Forecasting Rates of Volcanic Activity on Terrestrial Exoplanets and Implications for Cryovolcanic Activity on Extrasolar Ocean Worlds

Primitive atmospheres on rocky planets are formed from volatiles that are outgassed during their magma ocean phase. Rocky planets may experience numerous magma ocean phases during their early evolution; these phases may develop as a result of impacts (Tonks & Melosh 1993), potential energy release during core formation (Sasaki & Nakazawa 1986), or radiogenic heating (Elkins-Tanton 2012). Planetary evolution is fundamentally tied to the evolution of magma oceans as processes that take place during magma ocean crystallization determine a planet's tectonic behavior, initial chemical and thermal structure, and atmospheric composition (Massol et al. 2016; Ikoma et al. 2018; Kite et al. 2020). For example, degassing of the lunar magma ocean after the Moon-forming impact may have produced a short-lived metal atmosphere on the Moon's Earth-facing side; this early lunar atmosphere may be analogous to atmospheres that exist on young, close-in rocky exoplanets with global magma oceans (Saxena et al. 2017).

Notwithstanding, primitive atmospheres formed from magma ocean degassing are likely to be stripped from close-in planets by stellar winds and CMEs (Khodachenko et al. 2007; Kite et al. 2009). Volcanic outgassing will facilitate the formation of secondary atmospheres on these worlds. Conversely, a lack of volcanism, or infrequent volcanic events, would preclude a rocky planet from maintaining an atmosphere. On stagnant-lid planets, the lack of substantial CO₂-rich atmospheres would drive surface temperatures down, making them too cool to maintain liquid water at their surfaces, thereby reducing the widths of habitable zones in their respective systems (Kadoya & Tajika 2014; Noack et al. 2014, 2017; Abbot 2016). For this reason, geologically active planets that host frequent volcanic eruptions may represent more favorable habitable environments than planets on which volcanic events are sporadic or non-existent (Misra et al. 2015). In the case of cold, cryovolcanically-active exoplanets, the detection of eruptive activity could signify the presence of internal oceans and near-surface habitable niches, as is the case for several icy moons in our solar system. Volcanism is therefore intimately linked to planetary habitability.

Owing to their bright infrared flux and short orbital periods, volcanic exoplanets and exoplanets with global magma oceans may be among the most detectable and characterizable low-mass exoplanets in the coming decades (Henning et al. 2018). Recent studies have attempted to characterize volcanic activity on rocky extrasolar worlds using transmission and UV spectroscopy, and with eclipse observations. These studies have shown that while plasma tori surrounding volcanically-active exoplanets could be detected by Hubble Space Telescope UV observations (Kislyakova et al. 2019), eclipse observations could be used to characterize the variability in activity on highly volcanic exoplanets (Tamburo 2018). In addition, by modeling high-resolution alkali spectra, Oza et al. (2019) have demonstrated that volcanic activity may be prevalent on exomoons orbiting close-in gas giant exoplanets. In exploring the lifetime and spectral evolution of magma oceans on Earth-sized terrestrial planets, Hamano et al. (2015) found that the thermal and spectral evolution of magma oceans on these bodies is dependent upon their distance from the host star, with close-in planets producing strong enough thermal radiation to be detectable for the entire lifetime of the magma ocean, while thermal radiation from planets at farther orbital distances would sharply decrease within 10⁶ yr. Neglecting internal heat sources, Bonati et al. (2019) found that magma oceans may persist from 10²–3 × 10⁶ yr on newly formed planets. These authors also found that owing to their molten surfaces, young planets with magma oceans may be directly imaged at infrared wavelengths by space-based interferometers or high-resolution ground-based telescopes. Similarly, the high albedos of icy, cryovolcanically active worlds will make them far more detectable than rocky planets in reflected light (see Wolf 2017). Detections of volcanic activity on these far-away worlds would allow for the examination of their internal structures and compositions from afar.

