Exoplanet atmospheric observatory

An exoplanet atmospheric observatory characterizes planetary spectra through transit, eclipse, phase-curve, high-dispersion, or direct-imaging observations.

An atmospheric retrieval fits a parametrized radiative transfer model to an exoplanet transmission spectrum or exoplanet emission spectrum. Temperature, abundances, clouds and reference radius/pressure can trade off against one another. Posterior precision within one model is not proof that its structural assumptions are unique.

The Spitzer Space Telescope provided infrared observations of exoplanet secondary eclipses and exoplanet thermal phase curves. Its original mission included infrared spectroscopy; its warm mission retained the 3.6 and 4.5 micrometre imaging bands.

The Hubble Space Telescope provides ultraviolet, visible, and near-infrared transit and eclipse spectroscopy and established many early detections of atoms, aerosols, and atmospheric escape.

The Advanced Camera for Surveys is an imaging instrument on the Hubble Space Telescope, designed with separate wide-field, high-resolution and ultraviolet channels. Filters select spectral bands; dispersing elements also provide low-resolution spectroscopy. The different channels have different detectors and optical paths, so the instrument name alone does not specify the relevant wavelength coverage.^[ref]

For an exoplanet transit, a sequence of dispersed images can be divided into spectral channels, each yielding its own light curve. Comparing fitted exoplanet transit depths constructs an exoplanet transmission spectrum. Detector response and pointing variations must be calibrated along with the planetary signal. Historical observations should be interpreted using the channel and observing mode actually employed.

The Wide Field Camera 3 is an instrument on the Hubble Space Telescope with separate ultraviolet/visible and near-infrared channels. Its grisms disperse light onto the detector without an entrance slit, allowing a time series of stellar spectra during an exoplanet transit. The near-infrared channel uses two grisms covering different intervals; their combined coverage is broader than that of either alone.^[ref]

A spectral channel yields a light curve, and its fitted exoplanet transit depth contributes one interval of the exoplanet transmission spectrum. The resulting molecular bands can constrain water and the atmospheric scale height, subject to exoplanet clouds and reference-pressure degeneracy in transmission spectroscopy. Because the stellar photons dominate, stability and calibration are as important as collecting light.

The James Webb Space Telescope combines a 6.5-metre aperture with stable near- and mid-infrared spectroscopy. Its instruments measure transit, eclipse, and phase-curve spectra across multiple molecular bands.

The NIRISS single-object slitless spectroscopy mode measures time-series spectra from $0.6$ to $2.8\mu \mathrm{m}$. It supports exoplanet transmission spectra and eclipse observations, as described in the instrument guide.

NIRSpec provides multiple spectroscopic modes within $0.6$–$5.3\mu \mathrm{m}$. Its time-series modes can measure exoplanet transmission spectra and exoplanet secondary eclipses; individual settings have narrower coverage. See NASA's instrument overview.

MIRI's low-resolution spectroscopy includes a slitless time-series mode for exoplanet observations. Its useful coverage includes the $5$–$12\mu \mathrm{m}$ region, with longer-wavelength limits dependent on calibration and sensitivity. See the instrument guide.

Ariel is an ESA mission designed to survey the visible and infrared atmospheres of about one thousand planets through transit and eclipse spectroscopy.

The Extremely Large Telescope combines a 39-metre-class aperture, adaptive optics, direct imaging, and high-dispersion spectroscopy to characterize giant and terrestrial exoplanets.

The Very Large Telescope is an observatory array at Paranal with four $8.2$-metre Unit Telescopes and four movable $1.8$-metre Auxiliary Telescopes. The Unit Telescopes can observe independently; selected beams can also be combined in an astronomical interferometer. Combining beams coherently changes the angular information available and requires an appropriate observing instrument.^[ref]

For exoplanets, adaptive optics helps concentrate light for exoplanet direct imaging, while instruments such as CRIRES resolve molecular spectral lines. Spatial separation and Doppler shift provide complementary ways to distinguish a faint planet from its host star. The capability belongs to the telescope-plus-instrument combination, rather than to aperture size alone.

