Impact of rotation on synthetic mass-radius relationships of two-layer rocky planets and water worlds

TL;DR

This work demonstrates that rotation meaningfully alters mass-radius relations for rocky and water-world planets modeled with two-layer interiors, and provides fast, invertible fits R_{eq}(M,f,q) to quantify how core size and spin state shift observed radii. Using a Self-Consistent-Field approach with a modified polytropic EOS, the authors derive a unified framework to describe single-layer and two-layer planets, including a sigmoidal dependence on core size and a quadratic mass-term for the radius, while also detailing how transit-observation geometry (circular vs oblate) biases inferred structures. The study presents concrete results for LHS 1140 b, showing that rotation broadens the range of plausible interior configurations beyond spherical models and that perovskite-dominated interiors with modest rotation best reconcile the data, though two-layer alternatives remain viable within error bars. Overall, the methodology provides a practical tool for interpreting exoplanet densities in the presence of rotation, enabling more accurate inferences about core size, composition, and potential ocean layers in super-Earths and water worlds.

Abstract

We have analyzed the effects of rotation on mass-radius relationships for single-layer and two-layer planets having a core and an envelope made of pure materials among iron, perovskite and water in solid phase. The numerical surveys use the DROP code updated with a modified polytropic equation-of-state (EOS) and investigate flattening parameters $f$ up to $0.2$. In the mass range $0.1 M_\oplus < M < 10 M_\oplus$, we find that rotation systematically shifts the curves of composition towards larger radii and/or smaller masses. Relative to the spherical case, the equatorial radius $R_{eq}$ is increased by about $0.36f$ for single-layer planets, and by $0.30f$ to $0.55f$ for two-layer planets (depending on the core size fraction $q$ and planet mass $M$). Rotation is an additional source of confusion in deriving planetary structures, as the radius alterations are of the same order as i) current observational uncertainties for super-Earths, and ii) EOS variations. We have established a multivariate fit of the form $R_{eq}(M,f,q)$, which enables a fast characterization of the core size and rotational state of rocky planets and ocean worlds. We discuss how the observational data must be shifted in the diagrams to self-consistently account for an eventual planet spin, depending on the geometry of the transit (circular/oblate). A simple application to the recently characterized super-Earth candidate LHS1140b is discussed.

Figures (25)

• Figure 1: Typical configuration and notations for a rotating system made of ${\cal L}=2$ differenciated layers: the core (C) as layer 1, and the envelope (E) as layer 2. The polar radius is ${R_{\rm pol}} \equiv Z_2$ and the equatorial radius is $R_{\rm eq} \equiv R_2$. All layers rotate at the same rate $\Omega$ around the $Z$-axis.
• Figure 2: Normalized pressure ( left) and mass-density ( right) in color code ( top panels) close to the pole and to the equator for a two-layer, IP planet computed with the DROP code. The input/ouput parameters are in Tab. \ref{['tab:ref']}. Profiles along the rotation axis (spherical radius $r \equiv Z$) and along the equator ($r \equiv R$) are also shown ( bottom panels).
• Figure 3: Mass-radius relationships ( left axis) computed with the DROP code in the single layer case, for iron ( purple), perovskite ( green), and water ( cyan), and for three flattening parameters $f \in \{0,0.1,0.2\}$. Values of the input parameter $A$ are indicated along the curves for $f=0$; see Eq. \ref{['eq:biga']}. The domain where the fits are performed is indicated ( grey, shaded zone) as well as the absolute error ( right axis). The result obtained with ${\rho_{\rm s}}=1$ g/cm$^3$, $f=0$ and $A=0.6$ for water in Eq. \ref{['eq:mpeos']} is shown for comparison ( cyan, open circle). Also plotted are the empirical relationship valid in the range $1-8\, M_{\oplus}$ and built from the Earth's model by zsj16 ( black crosses), the data from Tab. B.1 of halde20 obtained from the AQUA-EOS for water assuming a $300$ K surface temperature ( thin, dotted lines), and the results by fmb07 for pure-water and pure-silicate planets ( open and filled squares), all obtained in the static case. See Sect. \ref{['sec:lhs1140b']} for a discussion about LHS 1140 b ( red dot).
• Figure 4: The central pressure $P_c$ in the same conditions as for Fig. \ref{['fig:mr_static_singlelayer']} for $f \in \{0,0.1,0.2\}$, and the classical estimate $P_0$ for spherical systems ( dashed line) according to Eq.\ref{['eq:p0']}.
• Figure 5: The rotation period $T$ of single-layer planets in the same conditions as for Fig. \ref{['fig:mr_static_singlelayer']} for $f=0.1$, together with the fit with Eq. \ref{['eq:periodfit']}. The period of the Maclaurin spheroid ( dotted lines) having with the same flattening and same mean mass density according to Eq.\ref{['eq:maclaurin0']} is also given.