Potential Melting of Extrasolar Planets by Tidal Dissipation

We consider a composite planet with a liquid core (e.g., melted via radioactive heating) and a rocky mantle. The heart of the mechanism stems from the fact that the volume-integrated deposition of energy from tidal stresses increases as the mantle thickness decreases. This feedback mechanism allows for a runaway process: As a thin shell surrounding the core melts, more total energy is deposited in the now-thinner mantle, which leads to further melting—as long as sufficient energy is deposited to raise the temperature above solidus. The feedback loop leads to saturation when melting of the mantle no longer increases the efficiency of tidal dissipation. As long as the tides are strong enough to melt the mantle faster than the body can radiate away the energy deposited, some of the tidally dissipated energy is released via widespread surface volcanism. We outline this process in Figure 1.

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The deformation of a two-layer body with a melted core and a solid mantle by an external potential was first calculated by Herglotz (1911) and updated by Harrison (1963). Zschau (1978) extended the analysis to parameterize the total tidal heating in a stratified and compressible body in the imaginary part of the Love number, ${\mathfrak{I}}({k}_{2})$. This analysis was extended by Peale & Cassen (1978), who first outlined the runaway melting mechanism with applications to Io (Peale et al. 1979). It should be noted that volcanism on Io has since been attributed to subtler effects such as partial melting (Lopes & Spencer 2007; Keane et al. 2023).

Beuthe (2013) derived the Love number, k₂, for a composite planet consisting of a liquid core and a rocky mantle (with rigidity μ), assuming the core and mantle have the same density, ρ. The result is

$$\begin{eqnarray}&&{k}_{2}=\displaystyle \frac{3}{2}{\left(1+{ \mathcal Z }(\eta )\displaystyle \frac{\mu }{\rho {{gR}}_{{\rm{P}}}}\right)}^{-1}.\end{eqnarray} \tag{ 1 }$$

In Equation (1), the parameter η = R_C/R_P is the liquid core radius, where R_C is the radius of the core, R_P is the total radius of the planet, and g is the surface gravitational acceleration. The function ${ \mathcal Z }(\eta )$ is derived in Equation (112) of Beuthe (2013):

$$\begin{eqnarray}&&{ \mathcal Z }(\eta )=12\left(\,\displaystyle \frac{19-75{\eta }^{3}+112{\eta }^{5}-75{\eta }^{7}+19{\eta }^{10}}{24+40{\eta }^{3}-45{\eta }^{7}-19{\eta }^{10}}\right).\end{eqnarray} \tag{ 2 }$$

A viscoelastic mantle can be modeled by a complex rigidity $\tilde{\mu }$. The modulus of the complex number gives the amplitude and the argument gives the phase lag between the perturbation force and the resulting deformation. For the remainder of this work, a quantity with a tilde indicates that it has both a real and imaginary component. Following the correspondence principle (Biot 1954), we can obtain the complex Love number ${\tilde{k}}_{2}$ by replacing the real rigidity of an elastic solid with the complex rigidity $\tilde{\mu }$ of the viscoelastic solid. The complex rigidity $\tilde{\mu }$ is highly uncertain, and depends on the rheology of the mantle.

The tidal heating rate, ${\dot{E}}_{\mathrm{Heat}}$, is related to ${\mathfrak{I}}({\tilde{k}}_{2})$ by ¹⁵

$$\begin{eqnarray}&&{\dot{E}}_{\mathrm{Heat}}=5{{\rm{\Omega }}}_{{\rm{P}}}\left(\displaystyle \frac{G\,{M}_{* }^{2}\,{R}_{{\rm{P}}}^{5}}{{a}^{6}}\right){\mathfrak{I}}({\tilde{k}}_{2})\,{{\rm{\Psi }}}_{0}\left(e,I,{\rm{\Delta }}{\rm{\Omega }}/{{\rm{\Omega }}}_{{\rm{P}}}\right).\end{eqnarray} \tag{ 3 }$$

In Equation (3), Ω_P is the rotation rate of the planet, M_* is the stellar mass, and Ψ₀(e, I, ΔΩ/Ω_P) is a dimensionless potential function of eccentricity e, obliquity I, and satellite rotation compared to the mean motion. For the case of zero obliquity (I = 0) and synchronous rotation (ΔΩ = Ω_P), the dimensionless potential function is given by Ψ₀ = 21e²/5. For this case the heating rate reduces to the form

$$\begin{eqnarray}&&{\dot{E}}_{\mathrm{Heat}}={{\rm{\Omega }}}_{{\rm{P}}}\left(\displaystyle \frac{G\,{M}_{* }^{2}\,{R}_{{\rm{P}}}^{5}}{{a}^{6}}\right){\mathfrak{I}}({\tilde{k}}_{2})\left(\displaystyle \frac{21}{2}{e}^{2}\right).\end{eqnarray} \tag{ 4 }$$

