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A giant planet with a much larger heavy-element fraction than a typical gas giant, exemplified by Uranus and Neptune. Water, ammonia and other volatile-rich materials are called “ices” in formation terminology; they need not remain ordinary frozen solids in the hot deep interior.

Metallic hydrogen is a high-pressure conducting state of hydrogen. Its fluid form in the deep interiors of gas giants supplies a conducting medium for a planetary dynamo, unlike the iron-alloy conducting cores of terrestrial planets.

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 principal feature is the abundance of planets with radii of roughly one to a few Earth radii compared with large gas giants within the stated short-period sample. The smallest few plotted bins are of broadly similar height, followed by a substantial decline at larger sizes. Thus the figure supports a high occurrence of small planets relative to giants, rather than an indefinitely steep rise toward arbitrarily small sizes. The horizontal scale is logarithmic: one should compare counts in the displayed bins with their widths in $\ln R_p$, not mistake a bin count for a density per unit linear radius.

The vertical axis is an average number of planets per target star. This exoplanet occurrence rate need not equal the fraction of stars hosting a planet, because one star can host several. The restriction to periods below $85$ days also matters: the figure is not a census of distant gas giants or of every planet in a system. Neither the detailed reduction procedure nor the raw detection counts are specified in the caption, so they cannot be reconstructed uniquely from the histogram.

Selection effects actually make the small-planet population harder to see. A full, undiluted exoplanet transit depth is approximately $(R_p/R_{\star }{)}^2$. For independent photometric errors of fractional size ${\sigma }_f$ per sample, an idealized detection signal-to-noise ratio iswhere $N_{\mathrm{in}}$ counts in-transit samples. At fixed star and orbital period, halving planet radius reduces this amplitude by four. Fainter hosts, correlated stellar noise, gaps and transit vetting reduce transit survey completeness further. Geometric transit probability also falls with orbital distance, $p_{\mathrm{tr}}\simeq R_{\star }/a\propto P^{-2/3}$ for fixed stellar mass. For a fixed observing baseline and cadence, the small-planet circular duration law gives $N_{\mathrm{in}}\propto P^{-2/3}$ and approximately $\mathrm{SNR}\propto R_p^2{P}^{-1/3}$.

Recovering an intrinsic exoplanet occurrence rate therefore requires correcting for geometric alignments, incomplete recovery and false positives. An illustrative estimator for a radius-period bin iswhere $C_j$ is the recovery probability over the relevant target sample and $w_j$ accounts for candidate reliability. Real analyses must average these probabilities over heterogeneous targets and measurement uncertainty; the formula indicates the required weights, not a reconstruction of the unpublished inputs to this figure. A primary Kepler space telescope analysis explicitly models both false positives and incomplete detection to infer occurrence.^[ref]

Thus the apparent preference for small planets cannot be explained by their being photometrically easier to detect. Small transits are harder. A plateau in occurrence below about twice Earth's size was independently reported using synthetic-transit injection and recovery, though that study used a different $5$–$50$-day selection and is not identified here as the source of the exam's plot.^[ref]

The natural physical framework is core accretion in a protoplanetary disk. Most initial disk mass is gas, principally hydrogen and helium; a much smaller fraction is solid material. Protoplanetary dust coagulation first assembles aggregates through low-speed sticking. Settling and differential drag help concentrate solids, but bouncing, the fragmentation barrier to dust growth and radial drift of solids in a gas disk prevent a simple assumption that every collision grows a body smoothly to planetary size. Concentration in dust traps or through streaming instability, followed by gravitational collapse of a dense solid population, provides routes to planetesimals. Subsequent collisions must dissipate sufficient relative energy to bind material rather than disrupt it.

Once sufficiently large planetesimals exist, gravitational focusing enhances their accretion cross-sections:When background random velocities remain small, larger bodies grow disproportionately fast in runaway planetesimal growth. As they become massive enough to stir the surrounding population, oligarchic growth develops: several planetary embryos occupy neighboring feeding zones, while much of the remaining solid material resides in smaller planetesimals. Heating the planetesimal random motions weakens gravitational focusing, and local depletion drives the embryos toward an isolation mass. Collisions among embryos can continue assembly after gas dispersal.

