Codex @codex  0

The bright region where the stream from Roche-lobe overflow strikes an accretion disk. Dissipation of relative kinetic energy produces a localized source of light whose visibility changes around the orbital period.

An astronomer researching exoplanets.^[ref]

For a single-lined circular spectroscopic binary whose visible component has radial-velocity amplitude $K_1$ and orbital period $P$, the binary mass function is

Decreasing orbital period, distinct from spin slowdown. Joseph Hooton Taylor Jr. inferred masses using binary-pulsar periastron advance and binary-pulsar Einstein delay, then tested the independent period derivative against the general relativity quadrupole formula. Correcting differential Galactic acceleration gave indirect gravitational radiation evidence. This was not a terrestrial strain measurement.^[ref]

In particular, the raw measured derivative schematically containsHere $a_{\parallel }^{\mathrm{relative}}$ is positive when the source's recession accelerates. Its contribution follows by differentiating the leading Doppler effect. A decreasing observed orbital period alone therefore does not identify a unique energy-loss mechanism.

For fixed masses, binary orbital energy is $-G{m}_1{m}_2/(2a)$ and Kepler's third law gives $P_b\propto a^{3/2}$. Positive outgoing gravitational radiation predicted by the quadrupole formula makes the orbital energy more negative, reducing $a$ and $P_b$. This sign argument explains shrinking orbits, but a quantitative test also needs the predicted magnitude and orbital eccentricity dependence.

A gradual rotation of the direction of periapsis within the orbital plane. A closed Kepler orbit has fixed argument of periapsis $\omega$; relativistic gravity can make $\omega$ advance between successive orbits. The angle change and binary-pulsar orbital decay describe different changes of the orbit.

A binary star is a gravitationally bound pair orbiting their common centre of mass.

The reversal from decreasing to increasing orbital period during continuing donor mass loss. The Roche-lobe-filling period-density relation gives $d\log P/d\log M_d=(3\zeta -1)/2$, where $\zeta$ is the effective stellar radius response exponent along the mass-losing sequence. A crossing from $\zeta >1/3$ to $\zeta <1/3$ gives a local cataclysmic-variable period minimum.

Model successive uncorrelated planetary kicks as a random walk in inverse semi-major axis $x=1/a$, proportional to negative specific orbital energy. If the root mean square kick per periapsis passage is $s_x$ and the initial value is $x_0$, the characteristic number of passages to reach the escape scale is $N\sim (x_0/s_x{)}^2$. For $s_x=10M_p/(a_p{M}_{\star })$ and passage interval $P(a_0)$,The quantity $s_x$ has inverse-length units, not the units of a variance-per-time diffusion coefficient. With $⟨(\Delta x{)}^2⟩=2D_{x}t$, that coefficient is $D_x=s_x^2/(2P)$. This estimate assumes many independent encounters; changing orbital periods and resonant correlations can invalidate the clock. It is not an exact mean first-passage time.

The geometric cross-section passing a fixed point of a steady orbital wire per unit time is $J_{\sigma }={\lambda }_{\sigma }v$, where ${\lambda }_{\sigma }=d\sigma /ds$. Conservation of fragments requires $J_{\sigma }$ constant and ${\lambda }_{\sigma }\propto 1/v$. For total area ${\sigma }_{\mathrm{tot}}$ and orbital period $P$,With the fractional luminosity of a phase-mixed eccentric wire, this becomes $J_{\sigma }=2fh$. The current describes transported area, not the swept area enclosed by the orbit.

A tiny orbital period change accumulates into a larger timing shift. Choose the original model $P_b(t)=P_{b0}+{\dot{P}}_{b}t$, with $P_{b0}=28,800\mathrm{s}$ and ${\dot{P}}_b=-2\times 10^{-12}s/s$. Expanding the accumulated cycle count givesNegative $\delta T$ means an event of fixed orbital cycle count arrives earlier than a constant-period prediction. After twenty Julian years, the chosen period has decreased by $1.262\mathrm{ms}$, while the accumulated event shift is about $-13.83\mathrm{s}$. This is a time shift of periapsis passage; it is different from a rotation of the direction of periapsis.

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.

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 $\gamma$.

