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Andrea M. Ghez measures stellar orbits around the supermassive black hole at the center of the Milky Way. Her team's Keck observations combined astrometry with radial velocities for the short-period star S0-2.

An angular orbit alone leaves a mass-distance degeneracy of an angular stellar orbit. Adding Doppler spectroscopy supplies a physical velocity scale, allowing central mass and Euclidean distance to be fitted together. Their 2008 analysis inferred roughly four million solar masses at a Euclidean distance of eight kiloparsecs, and examined how source confusion and assumptions about central motion affected the result.^[ref]

André Baranne designed precision astronomical spectrographs, including the ELODIE spectrograph, which was used for Michel Mayor and Didier Queloz's discovery of the planetary companion to 51 Pegasi. The instrument combined broad spectral coverage with precise radial velocity measurements.

The scientific goal was to distinguish tiny stellar Doppler shifts from instrumental changes. Its design and calibration supported long-term velocity stability sufficient for detecting giant planetary companions. The original instrument paper reports the construction, data treatment and measured performance, making the observing hardware part of the explanation of the discovery.^[ref]

An astrophysical black hole is an astronomical object interpreted as a black hole through evidence about its mass, compactness and interaction with surrounding matter or gravitational waves. A black hole of stellar mass can form after stellar core collapse; a supermassive black hole can grow through black-hole accretion and mergers. The event horizon prevents light from escaping from within, but matter outside can radiate and reveal the central gravity. For Sagittarius A*, astrometry and radial velocities of nearby stars constrain a very large mass inside their small orbits. Such dynamical evidence complements observations of accretion and of binary black holes, which probe different aspects of the black-hole interpretation.^[ref]

A stationary mass density disturbance can be balanced by a changed azimuthal velocity in a shearing sheet. The continuity and azimuthal equations give zero radial velocity, while the radial equation balances Coriolis force against pressure and gravity. This mode is separate from the density-wave pair in the softened Toomre dispersion relation and is lost if one divides every equation by frequency.

For plates $z=\pm h(t)$, incompressibility, impermeability and regularity on the axis determine a z-independent radial velocity $u_r=-r\dot{h}/(2h)$ and vertical velocity $u_z=z\dot{h}/h$. A possible azimuthal velocity requires additional rotational data; continuity alone does not determine it. For a smooth regular initial profile $U(r)$ independent of $z$, its azimuthal Euler equation transports it as $u_{\theta }(r,t)=\lambda U(\lambda r)$, where $\lambda =\sqrt{h/h_0}$. If the rotation is solid-body with angular velocity $\Omega (t)$, the vorticity equation gives $\Omega ={\Omega }_{0}h/h_0$. Along a particle path, $r^{2}h$ and $r^2\Omega$ are constant, illustrating conservation of angular momentum.

For thickness $h(r,t)$, radial velocity $u(r,t)$ and external pressure $p_{\mathrm{ext}}$, the leading Newtonian fluid stress tensor and incompressibility giveRadial force balance on an annular sector includes the inward projection of hoop traction and external pressure on the sloping broad faces. Together with conservation of mass it yields

A binary mass function combines the observed orbital period $P$ and radial velocity semi-amplitude $K_1$ of a visible member of a spectroscopic binary. For a circular Kepler orbit, $K_1=2\pi a_1\sin i/P$, where $a_1$ is the visible star's orbital semimajor axis about the centre of mass and $i$ the orbital inclination. The centre of mass relation $a_1=a{M}_2/(M_1+M_2)$ and Kepler's third law $4{\pi }^2{a}^3=G(M_1+M_2)P^2$ eliminate the unobserved separation:It has units of mass. The right side is at most $M_2$, so $f(M)\le M_2$ is a strict observational lower bound on the unseen companion's mass within this model. If $M_1$ is independently estimated, setting $\sin i=1$ and solving the equation gives a stronger minimum companion mass. An unknown orbital inclination generally prevents a unique pair of stellar masses from being inferred from one velocity curve alone.

For an eccentric Kepler orbit with measured orbital eccentricity $e$, the radial-velocity semi-amplitude is $K_1=2\pi a_1\sin i/[P\sqrt{1-e^2}]$, so the observable expression becomes $f(M)=P{K}_1^3(1-e^2{)}^{3/2}/(2\pi G)$. A fitted semi-amplitude, rather than an arbitrary instantaneous radial velocity, is required. This is why an unseen companion can be constrained dynamically even when its own spectral lines are absent.