Note that the conventional tidal Q is defined as

$$\begin{eqnarray}&&{\mathfrak{I}}({\tilde{k}}_{2})=| {\tilde{k}}_{2}| /Q,\end{eqnarray} \tag{ 5 }$$

where by construction in most cases ${\mathfrak{R}}({\tilde{k}}_{2})\gg {\mathfrak{I}}({\tilde{k}}_{2})$. Consider a simple model for $\tilde{\mu }$ of the form

$$\begin{eqnarray}&&\tilde{\mu }=\displaystyle \frac{\mu }{1+i\delta },\end{eqnarray} \tag{ 6 }$$

where μ and δ are real. For example, a Maxwell rheology has δ = ω_M /ω with ω_M the relaxation (Maxwell) frequency and the tidal frequency ω. This ansatz implies that

$$\begin{eqnarray}&&{\mathfrak{I}}({\tilde{k}}_{2})=\displaystyle \frac{3}{2}\displaystyle \frac{{ \mathcal Z }(\eta )\delta \hat{\mu }}{| 1+{ \mathcal Z }(\eta )\hat{\mu }{| }^{2}+{\delta }^{2}},\end{eqnarray} \tag{ 7 }$$

where the quantity $\hat{\mu }$ is defined as

$$\begin{eqnarray}&&\hat{\mu }=\displaystyle \frac{\mu }{\rho {{gR}}_{{\rm{P}}}}.\end{eqnarray} \tag{ 8 }$$

The corresponding Q-value is given by

$$\begin{eqnarray}&&\displaystyle \frac{1}{Q}=\displaystyle \frac{{ \mathcal Z }(\eta )\delta \hat{\mu }}{\sqrt{{\left({ \mathcal Z }(\eta )\delta \hat{\mu }\right)}^{2}+{\left(1+{ \mathcal Z }(\eta )\hat{\mu }+{\delta }^{2}\right)}^{2}}}.\end{eqnarray} \tag{ 9 }$$

In Figure 2, we show the imaginary part of the Love number, ${\mathfrak{I}}({\tilde{k}}_{2})$, as a function of liquid core radius η = R_C/R_P for several different values of $\hat{\mu }=\mu /\rho {{gR}}_{{\rm{P}}}$ and of δ.

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It is evident that the runaway melting operates for a range of $\hat{\mu }$ and δ. In Figure 3, we show the total tidal heating for a range of dimensionless rigidities and liquid core radii. These heating rates are calculated using Equations (4) and (7), where we assume δ ≪ 1. We plot the quantity ${ \mathcal E }(\eta )$, which we define as the ratio of the total heating as a function of liquid core radius to the unmelted case:

$$\begin{eqnarray}&&{ \mathcal E }(\eta )\equiv \displaystyle \frac{{\dot{E}}_{\mathrm{Heat}}(\eta )}{{\dot{E}}_{\mathrm{Heat}}(\eta =0)}.\end{eqnarray} \tag{ 10 }$$

Note that Figure 1 in Peale et al. (1979), which shows the melting mechanism for Io, corresponds to one horizontal slice of this plot. Note that the energy dissipated in the two-layer case is greater than the energy in the homogeneous case for a wide range of the parameter space, roughly when $\hat{\mu }\gtrsim 0.1$.

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In most of the parameter space shown in Figure 3, the melting mechanism operates because the normalized total heating increases with liquid core radius. This occurs when the derivative of the heating with respect to the liquid core radius is positive, or when $d{ \mathcal E }/d\eta \gt 0$. As a result, the runaway melting saturates when this derivative is zero. In Figure 4, we show this derivative for the same range of liquid core radius and rigidity in Figure 3. Note that the contour of $d{ \mathcal E }/d\eta =0$ indicates where the melting saturates and should correspond to the approximate location of the liquid core radius.

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However, it is critical to note that this is only an approximate estimation of the liquid core radius which relies entirely on the model assumptions. More detailed previous calculations estimate a more accurate equilibrium by balancing heat production from mechanisms including both tidal and radiogenic heating against heat loss via mechanisms such as heat piping, conduction, and convection (Barr et al. 2018; Dobos et al. 2019). In this work, we apply this idealized model to exoplanets for which there are little near-term prospects for accurately measuring the core radii. Therefore, we have not performed a more detailed estimation of the equilibrium state incorporating additional—yet unconstrained—heat production and loss mechanisms.

