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A phase fraction is the proportion of a mixture occupying a specified thermodynamic phase. State the basis: by mass, $x_{\beta }=m_{\beta }/{\sum }_{\alpha }{m}_{\alpha }$; by volume, divide the occupied volume by the total volume. These fractions need not agree when the mass densities differ. The latent heat heating curve uses the liquid fraction by mass, whereas liquid fraction in a mushy layer is defined by volume.

Photodissociation separates a molecule into fragments after photon absorption. For an isolated parent with a single absorbed photon, conservation of energy givesInternal excitation energies here are relative to the parent and product ground states used to define $D_0$; $E_{\mathrm{recoil}}$ is the total product translational kinetic energy in the parent's centre of mass frame. Measuring one product's kinetic energy is insufficient by itself. For two fragments of masses $m_A,m_B$, conservation of momentum gives $E_{\mathrm{recoil}}=E_A(m_A+m_B)/m_B$ when $A$ is detected. Unknown product excitation can conceal part of the available energy.

A modern teaching experiment slows gravity-driven motion enough to resolve it with a photogate. PASCO scientific's Inclined Plane Dynamics Set, ME-6966, Instruction Manual 012-10874A describes “Acceleration Down an Incline” on printed pages 15–16. Its apparatus is a GOcar on a PAStrack with an adjustable incline support, a bracket-mounted photogate, a cart-mounted picket fence, and an interface with recording software. For the interface method, the fence's $1\mathrm{cm}$ pattern is aligned to interrupt the beam. Recording starts before release from rest; the cart is caught at the bottom bumper. The procedure specifies three runs per angle, starting at $14^{\circ }$ and reducing by $2^{\circ }$, with the release position fixed. The readout is velocity against time; its slope estimates acceleration. Mean accelerations are plotted against $\sin \theta$, testing the ideal slope $g$. An added-mass comparison is also requested. These are protocol settings, not observed results.^[ref]

The cart undergoes rolling without slipping at its wheels, whereas the worked block undergoes contact slip. Even ideal wheel moment of inertia changes the cart prediction. For total cart mass $M$, wheel radii $r_i$ and axial moments of inertia $I_i$, rigid wheels with fixed axles and no losses giveTo derive it, combine translational kinetic energy $M{v}^2/2$ with wheel rotational kinetic energy ${\sum }_i{I}_i(v/r_i{)}^2/2$, then equate their rate of change to the release of gravitational potential energy $Mg\sin \theta v$. This is the multiwheel counterpart of rolling acceleration with rotational inertia. Neglecting wheel moment of inertia yields $a\simeq g\sin \theta$; treating a cart as a solid rolling cylinder would give a different correction.

Useful additional controls are repeated releases without a push, checking the fence alignment and spacing, and checking that the recorded velocity trace is straight over the fitted interval. A gently moving cart on a level track can expose losses or unintended tilt. These checks help distinguish an angle or timing error from neglected wheel moment of inertia, bearing friction, or drag force. The three-run spread measures repeatability but cannot by itself reveal a shared calibration error. The inspected protocol makes the gravity component measurable; it does not measure the sliding block's coefficient of kinetic friction.

A planar lamina is an ideal mass distribution confined to a plane, with surface mass density $\sigma$ and infinitesimal mass $dm=\sigma dA$. Its moment of inertia about an axis is the integral of squared perpendicular distance against this mass element. Three mutually perpendicular axes through one point, with two in the lamina's plane, obey the perpendicular axis theorem even when the density is nonuniform.

Planck units use fundamental constants to set natural units of mass, length and time. The displayed convention sets the ordinary Planck mass to one. The related reduced Planck units convention instead normalizes the reduced Planck mass.

A growing solid protoplanet dynamically important compared with the surrounding smaller planetesimals. The term identifies a formation stage, rather than a single sharply specified mass.

The scale $M_{\mathrm{th}}=h^3{M}_{\star }$ makes the Bondi radius comparable with the disk scale height. It organizes when planet-induced disturbances become nonlinear; it is not identical to an exact universal gap-opening mass.

The review led by Clément Baruteau discusses Lindblad and corotation torques, horseshoe drag, saturation and gap formation. Section2.1 covers low-mass planets; section2.2 discusses gap opening and more massive planets.

