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The particular, forced part of a complex inclination solution. For a fixed, circular, inclined inner planet and an exterior test ring, linear Laplace-Lagrange secular theory gives $\dot{w}=-i\nu (w-w_p)$, so $w_f=w_p$ and $w=w_p+(w(0)-w_p)e^{-i\nu t}$. The ring describes a circle about the planetary tilt in an Argand diagram; relative to the planet, its mutual inclination stays constant while its ascending node precesses.

A planet transfers angular momentum to disk material through its spiral wakes, tending to push gas away from its orbital annulus. A persistent deep gap requires this clearing to compete with pressure and disk transport. A gap can reduce local supply without completely shutting off gas streams or fixing the final planet mass.

General relativity describes gravity through the geometry of spacetime. Its metric tensor specifies distances, proper time measured by clocks, and the causal structure that limits communication. Matter and radiation affect this geometry through the Einstein field equations; freely falling test particles follow geodesics of the resulting geometry. A planet's orbit and a falling apple therefore belong to the same geometric story. The Newtonian limit recovers familiar Newtonian gravity when fields are weak and speeds are small compared with the speed of light. See David Tong's introduction to general relativity.

The most direct way to understand spacetime curvature is to compare neighboring freely falling bodies. Even when each body feels weightless, their separation can change: this is geodesic deviation, observed as a tidal force. Curvature concerns these relative measurements, rather than the shape of a drawing in an imagined surrounding space. A second accessible measurement is gravitational time dilation: identical clocks maintained at different heights accumulate different proper times. Together, tides and clocks turn an abstract geometry into laboratory questions.

Measurement and modeling of a planet’s shape, gravity field, orientation and deformation. Space geodesy can constrain both rigid plate motion and time-dependent solid-Earth loading response.

Geophysics applies physics and mathematics to the structure, composition, and evolution of planets, including Earth. It connects measurements with mechanisms: plate tectonics explains moving crust, the geodynamo explains a planetary magnetic field, and albedo connects reflected sunlight to a planetary surface energy balance. Even a small model can reveal a useful organizing principle, such as the competition between ice-ocean albedo feedback and stabilizing thermal radiation.

A disk-gas trajectory that turns from one side of a planet's orbital radius to the other in the corotating frame. Its angular-momentum exchange contributes to the corotation torque; repeated libration without restoring gradients can saturate that contribution.

For a population with randomized vertical phases, orbital inclination $I$, and $R_p\ll a_{p}I$, the effective volume probability near the planetary plane is proportional to ${\Sigma }_c/(a_{p}I)$. The collision time then scales as $I{Q}^{3/2}{M}_p^{-2/3}{a}_p^2$, at fixed stellar mass and planetary mass density. An individual inclined Kepler orbit whose orbital nodes do not intersect the planet's orbit need not collide at all.

Kepler's laws describe three connected features of an elliptic orbit: its shape, how fast it is traversed, and how its size determines its orbital period. Kepler's first law gives an ellipse with the attracting centre at one focus of a conic; Kepler's second law gives equal swept areas in equal times; Kepler's third law relates the squared orbital period to the cubed semi-major axis. Together they turn a drawing of a Kepler orbit into a prediction of where the bodies will be and when they will return. ^[ref]

Here the mathematical model is an isolated two-body problem in Newtonian gravity, with point masses $M,m>0$ and a bound, noncolliding elliptic orbit. Let $\mathbf{r}={\mathbf{r}}_m-{\mathbf{r}}_M$ be the relative displacement, $r=|\mathbf{r}|$ the separation, and $\mu =G(M+m)$ the gravitational parameter, where $G$ is the gravitational constant. Newton's law of universal gravitation then givesThe relative Kepler orbit has semi-major axis $a$ and orbital eccentricity $0\le e<1$. Both bodies move around their shared astronomical barycenter, the system's centre of mass. In its rest frame their displacements are ${\mathbf{r}}_m=M\mathbf{r}/(M+m)$ and ${\mathbf{r}}_M=-m\mathbf{r}/(M+m)$. Their individual semi-major axes therefore add to $a$; the relative ellipse's attracting focus of a conic represents zero separation, rather than a body fixed in space.

One example will connect all three laws. After nondimensionalization, choose $a=1$, $e=0.6$ and $\mu =1$. The semi-minor axis is $b=a\sqrt{1-e^2}=0.8$, while the periapsis distance and apoapsis distance areThus $a=(0.4+1.6)/2=1$ measures the ellipse's size even though the separation changes fourfold. The specific angular momentum will be $0.8$, the two endpoint speeds $2$ and $0.5$, and the orbital period $2\pi$. These are values in the chosen scaled units, not measurements of a particular planet.

