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For two approximately spherical stars approaching with speed $v_{\infty }$, contact at separation $R_1+R_2$ has gravitationally focused cross section $\pi (R_1+R_2{)}^2[1+2G(m_1+m_2)/((R_1+R_2)v_{\infty }^2)]$. This measures direct stellar encounters, not the probability of a collective encounter between their host galaxies.

A passing star perturbs a planetary or cometary system through its differential Newtonian gravitational field. A distant encounter is predominantly tidal; a sufficiently close one can substantially change specific orbital energy and angular momentum, eject a comet, or shift its periapsis into or out of the planetary region.

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.

Stellar intensity interferometry measures a star's angular diameter by comparing fluctuations of detected light intensity at separated collectors. Each collector has its own photodetector; an analog multiplier or a digital correlator compares the electrical outputs. Changing the interferometric baseline changes the sampled spatial optical coherence. An unresolved star produces correlated fluctuations; a resolved extended source supplies different mixtures of emission directions, reducing their correlation.

This makes a useful telescope out of a surprising observable: the fluctuations discarded by an ordinary average-brightness measurement. Optical beams need not travel to a common combiner. A relative wave phase perturbation $E_2\mapsto e^{i\varphi }{E}_2$ leaves $|E_2{|}^2$ unchanged, so pure phase errors do not directly change the local light intensity. Electronic delay alignment remains necessary; intensity changes from atmospheric scintillation, collection losses, and instrumental noise still require controls.

Let $\mathbf{b}$ join the collectors and let ${\mathbf{s}}_0$ be the unit line of sight toward the star. The relevant interferometric baseline is its projection onto the plane of the sky:The reference arrival-time difference has magnitude $|\mathbf{b}\cdot {\mathbf{s}}_0|/c$, with speed of light $c$. Correct that delay, with a sign fixed by channel order, before averaging the residual correlation at $\tau =0$. The physical collector separation equals $B$ only when it is perpendicular to ${\mathbf{s}}_0$; tracking a star can change the projection even for fixed collectors.

The bolometric luminosity of a star is its total radiated power integrated over all wavelengths. A band or spectral luminosity instead specifies a restricted spectral range. The Hertzsprung-Russell diagram compares total luminosity with effective temperature.

The useful idea is to weigh an unseen object through its effect on a visible tracer. A curved stellar orbit carries more information than a single fast-moving star: its changing position and velocity constrain a common centre, a length scale and a clock. The Galactic-centre stellar-orbit experiments combine astrometry with Doppler spectroscopy; the Galactic-centre point-mass orbit model below shows how the two measurements complement one another.

Stellar parallax is the apparent change in a star's viewing direction caused by the observer's changing position. Nearby stars shift more against a distant reference background. In an ideal circular Earth orbit, the conventional annual stellar parallax $p$ uses a baseline of one astronomical unit, approximately the Earth-Sun distance. Two opposite observing positions, about six months apart, are separated by approximately two astronomical units. Their maximum angular separation is $2p$, so $p$ is half that excursion, not the full January-to-July shift. The observed excursion depends on the projected baseline and observing dates. ^[ref].

For the perpendicular baseline in the diagram, a right triangle giveswhere $p$ in the final expression is in radians. Combining the small-angle approximation with the parsec convention givesThus $p=0.05$ arcseconds implies $d\simeq 20$ parsecs, while the ideal six-month excursion is $0.10$ arcseconds. Geometry supplies this distance without assuming a luminosity or a particular stellar atmosphere.

In practical astrometry, many observations separate the annual stellar parallax signal from proper motion, the star's own changing sky position. The Gaia DR3 catalogue reports stellar parallax and its uncertainty in milliarcseconds: $50\mathrm{mas}=0.05$ arcseconds. A catalogue angle is a fitted measurement with measurement error. ^[ref].

For a positive stellar parallax with small standard deviation ${\sigma }_p$, first-order propagation gives ${\sigma }_d/d\simeq {\sigma }_p/p$. Thus $p=50\mathrm{mas}$ with ${\sigma }_p=1\mathrm{mas}$ gives $d\simeq 20$ parsecs with ${\sigma }_d\simeq 0.4$ parsecs, assuming systematic offsets are negligible. This linear estimate requires ${\sigma }_p\ll p$; a reciprocal is nonlinear and does not generally preserve symmetric uncertainty intervals.

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.