In units where a circular planet has orbital radius and speed one, the particle's relative speed $u$ outside a close encounter satisfies $u^2=3-T$, to leading order in the planet-to-star mass ratio. This follows from the vis-viva equation and $u^2=v^2+1-2h_z$ at the encounter radius. A short conservative encounter changes the direction of this relative velocity while preserving its magnitude.

For carrier number density $n$, charge $q$ and effective mass $m$, the plasma frequency is ${\omega }_p^2=n{q}^2/({ϵ}_{0}m)$. Combining the second London equation with the magnetostatic Ampère law and Gauss's law for magnetism gives ${\nabla }^2\mathbf{B}=({\omega }_p^2/c^2)\mathbf{B}$. Therefore the decaying planar solution has length ${\lambda }_L=c/{\omega }_p=\sqrt{m/({\mu }_{0}n{q}^2)}$.

A pendulum of length $L$ in vertical gravity $g$ is weakly attracted horizontally by an added long circular cylinder of line density $\lambda$. Outside it, $g_h=2G\lambda /r$ at axis distance $r$. Small-angle equilibrium gives horizontal deflection $d\simeq L{g}_h/g=2GL\lambda /(rg)$. Removing the cylinder's mass reverses the change. It is a before/after differential estimate; finite length, a nonhorizontal radial direction and mechanical motion require corrections.

A point-mass gravitational lens is the ideal gravitational lens obtained by concentrating a positive mass $M$ at one location. Use geometrical optics, the weak-field approximation, small angles in radians, and a stationary lens whose deflecting region is short compared with the observer-lens and lens-source distances. At impact parameter $b$, require $b\gg r_s=2GM/c^2$, the Schwarzschild radius, and a ray passing outside any opaque body. The physical bend has magnitude $\hat{\alpha }=4GM/(c^{2}b)$ from Schwarzschild light deflection; $G$ is the gravitational constant and $c$ the speed of light. Multiple images can occur within this weak-field approximation: strong gravitational lensing does not require a ray to approach a black hole closely.^[ref]

Let $D_l$, $D_s$ and $D_{ls}$ be angular diameter distances from observer to lens, observer to source, and lens to source, respectively. They refer to the background geometry used in the thin gravitational lens equation. In nearby Euclidean geometry, $D_{ls}=D_s-D_l$; in cosmology, do not make that subtraction. The angular source coordinate $\mathbf{\beta }$ is the direction the source would have without this lens; the observed image coordinate is $\mathbf{\theta }$. Both are measured relative to the observer-lens optical axis.^[ref]

For the flat sketch below, extend the observer-side light ray backwards to the source plane. Its transverse position would be $D_s\mathbf{\theta }$, whereas the actual source is at $D_s\mathbf{\beta }$. Their separation is $D_{ls}\hat{\mathbf{\alpha }}$. Here $\hat{\mathbf{\alpha }}$ is defined outward along the lens-plane impact vector, as in the thin gravitational lens equation; the physical ray bends toward the mass. Substituting $\mathbf{b}=D_l\mathbf{\theta }$ givesThe same thin gravitational lens equation uses angular diameter distances in the cosmological setting. The factor $D_{ls}/D_s$ converts the physical bend into the reduced angular deflection; these are different angles.

Deflection changes one light ray's direction; focusing concerns relative directions of neighboring light rays and their bundle's cross section. To see the difference for a point-mass gravitational lens, send parallel incident light rays past the lens. A ray at impact parameter $b$ meets the optical axis approximately $f(b)=b/\hat{\alpha }=c^2{b}^2/(4GM)$ beyond the lens. Doubling $b$ quadruples this crossing distance. Thus the lens does not have a common focal length for all rays, although it can produce lensing magnification and an Einstein ring for a specified source-observer geometry.

The point-mass gravitational lens also illustrates why focusing must not be identified just with positive lensing convergence. Away from the central mass, differentiate the reduced deflection $\mathbf{\alpha }={\theta }_E^2\mathbf{\theta }/r^2$: its two-dimensional divergence is zero, so $\kappa =0$ there. Nevertheless its radial and tangential derivatives differ, giving lensing shear of magnitude ${\theta }_E^2/r^2$ and $\det A=1-{\theta }_E^4/r^4$. A vacuum ray bundle can be distorted and magnified. In the same ideal transparent geometrical optics model, surface brightness is preserved; the larger apparent solid angle supplies the increased received light.

