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The signed instantaneous contribution of apsidal precession to azimuthal motion during isotropic stellar mass loss is $\dot{\varpi }/\dot{\theta }=-(\dot{\mu }/\mu n)(1-e^2{)}^{3/2}\sin f/[e(1+e\cos f{)}^2]$. Its maximum absolute magnitude is small when $n|\mu /\dot{\mu }|\gg F(e)$, where $F(e)\sim 1/e$ near a circular orbit and $F(e)\to 3\sqrt{3}/4$ as $e\to 1$. The low-orbital eccentricity coordinate singularity can be avoided by evolving the eccentricity vector.

Time averaging over a Kepler orbit gives $⟨\cos f⟩=-e$, as follows from $dt=(1-e\cos E)dE/n$ and $\cos f=(\cos E-e)/(1-e\cos E)$. Consequently slow isotropic stellar mass loss preserves the secular orbital eccentricity. Conserved specific angular momentum then gives $\mu a=\mathrm{constant}$, so pericentre, apocentre and semimajor axis expand inversely with the remaining gravitational parameter.

In the isolated model, conservation of energy and angular momentum conservation reduce the exterior test particle's motion toHere $\epsilon$ is specific orbital energy and $h$ is specific angular momentum. The effective potential is ${\Phi }_{\mathrm{eff}}=h^2/(2r^2)-\mu /r$. A zero radial velocity is a radial turning point, and need not mean zero total speed: at the periapsis or apoapsis of a nonradial Kepler orbit, tangential velocity remains. Thus the vertical gap between $\epsilon$ and $-\mu /r$ represents total kinetic energy per unit mass, whereas the gap above ${\Phi }_{\mathrm{eff}}$ represents its radial part.

For $h>0$, the orbital eccentricity obeys $e^2=1+2\epsilon h^2/{\mu }^2$. Negative $\epsilon$ gives an elliptic orbit, including a circular orbit at its minimum allowed energy; zero $\epsilon$ gives a parabolic trajectory; positive $\epsilon$ gives a hyperbolic Kepler orbit. The effective potential tends to zero from below at infinity, so its outer region creates no extra positive escape barrier. This explains why escape velocity depends on energy even though angular momentum changes the path. The $h=0$ case is radial motion, rather than a nondegenerate parabola or hyperbola; inward radial motion can end in collision.

The difference between the angular momentum of circular coplanar orbits at fixed semi-major axes and the system's actual angular momentum along the reference normal. With circular momenta $L_j=m_j\sqrt{G{M}_{\star }{a}_j}$ it is ${\sum }_j{L}_j(1-\sqrt{1-e_j^2}\cos I_j)$. For small planar orbital eccentricities its quadratic part is $\frac{1}{2}{\sum }_j{L}_j|z_j{|}^2$. Its conservation in $\dot{z}=iAz$ requires $L_j{A}_{jk}=L_k{A}_{kj}$.

The apoapsis of an Earth-centred orbit. For an elliptic orbit its distance from the Earth's centre is $a(1+e)$, in terms of semimajor axis and orbital eccentricity.

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.

The geometric light-travel contribution from the pulsar's position within the binary orbit. To construct an eccentric Kepler orbit, use eccentric anomaly $E$, orbital eccentricity $e$, mean anomaly $M=2\pi (t-T_0)/P_b$, and argument of periapsis $\omega$. Here $T_0$ is passage through periapsis. Coordinates in the orbital plane are $X=a_1(\cos E-e)$ and $Y=a_1\sqrt{1-e^2}\sin E$. Rotate through $\omega$ and project by $\sin i$ to obtainThe first equation is Kepler's equation: equal time intervals give equal increments of $M$, not of $E$. Since $1-e\cos E>0$ for $0\le e<1$, each $M$ has a unique $E$ on the corresponding unwrapped orbit. A constant delay is absorbed into the arbitrary arrival-time origin. The circular limit is ${\Delta }_R=x\sin (M+\omega )$, so its amplitude measures $x$. For $e>0$, the delay's offset and amplitude also depend on $\omega$; its maximum is not generally $x$.

A phase-mixed population secularly forced by a distant eccentric planet has typical orbital eccentricity proportional to $e_{p}a/a_p$. Consequently a relative speed of order orbital eccentricity times Keplerian speed gives $v_{\mathrm{col}}^2\sim e_p^2(a/a_p)GM/a_p$. Comparing specific impact energy with a catastrophic disruption threshold gives a lower disruption radius $a/a_p\gtrsim Q_D^{\ast }{a}_p/(GM{e}_p^2)$, up to an order-one coefficient depending on velocity statistics and impact-energy convention. The planet mass sets the secular stirring time, but cancels from this energy scaling.

With constant mean motions and $P_c/P_b=(1+\delta )(p+q)/p$, mean conjunctions advance by $\Delta \Lambda =2\pi /(P_c/P_b-1)$ each synodic period. Tracking one branch every $q$ conjunctions and subtracting $2\pi p$ givesThe usual formula $-2\pi p\delta /(q{P}_b)$ keeps only first order in $\delta$. A labelled branch takes $q{P}_b(1+\delta )/(p|\delta |)$ to turn once, whereas the full set of $q$ interchangeable directions repeats after $P_b(1+\delta )/(p|\delta |)$. Finite orbital eccentricity introduces true-conjunction corrections.