Half the peak-to-peak line-of-sight orbital velocity range of a star in a single-companion Kepler orbit. It equals $K=2\pi a_{\star }\sin i/[P\sqrt{1-e^2}]$, where $a_{\star }$ is the stellar barycentric semi-major axis, $P$ the orbital period, $i$ the sky-plane orbital inclination, and $e$ the orbital eccentricity. The Doppler spectroscopy formula follows from Kepler's third law for the relative orbit, $a^3=G(M_{\star }+M_p)P^2/(4{\pi }^2)$.

The physical radius assigned to a star, with a specified surface convention.

The motion of a star around the astronomical barycenter caused by an orbiting companion. In an isolated two-body problem with relative semi-major axis $a$, the stellar semi-major axis is $a_{\star }=M_{p}a/(M_{\star }+M_p)$. Doppler spectroscopy measures its line-of-sight velocity and astrometry measures its projected position on the sky.

The stellar relaxation time is the time on which random gravitational encounters change a star's velocity by an amount comparable to its characteristic random velocity. A weak straight-line encounter at relative speed $u$ and impact parameter $b$ gives a kick $2Gm/(bu)$. Three-dimensional encounter counting gives $d⟨|\Delta \mathbf{v}{|}^2⟩/dt=8\pi G^2{m}^{2}n\log (b_{\max }/b_{\min })/u$. Definitions of relaxation, velocity averaging and system structure alter order-one coefficients; the robust spherical scaling is $t_{\mathrm{rel}}\sim v^3/(G^2{m}^{2}n\log \Lambda )\sim (N/\log N)t_{\mathrm{dyn}}$. Collisionless simulation requirements depend on the duration and permitted fractional relaxation, not a universal minimum particle number.

A stellar remnant is the object left after a star finishes its principal nuclear-burning evolution, such as a white dwarf, neutron star or black hole. Stellar remnants retain mass while often contributing little optical light, affecting an old stellar population's mass-to-light ratio.

The distribution of radiation from a star with wavelength or frequency, including its continuum and absorption lines. Continuum colors and temperature-dependent line strengths provide complementary estimates of effective temperature. Spectra of an unresolved binary star may need to be separated by eclipse observations or their different velocity shifts before the component effective temperatures can be inferred.

A stellar stream is a correlated collection of stars commonly originating in the disruption of a bound progenitor. Phase mixing can erase spatial contrast while leaving coherence in orbital integrals. Such fossil debris constrains a galaxy's past accretion, although not every stream is from a galaxy: star clusters also produce streams.

A highly luminous evolved star, associated with luminosity class I. Its location in a Hertzsprung-Russell diagram is distinct from the low-luminosity, compact white dwarf region.

Synchronous rotation means one prograde turn about the body's own axis per orbital period, measured against fixed directions in an inertial frame. This $1:1$ spin-orbit resonance approximately describes the Moon's synchronous rotation toward Earth. ^[ref].

The striking geometric idea is that one turn in space can keep one face toward the companion: a fixed orientation in the frame turning with the orbit is a changing orientation in an inertial frame. This distinction helps us predict illumination: a synchronously rotating exoplanet orbiting its illuminating star can have persistent day and night hemispheres, whereas the Moon, synchronized to Earth, receives changing illumination from a different body, the Sun.

The termination of the first-ascent red giant luminosity function near helium ignition provides a calibrated population-II distance indicator. In suitable passbands the tip luminosity varies relatively weakly over a useful old, metal-poor population range. Colour/composition corrections, foreground extinction, photometric completeness and contamination by other evolved stars must be controlled; an arbitrary brightest red star is not a reliable tip measurement.

Constant extra light makes an exoplanet transit appear shallower. If the host star supplies out-of-transit observed radiative flux $F_{\star }$ and an unresolved source supplies $F_c$, let $D=F_c/F_{\star }$. Then ${\delta }_{\mathrm{obs}}={\delta }_{\mathrm{host}}/(1+D)$. For $D=1$, a true exoplanet transit depth of $0.010$ appears as $0.005$; ignoring transit dilution gives a radius ratio $\sqrt{0.005}\simeq 0.07071$ instead of $0.10$, under the same uniform-disc assumptions.

For an edge-on orbit crossing the centre of a uniformly bright stellar disk at distance $r\gg R_{\star }$, projected speed is $h/r$ and chord time is $2R_{\star }r/h$. A steady cross-sectional-area current $J_{\sigma }=2fh$ therefore places area $4f{R}_{\star }r$ in front of the star, givingThis requires a geometrically narrow, optically thin wire with negligible overlap. A noncentral chord, limb darkening, finite angular extent and large optical depth change the estimate; it cannot predict dimming greater than one.