In this section, we calculate which currently known terrestrial exoplanets could exhibit volcanism driven by the melting mechanism outlined by Peale & Cassen (1978). Quick et al. (2020) calculated the combined tidal and radiogenic heating rates for 53 exoplanets with masses M_P < 8 M_⊕ and radii R_P < 2 R_⊕. Our results generally agree with their findings, although we use different selection criteria. The main difference is that we only consider "rocky" planets, defined here to have radii R_P ≲ 1.6 R_⊕ (Valencia et al. 2006, 2007; Dorn et al. 2015; Fulton et al. 2017). We also elected to not include radiogenic heating in these calculations, the levels of which are unconstrained in exoplanets. For a more detailed discussion, we refer the reader to the third paragraph in Section 5.

To evaluate the importance of heating, we write the heating rate in the scaled form

$$\begin{eqnarray}\begin{array}{rcl}{\dot{E}}_{\mathrm{Heat}} & = & \left(\,3.4\times {10}^{25}\,{\text{erg s}}^{-1}\right){\left(\displaystyle \frac{P}{1{\rm{d}}}\right)}^{-5}\\ & & \times \ {\left(\displaystyle \frac{{R}_{{\rm{P}}}}{{R}_{\oplus }}\right)}^{5}{\left(\displaystyle \frac{e}{{10}^{-2}}\right)}^{2}\left(\displaystyle \frac{{\mathfrak{I}}({\tilde{k}}_{2})}{{10}^{-2}}\right).\end{array}\end{eqnarray} \tag{ 11 }$$

We can compare this rate to that of insolation heating, which is given by

$$\begin{eqnarray}\begin{array}{rcl}{\dot{E}}_{\mathrm{Iso}} & = & \left(\,3.1\times {10}^{27}\,{\text{erg s}}^{-1}\right)\left(\displaystyle \frac{1-A}{1-{A}_{\oplus }}\right)\left(\displaystyle \frac{{L}_{* }}{{L}_{\odot }}\right)\\ & & \times \ {\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right)}^{-2/3}{\left(\displaystyle \frac{{R}_{{\rm{P}}}}{{R}_{\oplus }}\right)}^{2}{\left(\displaystyle \frac{P}{1{\rm{d}}}\right)}^{-4/3}.\end{array}\end{eqnarray} \tag{ 12 }$$

In Equation (12), M_⊙, M_*, L_⊙, and L_* are the solar and stellar mass and luminosity, respectively. The albedo of the planet is given by A, and the equation is normalized to the albedo of the Earth, A_⊕, which is approximately A_⊕ = 0.3.

In addition to the radius constraint, we only consider planets with finite eccentricity. Short-period planets are generally expected to have synchronized spin and zero obliquity because of the strong tidal torques. However, in the presence of misaligned companions, the spin of the planet may be captured into nontrivial "Cassini states" with finite obliquities—such obliquities can be long-lasting even in the presence of tidal alignment torque (Fabrycky et al. 2007; Millholland & Laughlin 2019; Su & Lai 2022).

As a final criteria, we conservatively only consider planets with surface temperatures less than 1000 K. Assuming that there is zero pressure at the surface, the condensation/evaporation temperatures of rocky, magnesium-silicate materials is T ≃ 1300 K (Lodders 2003; Sarafian et al. 2017).

In Figure 5 and Table 1, we show exoplanets that satisfy the following criteria: (i) are rocky, (ii) have nonzero eccentricity, and (iii) have low surface temperature—and may also have active volcanism from runaway melting. The transmission spectroscopy metric (TSM) shown in Table 1 was introduced by Kempton et al. (2018, Equation (1) in that paper) and is a metric of the observability of an exoplanet atmosphere via transmission spectroscopy. The tidal heating of each planet is calculated using Equation (11) and the mass is estimated with the mass–radius relationship in Chen & Kipping (2017) for all cases. The equilibrium temperature is calculated assuming a constant Earth-like albedo for all cases. We also show the ratio of the tidal heating (Equation (11)) to the stellar heating (Equation (12)). We normalize the heating parameters by ${\mathfrak{I}}({\tilde{k}}_{2})$, to indicate the uncertainty in the tidal properties of these bodies. In Table 1, we show relevant orbital and planetary properties for all of these planets.

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Table 1. Exoplanets with Properties That Are Amenable to Surface Volcanism

