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A giant planet with a much larger heavy-element fraction than a typical gas giant, exemplified by Uranus and Neptune. Water, ammonia and other volatile-rich materials are called “ices” in formation terminology; they need not remain ordinary frozen solids in the hot deep interior.

A planet dominated by rocky and metallic material rather than a massive H/He envelope. Earth and Mars are examples. A super-Earth is a mass category and need not be a terrestrial planet.

An Archimedean spiral has radius linear in polar angle, $r=a+b\theta$. For the special parametrization $(u\cos u,u\sin u)$, the curvature of a plane curve is $(u^2+2)/(1+u^2{)}^{3/2}$. Its parameter speed is $\sqrt{1+u^2}$, so it remains a regular curve at the origin.

This irreducible plane branch has parametrization $x=t^5$, $y=t^2$. Its only singular point is the origin. The resolution of the (2,5) cusp by two blowups makes its strict transform smooth in every characteristic.

Equality in the Cauchy-Schwarz and periodic Wirtinger bounds makes the arclength boundary parametrization a first harmonic with orthogonal equal-length coefficient vectors. The boundary is therefore a circle, and the enclosed domain is a disc.

For a fixed positive-area region $E_i$, its least-squares grey value is $c_i=|E_i{|}^{-1}{\int }_{E_i}g$. The minimum fidelity equals $\int g^2-{\sum }_i({\int }_{E_i}g{)}^2/|E_i|$. This eliminates the grey values before optimizing the partition geometry.

When grey values are fixed but region labels vary, a smooth interface satisfies $\beta \kappa =(c_j-g{)}^2-(c_i-g{)}^2$. The curvature uses the normal pointing out of region $i$, positive on an outward-oriented circle. It balances the change in data cost against the first variation of interface length. Fixing a full spatial image signal instead fixes its jumps and leaves no such boundary-relocation freedom.

An optical photodetector detector pixel is a spatially sampled photosensitive element. Counts from distinct detector pixels have independent photon shot noise in the ideal model, but charge transfer, electrical coupling and resampling can produce covariance.

Detector conversion gain is the number of collected electrons per analogue-to-digital unit. With independent Poisson distribution photoelectron counts, the shot-noise variance in count units is the mean count divided by the gain. Detected-photon and electron conventions agree only when each detected photon contributes one electron.

An analogue-to-digital unit is a unit of a digitized photodetector output after electronic conversion. Its relation to collected electrons is set by the detector conversion gain; an additive electronic offset must be removed before interpreting a count as a light signal.

Flat-field correction compensates stable multiplicative variations in photodetector response using calibrated illumination. Identically illuminated exposure ratios cancel the stable response in their mean, but their photon shot noise remains signal dependent.

A photodetector bias frame is an exposure of effectively zero integration time used to estimate the electronic offset and read noise. For independent identical bias reads, the variance of their difference is twice the single-read variance; subtracting an estimated mean bias removes the offset but adds the uncertainty of that estimate.

A direct transition from a solid phase to a gaseous phase. Heated interplanetary dust or a comet can lose material through this process, changing its size and radiation-pressure coefficient or destroying it before gravitational or drag-driven evolution is complete.

The first-order term means ${\sum }_i{p}_i(q_{i+1}-q_i)$ in a finite-dimensional integral. Specify the endpoints and an ordering prescription, then take the maximum time step to zero. Typical Euclidean paths are not classically differentiable, so the discrete prescription is essential.

Integrating each momentum in a quadratic Hamiltonian yields the configuration-space kinetic action and its time-slice normalization. A raw $dp$ measure gives $\sqrt{2\pi m/\Delta t}$; the normalized Fourier measure $dp/(2\pi )$ gives $\sqrt{m/(2\pi \Delta t)}$. Dropping the time dependence changes the quantum-mechanical propagator.

An integral curve of a vector field in phase space, representing the positions and momenta of a moving system. A periodic one-dimensional Hamiltonian orbit encloses the area used to define the action variable.

The symplectic area enclosed by a closed orbit in a two-dimensional phase plane. With the orientation of Hamiltonian motion it is $\oint pdq=2\pi I$, where $I$ is the action variable.

Velocity space is the space of velocity vectors at a fixed spatial position. Integrating a galactic distribution function over velocity space produces mass density; its normalized moments determine mean velocity and velocity dispersion.

The relative-phase drift $\dot{\theta }=\omega -ϵ\sin \theta$ describes competing frequency mismatch and phase coupling. For $0<\omega <ϵ$, the equilibrium points are $\arcsin (\omega /ϵ)$ and $\pi -\arcsin (\omega /ϵ)$, stable and unstable respectively. They merge in a saddle-node bifurcation at $\omega =ϵ$; greater mismatch gives running phase dynamics. Adding Gaussian white noise gives overdamped Langevin dynamics in a tilted washboard potential.

An event changing the unwrapped relative phase by one cycle, usually $2\pi$, compared with a phase-locked state. Thermally activated phase slips correspond to crossing neighboring barriers of a tilted washboard potential. Successive slips can produce long-term drift even while small fluctuations appear confined on shorter times.