Johannes Kepler published Kepler's first law and Kepler's second law in Astronomia Nova in 1609 and Kepler's third law in Harmonices Mundi in 1619. These publication dates are documented by ^[ref]. Their appeal is that three short statements organize both geometry and timing; Newtonian gravity supplies a common dynamical explanation.

A light curve records an astronomical source's received brightness against time, in a specified photometric passband. Its measured form is a time series of timestamps, brightness estimates, and uncertainties; a smooth $F(t)$ is a model of that record. Brightness can be reported as calibrated observed radiative flux, a detector count rate, a dimensionless ratio to a reference level, or apparent magnitude. These quantities require different units and calibration conventions. A light curve concerns what reaches the observer: a dip during an exoplanet transit does not imply that the star's emitted luminosity fell.

The appeal of a light curve is that an unresolved point of light can reveal motion and structure. In exoplanet transit photometry, repeated dips constrain the orbital period; under the uniform-disc, dark-planet, full-overlap assumptions, the exoplanet transit depth is the blocked area fraction $(R_p/R_{\star }{)}^2$. A toy one-percent dip therefore corresponds to a radius ratio of $0.10$. Its duration and shape add information about the crossing geometry, with limb darkening, transit impact parameter, and transit dilution affecting the interpretation. See exoplanet transit photometry for the geometry. The first figure instead follows the measurement: an instrumental trend and sky contribution can obscure a small physical dip.

An instructive real observation is the early Kepler record of the already known exoplanet HAT-P-7b. Welsh and colleagues analysed 33.5 days from 13 May to 15 June 2009, containing 15 complete exoplanet transits at an orbital period of about $2.20473$ days. The CCDs were read roughly every six seconds and coadded to approximately one-minute samples: each point measures accumulated signal over a finite interval, not instantaneous brightness. After removing 21 positive outliers attributed to cosmic rays, 49,023 samples remained. The authors estimated a scatter of about $150$ parts per million per one-minute sample. A gradual $0.13\%$ count decrease, suspected to arise from focus drift with a fixed extraction aperture, was removed with a cubic polynomial fitted while masking the exoplanet transits. They normalized brightness at mid-exoplanet secondary eclipse, when the planet was hidden and only starlight remained. ^[ref].

What does the instrument actually measure? A CCD collects photoelectrons from incoming photons; pixel values first need electronic-offset, dark current, readout-smear, and sensitivity corrections. ^[ref]. Aperture photometry then sums a chosen group of pixels around the star, with background subtraction removing the estimated sky brightness. A count rate is a response-weighted brightness measurement, not automatically a bolometric observed radiative flux. Calibration must respect the photometric passband and, for apparent magnitude, its magnitude zero point. Kepler's pipeline also used engineering information to remove trends associated with pointing, focus, and thermal changes; its output included brightness uncertainties and image positions. ^[ref].

Calibration and interpretation need separate checks. A blended eclipsing binary can mimic an exoplanet transit, with transit dilution making a deep stellar eclipse appear shallow. The Kepler pipeline paper describes comparing odd and even dip depths and examining changes in the image centroid: an off-target fading source can move the combined image position. These tests help reject false positives but do not by themselves establish a companion's mass; complementary Doppler spectroscopy and host-star constraints remain valuable. ^[ref].

Even a real exoplanet's signal can have several contributors. Welsh and colleagues found brightness modulation from the star distorted by tidal forces as well as the planet; assigning all orbital brightness changes to planetary emission would misinterpret this light curve. ^[ref]. Starspots, limb darkening, and transit dilution likewise belong in the relevant physical model. A flexible trend fitted through a dip could absorb that dip; the masked fit in the real example illustrates why the fitting procedure is part of the measurement. Exposure-averaged light curves and phase folding of a light curve make two further limitations explicit.

The parameter $u$ in the linear limb-darkening law specifies its contrast between disk centre and the limiting edge. Its interpretation requires the chosen law and interval of frequency or wavelength. A limb-darkening coefficient fitted to an exoplanet transit may trade off against the transit impact parameter and planet radius; it is not a direct measurement of one layer's temperature.

A rotating gravitational perturbation drives an epicyclic disk response where $m(\Omega -{\Omega }_p)=\pm \kappa$. The launched density waves carry angular momentum. For a planet, inner and outer waves exert torques of opposite signs, and their imbalance contributes to migration.