CRIRES is a high-resolution infrared spectrograph. Its upgraded form supports molecular spectroscopy of exoplanet atmospheres and measurements based on the Doppler effect. See ESO's instrument overview.

Exoplanet direct imaging separates planetary light from the host star's light on the sky. The two central obstacles are small angular separation and a large brightness contrast. Diffraction spreads stellar light into the surrounding image, and imperfect optics add structure that can resemble or hide a faint companion. Adaptive optics reduces blurring from the terrestrial atmosphere, improving the concentration and stability of the image.

Young giant planets on wide orbits are favourable targets because their intrinsic thermal radiation can be substantial. Measuring their light at several wavelengths constrains the exoplanet atmosphere; repeated positions trace the orbit. Brightness alone does not directly measure mass: converting it into mass requires an age and an evolutionary cooling model. Doppler spectroscopy or orbital dynamics can supply independent constraints.

Exoplanet transit photometry measures the decrease in a star's observed radiative flux during an exoplanet transit. Define $F_0$ as the unobscured star's observed radiative flux and $F(t)$ as the value at time $t$. For a dark, opaque exoplanet crossing a uniformly bright circular stellar disc, a fully superimposed planetary disc blocks exactly the fractionHere $R_p$ and $R_{\star }$ are the planet and star radii. This exoplanet transit depth determines a radius ratio, $R_p/R_{\star }=\sqrt{\delta }$; an absolute planet radius requires an independently estimated stellar radius. For $R_p/R_{\star }=0.10$, $\delta =0.010$: a one-percent dip, with $F_{\mathrm{full}}/F_0=0.990$. Halving the radius ratio to $0.05$ reduces the exoplanet transit depth to $0.0025$, a quarter as much.

For the central exoplanet transit illustrated below, the transit impact parameter is $b=0$. Write $k=R_p/R_{\star }<1$ and let $z$ be the projected centre-to-centre separation in units of $R_{\star }$. The light curve is $F/F_0=1-A(z)/\pi$, where $A(z)$ is the overlap area of circles of radii $1$ and $k$. There is no overlap when $z\ge 1+k$; the planetary disc is fully inside when $z\le 1-k$, giving $A=\pi k^2$. Between those limits, the overlap area grows or shrinks continuously. Thus even a uniform star has curved ingress and egress, with a flat bottom only during full overlap. The illustration uses approximately constant transverse velocity over the short crossing, so $z=|ut|/R_{\star }$ with mid-transit at $t=0$. Analytic models also handle limb darkening. ^[ref].

The simple exoplanet transit depth formula assumes constant stellar brightness, no limb darkening or starspots, negligible light from the exoplanet, and no transit dilution by other light. It describes full overlap in a non-grazing exoplanet transit, requiring $b<1-k$ for a full-overlap interval. A grazing exoplanet transit never hides the whole planetary disc against the star, and limb darkening weights different blocked areas differently. Repeating dips are valuable evidence, but eclipsing binary stars and starspots can imitate parts of the signal, so confirmation and joint modelling matter.

The probability that a genuinely transiting planet with specified parameters is recovered by a survey and passes its selection procedure. It includes light curve noise, observing gaps, search algorithms and vetting. It differs from geometric transit probability; synthetic-transit injection and recovery can estimate it.

The fractional decrease in observed radiative flux during an exoplanet transit, relative to the out-of-transit baseline. For a uniform star, a dark planet, full disc overlap, and no transit dilution, the exoplanet transit depth is $(R_p/R_{\star }{)}^2$. For a nonuniform star it depends on the blocked specific intensity as well as the radius ratio.

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.