Name	POrb	e	RP	MP	Teq	L⋆	M⋆	D	J	${\dot{E}}_{\mathrm{Heat}}/{\mathfrak{I}}({\tilde{k}}_{2})$	${\dot{E}}_{\mathrm{Iso}}$	TSM	e References
	(days)		(R⊕)	(M⊕)	(K)	(L⊙)	(M⊙)	(pc)	(Mag)	(erg s−1)	(erg s−1)
HD 136352 b	11.58	${0.079}_{-0.053}^{+0.068}$	1.66	3.41	828.0	1.04	0.87	14.7	4.3	1.3 × 1025	3.7 × 1026	137	RV; Kane et al. (2020)
HD 219134 c	6.77	0.062 ± 0.039	1.51	2.88	715.0	0.26	0.81	6.5	4.0	7.1 × 1025	1.7 × 1026	223	RV; Gillon et al. (2017)
HD 219134 f	22.72	0.148 ± 0.047	1.31	2.27	477.0	0.26	0.81	6.5	4.0	4.7 × 1023	2.5 × 1025	124	RV; Gillon et al. (2017)
HD 23472 d	3.98	${0.069}_{-0.046}^{+0.053}$	0.75	0.35	857.0	0.24	0.67	39.0	7.9	3.8 × 1025	8.6 × 1025	55	RV; Barros et al. (2022)
HD 23472 e	7.91	${0.056}_{-0.038}^{+0.054}$	0.82	0.47	681.0	0.24	0.67	39.0	7.9	1.3 × 1024	4.1 × 1025	42	RV; Barros et al. (2022)
HD 23472 f	12.16	${0.048}_{-0.034}^{+0.044}$	1.14	1.54	590.0	0.24	0.67	39.0	7.9	5.6 × 1023	4.4 × 1025	30	RV; Barros et al. (2022)
HD 260655 b	2.77	${0.039}_{-0.023}^{+0.043}$	1.24	2.07	649.0	0.04	0.44	10.0	6.7	9.2 × 1026	7.7 × 1025	143	RV; Luque et al. (2022)
HD 260655 c	5.71	${0.038}_{-0.022}^{+0.036}$	1.53	2.97	510.0	0.04	0.44	10.0	6.7	6.8 × 1025	4.5 × 1025	148	RV; Luque et al. (2022)
K2-116 b	4.66	${0.06}_{-0.053}^{+0.142}$	0.69	0.26	762.0	0.18	0.69	49.4	8.6	8.7 × 1024	4.5 × 1025	41	T; Dressing et al. (2017)
K2-128 b	5.68	${0.23}_{-0.19}^{+0.22}$	1.42	2.61	671.0	0.15	0.71	114.6	10.5	1.8 × 1027	1.2 × 1026	10	T; Dressing et al. (2017)
K2-129 b	8.24	${0.13}_{-0.11}^{+0.21}$	1.04	1.12	372.0	0.01	0.36	27.8	9.7	1.8 × 1025	5.9 × 1024	33	T; Dressing et al. (2017)
K2-136 b*	7.98	${0.14}_{-0.11}^{+0.12}$	1.01	1.02	612.0	0.17	0.74	59.2	9.1	2.2 × 1025	4.1 × 1025	20	RV; Mayo et al. (2023)
K2-136 d*	25.58	${0.071}_{-0.049}^{+0.063}$	1.57	3.07	415.0	0.17	0.74	59.2	9.1	1.5 × 1023	2.1 × 1025	17	RV; Mayo et al. (2023)
K2-266 c	7.81	${0.042}_{-0.03}^{+0.043}$	0.7	0.28	608.0	0.15	0.69	77.6	9.6	3.6 × 1023	1.9 × 1025	19	T; Rodriguez et al. (2018)
K2-72 b	5.58	${0.11}_{-0.09}^{+0.20}$	1.08	1.28	442.0	0.01	0.27	66.4	11.7	1.1 × 1026	1.3 × 1025	18	T; Dressing et al. (2017)
K2-72 c	15.19	${0.11}_{-0.09}^{+0.20}$	1.16	1.65	317.0	0.01	0.27	66.4	11.7	1.1 × 1024	3.9 × 1024	13	T; Dressing et al. (2017)
K2-72 d	7.76	${0.11}_{-\mathrm{0.0.09}}^{+0.21}$	0.73	0.31	412.0	0.01	0.22	66.4	11.7	3.0 × 1024	4.3 × 1024	44	T; Dressing et al. (2017)
