Popcorn Planets: Ultra-Low Density Exoplanets

• Popcorn planets are exoplanets with Neptune-like masses and Jupiter-like radii, resulting in exceptionally low bulk densities.
• Precision radial velocity and transit data combined with MCMC analysis establish tight eccentricity limits that rule out significant tidal heating.
• The findings prompt alternative explanations, including obliquity tides, Ohmic dissipation, and unique internal structures, for the observed inflated radii.

A population of exoplanets—designated “popcorn planets”—has emerged, defined by masses comparable to Neptune’s $(0.05$–$0.2\,M_\mathrm{J})$, radii near those of Jupiter ($\sim0.8$–$1\,R_\mathrm{J}$), and exceptionally low bulk densities $(\rho_p \lesssim 0.3\,\mathrm{g\,cm}^{-3})$. With equilibrium temperatures $T_\mathrm{eq} \lesssim 1000\,\mathrm{K}$, these bodies are cooler than classical hot Jupiters, implying negligible irradiation-driven inflation. Observational and theoretical investigations focus on their anomalously large radii and potential sources of internal heating, drawing attention due to their tension with standard planetary evolution models.

1. Definition and Observational Archetypes

Popcorn planets are characterized by a Neptune-like mass range but exhibit radii close to Jupiter’s, yielding extremely low mean densities. Archetypes of this class, all on orbits of 5–7 days, include WASP-107 b ($M_p=0.1039\pm0.0039\,M_\mathrm{J}$, $R_p=0.935\pm0.023\,R_\mathrm{J}$, $$\rho_p=0.157^{+0.012}_{-0.014}\,\mathrm{g\,cm}^{-3}$$), TOI-1173 b ($M_p=0.0890\pm0.0032\,M_\mathrm{J}$, $R_p=0.783\pm0.025\,R_\mathrm{J}$, $$\rho_p=0.230^{+0.019}_{-0.026}\,\mathrm{g\,cm}^{-3}$$), and HAT-P-18 b ($M_p=0.1747\pm0.0053\,M_\mathrm{J}$, $R_p=0.955^{+0.032}_{-0.030}\,R_\mathrm{J}$, $$\rho_p=0.249^{+0.024}_{-0.026}\,\mathrm{g\,cm}^{-3}$$). Their radii are anomalously inflated for this mass regime, and classical hot Jupiter mechanisms are insufficient to explain them given their lower incident flux.

Planet	Mass $M_p$ ($M_\mathrm{J}$)	Radius $R_p$ ($R_\mathrm{J}$)	Density $\rho_p$ ($\mathrm{g\,cm}^{-3}$)
WASP-107 b	$0.1039\pm0.0039$	$0.935\pm0.023$	$0.157^{+0.012}_{-0.014}$
TOI-1173 b	$0.0890\pm0.0032$	$0.783\pm0.025$	$0.230^{+0.019}_{-0.026}$
HAT-P-18 b	$0.1747\pm0.0053$	$0.955^{+0.032}_{-0.030}$	$0.249^{+0.024}_{-0.026}$

2. Eccentricity-Tide Heating Hypothesis

The leading explanation for radius inflation in this regime has been dissipation of tidal energy from ongoing orbital eccentricity. The underlying mechanism posits that a nonzero orbital eccentricity ($e$) periodically deforms the planetary figure, generating a time-variable tidal distortion. The work against internal friction dissipates energy as heat within the planetary interior,

$$L_\mathrm{tide} = \frac{21}{2} \frac{k_2}{Q} \frac{G M_*^2 R_p^5 e^2}{a^6}$$

or, employing the modified tidal quality factor $Q_p' = Q_p / k_2,$

$$L_\mathrm{tide} = \frac{21}{2} \frac{1}{Q_p'} \frac{G M_*^2}{R_p} \left(\frac{R_p}{a}\right)^6 n e^2$$

where $k_2$ is the degree-2 Love number (dimensionless measure of tidal response), $Q_p$ is the planetary tidal quality factor, $a$ is the orbital semimajor axis, $M_*$ stellar mass, $R_p$ planetary radius, $n=2\pi/P$ the mean motion, and $e$ the orbital eccentricity. Empirically, gas giants typically have $Q_p' \sim 10^5$–$10^6$, while Neptune exhibits $Q_p' \sim 10^5$.

This framework requires $e\gtrsim 0.1$ for inflation to be significant, given canonical $Q_p'$ values, and is sensitive to the details of orbital and internal structure.

3. Precision Radial Velocity Constraints

A dedicated radial-velocity (RV) campaign with the MAROON-X spectrograph on Gemini-North targeted the three archetypal systems. The observational strategy sampling included:

• WASP-107: 20 exposures of 600 s
• TOI-1173: 30 exposures of 900 s
• HAT-P-18: 24 exposures of 600 s

Median instrumental RV precisions per epoch, obtained in blue and red channels, ranged from $0.66$ to $2.4$ m/s. Spectra were reduced using the MAROON-X pipeline, RVs extracted with SERVAL, and a joint Markov Chain Monte Carlo (MCMC) fit (Hamiltonian NUTS sampler as implemented in PyMC) combining new RVs, archival RVs, and TESS transits. The exoplanet package was used for probabilistic modeling. Instrumental offsets and jitter were fit per dataset. Eccentricity was parameterized as $\sqrt{e} \cos\omega$, $\sqrt{e} \sin\omega$ to ensure a uniform prior in $e$.