A starspot is a relatively cool, dark region on a star's surface associated with a magnetic field. Starspots change the light curve as the star rotates. A planet crossing a starspot blocks less light than it would from an unspotted region, producing a local upward anomaly during an exoplanet transit; unocculted starspots can instead deepen the normalized exoplanet transit depth. They can also distort spectral lines and imitate a Doppler spectroscopy signal.

A departure of transit arrival times from a constant-period ephemeris can reveal gravitational interactions with other bodies. Dynamical fits may constrain an exoplanet mass independently of transit depth, providing a mass to combine with radius for exoplanet interior models.

For two nearly circular Kepler orbits near a first-order $(j+1):j$ period ratio, the slow resonant argument circulates on the displayed superperiod when longitude of periapsis precession is negligible. If $P_2/P_1=((j+1)/j)(1+\Delta )$, then $P_t=P_1(1+\Delta )/(j|\Delta |)$. Within resonant-argument libration, this circulation formula is replaced by an amplitude-dependent libration period.

Near a first-order $(j+1):j$ orbital resonance, a weak periodic resonant orbital perturbation can accumulate a large transit-timing variation. Let subscripts 1 and 2 denote inner and outer planets, define ${\mu }_2=M_2/M_{\star }\ll 1$, $\alpha =a_1/a_2$, and writeHere $n_i=2\pi /P_i$ are mean motions. Approximate the inner planet's disturbing potential per unit mass bywhere $f$ is the dimensionless coefficient of this chosen resonant harmonic, $e$ is its effectively constant free orbital eccentricity, ${\lambda }_i$ are mean longitudes and ${\varpi }_1$ is longitude of periapsis. Neglect precession and suppose the resonant argument circulates approximately uniformly.

The leading semimajor-axis equation gives $|{\dot{a}}_1|/a_1\sim 2j{\mu }_2\alpha n_1|f|e$. Since $\delta n_1/n_1=-3\delta a_1/(2a_1)$, one integration supplies a factor $1/|\omega |$ in the mean motion perturbation, and a second supplies it in the mean longitude. Converting the phase perturbation to a timing amplitude givesDropping $\Delta$ against one in the numerator gives the displayed near-resonant estimate. Its $e/{\Delta }^2$ enhancement assumes this term dominates direct phase and forced-eccentricity effects. Close enough to resonance, resonant-argument libration and changes of the free orbital eccentricity invalidate the expansion; the divergence does not predict arbitrarily large transit times.

In leading mean-conjunction geometry a common observer phase $x$ for two resonant arguments requires $qx={\varphi }_1$ and $nx={\varphi }_2$ modulo $2\pi$. With $d=\gcd (q,n)$, the exact modular criterion isNecessity follows by subtracting the two congruences after multiplication; sufficiency follows from the integer linear combinations of $q$ and $n$, whose set is $d\mathbb{Z}$. For exterior centres $\pi$ and interior centres $0,\pi ,0$ at orders one to three, only $q=n=2$ satisfies it. This is phase compatibility, not a guarantee of an observed exoplanet transit: timing, observer orientation, resonant-argument libration widths and true-versus-mean longitude must also be considered.

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.

An exoplanet passes in front of its host star as seen by an observer and blocks part of the stellar flux. Exoplanet transit photometry infers geometry and radii from the resulting light curve. A true alignment involves the true anomaly, not generally the mean anomaly.

The transit radius is the effective radius inferred from the fraction of stellar radiative flux blocked during an exoplanet transit. For a uniformly bright star, a completely overlapping opaque planet gives transit depth $\delta =(R_{\mathrm{tr}}/R_{\star }{)}^2$. Thus a one-percent depth corresponds to a radius ratio of about one tenth.

Real inference accounts for limb darkening, transit geometry and contaminating light. A planetary atmosphere can make the effective occulting area depend on wavelength: a grazing ray traverses a long slant path, so absorption need not correspond to one sharply defined material surface. A quoted transit radius therefore belongs to a measurement band and a stated model.^[ref]

The probability that an isotropically oriented circular orbit produces an exoplanet transit, assuming $a>R_{\star }+R_p$. Uniformity of $|\cos i|$ gives the displayed result. Orbital eccentricity, a restriction to full rather than grazing crossings, or a non-isotropic orientation distribution changes the expression.