K2-72 e	24.17	${0.11}_{-0.09}^{+0.20}$	0.82	0.48	283.0	0.01	0.22	66.4	11.7	1.8 × 1022	1.2 × 1024	28	T; Dressing et al. (2017)
K2-91 b	1.42	${0.09}_{-0.07}^{+0.17}$	1.1	1.37	689.0	0.01	0.29	62.6	11.4	7.6 × 1028	7.7 × 1025	44	T; Dressing et al. (2017)
Kepler-138 b	10.31	0.020 ± 0.009	0.59	0.15	406.0	0.04	0.51	66.9	10.3	8.3 × 1021	2.7 × 1024	20	RV; Piaulet et al. (2023)
Kepler-138 c	13.78	${0.017}_{-0.007}^{+0.008}$	1.58	3.13	367.0	0.04	0.52	66.9	10.3	1.9 × 1023	1.3 × 1025	16	RV; Piaulet et al. (2023)
Kepler-138 d	23.09	0.010 ± 0.005	1.26	2.13	309.0	0.04	0.52	66.9	10.3	1.6 × 1021	4.1 × 1024	10	RV; Piaulet et al. (2023)
Kepler-444 c	4.55	${0.31}_{-0.15}^{+0.12}$	0.5	0.08	900.0	0.37	0.76	36.4	7.2	5.1 × 1025	4.5 × 1025	88	T; Campante et al. (2015)
Kepler-444 d	6.19	${0.18}_{-0.12}^{+0.16}$	0.53	0.1	812.0	0.37	0.76	36.4	7.2	5.0 × 1024	3.4 × 1025	76	T; Campante et al. (2015)
Kepler-444 e	7.74	${0.10}_{-0.07}^{+0.20}$	0.42	0.04	778.0	0.37	0.62	36.4	7.2	1.6 × 1023	1.8 × 1025	135	T; Campante et al. (2015)
Kepler-444 f	9.74	${0.29}_{-0.19}^{+0.20}$	0.74	0.33	698.0	0.37	0.76	36.4	7.2	7.2 × 1024	3.7 × 1025	54	T; Campante et al. (2015)
L 98-59 b	2.25	${0.103}_{-0.045}^{+0.117}$	0.8	0.44	549.0	0.01	0.31	10.6	7.9	2.0 × 1027	1.7 × 1025	173	RV; Demangeon et al. (2021)
L 98-59 c	3.69	${0.103}_{-0.058}^{+0.045}$	1.35	2.39	461.0	0.01	0.31	10.6	7.9	2.3 × 1027	2.4 × 1025	128	RV; Demangeon et al. (2021)
L 98-59 d	7.45	${0.074}_{-0.046}^{+0.057}$	1.57	3.09	369.0	0.01	0.31	10.6	7.9	7.7 × 1025	1.3 × 1025	124	RV; Demangeon et al. (2021)
TOI-1266 c	18.8	0.04 ± 0.03	1.56	3.06	315.0	0.03	0.45	36.0	9.7	2.1 × 1023	6.8 × 1024	25	RV; Demory et al. (2020)
TOI-2096 b	3.12	${0.15}_{-0.12}^{+0.27}$	1.24	2.08	447.0	0.01	0.23	48.5	11.9	7.6 × 1027	1.7 × 1025	33	RV; Pozuelos et al. (2023)
TOI-270 b	3.36	0.034 ± 0.025	1.25	2.09	525.0	0.02	0.4	22.5	9.1	2.7 × 1026	3.0 × 1025	51	RV; Van Eylen et al. (2021)
TOI-700 b	9.98	${0.081}_{-0.058}^{+0.095}$	1.01	1.01	394.0	0.02	0.42	31.1	9.5	2.4 × 1024	6.1 × 1024	29	T; Rodriguez et al. (2020)
TOI-700 d	37.42	${0.111}_{-0.078}^{+0.140}$	1.14	1.57	246.0	0.02	0.41	31.1	9.5	1.1 × 1022	1.4 × 1024	17	T; Rodriguez et al. (2020)
TOI-700 e	27.81	${0.059}_{-0.042}^{+0.057}$	0.95	0.82	271.0	0.02	0.41	31.1	9.5	5.6 × 1021	1.4 × 1024	21	T; Gilbert et al. (2023)
LP 791-18 d	2.75	${0.0015}_{-0.00014}^{+0.00014}$	1.03	0.90	395.5	0.002	0.14	26.7	11.6	5.6 × 1023	2.8 × 1024	71	RV; Peterson et al. (2023)