For the separation of a two-dimensional central-potential eigenstate, the guidance equation gives zero radial velocity and constant-radius circular motion, with circulation $2\pi \hslash k/m$. A real superposition of the degenerate $k$ and $-k$ eigenstates has zero current away from nodes instead. The speed depends on the actual wavefunction, not just its energy.

For unit channel width, incompressible flow towards a corner withdrawal point has $\psi =0$ on the lower wall and $\psi =-m$ on the upper and left walls, with $\psi \to -my$ far downstream. For irrotational flow, separation of variables gives the displayed harmonic correction: the Fourier sine coefficients of $-m(1-y)$ are $-2m/(n\pi )$. Near the corner, $\psi \sim -2m\theta /\pi$ and radial velocity is $-2m/(\pi r)$, whose integrated inward flux across a quarter-circle is $m$.

Daniel C. Fabrycky coauthored research with Rebekah I. Dawson on aliasing in exoplanet radial velocity observations. The work connects an observation window with spurious spectral peaks, helping distinguish a candidate orbital period from a sampling artifact.^[ref]

Didier Queloz and Michel Mayor detected a planetary companion to the solar-type star 51 Pegasi through periodic changes in its radial velocity. Their 1995 observations used the ELODIE spectrograph to measure the stellar Doppler shift.

The signal repeated in about four days, implying a close companion rather than a distant analogue of Jupiter. The inferred minimum companion mass was planetary. Because radial velocity measures only the line-of-sight component of the reflex motion, the orbital inclination remains part of the mass inference.^[ref]

For a detached double-lined binary, radial velocities supply an orbital length scale times $\sin i$, while eclipse geometry supplies inclination and relative radii. Together they determine physical stellar radii. Comparing them with angular radii from a calibrated surface-brightness/colour relation gives distance. The orbital scale is geometric, but the angular-radius calibration, extinction and third light add astrophysical assumptions.^[ref]

Geoffrey W. Marcy worked on the radial velocity method for finding exoplanets, including precision iodine-cell Doppler calibration. His paper with R. Paul Butler combined a wide recorded stellar radiation spectrum with reference absorption lines introduced directly into the stellar beam.

Measuring a small Doppler shift requires distinguishing real stellar motion from instrumental changes. The superimposed reference constrains the spectrograph's wavelength scale and point spread function during the exposure, allowing the stellar shift to be fitted with those effects included. The paper reported year-long tests of the technique on three solar-type stars.^[ref]

Let a particle enter a uniform solenoid at transverse position $u_0=x_0+i{y}_0$ with zero radial velocity after an ideal fringe. Define signed $\Omega =qB/(2\gamma m)$ and initial field-free speed $v$. Its Lorentz force trajectory inside isThe transverse motion is a circle about the guiding centre $u_0/2$ at cyclotron angular frequency $2|\Omega |$. Cylindrical radius is $|u_0\cos \Omega t|$; its apparent azimuthal jumps when the orbit crosses the axis are coordinate singularities. Forward entry requires $v>|\Omega u_0|$ in this ideal model.

Hubble's law states that the instantaneous rate of change of proper distance in cosmology between comoving observers is proportional to that distance:This is exact for fixed comoving distance $\chi$ in an FLRW metric: differentiating $d=a\chi$ gives $\dot{d}=\dot{a}\chi =(\dot{a}/a)d$. The dot uses cosmic time, and both distances are compared on spatial slices at the same cosmic time. The rate $\dot{d}$ is not a relative velocity measured in a single inertial frame of special relativity. Thus the instantaneous relation is not an exact observed redshift-distance formula. The associated Hubble flow describes the ideal expansion; peculiar velocities describe departures from it.

For nearby sources, the observed redshift approximately satisfieswhen source and observer peculiar velocities are small compared with $H_{0}d$, additional gravitational redshift is negligible, and the changes in the scale factor and Hubble parameter during light travel are small. Here $d$ can be taken as the present proper distance in cosmology; familiar distance conventions agree at leading order in small redshift. Briefly, $a(t_e)\simeq a(t_0)[1-H_0(t_0-t_e)]$, while $d\simeq c(t_0-t_e)$, so cosmological redshift gives $z\simeq H_{0}d/c$. At greater redshift, use an expansion history and an explicit distance convention instead.

Stipulate the illustrative Hubble constant $H_0=70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ and speed of light $c=299792.458\mathrm{km}{\mathrm{s}}^{-1}$. A $\mathrm{Mpc}$ is a million parsecs. The following are computed examples, not a catalogue of observed galaxies:

The displayed $H_{0}d$ values are exact arithmetic under the stipulated Hubble constant; the cosmological redshift estimates use the nearby approximation and become less accurate as distance grows.