The first-to-fourth-contact duration of an exoplanet transit by a spherical opaque planet of radius ratio $k=R_p/R_{\star }$ on a circular orbit. The transit impact parameter is $b=(a/R_{\star })\cos i$. The formula includes finite planet size and orbital curvature; it reduces to the transit duration of a small planet when $k\to 0$ and $R_{\star }/a\ll 1$. It describes contact geometry, not a uniquely specified light-curve threshold in the presence of limb darkening.

The total duration $T_{14}$ runs from first to fourth contact; the fully superimposed duration $T_{23}$ runs from second to third contact. Circular-orbit formulas contain $1+k$ for the former and $1-k$ for the latter. Confusing the two changes inferred impact parameter, even if the period and transit depth are unchanged.

For a circular orbit, define $k=R_p/R_{\star }$, $A=a/R_{\star }$ and $s=\sin (\pi T_{14}/P)$. Inverting contact geometry gives $b^2=[(1+k{)}^2-A^2{s}^2]/(1-s^2)$. The inferred value is conditional on the stellar scale, the total-duration convention and the orbital model; it must satisfy $0\le b<1+k$. A non-grazing complete crossing additionally has $b<1-k$.

The orbital speed near transit differs from the circular value by approximately $(1+e\sin \omega )/\sqrt{1-e^2}$. At otherwise fixed geometry, duration is shortened by the inverse of this factor. Without eccentricity and periapsis constraints, a measured duration need not select one unique impact parameter.

Assume Kepler's third law, a negligible planet mass, and $M_{\star }/M_{\odot }=R_{\star }/R_{\odot }$. Define $A_{\odot }=[(GM{)}_{\odot }{P}^2/(4{\pi }^2){]}^{1/3}/R_{\odot }$. The total duration of a circular planetary transit determines $a/R_{\star }$, giving the displayed stellar radius. Unknown orbital eccentricity, transit dilution or a different stellar mass-radius calibration introduces additional uncertainty.

The minimum projected planet-star separation during an exoplanet transit, divided by the stellar radius. It determines the stellar chord length and hence the transit duration of a small planet.

For two exactly coplanar circular orbits around the same star, transit impact parameters scale as semi-major axis, hence as $P^{2/3}$. Any overlap by the outer planet requires $b_2<1+k_2$. In the point-planet limit the period ratio is bounded by $b_1^{-3/2}$. Tangency has zero duration, so this is a geometric supremum; practical detection also requires sufficient signal-to-noise ratio and survey coverage.

For a small planet on a nearly circular Kepler orbit, a stellar chord at normalized transit impact parameter $b$ has length $2R_{\star }\sqrt{1-b^2}$. Dividing by transverse orbital speed $2\pi a/P$ gives the duration. A central transit has the longest duration within this approximation. Eliminating $a$ with Kepler's third law yields $P/D^3\ge {\pi }^{2}G{\rho }_{\star }/3$ for known mean stellar mass density. Finite planet radius, appreciable orbital eccentricity, and $R_{\star }/a$ corrections change the bound.

For an opaque or detectably extinguishing spherical debris cloud of radius $R_c=\eta R_H$ much larger than the stellar radius, a central exoplanet transit lasts approximately $2R_c/(na)$. Combining this with the Hill radius gives the displayed mass estimate. For cloud impact distance $s$, replace $R_c$ by $\sqrt{R_c^2-s^2}$: the central formula then gives a lower mass estimate unless the chord geometry is known.