Notes. The parameters shown are orbital period P_Orb, eccentricity e, radius R_P, planetary mass M_P, planetary equilibrium temperature T_eq, stellar luminosity L_*, stellar mass M_*, effective temperature of the surface of the planet T_Eq, system distance D, J magnitude J, tidal heating ${\dot{E}}_{\mathrm{Heat}}/{\mathfrak{I}}({\tilde{k}}_{2})$, insolation heating ${\dot{E}}_{\mathrm{Iso}}$, transmission spectroscopy metric (TSM; Kempton et al. 2018), and the reference paper for the eccentricity value. Note that the value listed ${\dot{E}}_{\mathrm{Heat}}/{\mathfrak{I}}({\tilde{k}}_{2})$ corresponds to the case of ${\mathfrak{I}}({\tilde{k}}_{2})=1$. For reference, the tidal heating of Io is $\simeq 1.6\times {10}^{21}/{\mathfrak{I}}({\tilde{k}}_{2})$ erg s⁻¹. The table is organized by the planetary systems, so sets of planets within the same system are shown adjacent. The reference for e is provided in the last column, along with how the e was derived (RV = radial velocity or radial velocity + transit fits; T = transit fits).

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Some of these planets have ∼5 orders of magnitude more tidal heating per unit mass than Io—assuming that the tidal quality factors are the same. However, even if the quality factor (or Love number accordingly) were orders of magnitude different from that of Io, there may be sufficient tidal heating to trigger melting—or partial melting—and volcanism in the most dramatic cases. Moreover, for the extreme cases, the tidal heating flux modulo the quality factor exceeds the instellation heating by a few orders of magnitude, or ${\dot{E}}_{\mathrm{Heat}}/{\mathfrak{I}}({\tilde{k}}_{2})/{\dot{E}}_{\mathrm{Iso}}\sim 1\mbox{--}{10}^{3}$ (Figure 5). Treating the tidal heating as an extra luminosity source, the equilibrium temperatures scale as ${\dot{E}}^{1/4}$. For ${\mathfrak{I}}({\tilde{k}}_{2})\sim 1\mbox{--}100$ the additional heating could be sufficient to raise the equilibrium temperatures by a factor of 1–5. For L 98-59 b, where ${\dot{E}}_{\mathrm{Heat}}/{\mathfrak{I}}({\tilde{k}}_{2})/{\dot{E}}_{\mathrm{Iso}}\sim {10}^{2}$ and assuming ${\mathfrak{I}}({\tilde{k}}_{2})\sim 10$, the equilibrium temperature is increased to ∼1000 K.

Moreover, the inferred heating rate per unit volume of radioactive nuclei in our solar system is ∼0.006 erg s⁻¹ cm⁻³, which corresponds to a heating rate of $\dot{E}\sim 6.5\times {10}^{24}$ erg s⁻¹ (Lodders 2003). This radioactive heating per unit volume in the early solar system is known to melt objects with sizes ≳20 km (Figure 4 and Equation (34) in Adams 2021). Therefore, exoplanets with ${\dot{E}}_{\mathrm{Heat}}/{\mathfrak{I}}({\tilde{k}}_{2})\sim {10}^{27}$ erg s⁻¹ should experience interior melting to some extent even if ${\mathfrak{I}}({\tilde{k}}_{2})\sim 100$.

It is worth noting that the effective surface temperatures could be augmented by the tidal heating (Becker et al. 2023). However, even in the most dramatic case in Figure 5, if Q ∼ 10², the tidal heating is still comparable to the stellar heating. It is also worth noting that we do not incorporate the melting of the core in the mass estimate in Figure 5. This choice is partially motivated because the rigidity (which sets the liquid core radius as in Figure 4) is unconstrained. Moreover, the total dissipation is typically not altered by an order of magnitude due to core melting (Figure 3). The planet with the highest levels of heating (assuming constant tidal properties for all planets) is TOI-2096 b, a super-Earth close to a 2:1 mean motion resonance with an exterior companion (Pozuelos et al. 2023).

A promising system for exhibiting volcanism is the L 98-59 multiplanet system (Kostov et al. 2019; Demangeon et al. 2021). This small M-dwarf harbors four planets, and the outer three planets are close to a 2: 4: 7 mean motion resonance, reminiscent of the Laplace resonance in the Galilean satellites. It is possible that the nonzero eccentricities of the planets—despite the significant tidal heating—could be explained by forced eccentricity from the mean motion resonance. Measurements obtained with the Hubble Space Telescope (HST) of L 98-59 b were consistent with no atmosphere (Damiano et al. 2022; Zhou et al. 2022). However, Barclay et al. (2023) presented preliminary and marginal evidence of an atmosphere on L 98-59 c with HST measurements, although this has since been debated (Zhou et al. 2023).

Another intriguing system is Kepler-444 (Campante et al. 2015), an old triple-star system whose main star hosts five small rocky exoplanets (Winters et al. 2022). However, Stalport et al. (2022) performed a dynamical analysis of the system and found that none of the planets are in low-order two or three planet mean motion resonances.

It is tempting to replicate the analysis performed by Peale et al. (1979) for Io with these planets that appear to have the prerequisites for volcanism and that are in mean motion resonances. Peale et al. (1979) estimated the quality factor of Io by equating the increase in Io's mean motion from tidal dissipation (Equation (8) in that work) to the decrease in mean motion from tidal transfer from Jupiter's rotation (Equation (9) in that work). However, in the Jupiter-Io case, the rotation period of Jupiter is faster than the orbital period of Io so the two tidal forces act in opposite directions. However, the stellar rotation rates are likely slower than the orbital periods of the planets. Therefore, the tidal forces will operate coherently and—somewhat disappointingly—the quality factor of these planets cannot be estimated in a similar fashion.