The regular $n=5$ Lane-Emden equation solution has no finite first zero. Its Lane-Emden mass formula gives $m(\xi )=4\pi {\rho }_c{\alpha }^3{\xi }^3/[3(1+{\xi }^2/3{)}^{3/2}]$, so its infinite-radius total mass is $4\pi \sqrt{3}{\rho }_c{\alpha }^3$. Its enclosed mean mass density is ${\rho }_c(1+{\xi }^2/3{)}^{-3/2}$ and tends to zero. A finite externally confined core instead requires a specified truncation pressure.

Potential energy assigns energy to a system's arrangement, so a change of arrangement accounts for the work done by a conservative force. For a time-independent potential energy $V(\mathbf{r})$,The gradient points toward increasing potential energy; the force points toward decreasing potential energy. Only differences matter: adding a constant to $V$ changes neither the force nor its work. This endpoint rule follows by substituting $-\nabla V$ into the line integral for work. ^[ref]

Near Earth's surface, stipulate constant gravitational acceleration $g=9.81m/{\mathrm{s}}^2$ and measure height $h$ upward from a chosen zero. Then gravitational potential energy is $V=mgh$. Raising a $2\mathrm{kg}$ mass by $0.5\mathrm{m}$ increases its gravitational potential energy by $9.81\mathrm{J}$, while gravity does $-9.81\mathrm{J}$ of work. These are computed model values, not measurements of a particular lift. An ideal spring obeying Hooke's law instead has $V=k{x}^2/2$, with displacement $x$ from its relaxed state and spring constant $k$: $k=100N/m$ and $x=0.2\mathrm{m}$ give $V=2\mathrm{J}$ relative to $x=0$. ^[ref]

If otherwise comparable isolated planets of one mass follow $L_{\mathrm{int}}=C{t}^{-\alpha }$, their intrinsic luminosity ratio is $L_{\mathrm{int}}(t_1)/L_{\mathrm{int}}(t_2)=(t_2/t_1{)}^{\alpha }$. The normalization depends on composition and initial entropy. Strong stellar irradiation, deep heating, or accretion can invalidate extrapolating the same cooling law.

Physics of the Earth and Planetary Interiors 25, 297–356 (1981). Joint inversion of travel times, normal modes, mass and inertia for reference Earth structure. doi.org/10.1016/0031-9201(81)90046-7 lweb.cfa.harvard.edu/~lzeng/papers/PREM.pdf

For $p=a_T^2\rho$, spherical hydrostatic equilibrium admits $\rho =a_T^2/(2\pi G{r}^2)$ with $M(r)=2a_T^{2}r/G$. Matching an external pressure $p_0$ fixes $r_s=a_T^2/\sqrt{2\pi G{p}_0}$. The central density is singular although the enclosed mass tends to zero; this is one branch, not every finite-centred isothermal cloud.

Mechanical work transferred by a pressure acting through a displaced volume. In steady incompressible flow, the Bernoulli equation contains $p/\rho$ as flow work per unit mass; it is not an independently conserved internal energy of an isolated parcel.

A primary standard supplies a reliably known reacting inventory from which a titrant concentration can be determined.

The defining advantage is known high purity: the weighed portion can be assigned a known amount of substance. A titrant whose concentration is established against that inventory is a secondary standard solution in IUPAC terminology.^[ref]

An accurate balance determines total mass, including any unwanted water or other material. To infer reacting inventory, one also needs known composition, stability during storage and weighing, complete transfer and dissolution, and a sufficiently complete chemical reaction of known stoichiometry. A finely resolved balance cannot establish these chemical facts. Potassium hydrogen phthalate is useful because a suitably prepared portion provides a known acid inventory for sodium hydroxide titrant standardization; the KHP standardization protocol below shows how that claim is put into practice.

Preparation belongs to the particular material and method. For example, the MIT handout specifies drying potassium hydrogen phthalate at $110{}^{\circ }\mathrm{C}$ for $1.5$ hours or overnight, cooling in a desiccator, and weighing by difference; its footnote identifies removal of superficial moisture as the purpose. This is a condition of that potassium hydrogen phthalate procedure, not a universal instruction to heat every primary standard in the same way.^[ref]