Hubble's law turns a nearby observed redshift into an approximate distance indicator and makes the uniformity of cosmic expansion tangible: twice the proper distance in cosmology means twice the instantaneous separation rate. Recentring the ideal Hubble flow on another comoving observer preserves the same proportionality, as illustrated under scale factor, so recession in every direction does not select a central observer.

In his 1929 PNAS paper, Edwin Hubble presented a relation between the distances and radial velocities of extragalactic objects. His fits gave coefficients of $465$ and $513\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ for 24 individual objects and nine groups respectively, and he used about $500\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ as a rounded representative value. Those historical fits used the paper's distance estimates; they are distinct from the stipulated modern-scale illustration above. ^[ref].

Jerome A. Orosz coauthored a dynamical model of Cygnus X-1 that combines optical light curves, companion-star radial velocities, rotational broadening and a distance constraint to infer the black hole's mass.^[ref]

The measurements constrain different parts of the binary geometry. Radial motion alone leaves an inclination dependence; stellar size and light-curve information help determine the scale and orientation. The quoted paper is a specific historical analysis, so its parameter estimates should not be mistaken for an automatically current calibration.

Michel Mayor develops precise stellar radial velocity measurements and their use in searches for unseen companions. With Didier Queloz, he reported a planetary companion to 51 Pegasi in 1995.

The host star's regularly changing Doppler shift revealed its reflex motion around a shared center of mass. The approximately four-day orbital period showed that a giant planet could exist much closer to its host star than the giant planets of the Solar system. The discovery made this previously unexpected orbital arrangement an observational constraint on planet formation.^[ref]

For a smooth axisymmetric flow, write $u_z(r,z)=w(z)+O(r^2)$ near the axis. The incompressible flow condition is $r^{-1}{\partial }_r(r{u}_r)+{\partial }_z{u}_z=0$. Integrating radially and excluding a singular line source at the axis gives $u_r=-r{w}^{\prime }(z)/2+O(r^3)$. The signed axial velocity is essential: specifying only its magnitude does not fix the sign of the radial velocity.

An astronomical interferometer combines light from collectors separated by projected interferometric baseline $\mathbf{B}$. For a distant, narrowband, spatially incoherent source, the Van Cittert-Zernike theorem gives normalized complex visibilityThe resolving scale is $\lambda /B$, while the individual collector and beam combiner restrict the field of view. Delay lines compensate geometric path differences. To see broadband fringes, residual optical path length mismatch must lie within the coherence length of order ${\lambda }^2/\Delta \lambda$; fringe tracking follows rapidly changing atmospheric piston in interferometry. Baseline changes and Earth-rotation aperture synthesis sample additional spatial frequencies.

For an ideal uniform circular star of full angular diameter $\Theta$,Real limb darkening changes the conversion. Several baselines constrain binaries, stellar radii and circumstellar structure; orbital angular separations combined with radial velocities can give physical orbital scales. calibration stars test contrast losses, alongside wavelength, photometric imbalance, polarization and bandwidth controls. For three collectors the closure phase cancels telescope-based phase errors: ${\varphi }_{12}+{\varphi }_{23}+{\varphi }_{31}$ retains source asymmetry information. Sparse baseline coverage still leaves image/model ambiguities.

In stellar intensity interferometry, thermal-light correlations instead give $g^{(2)}(0)-1=|g^{(1)}{|}^2$ for an ideal single-polarization resolved temporal mode. The Sirius experiment correlated separate detector outputs rather than optically combining the beams.^[ref] Pure atmospheric phase shifts do not alter local intensities, but finite detector bandwidth dilutes the correlation and photon statistics limit sensitivity; the two-detector observable also loses the complex phase.

Let $u(r)>0$ denote the inward radial velocity. Steady spherical symmetry and mass conservation giveThe radial Euler momentum equation for the globally isothermal equation of state $p=c_s^2\rho$ isThe logarithmic derivative of mass conservation is ${\rho }^{\prime }/\rho =-u^{\prime }/u-2/r$. Substitution gives the Isothermal Bondi equationAt a smooth sonic point, $u=c_s$ makes the coefficient of $u^{\prime }$ vanish, so the right-hand side must vanish too. ThereforeThe solution that crosses this critical point of the isothermal Bondi equation continuously is the transonic accretion solution.