A stellar population is an ensemble of stars described by its formation times, initial masses, composition and present stellar evolution states. Number fractions must be distinguished from mass-weighted fractions. Shared formation times can correlate the states of components of a binary star even when their initial masses have independence.

Outside a spherical star, its gravitational acceleration is $-G{M}_{\star }\mathbf{r}/r^3$. It provides the dominant central force for Kepler orbits in a star-dominated planetary system.

A statistic is a measurable function of observed data whose definition does not involve an unknown parameter. It may be scalar or vector valued. Sufficient statistics summarize all information about a parameter in the conditional-distribution sense.

A nonstationary spacetime lacks the stationary time-translation symmetry in the domain being considered. Without a preferred suitable timelike Killing vector field, the classical geometry alone does not determine a unique positive-frequency solution subspace for quantization.

For a smooth real phase with an isolated nondegenerate interior critical point $q$, smooth amplitude $a$, and no competing boundary contribution, the leading contribution is $2\pi {\lambda }^{-1}a(q)|\det H{|}^{-1/2}{e}^{i\lambda \varphi (q)+i\pi \mathrm{sgn}(H)/4}$, where $H$ is the Hessian matrix. Orthogonal diagonalization gives two one-dimensional Fresnel integrals. A degenerate critical point or a stationary boundary needs its own local analysis.

In an integral $\int a(q)e^{i\lambda \beta (q)}dq$, the real-valued function $\beta$ is the phase. Its stationary points satisfy $\nabla \beta =0$; its Hessian matrix determines the local quadratic oscillation and the signature factor in stationary phase method. Degenerate stationary points and boundaries require separate analysis.

If the first two phase derivatives vanish at an endpoint but the third does not, an oscillatory contribution is concentrated on a width proportional to $N^{-1/3}$. For a smooth nonzero endpoint oscillatory integral amplitude, rescaling reduces the leading contribution to a cubic oscillatory integral. Its constant follows from the Gamma function Fourier integral. This is a degenerate form of the stationary phase method, and a quadratic formula is insufficient.

For $u,v\ge 0$ with $u+v>0$, multiplication by the sum of the square roots givesThis identity exposes small differences hidden by cancellation. For example, $\sqrt{n^2+n}-n=1/(\sqrt{1+1/n}+1)\to 1/2$. It also converts normalized square-root increments into terms comparable with the harmonic series.

For a spherically symmetric gradient field regular at the origin and satisfying Poisson equation, the divergence theorem on a ball determines its radial magnitude from the enclosed source integral. It depends only on the source at smaller radii. Regularity excludes the source-free but origin-singular addition $Cx/|x{|}^3$; without that condition, solving the local equation on a punctured domain leaves this extra term undetermined.

Here $\mu =\cos \theta$ and $a_{\ell ,m}=\sqrt{({\ell }^2-m^2)/[(2\ell +1)(2\ell -1)]}$. This follows by combining the associated Legendre function recurrence $(2\ell +1)\mu P_{\ell }^m=(\ell -m+1)P_{\ell +1}^m+(\ell +m)P_{\ell -1}^m$ with the normalized spherical harmonic constants. It gives the adjacent-multipole couplings in a photon Boltzmann hierarchy.

A spin-weight-$s$ field on an oriented sphere acquires a phase under rotation of its local tangent-frame basis. Spin-weighted spherical harmonics form the angular basis for such fields. The polarization combinations $Q\pm iU$ carry opposite spin weights of magnitude two; ordinary scalar functions have spin weight zero.

For orthonormal complex spherical harmonics, $P_{\ell }(\hat{\mathbf{n}}\cdot \hat{\mathbf{m}})=4\pi {\sum }_{m=-\ell }^{\ell }{Y}_{\ell m}(\hat{\mathbf{n}})Y_{\ell m}^{\ast }(\hat{\mathbf{m}})/(2\ell +1)$. This converts an angle between two directions into a separable sum. Together with Orthogonality of Legendre polynomials, it projects rotationally invariant scattering kernels onto angular multipoles.

A regular solid harmonic is $r^l{Y}_l^m(\theta ,\varphi )$: it is homogeneous of degree $l$ and solves the Laplace equation. Thus $\mathbf{r}\cdot \nabla F=lF$ and ${\nabla }^{2}F=0$. These identities turn angular spherical harmonic expansions of displacement, density and Newtonian gravitational potential into radial equations.

The integral of three spherical harmonics on the sphere. In the conventional complex basis it is real and equals a product of two Wigner 3j symbols and $\sqrt{(2{\ell }_1+1)(2{\ell }_2+1)(2{\ell }_3+1)/(4\pi )}$. It vanishes unless the multipoles satisfy a triangle condition, ${\ell }_1+{\ell }_2+{\ell }_3$ is even, and $m_1+m_2+m_3=0$. It carries the angular geometry of the reduced CMB bispectrum.

The traceless symmetric part of the galaxy velocity-gradient matrix changes shape without isotropic expansion. An antisymmetric part instead represents rigid rotation. A matrix proportional to the identity plus an antisymmetric matrix has no shear.

With complex inner products linear in the first argument, the adjoint operator of the source-to-far-field operator at fixed frequency isIt is a normalized Herglotz wave function. Complex conjugation reverses the phase sign, while retaining the real factor $-1/(4\pi )$.

The adjoint source-to-far-field operator composed with the source-to-far-field operator at fixed frequency has volume kernel $K(\mathbf{r},{\mathbf{r}}^{\prime })=(4\pi {)}^{-1}\mathrm{sinc}(k|\mathbf{r}-{\mathbf{r}}^{\prime }|)$, where $\mathrm{sinc}(s)=\sin (s)/s$ and $\mathrm{sinc}(0)=1$. This follows from ${\int }_{S_1}{e}^{ik\mathbf{\theta }\cdot \mathbf{d}}dS=4\pi \mathrm{sinc}(k|\mathbf{d}|)$. Its smoothing action helps explain the unstable normal equation for a linear inverse problem.

The mechanically strong outer shell of a rocky planet, including crust and the colder upper mantle. It can form mobile plates in plate tectonics or a relatively stagnant lid. Its temperature, water content and damage affect its ability to yield.

A transient release of stored mechanical energy in the solid planet, often associated with fault motion. Plate tectonics is a major terrestrial driver, but earthquakes can also occur without an active global plate system.

Transport and eruption or intrusion of molten rock and associated gases. Subduction, spreading and mantle upwellings can supply melt; volcanism does not require globally active plate tectonics. Outgassing couples interior evolution to atmospheric inventories.