For bound Kepler orbits, the Delaunay variables obey $0<G<L$, $-G<H<G$, and three periodic angles. A canonical transformation preserves phase space volume. At fixed $L$, integrating $H$ supplies the measure $2GdG$. Since $G=L\sqrt{1-e^2}$, the fraction with orbital eccentricity less than $e$ is $L^{-2}{\int }_{L\sqrt{1-e^2}}^{L}2GdG=e^2$. Thus the probability density function is $2e$, independently of any normalizable energy-only phase-space distribution function. A Boltzmann distribution is unnecessary. Even the time-dependent density $F(E(L))[1+\eta \cos (l-n(L)t)]$, $|\eta |<1$, has this same marginal because integrating the mean anomaly removes its oscillatory part; an eccentricity marginal alone does not imply equilibrium.

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.

For resonant trapping of dust in an exterior $(p+q):p$ mean-motion resonance, let Poynting–Robertson drag have parameter $A$. Holding semi-major axis $a$ fixed and retaining secular rates through second order in orbital eccentricity givesFor $5p>q$, squared orbital eccentricity approaches this equilibrium exponentially in the truncated model. If the predicted value is not small, higher-order terms are needed for quantitative dynamics. Finite planetary torque also imposes a minimum eccentricity for balance, which alone does not guarantee capture during migration.

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 $\gamma$.

If retained Lagrange planetary equations have $\dot{a}=-2(p+q)a{e}^q{C}_r\sin \varphi$ and $\dot{e}=-q{e}^{q-1}{C}_r\sin \varphi$, then $d(e^2)/d\log a=q/(p+q)$. Thus $e^2-[q/(p+q)]\log (a/a_{\mathrm{ref}})$ is constant. A resonantly oscillating orbital eccentricity–semi-major axis trajectory retraces one curve rather than enclosing area in that projection.

A first-order resonant orbital perturbation changes the semi-major axis and therefore the mean motion. Integrating the sinusoidal mean-motion variation over the near-resonant transit-timing superperiod gives a mean-longitude amplitude proportional to ${\mu }_{2}e/{\Delta }^2$. Converting phase to transit time gives the displayed amplitude ratio. It assumes small perturbations, circulating resonant argument, effectively fixed free orbital eccentricity, and dominance of this enhanced term over direct phase and forced-eccentricity contributions. The formal divergence as $\Delta \to 0$ signals failure of this expansion, not infinitely large transit times.

For an attractive Kepler orbit with gravitational parameter $GM$, velocity $\mathbf{v}$ and specific angular momentum $\mathbf{h}=\mathbf{r}\times \mathbf{v}$, the eccentricity vector isIts magnitude is the orbital eccentricity, and it points toward periapsis for a noncircular orbit. It is a normalized attractive Laplace-Runge-Lenz vector; it remains constant under the pure inverse-square force.

An orbit with nonzero orbital eccentricity. A bound Kepler orbit with $0<e<1$ is a noncircular ellipse; $e=1$ and $e>1$ describe the parabolic and hyperbolic unbound cases. A semi-major axis fixed by Kepler's third law need not equal the time average of the orbital radius.

Take an illustrative elliptic Kepler orbit with semi-major axis $a=1$, orbital eccentricity $e=0.6$ and gravitational parameter $\mu =1$, in consistent dimensionless units. Its semi-minor axis is $b=a\sqrt{1-e^2}=0.8$, mean motion is $n=\sqrt{\mu /a^3}=1$, and specific angular momentum magnitude is $h=\sqrt{\mu a(1-e^2)}=0.8$. About the attracting focus,The two sectors below each span $\Delta t=0.5$, giving $\Delta A=0.2$. Near periapsis the short radius sweeps a wide angle; near apoapsis the long radius sweeps a narrow angle. Constant areal velocity therefore does not require constant angular velocity or constant speed.

To compute the sectors without mistaking equal angles for equal times, parametrize the ellipse by eccentric anomaly $E$:This follows by integrating $\frac{1}{2}(xdy-ydx)$. Kepler's equation gives $E-e\sin E=M=n(t-t_{\mathrm{p}})$, with mean anomaly $M$ measured from periapsis time $t_{\mathrm{p}}$. Consequently $\Delta A=abn\Delta t/2=h\Delta t/2$. The displayed sectors use mean anomaly intervals $[-0.25,0.25]$ and $[\pi -0.25,\pi +0.25]$, rather than equal eccentric anomaly increments.

With a fixed planetary pericentre, the linear secular equation $\dot{z}=iA(z-Z_f)$ has solution $z=Z_f+(z(0)-Z_f)e^{iAt}$. An initially circular orbit follows $z=Z_f(1-e^{iAt})$, a counterclockwise circle in the complex plane centered at the forced eccentricity $Z_f$, with free eccentricity amplitude $Z_f$. Its maximum orbital eccentricity is $2Z_f$ and its phase-averaged mean orbital eccentricity is $4Z_f/\pi$.

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.