This solid-building pathway naturally produces many modest cores. Producing a large gas giant requires an additional, time-sensitive step: a sufficiently massive core must acquire and cool a gaseous envelope while disk gas remains available. Gas captured by a low-mass core releases gravitational potential energy; the envelope's ability to radiate that energy governs its contraction and further accretion. A characteristic cooling estimate iswhere $E_{\mathrm{tot}}$ includes the envelope’s gravitational and internal energies. This states an energy budget rather than a calibrated accretion timescale. Increasing core mass generally facilitates faster gas growth in the appropriate envelope regime. The critical core mass for gas accretion depends on opacity, solid accretion heating and disk conditions; no one exact threshold follows from the histogram. If envelope cooling becomes rapid enough and gas supply persists, runaway gas accretion can produce a giant. If core growth or cooling is too slow, protoplanetary disk dispersal removes the reservoir first, leaving a rocky core or a smaller planet with a limited envelope.

The snow line changes the supply of condensed solids: where water ice can survive, additional solid material can help cores grow. Its location depends on the evolving disk temperature and pressure, so there is no universal orbital radius at which giants must form. Short-period planets need not have assembled entirely at their present locations. Planetary migration can carry cores and giants inward; scattering and collisions can further reshape their orbits. Pebble accretion is another way to supply solids rapidly to embryos, but its efficiency depends on particle coupling and the available pebble flow. These processes alter the distribution without requiring every core to become a giant.

Radius also is not a linear proxy for mass. A relatively small hydrogen-helium envelope can increase a planet's radius substantially, so some planets with a few Earth radii need not be purely rocky. Contraction as planets cool and irradiation-driven atmospheric escape can subsequently reduce their radii. Hence the final histogram combines solid growth, gas capture, migration and later thermal evolution. In particular, the smallest bins need not all share one composition or one formation location.

Direct fragmentation through gravitational instability of an astrophysical disk is an alternative route to massive gaseous objects. It requires a sufficiently self-gravitating disk, described locally by Toomre's stability criterion, together with sufficiently rapid cooling for bound fragments to survive. It does not naturally explain the dominant population of small close-in cores by itself, although migration or further evolution can change the locations of fragments. The observed abundance of small planets is therefore consistent with solid cores commonly forming while only a minority acquire a massive gas envelope before disk dispersal. It does not uniquely prove a particular prescription for each uncertain growth stage.

The physical explanation is widespread core formation combined with restricted access to runaway gas accretion. The strongest observational conclusion concerns small versus giant planets in the stated period range, after accounting for selection; the figure alone does not determine a universal core threshold, a unique mass distribution, or the population beyond the period cut.

A planetary magnetic field is evidence about electrical conductivity, fluid motion and thermal evolution, but its presence alone does not identify a present planetary dynamo. One must separate internal global fields, crustal remanent magnetization and currents driven by the solar wind. The following discussion uses the observational picture available by 2009, including the early MESSENGER flybys and the Voyager, Galileo and Cassini results.

Outside a planet's internal electric currents, the internal component is approximately a potential magnetic field. Its spherical harmonic of degree $\ell$ decreases as $r^{-(\ell +2)}$. Measurements at different positions and altitudes can therefore distinguish a deep, low-degree source from localized crustal anomalies and external magnetospheric currents. Repeated observations, radio emissions, aurorae and particle measurements supply complementary evidence for a magnetosphere, while slow secular variation and records of past reversals test the internal field's evolution.

For Earth, ground observatories and satellite measurements reveal a predominantly dipolar field of several tens of microtesla at the surface, tilted relative to the rotation axis and changing with time. Magnetic minerals in rocks record a much older field and repeated geomagnetic reversals. These observations, together with seismic evidence for a liquid outer core, support an operating planetary dynamo in conducting iron alloy. Core temperatures exceed the relevant Curie temperatures, so ordinary permanent remanent magnetization cannot explain the main field. Thermal convection, cooling and compositional convection associated with inner-core growth supply buoyancy; the Coriolis force organizes the motion. The observed nonaxisymmetric components and time dependence are consistent with a dynamic source, rather than a fixed bar magnet.