It is possible that the eccentricities of these planets are maintained via the mean motion resonances. For example, the planet LP 791-18 d has nonzero forced eccentricity, which led Peterson et al. (2023) to claim that it would have active volcanism. Unfortunately, the value of the forced eccentricity does not provide useful constraints on the tidal properties. To illustrate this, Equation (17) in Lithwick & Wu (2012) can be solved, by noting that $| {z}_{1}\simeq \mu {n}_{2}/{({\gamma }^{2}+{(2{n}_{2}{\rm{\Delta }})}^{2})}^{1/2}\leqslant \mu {n}_{2}/\gamma |$ using the notation in that paper and rearranging, to yield the following constraint for planets in mean motion resonances:

$$\begin{eqnarray}\begin{array}{ccc}{\gamma }_{e} & \leqslant & \displaystyle \frac{{M}_{1}}{{M}_{\ast }}\displaystyle \frac{{n}_{2}}{{e}_{1}}\simeq (0.08\,{{\rm{y}}{\rm{r}}}^{-1})\left(\displaystyle \frac{{M}_{1}}{2.2\,{M}_{\oplus }}\right)\\ & & \times \text{}\left(\displaystyle \frac{0.27\,{M}_{\odot }}{{M}_{\ast }}\right)\left(\displaystyle \frac{{P}_{2}}{7.45\,{\rm{d}}}\right)\left(\displaystyle \frac{0.1}{{e}_{1}}\right).\end{array}\end{eqnarray} \tag{ 13 }$$

In Equation (13) γ_e is the damping rate of eccentricity for a pseudo-synchronized planet γ_e = (1/e)d e/dt, and the subscripts indicate the inner and outer of two planets in a mean motion resonance. On the right-hand side of Equation (13), we have substituted values for L 98-59 c and d (which are close to a 2:1 resonance). Typical values of the damping rates from tidal dissipation are much smaller than this, and for L 98-59 c as an example with planetary parameters from Demangeon et al. (2021):

$$\begin{eqnarray}&&\begin{array}{l}{\gamma }_{e}=-\displaystyle \frac{21}{2}{\mathfrak{I}}({\tilde{k}}_{2})\,n\left(\,\displaystyle \frac{{M}_{\ast }}{{M}_{{\rm{P}}}}\right){\left(\displaystyle \frac{{R}_{{\rm{P}}}}{a}\right)}^{5}\\ \simeq \,(6.8\times {10}^{-6}\,{{\rm{y}}{\rm{r}}}^{-1})\text{}{\mathfrak{I}}({\tilde{k}}_{2})\left(\displaystyle \frac{2.4\,{M}_{\oplus }}{{M}_{{\rm{P}}}}\right)\\ \times \,\left(\displaystyle \frac{{M}_{\ast }}{0.31\,{M}_{\odot }}\right){\left(\displaystyle \frac{{R}_{{\rm{P}}}}{1.35\,{R}_{\oplus }}\right)}^{5}\\ \times \,{\left(\displaystyle \frac{0.0304\,{\rm{a}}{\rm{u}}}{a}\right)}^{5}\left(\displaystyle \frac{3.69\,{\rm{d}}}{P}\right).\end{array}\end{eqnarray} \tag{ 14 }$$

Unfortunately, this does not provide a useful constraint on the tidal properties.

We generate atmospheric models of the three planets in the L 98-59 system using Aurora (Welbanks & Madhusudhan 2021). Aurora numerically computes the line-by-line radiative transfer in transmission geometry assuming hydrostatic equilibrium. We compute these models using 100 layers uniformly spaced in log₁₀(P) between 10² bar and 10⁻⁹ bar, at a constant resolution of R = 10⁵ between 0.5 μm and 15 μm. We consider three main atmospheric scenarios, ranging from CO₂- to SO₂-rich, with trace amounts of H₂O, H, and He (see Table 2). For these chemical species, we use line lists from Rothman et al. (2010) for H₂O and CO₂, from Underwood et al. (2016) for SO₂, and from Richard et al. (2012) for H₂-H₂ and H₂-He collision-induced absorption (mostly negligible in the chosen scenarios). These initial models consider cloud-free atmospheres only, as we aim to inform the observational prospects of these optimistic scenarios. Future investigations will consider the effects of different cloud and haze species, as well as other chemical scenarios.

We consider three atmospheric scenarios, summarized in Table 2. First, we consider a CO₂-rich atmosphere composed of 98% CO₂, 5% SO₂, 1% H₂O, and a mixture of H₂ and He in the remaining percentage, assuming a He/H₂ solar ratio of 0.17 (Asplund et al. 2009). Second, we consider a scenario where the atmosphere is composed of 50% SO₂, 48% CO₂, 1% H₂O, and a 1% solar mixture of H₂ and He. Finally, we consider a scenario of SO₂ dominated with a 98% SO₂ composition with 1% H₂O and 1% H₂ and He. The first scenario aims to assess the observability of a composition loosely similar to a Venus-like CO₂-rich atmosphere, while the third scenario considers a scenario loosely similar to Io, with an SO₂-rich atmosphere. The second scenario is an intermediate case between these two extremes.