Doppler spectroscopy detects an exoplanet through the host star's changing Doppler shift. Both bodies orbit their shared astronomical barycenter: the star undergoes stellar reflex motion even when the exoplanet itself is unseen. Its velocity along the line of sight shifts its spectral lines; at speeds small compared with the speed of light, $\Delta \lambda /\lambda \simeq v_r/c$, with positive $v_r$ for recession. Repeated measurements can fit a Kepler orbit, rather than merely detect one shifted spectrum. Starspots and other changes in stellar spectral lines must be distinguished from actual orbital motion.

For one companion in an isolated two-body problem, the stellar radial-velocity semi-amplitude isHere $G$ is the gravitational constant, $P$ the orbital period, $e$ the orbital eccentricity, and $M_{\star },M_p$ the star and exoplanet masses. The orbital inclination $i$ is measured relative to the sky plane: $i=0$ is face-on and $i=90^{\circ }$ is an edge-on orbit. The exact binary mass function measured from the fitted orbit isWith known $M_{\star }$ and $M_p\ll M_{\star }$, this is usually reported as $M_p\sin i$. Taking $\sin i=1$ gives the minimum companion mass; a smaller orbital inclination allows a larger mass. A detected exoplanet transit constrains $i$ close to $90^{\circ }$ for a compact planet orbiting well outside the star's surface, substantially resolving this ambiguity. Combining Doppler spectroscopy and exoplanet transit photometry then supplies a mass and radius for exoplanet interior studies.

For a computed circular example, let $q=M_p/M_{\star }=0.001$ and choose a relative orbital speed $V_{\mathrm{rel}}=100\mathrm{km}{\mathrm{s}}^{-1}$: this is the speed of the planet relative to the star, not its speed about the astronomical barycenter. Conservation of momentum givesThe barycentric velocities point in opposite directions, so their speeds add to $V_{\mathrm{rel}}$. Here $e=0$ and $K=V_{\star }\sin i$: $K=99.9001\mathrm{m}{\mathrm{s}}^{-1}$ at $i=90^{\circ }$, $49.9500\mathrm{m}{\mathrm{s}}^{-1}$ at $i=30^{\circ }$, and zero at $i=0$. The star still orbits in the face-on case, but its orbital velocity has no line-of-sight component. The plotted circular signal is $v_r(t)-\gamma =K\sin (2\pi t/P)$ for a chosen phase origin and constant system velocity $\gamma$.

An iodine cell places reference absorption on the same beam as stellar lines. A template and known cell transmission constrain stellar shift and instrumental line shape together. The observed spectrum is modelled by multiplying the stellar and cell spectra before instrumental convolution.

At Lick Observatory, the Hamilton spectrograph records starlight with an imprinted iodine reference. The reference tracks instrumental wavelength and profile changes during the science exposure. Barycentric correction and a time series of relative stellar shifts then support a dynamical companion model.

A transit radius $R_p$ and an inclination-constrained mass upper limit $M_{\max }$ imply ${\bar{\rho }}_p<M_{\max }/(4\pi R_p^3/3)$. The inferred density bound depends on the adopted stellar radius, transit dilution, orbital model and how the velocity detection threshold is defined. A nondetection threshold without a likelihood model does not by itself specify a confidence level.

A stellar binary identified through periodic Doppler shifts of spectral lines. A single-lined low-inclination binary can share a small projected velocity amplitude with an edge-on planetary system; the companion luminosity, line shapes and independent inclination constraints help distinguish them.

Doppler spectroscopy measures the line-of-sight component of a host star's stellar reflex motion, rather than the full companion mass. Let the measured orbital period, orbital eccentricity and stellar radial-velocity semi-amplitude be $P,e,K$, with host mass $M_{\star }$ and sky-plane orbital inclination $i$. The exact two-body relation isCubing and rearranging defines the observed binary mass functionFor fixed $M_{\star }$, the function $M_p^3/(M_{\star }+M_p{)}^2$ is strictly increasing: its logarithmic derivative is $3/M_p-2/(M_{\star }+M_p)>0$. Since $\sin i\le 1$, the smallest compatible companion mass occurs for an edge-on orbit, $i=\pi /2$.