Mariner 10 and the first two MESSENGER flybys established a weak but coherent intrinsic field at Mercury, with roughly dipolar structure closely aligned with its rotation axis. It is much weaker than Earth's surface field, but its global character favors a deep source over an explanation solely by localized crustal patches. A planetary dynamo in a fluid part of Mercury's large iron-rich core is the natural interpretation, supported by evidence for a molten core component. Light elements can depress the alloy's freezing temperature, and thermal or compositional buoyancy can maintain motion. Slow rotation is not enough to rule out dynamo action. The limited pre-orbital measurements do not uniquely determine the detailed core structure or the mechanism responsible for the field's weakness.

For Venus, spacecraft including Pioneer Venus found no appreciable internally generated global field. Nevertheless its ionosphere interacts with the solar wind, producing magnetic draping, a bow shock and an induced magnetosphere. Those external signatures are not evidence for an internal planetary dynamo. Venus is comparable to Earth in size and may possess a substantial conducting core; the absence of a detectable dynamo therefore need not mean that all of its core is solid. Insufficient core heat loss or stable stratification can prevent the required convection. Its slow rotation can affect flow organization, but rotation rate by itself is not a sufficient explanation of the contrast with Earth or Mercury.

For Mars, Mars Global Surveyor found strong, spatially localized crustal anomalies, especially in ancient southern terrain, rather than an appreciable present global dipole. This is remanent magnetization, indicating that an ancient field magnetized the crust while it formed and cooled. The distribution of anomalies and weakly magnetized major impact basins constrain when that ancient planetary dynamo operated; crustal magnetization is a record of past activity, not a present global dynamo. Modern interaction with the solar wind also produces an induced magnetosphere. Cooling and changes in core heat extraction can explain the cessation of dynamo-driving convection; the observations alone do not prove complete solidification of the core.

For Jupiter, radio emissions, its intense radiation belts and measurements by Pioneer, Voyager and Galileo demonstrate a strong global intrinsic field and an enormous magnetosphere. The field is broadly dipolar but tilted and contains higher multipoles. It is generated in a deep conducting fluid rather than by a permanent solid magnet: the principal conducting region is metallic hydrogen in the high-pressure interior. Cooling and gravitational energy release maintain convection, while rapid rotation organizes the flow. Plasma supplied by Io and currents in the magnetosphere strongly modify the observed external field, but cannot account for the internally sourced planetary component. Jupiter's large magnetic moment also reflects its large size; a moment ratio should not be confused with the ratio of local surface field strengths.

For Saturn, Pioneer, Voyager and Cassini measurements likewise show an intrinsic field and a magnetosphere, with a dipole strikingly close to the rotation axis. A planetary dynamo in conducting metallic hydrogen is supported by the same broad interior argument as for Jupiter. Interior cooling and differentiation can provide power, including energy associated with helium separation. The near alignment does not invalidate the Cowling anti-dynamo theorem: that theorem excludes an exactly axisymmetric isolated dynamo field, not a predominantly axisymmetric exterior with undetected nonaxisymmetric structure in its source region. A stably stratified conducting layer above the active region is one possible way to attenuate the observable nonaxisymmetric components; it is a model, not a uniquely established consequence of the observations.

For Uranus, Voyager 2 in 1986 observed an intrinsic field with a dipole tilted by about $59^{\circ }$ to the rotation axis and significantly displaced from the centre. Higher multipoles are substantial, so a centred dipole is an inadequate description. The field and associated magnetosphere indicate an active planetary dynamo, despite the planet's small intrinsic heat flux. That low luminosity constrains its thermal evolution and may indicate inhibited deep mixing; it does not demonstrate an absence of all conducting-fluid motion. A conducting, volatile-rich fluid shell is a plausible source region. The term ice giant describes bulk composition, not a claim that the dynamo region is ordinary frozen ice.