The innermost planet has a large metallic core and a weak but global intrinsic magnetic field. Its observed field supports an active planetary dynamo in a fluid conducting part of that core. Slow rotation does not by itself exclude dynamo action.

A natural satellite is a naturally occurring body orbiting a planet or another nonstellar body. The Moon is Earth's natural satellite; its synchronous rotation provides a familiar example of tidal locking.

By Charles F. Gammie. The2001 local razor-thin cooling model distinguishes self-regulated gravitational turbulence from fragmentation. Its numerical cooling threshold is model-specific; it is not a universal planet-formation constant.

The difference between star-plus-planet light outside secondary eclipse and star-only light during eclipse isolates the planet's dayside spectrum. Thermal radiation probes its temperature-pressure structure and opacity, while optical reflected light probes scattering and albedo. Contrast and the subtraction of stellar photons limit sensitivity.

A stable sinusoidal radial-velocity semi-amplitude can be stellar reflex motion caused by a companion on a nearly circular orbit. The amplitude is compatible with a giant planet; a low-mass star or brown dwarf viewed nearly face-on can mimic the same projected motion, but requires an increasingly unlikely orientation. Starspots and magnetically suppressed convection can produce a rotational stellar activity signal on a ten-day timescale; contemporaneous photometry, activity indicators and stellar spectral-line bisector and other profile changes distinguish these from a clean common Doppler shift. Solar-like stellar oscillations ordinarily have much shorter periods, so a coherent ten-day oscillation of an otherwise ordinary solar-type dwarf is less plausible. Instrumental drift, sampling aliases and blended spectra also need checking. A sinusoid alone does not establish a planetary origin, and the data do not assign numerical probabilities to these alternatives.

For the minimum companion mass from radial velocity, take a circular two-body orbit and initially $M_{\star }=M_{\odot }$. The exact relation and its low-mass limit areWith $\sin i=1$,Here $a=[G{M}_{\star }{P}^2/(4{\pi }^2){]}^{1/3}$ is the relative orbital semi-major axis. Nonzero orbital eccentricity multiplies the inferred $M_p\sin i$ by $\sqrt{1-e^2}$ at fixed measured $K,P$, but the sinusoidal model motivates $e=0$ here.

For the radial-velocity sensitivity limit at fixed planet mass, use $K=M_p\sin i\sqrt{G/(M_{\star }a)}$ and interpret the stated detection scale as a semi-amplitude threshold $K_{\min }$. The largest distance occurs for $\sin i=1$:For an inclination $i$, this boundary is smaller by ${\sin }^{2}i$. The result assumes adequate time coverage and neglects the stipulated stellar noise. If the detection scale instead denotes a full peak-to-peak excursion, set $K_{\min }$ to half that scale; $a_{\max }$ then increases fourfold.

The stellar-mass scaling of radial-velocity planet estimates changes these inferences because Kepler's third law and the stellar reflex response depend on the stellar mass. At fixed $K,P$, $M_p\sin i\propto M_{\star }^{2/3}$ and $a\propto M_{\star }^{1/3}$, whereas the fixed-mass detectability boundary scales as $M_{\star }^{-1}$. ThereforeThe observed period and amplitude are unchanged. The stellar radius does not enter these leading dynamical estimates, but is crucial for the exoplanet transit inference.

For the transiting companion, assume an opaque spherical planet, negligible transit dilution and negligible limb darkening. Its radius ratio is $k=R_p/R_{\star }=\sqrt{\delta }=0.03873$, henceUsing the revised stellar mass and a circular orbit gives $a=0.01850\mathrm{au}$ and $A=a/R_{\star }=7.957$. Interpret the duration as the total first-to-fourth-contact duration $T_{14}$. The impact parameter from total transit duration follows from the total duration of a circular planetary transit:This is noncentral but not grazing: $b<1-k$. The small-planet straight-chord approximation gives $b\simeq 0.845$; retaining finite $R_p$ and orbital curvature gives the stated value. A duration measured between other contacts needs the corresponding contact formula. Unknown orbital eccentricity and argument of periapsis change the transverse speed and prevent a unique unconstrained $b$; the numerical estimate is conditional on the stated circular model.^[ref]

Because the transit constrains $\sin i\simeq 0.994$, the RV nondetection now bounds the actual planetary mass, rather than only its projected mass. The bulk-density upper limit from a transit and an RV nondetection isThese bounds assume the instrumental threshold can be applied to a fitted signal at the transit period; no statistical confidence level is specified. Such a large radius at this low mass is inconsistent with an ordinary compact rock-and-iron body. A low-density volatile-rich planet, possibly with an extended H/He envelope, is indicated under the planetary interpretation. The observations do not uniquely determine interior composition, atmospheric abundance or formation history; blended eclipses and activity still require validation. There is no luminosity or effective temperature supplied for the host, so a numerical planetary equilibrium temperature cannot be inferred from these data alone.