Only when $M_p\ll M_{\star }$ does this simplify toA small inferred projected mass alone therefore does not exclude a more massive companion at small orbital inclination. An exoplanet transit constrains orbital inclination and helps remove this degeneracy. When the companion mass is appreciable, solve the exact binary mass function instead of inserting a total-mass correction on the wrong side of the approximate formula.

At fixed observed period and semi-amplitude, $M_p\sin i\propto M_{\star }^{2/3}$ and $a\propto M_{\star }^{1/3}$. At fixed planet mass and amplitude threshold, the largest circular orbit instead scales as $M_{\star }^{-1}$. A revised stellar radius affects transit-derived planet radii but not these leading dynamical relations.

For a circular low-mass companion and a semi-amplitude threshold $K_{\min }$, $a_{\max }=G{M}_p^2{\sin }^{2}i/(M_{\star }{K}_{\min }^2)$. The edge-on case is the largest geometrically accessible distance. Adequate baseline and cadence are separate requirements, and a full peak-to-peak detection threshold is twice the corresponding semi-amplitude.

The correction for an observer's motion relative to the Solar System barycenter along the target line of sight. Precision work also requires accurate timing and an appropriate relativistic convention.

Built by André Baranne, Michel Mayor, Didier Queloz and colleagues, the ELODIE fiber-fed spectrograph carried light from Haute-Provence's 1.93-metre telescope through optical fibers. Its echelle grating dispersed light in high diffraction orders; an optical prism and grism cross-disperser separated them perpendicularly on a charge-coupled device. This cross-dispersed echelle spectrograph achieved spectral resolving power $R=42000$. A simultaneous thorium reference tracked instrumental drift. Wavelength calibration mapped detector positions to wavelengths; spectral cross-correlation combined many spectral lines into a fitted profile. A barycentric velocity correction removed the observer's motion.^[ref]

Why can a spectrograph measure a Doppler shift much smaller than one spectral resolution element? Resolution concerns separating neighbouring features, whereas locating the centre of a known profile uses information across that profile and across many spectral lines. For a chosen example, at $\lambda =500\mathrm{nm}$ a recession velocity of $60\mathrm{m}{\mathrm{s}}^{-1}$ givesAt $R=42000$, one spectral resolution element corresponds to $c/R\simeq 7.14\mathrm{km}{\mathrm{s}}^{-1}$, about $119$ times larger. A well measured centre can move by a small fraction of that width. More photons and more useful spectral lines improve the estimate, but do not remove wavelength calibration errors or stellar activity.

A simple mathematical picture of spectral cross-correlation makes the readout concrete. Define $x=c\log (\lambda /{\lambda }_0)$, with fixed reference wavelength ${\lambda }_0$, and let $d(x)$ be continuum-normalized line absorption. For small velocities, a common Doppler shift approximately translates $d(x)$. A positive absorption mask $m(x)$ then givesIf $d(x)=m(x-v_r)$, the maximum occurs at $u=v_r$; a convention correlating the received observed radiative flux instead produces a dip. A fitted correlation centre is an effective collective line position, not a claim that each individual spectral line has been resolved into a narrower feature. The lamp tracks instrumental drift, while the barycentric velocity correction removes the observer's changing line-of-sight velocity; neither removes changes in the star itself.

