physical polarization vectors are transverse to $k$ and have norm $-1$ in metric $(+---)$. In the rest frame their sum is $\mathrm{diag}(0,1,1,1)$; Lorentz covariance then gives the displayed expression for $k^2=m^2$. This includes the longitudinal mode. It cannot be specialized to $m=0$ by setting its denominator to zero.

For a Schwartz function on the real numbers, its periodization $P_f$ is a smooth periodic function of period of a function one. Rapid decay gives uniform convergence on $[0,1]$ of the defining series and each differentiated series. Its $k$th Fourier coefficient isThe change of variable $u=x+n$ uses $e^{2\pi ikn}=1$. Rapidly decreasing Fourier coefficients give the absolutely and uniformly convergent Fourier series $P_f(x)={\sum }_k\hat{f}(k)e^{2\pi ikx}$. Evaluation at $x=0$ proves the Poisson summation formula.^[ref]

The same identity explains aliasing in the Nyquist–Shannon sampling theorem: applying the Poisson summation formula to the Schwartz function $x\mapsto f(x)e^{-2\pi i\xi x}$ givesThe sampled spectrum is periodic in frequency, and shifted copies of the continuous Fourier transform add together. When those copies overlap, the samples cannot separate their contributions. The Nyquist–Shannon sampling theorem adds a bandlimited function hypothesis to make reconstruction possible.

For a radial Dirichlet solution on a fixed three-dimensional shell $a<r<b$, $(r^2{U}^{\prime }{)}^{\prime }=r^{2}F$. Thus $r^2{U}^{\prime }=C_0+{\int }_a^r{s}^{2}F(s)ds$, and $U(a)=U(b)=0$ bounds $|C_0|$ by the supremum of that integral. Cauchy-Schwarz inequality in the radial volume measure bounds the integral by $C\Vert f{\Vert }_2$. Since $r\ge a>0$, the displayed gradient bound follows. The estimate exploits radiality and does not extend to arbitrary forcing in three dimensions.

The Strehl ratio compares the peak of an observed point spread function with the ideal diffraction-limited system peak for the same pupil and total flux. It measures how strongly wavefront errors redistribute light away from the central peak.

A diffraction-limited system has an optical response close to the ideal diffraction response for its specified optical pupil, wavelength and optical focus. Improving the optical aberrations further makes little difference to that response. This describes the optics: a poorly sampled or noisy detector can still deliver an image with much worse usable diffraction-limited resolution.

The optical pupil determines which wave amplitudes and wave phases contribute to an image point. In a scalar, paraxial approximation at the focal plane of an optical lens in air, write the pupil transmission as $T(u,v)e^{i\varphi (u,v)}$, after removing the ideal focusing phase. The amplitude spread function is proportional to its scaled Fourier transform:Here $(u,v)$ are pupil coordinates, $(x,y)$ are image coordinates, $f$ is focal length, and $\lambda$ is the wavelength in air. For a uniformly transmitting, unobscured circular optical pupil at ideal optical focus, the result is an Airy disk with surrounding rings. A blocked centre, support structures, or nonuniform transmission changes the ideal point spread function; “diffraction limited” does not specify one universal spot shape. The existing Fourier model of a Lyot coronagraph illustrates how pupil and focal masks deliberately alter this response.^[ref]

Optical aberrations add pupil wave phase errors. Defocus contributes a quadratic pupil wave phase in the paraxial approximation, redistributing light intensity from the best-focused spot. The Strehl ratio compares the peak with the ideal response for the same pupil and total radiant flux. A broad optical spectrum produces a weighted sum of wavelength-dependent point spread functions; chromatic aberration can additionally change optical focus with wavelength. Thus the operating conditions belong with any claim that a system is diffraction limited.

For a hyperbolic Möbius transformation represented by $A\in {\mathrm{SL}}_2(\mathbb{R})$, the infimum of $d(z,Az)$ is attained on its axis and equals $2\mathrm{arcosh}(|\mathrm{tr}A|/2)$. It is the length of the corresponding closed geodesic in a quotient.