The numerical conversions above use externally documented $(GM{)}_{\odot }=1.3271244\times 10^{20}{\mathrm{m}}^3{\mathrm{s}}^{-2}$, $R_{\odot }=6.957\times 10^8\mathrm{m}$, $M_{\oplus }=5.9722\times 10^{24}\mathrm{kg}$, $R_{\oplus }=6.371\times 10^6\mathrm{m}$, $M_J=1.89813\times 10^{27}\mathrm{kg}$, $G=6.67430\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ and $1\mathrm{au}=149597870700\mathrm{m}$. They are conversion constants, not additional measured system parameters.^[ref][ref]

Core accretion builds a solid core and then a massive gaseous envelope before protoplanetary disk dispersal. The solid and gas stages overlap; this is not direct collapse of a whole disk patch.

In a protoplanetary disk, refractory grains and, beyond a snow line, condensed ices provide solids. Settling and protoplanetary dust coagulation build aggregates, but fragmentation, bouncing and radial drift of solids in a gas disk can interrupt simple sticking. Concentration processes such as the streaming instability can produce dense particle structures that collapse into planetesimals. Subsequent collisions build planetary embryos; alternatively, embryos can efficiently capture drifting particles through gas-assisted pebble accretion. Neither the initial planetesimal population nor unrestricted growth through every size should be assumed without a mechanism.

For gravitationally focused growth of a planetary core, the two-body capture cross-section isAn illustrative thin-disk estimate is ${\dot{M}}_c\sim {\Sigma }_s\Omega {\sigma }_{\mathrm{coll}}$, up to geometry and velocity-distribution factors. At fixed solid density and random speed, strong gravitational focusing gives ${\dot{M}}_c\propto M_c^{4/3}$: runaway planetesimal growth. As large embryos stir the smaller bodies, encounter speeds increase and focusing weakens, giving oligarchic growth. Consuming the local feeding zone leads toward an isolation mass; radial supply, migration and mergers can extend growth. Snow lines can increase the solid reservoir, but a distant core also grows more slowly because orbital times are longer.^[ref]

A growing core binds gas over a scale limited by the smaller of its Bondi radius and Hill radius. The envelope approximately satisfiesAn equation of state, disk boundary conditions, and radiative or convective energy transport close the structure. In an optically thick radiative region,Here $\kappa$ is opacity; an unstable radiative gradient instead drives convection. Solid accretion releases roughly $L_s\sim G{M}_c{\dot{M}}_c/R_c$, heating and supporting the envelope. Gas capture consequently depends on cooling, not just whether the potential can bind gas instantaneously.

The envelope loses energy on a Kelvin-Helmholtz cooling time, $t_{\mathrm{KH}}\sim |E_{\mathrm{env}}|/L$. Lower grain opacity, higher core mass, or reduced solid-accretion heating can accelerate contraction. The critical core mass for gas accretion is therefore model dependent; “of order ten Earth masses” is a common scale, not a universal switch. As envelope self-gravity becomes important and the gas and solid masses become comparable, contraction permits still more gas capture, producing runaway gas accretion. The classical sequence has a rapid solid-growth phase, a slower envelope-growth phase, and accelerated gas accumulation.^[ref]

The decisive race is $t_{\mathrm{core}}+t_{\mathrm{envelope}}\lesssim t_{\mathrm{disk}}$. Failure to cool or obtain a sufficiently large core before gas removal leaves a smaller solid or ice-rich planet. Successful growth is ultimately limited by disk supply, gap opening by a planet, and dispersal; a gap does not completely seal off accretion. Angular-momentum exchange drives planetary migration, so the observed orbit need not be the formation radius. Accretion-shock radiation and subsequent contraction set the young giant's entropy and luminosity; final core mass, envelope enrichment and orbital location depend on the whole growth history, not on one fixed threshold. The necessary cooling, supply and orbital constraints make the model testable rather than an assertion that every embryo must become a gas giant.

The Hill radius compares planetary gravity with the differential stellar tide. Expanding the stellar tide near a circular orbit gives $G{M}_p/r_H^2\simeq 3{\Omega }^2{r}_H$, soIt sets the size of the planet's tidal neighbourhood, controls encounter and feeding zone scales, and limits the region within which material can remain bound against stellar shear. It is not itself a solid surface or a guarantee of capture.