Michel Mayor and Didier Queloz began observing 51 Pegasi in September 1994 and revisited its repeating Doppler spectroscopy signal in 1995. Their circular fit used 35 measurements: orbital period $4.2293\pm 0.0011$ days, stellar radial-velocity semi-amplitude $59\pm 3\mathrm{m}{\mathrm{s}}^{-1}$, and residual scatter $13\mathrm{m}{\mathrm{s}}^{-1}$ after epoch-dependent velocity offsets. Apparently stable comparison stars checked the instrumental zero point. Low chromospheric stellar activity and an inferred roughly 30-day stellar rotation period argued against starspots; light curves and the spectral line bisector tested stellar pulsation. These checks favoured orbital motion but had finite sensitivity. The reported $M_p\sin i=0.47\pm 0.02$ Jupiter masses was not a true mass. Stronger upper bounds used stellar-spin information and assumed spin–orbit alignment, limiting a nearly face-on binary star interpretation conditionally.^[ref]

The hot Jupiter matters both for planetary migration and for how exoplanets are found. At fixed small companion mass, orbital inclination and orbital eccentricity, the stellar radial-velocity semi-amplitude scales as $K\propto P^{-1/3}$. Comparing chosen orbital periods of $4$ days and $12$ years gives an amplitude ratio of about $10.3$: a close companion can be much easier to detect and complete many orbits during one observing season. Finding a giant planet close to its star also asks how its present orbit relates to where it formed. Doppler spectroscopy supplies a dynamical constraint; it does not by itself establish a formation pathway or measure the exoplanet's radius.

A repeating Doppler shift contains more than an amplitude. For an isolated two-body problem, let $\nu$ be the true anomaly, ${\omega }_{\star }$ the stellar argument of periapsis, $\tau$ the periapsis time and $\gamma$ the constant system line-of-sight velocity. The companion's argument of periapsis differs by $\pi$ from ${\omega }_{\star }$, so mixing the two conventions reverses the orbital signal. With positive velocity for recession, choose the stellar line-of-sight coordinate positive away from the observer. Kepler's equation supplies the clock:Here $M$ is the mean anomaly and $E$ the eccentric anomaly. The time coordinate is uniform in $M$, not in $\nu$: an eccentric Kepler orbit runs fastest near periapsis.

To derive the velocity law without guessing a sinusoid, write the stellar orbital coordinates as $X=a_{\star }(\cos E-e)$ and $Y=a_{\star }\sqrt{1-e^2}\sin E$. Project toDifferentiating $z$ and substituting the true anomaly identities givesThe constant $e\cos {\omega }_{\star }$ term makes the time average of the orbital velocity zero: $z$ returns to its original value after one orbital period. For $e=0$ this reduces to a sinusoid with an arbitrary phase origin; periapsis itself is then undefined. For $e>0$, the largest and smallest velocities are $\gamma +K(e\cos {\omega }_{\star }\pm 1)$, so $K$ is still half the peak-to-peak range, although the extrema need not be symmetric about $\gamma$.

For an original worked example, choose $M_{\star }=1M_{\odot }$, $P=10$ days, $K=50\mathrm{m}{\mathrm{s}}^{-1}$, $e=0.4$ and ${\omega }_{\star }=60^{\circ }$. Taking one day as $86400\mathrm{s}$ and $G=6.67430\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, the binary mass function is $f=1.98270\times 10^{20}\mathrm{kg}$. Its units check directly: $[P{K}^3/G]=\mathrm{kg}$. With adopted solar mass $M_{\odot }=1.98847\times 10^{30}\mathrm{kg}$ and Jupiter mass $M_J=1.89813\times 10^{27}\mathrm{kg}$, solving the exact binary mass function gives

All three systems have the same chosen $P,K,e,{\omega }_{\star },\tau$ and therefore the same Doppler spectroscopy time series, despite different true masses. Exactly what remains fixed is $M_p\sin i/(M_{\star }+M_p{)}^{2/3}$; $M_p\sin i$ alone is only the small-companion approximation. The $90^{\circ }$ value is the minimum mass. For this example the extrema are $+60$ and $-40\mathrm{m}{\mathrm{s}}^{-1}$ relative to $\gamma$, and the periapsis value is $+35\mathrm{m}{\mathrm{s}}^{-1}$. This is a chosen eccentric system, not a reconstruction of 51 Pegasi b.

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.

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, $a^3=G(M_{\star }+M_p)P^2/(4{\pi }^2)$.