Planetary Radii Ordering

• Planetary radii ordering is a systematic arrangement of planet sizes based on mass, composition, and orbital position.
• It integrates formation theory, empirical mass–radius relations, and stellar characteristics to classify planets into rocky, volatile-rich, and giant regimes.
• Observed trends, including inner planets generally being smaller and deviations near the radius gap, highlight the roles of migration and stochastic events.

Planetary radii ordering refers to the systematic arrangement and distribution of planet sizes within individual planetary systems and across the observed population. It encompasses both the ordinal sequence of radii within a system (e.g., inner versus outer planets), the statistical distribution of radii as a function of planet mass, and the physical mechanisms and evolutionary pathways leading to these patterns. Central observables include the relationship between planet radius, mass, composition, orbital distance, host star properties, and the collective architecture of multi-planet systems. The following sections provide a comprehensive analysis of planetary radii ordering, integrating formation theory, empirical scaling laws, compositional transitions, effects of migration and atmospheric loss, and recent observational constraints.

1. Formation Theory and the Mass–Radius Diagram

The fundamental structure of planetary radii ordering is set during planet formation, predominantly through the core accretion model (Mordasini et al., 2011, Mordasini et al., 2012). In this paradigm, planets begin as solid, heavy-element embryos that grow via planetesimal accretion. Once a critical core mass (typically 10–15 Mₑ) is achieved, runaway gas accretion begins, forming a massive envelope and producing the marked transition in radii observed in the mass–radius (M–R) diagram.

The accretion rate for the solid core is described by the Safronov-type equation:

$$\frac{dM_z}{dt} = \Omega\,\Sigma_p\,R_{\rm capt}^2\,F_G$$

where $\Omega$ is the orbital frequency, $\Sigma_p$ the disk planetesimal surface density, $R_{\rm capt}$ the (envelope-boosted) capture radius, and $F_G$ the gravitational focusing factor.

Population synthesis simulations, varying disk properties and metallicity, quantitatively reproduce the M–R diagram: low-mass planets are compact and dense (core-dominated), while high-mass planets have large radii (envelope-dominated), except for "bloated" Hot Jupiters requiring additional inflation mechanisms (Mordasini et al., 2011). Case studies (e.g., Jupiter) confirm that core accretion models can match observed mass, radius, and luminosity, and that radii ordering is dictated by timing of gas accretion and disk conditions.

2. Empirical Mass–Radius Relations and Classification

Exoplanet populations are empirically ordered via broken power-law fits to the M–R data, revealing distinct compositional transitions (Müller et al., 2023, Otegi et al., 2019, Lozovsky et al., 2018, Unterborn et al., 2022). Recent analyses (Müller et al., 2023) extract three primary regimes:

• Small (rocky) planets: $M \lesssim 4.4\,M_\oplus$, $R \propto M^{0.27}$, representing compact, core-only objects.
• Intermediate-mass (volatile-rich) planets: $4.4 \lesssim M \lesssim 127\,M_\oplus$, $R \propto M^{0.67}$, reflecting H–He envelope accretion.
• Giant planets: $M \gtrsim 127\,M_\oplus$, $R \propto M^{-0.06}$, indicating electron degeneracy and near-constant radii.

The radius–density relation identifies a sharp transition at $R \approx 1.6$–$1.64\,R_\oplus$ (Müller et al., 2023, Lozovsky et al., 2018), demarcating rocky and volatile-rich compositions. Threshold radii at $1.6\,R_\oplus$ and $2.6\,R_\oplus$ delimit rocky/ice worlds and mini-Neptunes, respectively (Lozovsky et al., 2018). Classification schemes employing the "Nominally Rocky Planet Zone" (NRPZ) (Unterborn et al., 2022) further divide planets into nominally rocky, super-Mercury (Fe-rich, high-density), and volatile-enriched regimes, using mass and radius measurements alone.

Category	Mass Range ($M_\oplus$)	Radius Power-law	Notable Physical Mechanism
Rocky/Small	$\lesssim$4.4	$R\propto M^{0.27}$	Core accretion, no gas envelope
Volatile-rich/Intermediate	4.4–127	$R\propto M^{0.67}$	H–He accretion, compositional diversity
Giant	$>$127	$R\propto M^{-0.06}$	Electron degeneracy, saturated radius

Such empirically derived relations delineate the ordering of planet radii by mass and composition across the observed population and provide a statistical foundation for compositional inference.

3. Effects of Composition, Structure, and Stellar Properties

Ordering also depends on core mineralogy and chemical composition, set by accreted solids and disk chemistry (Alessi et al., 2020, Unterborn et al., 2022). Inner disks favor refractory, high-density cores; outer disks (beyond the ice line) yield ice-rich, lower-density cores, inflating the bare core radius. Gas accretion, envelope metallicity, and atmospheric loss via photoevaporation (energy-limited escape, e.g. $\dot{M}_{loss} = \epsilon\pi F_{XUV} R_p^3/(GM_pK)$) introduce further radius diversity.

Stellar host parameters (type, metallicity) influence radii ordering in multi-planet systems (Lozovsky et al., 18 Aug 2025). Systems around metal-rich stars preferentially exhibit well-ordered architectures with more pronounced inner/small–outer/large size gradients. This is plausibly tied to enhanced core growth, outer disk envelope accretion, and differential migration outcomes.