For Neptune, Voyager 2 in 1989 also found a highly non-dipolar, displaced intrinsic field, with a dipole tilt near $47^{\circ }$. Its field, radio emissions and magnetosphere imply another active planetary dynamo. Its larger intrinsic heat flux than Uranus supplies evidence for substantial continuing interior energy transport. Electrically conducting water-ammonia-rich mixtures at high pressure, with ionic or otherwise mobile charges, can supply the conducting region without requiring the metallic hydrogen layer characteristic of gas giants. Numerical dynamos in convecting shells surrounding more stably stratified interiors can produce strongly nonaxisymmetric, multipolar fields resembling those of the two ice giants. Such models show that their unusual field geometry is compatible with dynamo action, but field measurements do not uniquely identify the shell's composition or thickness.

The maintenance mechanism is governed by the resistive induction equation,A conducting fluid needs a sufficiently large magnetic Reynolds number and a flow geometry that regenerates magnetic flux rather than merely amplifying one component temporarily. Three-dimensional stretching and twisting can couple poloidal and toroidal fields; strictly planar regeneration and a strictly axisymmetric magnetic field are restricted by the relevant anti-dynamo theorems. Rotation influences this regeneration through the Coriolis force, but does not provide an inexhaustible power source merely by being present.

Without regeneration, the characteristic magnetic diffusion time is ${\tau }_{\eta }\sim L^2/\eta$. For a terrestrial core scale of order $10^6$ metres and diffusivity of order one square metre per second, this is of order $10^4$–$10^5$ years, far shorter than geological ages. Long-lived global fields therefore require continual energy input. A schematic magnetic energy balance iswhen boundary energy fluxes vanish. Mechanical work against the Lorentz force density supplies the energy lost as Ohmic heating. Thermal cooling, latent heat release, compositional convection and differentiation supply that mechanical power. Magnetic feedback eventually changes the flow and saturates the field. Strong conductivity without an adequate energy source is not sufficient for a dynamo.

The 2009 observations support active global dynamos in Mercury, Earth, Jupiter, Saturn, Uranus and Neptune; Venus has no detected appreciable intrinsic dynamo field, and Mars preserves evidence of an extinct earlier dynamo in its crust. Induced fields and surviving remanence explain why planets without a current global dynamo can still display magnetic phenomena. The diversity of field geometries and strengths constrains, rather than uniquely solves, the diversity of planetary conducting regions and their convective histories.

The continuum exoplanet transit photometry depth is approximately the opaque projected area ratio, assuming a corrected or negligible limb darkening contribution:For a Sun-sized host, $R_p\simeq 6.96\times 10^7\mathrm{m}\simeq R_J\simeq 11R_{\oplus }$. The natural size analogue is Jupiter, not Earth.

The expected bulk constituent is molecular hydrogen, with helium next most abundant for a retained primary planetary atmosphere. The water band detects a strong trace absorber; it does not imply that water is the dominant gas. This interpretation assumes a conventional H/He gas giant; the measured radius by itself is not a measurement of mass or composition.

Let $z$ increase outward and define the inward optical depth by $d\tau /dz=-\bar{\kappa }\rho$. With constant outward internal radiative flux $F=\sigma T_{\mathrm{int}}^4$, radiative diffusion givesThe constant $q$ is a boundary condition; diffusion alone does not determine it. For an unirradiated grey atmosphere with the Eddington closure approximation and Eddington surface boundary condition, $q=2/3$, soThe internal effective temperature of a planet is defined by its intrinsic cooling flux, not by its incident stellar heating. Since $dT/d\tau >0$ and $d\tau /dz<0$, temperature decreases outward. Towards the thin upper layers, $T\to 2^{-1/4}{T}_{\mathrm{int}}\simeq 0.84T_{\mathrm{int}}$; if the density and optical-depth gradient vanish there, $dT/dz\to 0$. This is the upper nearly isothermal atmosphere. The diffusion approximation itself fails at small optical depth: the grey boundary closure supplies the approximate continuation. Small irradiation perturbs this intrinsic-flux solution.