4. Upper Limits on Orbital Eccentricity

The joint RV plus transit analysis established stringent $95\%$ confidence upper limits:

• TOI-1173 b: $e < 0.037$
• WASP-107 b: $e < 0.052$
• HAT-P-18 b: $e < 0.030$

These upper limits are $1$–$2$ orders of magnitude below the $e\gtrsim0.1$ required to power substantial inflation via tidal dissipation. This result decisively rules out eccentricity-tide heating as the primary source of inflation, except under extremely unphysical assumptions for the dissipation efficiency.

5. Energetic Constraints on Tidal Heating

Assuming a representative $Q_p' = 10^5$, and taking the 68% upper limit on $e$ for each system, the calculated maximal tidal luminosities are:

• TOI-1173 b: $$L_\mathrm{tide} \lesssim 4\times10^{23}\ \mathrm{W}$$
• WASP-107 b: $$L_\mathrm{tide} \lesssim 6\times10^{24}\ \mathrm{W}$$
• HAT-P-18 b: $$L_\mathrm{tide} \lesssim 2\times10^{24}\ \mathrm{W}$$

In contrast, interior heat flux requirements for the observed radii are on the order of $L_\mathrm{int} \sim 10^{26}\ \mathrm{W}$ (e.g., for WASP-107 b, $T_\mathrm{int}\approx460\,\mathrm{K}$ from JWST $CH_4$-depletion studies yields $$L_\mathrm{int} \approx 4\pi R_p^2 \sigma_\mathrm{SB} T_\mathrm{int}^4 \sim$$ few $\times 10^{26}\ \mathrm{W}$; even $T_\mathrm{int}\approx350\,\mathrm{K}$ gives $L_\mathrm{int}\sim10^{26}\ \mathrm{W}$). Thus, the tidal heating rates obtainable with plausible $Q_p'$ are at least two orders of magnitude too small.

Reconciling the required $L_\mathrm{int}$ with $L_\mathrm{tide}$ would require $Q_p' \sim 10^2$–$10^3$, values observed for terrestrial bodies, not gas or ice giants. Such low $Q_p'$ would also lead to circularization of the orbit on timescales $\ll 10$ Myr, inconsistent with the $\sim$Gyr system ages in the absence of an extremely short-lived evolutionary stage. Consequently, ongoing eccentricity tide heating is robustly excluded as the dominant inflation mechanism.

6. Alternative Mechanisms and Evolutionary Scenarios

Multiple alternative explanations are posited for popcorn planet inflation:

• Obliquity Tides: Sustained obliquity ($\varepsilon$) induces tidal heating proportional to $\sin^2\varepsilon$. For WASP-107 b, an obliquity $\varepsilon \gtrsim 40^\circ$ can provide $L \sim 10^{27}$ W, sufficient to explain the observed radius (Millholland & Laughlin 2020).
• Ohmic Dissipation: Ionized atmospheric winds interacting with an intrinsic magnetic field can generate $L \sim 10^{26}$–$10^{27}$ W, subject to uncertain wind speeds, conductivities, and planetary $B$-fields (Batygin & Stevenson 2025).
• Structural Explanations: Anomalously low core masses ($M_\mathrm{core} \lesssim 5\,M_\oplus$) and large hydrogen-helium envelopes ($f_\mathrm{env}\gtrsim 60$–$80\%$) can yield inflated radii absent extra heat, or standard envelope fractions ($10$–$30\%$) may suffice if accompanied by significant internal heat.
• Thermal Evolution (“Delayed Cooling”): Radius “deflation” following cessation of prior heating proceeds over $\sim 100$ Myr timescales (Thorngren et al. 2021). A plausible implication is that a remnant inflation signature persists if historical heating occurred recently in a planet’s evolutionary history.

7. Implications for Eclipse Timing and Future Observations

Timing of secondary eclipses, crucial for scheduling JWST thermal emission spectroscopy, is sensitive to the quantity $e\cos\omega$ according to the phase shift formula $\Delta t_\mathrm{ecl} \approx (2P/\pi)e\cos\omega$. Previously poorly constrained $e\cos\omega$ values produced eclipse window uncertainties of $\gtrsim 12$ hours. The new joint RV+transit constraints reduce 95% confidence windows to $\pm1.7$ hr (TOI-1173 b), $\pm2.4$ hr (WASP-107 b), and $\pm1.2$ hr (HAT-P-18 b), thereby facilitating confident planning and execution of JWST and other thermal-emission observations.

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Popcorn planets—defined by Neptune masses and Jupiter radii—represent a distinct class of ultra–low-density exoplanets for which ongoing eccentricity tidal dissipation is now conclusively ruled out, thanks to stringent radial-velocity constraints on their orbital eccentricities and robust energetic arguments assuming canonical tidal quality factors. The physical origin of their anomalous radii remains an open question, invoking alternative mechanisms including obliquity tides, Ohmic dissipation, distinct envelope structures, or delayed cooling. These advances in orbital parameter estimation also yield immediate practical benefits for thermal observation scheduling, positioning popcorn planets as key targets for further atmospheric and structural investigations (Yee et al., 11 Nov 2025).