Within the wavelength coverage of JWST, SO₂ has absorption features at 2.5, 4.05, 7.34, and 8.68 μm. We use the aforementioned models for each of the L 98-59 planets and simulate observations of their atmospheres with NIRISS/SOSS, NIRSPec/G395H, and MIRI/LRS. To do this, we run our models through pandexo 2.0, an open-source Python package for generating instrument simulations for HST and JWST (Batalha et al. 2022). We assumed the transit duration, T₁₄, presented in Demangeon et al. (2021) and assumed an out-of-transit duration of 3 × T₁₄. The resulting simulated transmission spectra are presented in Figure 7.

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The SO₂ absorption features at λ = 4.05 and 7.34 μm are visible by eye in each of the simulated spectra, while the SO₂ absorption feature at λ = 2.5 and 8.68 μm is likely to be overwhelmed by either H₂O absorption or instrument noise. We find a maximum ∼10 ppm difference between the 5% and 98% SO₂ atmospheres at λ = 4.05 and 7.34 μm for all three planets. While the depth of the SO₂ does not change significantly across all three scenarios, the depths of other molecules, such as CO₂, do. Therefore, it may be possible to distinguish these atmospheric compositions via broad-wavelength observations, which will allow for accurate measurements of the abundances of other species.

We determine the detectability of each SO₂ absorption feature as a function of the number of transits to observe, n. We use Equation (5) of Lustig-Yaeger et al. (2019) in order to calculate the expected signal-to-noise (S/N) of each feature, which is given by

$$\begin{eqnarray}&&\langle {\rm{S/N}}\rangle =\sqrt{\displaystyle \sum _{i=1}^{{N}_{\lambda }}{\left(\displaystyle \frac{{m}_{1,i}-{m}_{2,i}}{{\sigma }_{i}}\right)}^{2}}.\end{eqnarray} \tag{ 15 }$$

In Equation (15), m₁ is the transmission model, m₂ is a featureless transmission model, and σ is the error on the observations. The subscript i for each of these parameters indicates that the residual is computed at discrete resolution elements and N_λ is the total number of resolution elements. We define the featureless transmission model as the best-fitting constant planet radius with wavelength. We evaluate the 〈S/N〉 as λ = [2.45, 2.55], [3.92, 4.20], and [7.0, 7.9], which broadly covers all three SO₂ features. We repeat this calculation for n = 1, 2, 5, 10, and 20 transits observed with each JWST instrument. The results for L 98-59 bcd with NIRISS/SOSS, NIRSpec/G395H, and MIRI/LRS are presented in Figure 8.

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Using the metric above, we find that a significant (i.e., >3σ) preference for an atmosphere with SO₂ over a featureless spectrum ("flat line") with NIRISS/SOSS in under 20 transits observed is unlikely. On the other hand, we find that it is likely to prefer an atmosphere with some amount of SO₂ for all three L 98-59 planets in 3–5 transits if observed with NIRSpec/G395H. It is possible to prefer an atmospheric model with SO₂ in just three transits of L 98-59 c with NIRSpec/G395H. In order to detect an atmosphere composed of 98% SO₂ for 98-59 bcd, 5–10 transits would need to be observed. While there is more SO₂ in the atmosphere, this results in an atmosphere with a higher mean molecular weight, and therefore the absorption features are more muted (de Wit & Seager 2013).

Preferring SO₂ over a featureless spectrum with MIRI/LRS may be feasible with 5–10 transits for L 98-59 bc, and 10–20 transits for L 98-59 d. Observing multiple SO₂ features with different instruments would provide the greatest confidence in the planetary nature of the absorption feature (Dyrek et al. 2023; Alderson et al. 2023; Rustamkulov et al. 2023). Furthermore, broad-wavelength coverage would allow us to better constrain the abundances of other species such as H₂O and CO₂ and will provide insights into the relative abundance of SO₂. Assessing the detection of a chemical species within a Bayesian framework would require performing atmospheric retrievals and an in-depth model comparison and criticism (see, e.g., Welbanks et al. 2023, for a discussion). Such exploration with atmospheric retrievals would enable an assessment of the ability to infer the abundance of SO₂ and other trace species. This exploration is beyond the scope of this paper, and we reserve it for a follow-up study.

Five transits of L 98-59 b, one transit of L 98-59 c, and two transits of L 98-59 d will be observed with NIRSpec/G395H during JWST Cycles 1 and 2 (GO 2521, 3942, 4098; GTO 1201). The observations of L 98-59 cd will likely be insufficient for any significant detection of SO₂; however, if the atmosphere of L 98-59 b has any amount of SO₂, it is possible these observations would be able to detect the absorption feature at 4.05 μm at ≥3σ. More transits of L 98-59 cd would need to be observed to search for evidence of a SO₂-rich atmosphere.