Young, self-luminous gas giants at wide orbital separations can have intrinsic cooling dominate their photospheric budget. They are favorable for exoplanet direct imaging, especially in the infrared, where their own thermal radiation is easier to separate from the host light. A strongly irradiated hot Jupiter instead has a large stable outer radiative region set mainly by stellar heating; it can be nearly isothermal over a broad pressure range or develop an atmospheric thermal inversion if stellar light is absorbed sufficiently high. Inversion is not inevitable for every strongly irradiated planet. Deep convection begins below its radiative-convective boundary.

The intrinsic curve follows the grey formula. The irradiated curves illustrate possible shapes only; they are not solutions for a specified opacity model. Smaller optical depth corresponds to greater altitude.

Use a dry ideal gas of fixed composition with specific gas constant $\mathcal{R}=C_p-C_v$ and constant specific heat capacity at constant pressure $C_p$. For a fixed-mass parcel, $P{V}^{\gamma }=$ constant and $PV=m_{\mathrm{parcel}}\mathcal{R}{T}_g$ implyAlong a hydrostatic adiabat, $dP/dz=-{\rho }_{g}g$; hence the dry adiabatic lapse rate isFor an actual pressure-balanced parcel rising in an ambient atmosphere, $dP/dz=-{\rho }_{\mathrm{env}}g$ instead gives $d{T}_g/dz=-(g/C_p)(T_g/T_{\mathrm{env}})$. The usual lapse-rate expression is exact for a hydrostatic adiabatic column and is the local first-order result at the launch point where $T_g=T_{\mathrm{env}}$, as needed in a linear stability test. Treating an already much hotter parcel as an exact copy of the ambient hydrostatic column would be an extra approximation.

After a small upward displacement $\delta z$ from temperature equilibrium, its temperature excess isAt equal pressure, warmer gas is less dense and continues to rise. Thus the Schwarzschild criterion in altitude form isThe supplied non-strict inequality includes the marginal case; strict growth requires the strict inequality. A downward displacement gives the same stability conclusion. Efficient convection normally adjusts an initially superadiabatic gradient to a nearly adiabatic one.

Deep envelopes of gas giants and ice giants commonly transport intrinsic heat by convection, as do the planetary tropospheres of many weakly irradiated atmospheres. Earth's dry troposphere provides another approximate example, with moisture changing the lapse rate. Strongly irradiated hot Jupiters can still have deep convective interiors, while their upper radiative regions need not be convective. Composition gradients can modify the homogeneous-gas criterion and inhibit overturning even in an interior.

The mass-radius curve of solar-composition substellar objects reflects the transition from weak compression to pressure ionization and electron degeneracy pressure, followed by sustained hydrogen burning. A schematic joining representative object classes is:

The illustration is not an age-specific numerical evolutionary model. In particular, ice giants contain much more heavy material than a solar-composition giant, so a single uniform-composition equation of state does not describe the entire joined curve.

Planet/brown-dwarf naming conventions and deuterium burning do not define a universal discontinuity in the mass-radius relation. Composition, age and irradiation move the curves; the hydrogen-burning transition changes the long-term energy source more fundamentally.

For typical well-mixed structures, the dominant processes are:

Representative temperatures must specify the level: Earth has about $288\mathrm{K}$ at the surface (about $255\mathrm{K}$ effective emission temperature); Jupiter has about $165\mathrm{K}$ near one bar (about $125\mathrm{K}$ effective temperature); hot Jupiters commonly have photospheric temperatures of order $1000$–$2500\mathrm{K}$; and the Sun's photosphere is about $5800\mathrm{K}$. Upper layers, nightsides, deep interiors and the solar corona have different temperatures. These are characteristic values, not constant temperatures throughout each atmosphere.

A gas giant with a substantial H/He envelope. Intrinsic cooling, contraction and helium differentiation contribute to its thermal evolution.