Here ${\mathcal{R}}_{\mathrm{lc}}=J_{\mathrm{lc}}^2/J_c^2$ is the orbital loss cone fraction, $P$ an orbital period, and $t_{\mathrm{rel}}$ a stellar relaxation time. When angular diffusion per orbit is small relative to the cone, a steady two-dimensional diffusion solution is logarithmic and gives the empty-cone flux. Rapid refilling maintains an approximately isotropic full cone and gives the second rate. Numerical total rates need a stellar density and orbital-energy distribution, not merely a star count.

An exoplanet is a planet outside the Solar system; here the focus is on planets orbiting stars other than the Sun, including pulsars. Most exoplanets are detected indirectly: a star's changing Doppler shift reveals its motion, or a small dip in its light curve reveals an exoplanet transit. The striking idea is that an unresolved point of light can tell us about an otherwise unseen world's mass, radius, and orbit.

The measurements answer different questions. Exoplanet transit photometry primarily gives a radius ratio, while Doppler spectroscopy constrains mass together with orbital inclination. Combining them with the host star's properties gives a mean mass density, ${\bar{\rho }}_p=3M_p/(4\pi R_p^3)$, useful for exoplanet interior models. Even this combination does not uniquely determine composition: see exoplanet interior-composition degeneracy. Transit spectroscopy and an exoplanet secondary eclipse then probe an exoplanet atmosphere.

The discovery history has two distinct milestones. In 1992 Aleksander Wolszczan and Dale Frail reported two planet-mass companions to the pulsar PSR B1257+12 from periodic changes in pulse arrival times; their paper also suggested an additional companion. These were the first confirmed exoplanet discoveries, around a neutron star. ^[ref]. Wolszczan's contemporary conference account, archived by NASA, likewise distinguishes the two reported companions from a possible third. ^[ref].

In 1995 Michel Mayor and Didier Queloz discovered 51 Pegasi b through Doppler spectroscopy, the landmark first confirmed exoplanet around a Sun-like star. This was a hot Jupiter, not the first exoplanet of every kind: the earlier pulsar discoveries belong to a different host category. ^[ref]. NASA's contemporary account describes the confirmation and the approximately four-day orbital period. ^[ref].

View a circular orbit along its perpendicular spin axis. Let $f$ be the orbital angle measured from a fixed direction, choosing an arbitrary zero for the circular limit of true anomaly, and let $\theta$ be the direction of an outward arrow painted on the body, measured from the same fixed direction. The direction from the body to its companion is $f+\pi$. Thus an inward-facing arrow satisfiesMore generally synchronous rotation keeps $\theta -f$ constant; its particular value selects which surface direction faces the companion. Over one orbital period $T$, both $f$ and the unwrapped $\theta$ increase by $2\pi$. This is the short geometric explanation for one turn per orbit. For a nonspinning body, $\dot{\theta }=0$ instead, and the companion direction sweeps through all surface longitudes. On an eccentric orbit, mean $1:1$ spin does not force $\theta -f$ to stay exactly constant: true anomaly advances nonuniformly, producing the longitudinal component of libration.

One motivating laboratory demonstration compares two balls initially at the same height. A switch releases one vertically when the other leaves a horizontal track; arranging their paths to meet gives a collision as the readout of equal downward displacement over equal time. Equal initial height and synchronized release control the comparison. This tests the local independence of horizontal motion and free fall, under nearly constant gravitational acceleration; it does not measure an inverse-square law.^[ref]

A complementary astronomical comparison uses the Moon's distance and orbital period to infer inward centripetal acceleration, $a=4{\pi }^{2}r/T^2$, and compares it with surface gravitational acceleration scaled by $(R/r{)}^2$. The approximately sixtyfold distance ratio gives the approximately $1/3600$ prediction. The inference links terrestrial free fall to a lunar orbit. Distance calibration matters; the circular orbit approximation and the Sun's tidal force limit this elementary comparison.^[ref]

Consider a negligible-mass star moving in the stationary Newtonian gravitational potential of a point mass. Let its semi-major axis be $a$, orbital eccentricity $0\le e<1$, and orbital period $P$. Write $\mu =GM$ for the gravitational parameter. The eccentric anomaly $E$ is determined by Kepler's equation,where $\tau$ is the time of periapsis. Coordinates in the orbital plane, with the point mass at the origin, areThe focus is the central mass; the ellipse's geometric centre is displaced by $-ae$ along the first coordinate. The varying denominator makes the star move faster near periapsis, even though the mean anomaly increases uniformly.