Tidal heating converts mechanical energy of tidal deformation into internal heat through tidal dissipation. Sustained orbital eccentricity or a spin axis tilted relative to the orbital plane's perpendicular can maintain changing tidal deformation and heat an exoplanet interior. Thus synchronous rotation alone does not eliminate tidal heating.

Tidal locking settles a body's spin into a sustained relation with its orbital motion through tidal forces and tidal dissipation. The simplest case is synchronous rotation, a $1:1$ spin-orbit resonance. On an ideal circular orbit, with spin axis perpendicular to the orbital plane, the body keeps the same face toward its companion, as the Moon approximately does toward Earth. ^[ref].

This article retains the broader sense of tidal locking as capture into a spin-orbit resonance; the term is also commonly used specifically for synchronous rotation. Tidally evolved rotation need not be $1:1$: Mercury has a $3:2$ spin-orbit resonance with the Sun, turning three times in two orbital periods. The distinction matters when inferring illumination or climate from a spin period. ^[ref].

A rogue planet has no bound host star. Without stellar irradiation, its luminosity can still be supplied by internal cooling and Kelvin-Helmholtz contraction; its emitting temperature need not equal the negligible irradiation-based planetary equilibrium temperature.

Differentiating $\mathbf{r}=r\hat{\mathbf{r}}$ twice and using $\dot{\hat{\mathbf{r}}}=\dot{\theta }\hat{\mathbf{\theta }}$ and $\dot{\hat{\mathbf{\theta }}}=-\dot{\theta }\hat{\mathbf{r}}$ gives the displayed acceleration. For a central force, its angular component vanishes, so $r^2\dot{\theta }$ is constant and expresses conservation of angular momentum.

A photometric passband is the wavelength-dependent response used for a brightness measurement. An energy-weighted measurement is $I=\int {\eta }_{\lambda }{F}_{\lambda }d\lambda$, with spectral flux density $F_{\lambda }$. Photon-counting conventions can instead include a factor $\lambda /(hc)$; the calibration must use the same convention as the measurement.

For source counts $S$ and independent background counts $B$ with Poisson distributions, the ideal counting standard deviation is $\sqrt{S+B}$, so the signal-to-noise ratio is $S/\sqrt{S+B}$. Detector noise and uncertainty in the estimated background add further variance. A larger sky aperture raises $B$, while better throughput raises $S$; both affect the faintness limit of a spectrograph.

A charge-coupled device stores photoelectrons in potential wells and transfers the charge through clocked stages to an output amplifier. The transfer differs from independently addressed pixels in a hybrid infrared detector.

In an imaging charge-coupled device, absorbed photons produce mobile charge carriers in a semiconductor; collection stores an electric charge packet associated with each optical detector pixel. Closely spaced MOS capacitors allow applied electric potential differences to move the packet from site to site. Willard Sterling Boyle and George Elwood Smith described storage, transfer and detection as distinct operations, including imaging as one application.^[ref]

The useful architectural trick is that many optical detector pixels can send their electric charge to a shared output electronic amplifier. This makes repeatable conversion possible but makes fidelity depend on every transfer along the route. An active-pixel sensor made with CMOS technology instead uses circuitry at pixels to convert or buffer their signal and select it for readout; independent pixel addressing does not require transporting each packet through the whole image. This compares readout architectures, not whether either technology can use a photodiode.^[ref]

Quantum efficiency is the mean fraction of incident photons producing collected signal carriers. Noise-equivalent quantum efficiency can be lower when amplification adds fluctuations; this does not imply a corresponding loss of actual photon absorption.

Independent photon arrivals have a Poisson distribution: a mean count $\mu$ has variance $\mu$ and standard deviation $\sqrt{\mu }$. Subtraction of independent counting measurements adds their variances.

Read noise is uncertainty added by measuring detector charge or voltage. It is often expressed as an equivalent number of input electrons per pixel per read.

Background subtraction estimates a source contribution by removing a sky or detector-background measurement. An independent background estimate adds photon shot noise; a mismatched background adds bias of an estimator.