The Bondi radius compares planetary potential energy per unit mass with the gas thermal scale,This convention uses the ambient sound speed; definitions based on escape speed can differ by a factor of two. Inside $r_B$, gravity can substantially perturb thermal gas. Gas still needs a way to dissipate energy and angular momentum to accrete; a geometric boundary alone does not supply a gas-capture rate.

For an embedded hydrostatic envelope a useful outer scale is $r_{\mathrm{out}}\sim \min (r_B,r_H)$, subject to disk thickness and gas recycling. Let $h=H/a=c_s/(a\Omega )$ and $q=M_p/M_{\star }$. The Bondi-Hill crossover in a protoplanetary disk obeysSmall cores are usually thermally limited, whereas larger ones become tidally limited. These radii also organize pebble accretion: gas drag can turn an encounter into capture, with drift-dominated and shear-dominated regimes depending on particle stopping and encounter times. There is no universal accretion efficiency fixed by $r_H$ or $r_B$ alone.^[ref]

Type I migration is orbital angular-momentum exchange between an embedded low-mass planet and its disk without a deep gap. The planet excites spiral density waves near Lindblad resonances; the disk's reaction torques act back on the planet. The outer wake usually supplies a negative torque and the inner wake a positive one, with their small imbalance often driving inward migration. Gas on horseshoe orbits in a planetary disk near corotation adds a corotation torque that depends on entropy and vortensity gradients.

For a thin disk, define $q=M_p/M_{\star }$, $h=H/a$ andThe coefficient $C$ is signed and depends on disk gradients, thermodynamics and diffusion; it is not a universal constant. For a circular orbit with fixed planet mass, $L_p=M_p{a}^2\Omega \propto a^{1/2}$. The Type I migration timescale from disk torque follows from ${\dot{L}}_p=\Gamma$:Thus a more massive embedded core generally migrates faster within this regime, unlike a picture in which migration starts only after giant-planet formation.

In a simple locally isothermal smooth disk the net torque is commonly negative, potentially removing cores before they acquire gas. In a non-isothermal disk an unsaturated entropy-related corotation torque can balance or exceed the negative wave torque, giving a planet trap or outward migration. Viscosity and thermal diffusion help maintain the background gradients across the horseshoe region; without adequate replenishment corotation-torque saturation can suppress that contribution. The linear low-mass scaling requires a small disturbance, roughly $q\ll h^3$; larger planets, gap formation and strong disk feedback require a different treatment. Type I migration is therefore neither universally inward nor literally viscous advection of the planet.^[ref]

In transit spectroscopy, starlight passes tangentially through the planet's terminator atmosphere during primary transit. Wavelength-dependent opacity changes the effective occulting radius. For a thin atmosphere with an absorption feature spanning $N_H$ atmospheric scale heights,Here $\mu$ is the mean particle mass in units of $m_H$, not a chemical potential. Large planets with warm low-gravity, low-molecular-weight atmospheres against small stars have larger signals. A spherical isothermal atmosphere has tangential column enhancement of order $\sqrt{2\pi R_p/H}$ over a vertical column, explaining sensitivity to tenuous high-altitude gas. Molecular bands, atomic lines, scattering slopes and clouds constrain the terminator, but cloud opacity, reference pressure and abundance-temperature degeneracies limit unique retrievals.^[ref]

Occultation spectroscopy instead subtracts the star-only spectrum during secondary eclipse from the star-plus-planet spectrum immediately outside eclipse. It measures the dayside planet/star flux ratio. In the blackbody approximation the thermal eclipse depth isLonger infrared wavelengths generally improve contrast for a cooler planet; molecular absorption or emission depends on the vertical temperature gradient. At optical wavelengths reflected light can dominate, with full-phase contrast $A_g(\lambda )(R_p/a{)}^2$, where $A_g$ is geometric albedo. Eclipse spectra therefore probe dayside emission or reflection, rather than the transit's slant absorption. Clouds and temperature inversions affect interpretation.^[ref]

For either measurement, the photon-limited signal-to-noise ratio scales as the spectral feature amplitude times $\sqrt{N_{\gamma }}$, with a penalty for subtracting finite in- and out-of-event samples. Finer spectral bins reduce photons per bin; bright hosts, repeated events and stable instruments improve sensitivity. Detector systematics, telluric absorption for ground observations, and variable starspots can limit the practical gain. Space observations remove telluric contamination, not stellar variability or instrumental drift. Neither technique directly measures a unique bulk composition or samples all longitudes of the planet.