Use of direct stellar radius measurements (e.g., interferometry via the CHARA Array (Zielinski et al., 2012)) improves the accuracy of planet radii derived from transit depths ($dF = (R_p/R_*)^2$), reducing systematic errors in radii ordering and compositional classification.

4. Architectures in Multi-planet Systems: Statistical Ordering and System Patterns

Comprehensive studies of multi-planet systems reveal robust order in sizes and spatial configuration. Observationally, the majority of adjacent planet pairs have the inner planet smaller than the outer (Ciardi et al., 2012, Lozovsky et al., 18 Aug 2025):

$\frac{R_{\mathrm{in}}}{R_{\mathrm{out}}} < 1$

This ordering is most distinct in pairs where at least one planet is Neptune-sized ($R_p \gtrsim 3\,R_\oplus$), with $\sim68\%$ of systems exhibiting this trend versus a random $50\%$ expectation.

The empirical scaling between inner-to-outer planet size ratio and period ratio can be parameterized as

$$\frac{R_{\mathrm{in}}}{R_{\mathrm{out}}} \propto \left(\frac{P_{\mathrm{in}}}{P_{\mathrm{out}}}\right)^{\beta}$$

where $\beta$ is an empirically determined exponent. Variations in multiplicity (number of planets), stellar type/metallicity, and pair position (e.g., "ab" versus "bc" pairs in three-planet systems) modulate the strength and steepness of the ordering (Lozovsky et al., 18 Aug 2025).

System Feature	Ordering Trend
Multiplicity (2–4)	Steepest for inner pairs, persists in all
Stellar Metallicity	More pronounced in metal-rich systems
Stellar Type	Most visible for G–K stars; persists overall
Resonant Pairs	No statistically distinct ordering compared to non-resonant pairs

Notably, planet pairs in orbital resonance do not display significantly different size ratios compared to non-resonant pairs (Lozovsky et al., 18 Aug 2025). This challenges models positing that resonant capture leads to strong size ordering and suggests that dynamical instabilities or post-resonant evolution may erase such differences.

5. Deviations, the Radius Gap, and Stochastic Evolution

The general trend of self-similar ordering ("peas-in-a-pod") is demonstrably interrupted in special cases. Most notably, in systems containing a planet in the radius gap (near $1.8\,R_\oplus$), adjacent pairs show a significant deficit of size ratios near unity and a statistically significant prevalence of reverse size-ordering (larger inner, smaller outer), with peaks at ratios of $\sim0.7$–$0.8$ and $\sim1.3$ (Chance et al., 3 Oct 2024). This deviation is quantified using Poisson likelihood statistics, establishing a $3$–$4\sigma$ break from generic self-similarity.

Additionally, these gap-adjacent pairs are disproportionately likely to be near mean-motion resonance (especially 3:2 or 2:1), and also lack very closely spaced (<1.4 period ratio) companions. The evidence suggests that formation and post-formation processes for radius gap planets involve stochastic events, such as late giant impacts, that both strip atmospheres and dynamically reconfigure the system.

A plausible implication is that, while smooth processes like photoevaporation or core-powered mass loss can generate the global "radius valley," the detailed local ordering and dynamical spacing of gap planets reflect the effects of stochastic perturbations on planetary architecture.

6. Theoretical Implications and Observational Constraints

The observed planetary radii ordering provides stringent constraints for planet formation and migration models. Formation theory must accommodate the steep rise in radius from core-dominated to envelope-dominated regimes, account for the bimodal radius distribution (with peaks at small radii and at $\sim1\,R_J$ for gas giants (Mordasini et al., 2012)), and explain the maximum rocky core mass threshold (approx. $4.4$–$25\,M_\oplus$) before significant H–He accretion ensues (Otegi et al., 2019, Müller et al., 2023).

The fact that smaller inner planets are found robustly in multi-planet systems aligns with models where larger bodies accrete far from the star and migrate inward, while inner planets lose envelopes to irradiation and/or photoevaporation (Ciardi et al., 2012, Alessi et al., 2020). The lack of distinct radius-ordering in resonant pairs or in radius gap neighbors points to a dynamical evolution that can erase initial resonant signatures or reorder sizes via impacts or instability.

Direct measurement improvements (e.g., via CHEOPS and CHARA (Bonfanti et al., 2021, Zielinski et al., 2012)) and comprehensive data-driven modeling (e.g., neural network regression/classification (Sandford et al., 2021)) enhance the precision and scope of radii ordering studies, revealing predictable patterns and groupings.

7. Future Directions and Limitations

Quantitative ordering of planetary radii continues to evolve as larger, more precise datasets become available (TESS, CHEOPS, PLATO, RV follow-up), and as interior structure and atmospheric models improve. Remaining challenges include:

• Degeneracies in inferred composition from mass/radius alone (Lozovsky et al., 2018).
• Sensitivity to mean molecular weights, temperature, envelope metallicity, and core fraction (Lozovsky et al., 2018, Unterborn et al., 2022).
• Uncertainties in dynamical histories, particularly in interpreting architectures shaped by migration, instability, and collisions (Chance et al., 3 Oct 2024, Fang et al., 2013).

The systematic and statistically robust ordering of planetary radii—expressed through size ratios, breakpoints in M–R diagrams, and architectural trends—is now recognized as a critical observable constraining planet formation, migration, and long-term dynamical evolution (Lozovsky et al., 18 Aug 2025). These patterns enable both compositional classification and provide diagnostic power for testing and refining theoretical models in planetary system science.