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A regime in which collisions disrupt aggregates faster than sticking can build larger bodies. The outcome depends on collision speed, porosity and material properties; it is distinct from loss of intact aggregates through radial drift of solids in a gas disk.

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.

Sub-Keplerian rotation means that the gas rotates more slowly than a pressure-free particle on a circular orbit at the same cylindrical radius. Let $v_K=(G{M}_{\star }/r{)}^{1/2}$. Neglecting radial acceleration, radial force balance in the gas isAn outwardly decreasing pressure, $d{P}_g/dr<0$, supplies part of the inward-gravity balance through an outward pressure force. Less centrifugal support is needed, so $v_{g\varphi }<v_K$. Using $c_s^2=P_g/{\rho }_g$, define the pressure-gradient parameter of a protoplanetary disk byThenFor a thin disk and a smooth radial pressure profile, $\eta$ is small, of order $(H/r{)}^2$. The factor of two is part of this explicit convention; changing the convention for $\eta$ changes the drift formula correspondingly.

A solid particle feels essentially no continuum gas-pressure support. Left alone, it would orbit at $v_K$, so in the gas it feels a headwind. Drag removes angular momentum, causing radial drift of solids in a gas disk. Define Stokes number $\mathrm{St}={\Omega }_K{t}_s$ using the aerodynamic stopping time $t_s$. In a dilute test-particle limit with linear drag, negligible radial gas flow and small departures from a circular Keplerian orbit, let $u_{\varphi }=v_{p\varphi }-v_K$. The steady local radial and azimuthal balances areSolving givesThe sign is inward for the usual negative pressure gradient. Very small particles, $\mathrm{St}\ll 1$, closely follow the gas and have a small relative drift $v_{pr}\simeq -2\eta v_K\mathrm{St}$. Very weakly coupled large bodies, $\mathrm{St}\gg 1$, also drift slowly, with $v_{pr}\simeq -2\eta v_K/\mathrm{St}$. The fastest pressure-driven inward drift occurs at $\mathrm{St}=1$, where $|v_{pr}|=\eta v_K$. The corresponding physical grain or aggregate size depends on gas density, orbital radius and the drag regime; “metre-size barrier” is a conventional description of some disk models, not a universal size.

The drift timescale in this model isIt can be much shorter than the disk gas lifetime, threatening loss of the material needed to make planetesimals. Growth through the vulnerable coupling range must compete with this loss, while differential drift generates collisions that may stick, bounce or fragment. Radial redistribution also changes solid-to-gas ratios and moves material across condensation fronts. These are consequences for planet formation, rather than merely a small correction to an orbital frequency.

The same mechanism can help concentrate solids. At a local radial pressure maximum, $d{P}_g/dr=0$, so this component of drift vanishes. On the inside, a positive pressure gradient gives outward drift; on the outside, a negative gradient gives inward drift. Particles therefore converge toward a dust trap. Turbulent diffusion and inward gas advection can limit retention. With a nonzero radial gas speed $v_{gr}$, the test-particle result generalizes to $v_{pr}=[v_{gr}-2\eta v_K\mathrm{St}]/(1+{\mathrm{St}}^2)$, so gas advection need not vanish even at a pressure maximum.

Finally, if local solid loading becomes large, backreaction on the gas can no longer be neglected. Coupled gas-particle drift can support streaming instability and strong solid concentrations, potentially enabling gravitational collapse into planetesimals. The simple test-particle drift formula is a useful starting point, but not a description of that nonlinear high-loading regime. Pressure support creates both a drift obstacle to growth and opportunities for concentration where the pressure structure changes.