While volcanism can indeed induce habitable environments on low-mass exoplanets (e.g., Ramirez & Kaltenegger 2017), excessive volcanism (see Demory et al. 2015) would produce unstable surface environments that could render exoplanets incapable of hosting life (Jackson et al. 2008). The frequency and relative strength of volcanic activity on extrasolar planets will therefore directly affect their habitability. The prevalence of volcanic activity on terrestrial exoplanets is intimately linked to the thermal states of these worlds (Dorn et al. 2018). In order to gauge the habitability of terrestrial exoplanets, including the potential for reservoirs of liquid water on volatile-rich planets, it is therefore necessary to constrain the amount of volcanic or cryovolcanic activity that may be occurring at their surfaces. While the identification of gas and dust produced by active eruptions is one way to identify volcanically active extrasolar worlds (see Guenther et al. 2011; Misra et al. 2015; Arkhypov et al. 2019), our first step is to determine which exoplanets are likely to be volcanically or cryovolcanically active based on their internal heating rates. Dorn et al. (2018) explored volcanic outgassing of CO₂ on rocky exoplanets with stagnant lids that ranged from 1 to 8M_⊕. They found that planets between 2 and 3 M_⊕ are most likely to be volcanically active. Kite et al. (2009) explored rates of volcanism as a function of planet mass and time for terrestrial exoplanets between 0.25 and 25M_⊕. However, their analysis did not consider the effects of tidal heating on volcanic activity.

Here, we provide rough estimates for the total internal heating rates of 53 low-mass exoplanets, with masses between 0.086 and 8M_⊕ and radii between 0.54 (Mars-sized) and 2R_⊕ assuming that these planets are primarily heated by radiogenic and tidal sources. We then use these results to constrain rates of (cryo)volcanic activity at their surfaces, employing activity rates for the planets and moons in our solar system as a baseline. We have surveyed exoplanets that are up to 8M_⊕ and 2R_⊕ in an effort to ensure that we capture Corot-7b, which is likely very volcanically active (Barnes et al. 2010; Léger et al. 2011), along with recently discovered planets such as Pi Mensae c (Huang et al. 2018), and low-density exoplanets such as Kepler 60 c & d (Jontof-Hutter et al. 2016) in our analyses. Notwithstanding, the overwhelming majority of the exoplanets we have surveyed have radii ≤1.7R_⊕ (Table 1). This was done to ensure that the exoplanets considered here are indeed super-Earths rather than mini-Neptunes (Owen & Wu 2017; Fulton & Petigura 2018; Mordasini 2020).

Table 1.  Solar System and Extrasolar Planetary Parameters [1 W = 10⁷ erg s⁻¹] SI = Super-Io; Mow = Magma Ocean World; EVP = Exo-Venus Planet; SE = Super-Europa/Super-Enceladus Planet; ST = Super-Triton

World	RP (${R}_{\oplus })$	MP (${M}_{\oplus })$	ρ (kg m−3)	g (m s−2)	${T}_{\mathrm{EFF}/}{T}_{{\rm{S}}}$ a (K)	System Age (Gyr)	HRadiogenic (W)	HTidal (W)	HTotal (W)	Type	References
Earth	1	1	5500	9.8	255 (TEFF) 288(TS)	4.5	4 × 1013	⋯b	4.7 × 1013	rocky	Turcotte (1995), Turcotte & Schubert (2002)
Venus	0.82	0.95	5240	8.87	260 K (TEFF) 737 (TS)	4.5	2.91 × 1013	⋯	2.91 × 1013	rocky	Turcotte (1995), Schubert et al. (1997)
Early Venus	0.82	0.95	5240	8.87	?	4	2 × 1014	⋯	2 × 1014	rocky	Turcotte (1995)
Io	0.286	0.015	3528	1.8	110	4.5	3.7 × 1011	1 × 1014	1 × 1014	rocky	Yoder & Peale (1981), Schubert et al. (2004), Hussmann & Spohn (2004)
Europa	0.245	0.008	3000	1.31	100	4.5	2.1 × 1011	1 × 1012	1.21 × 1012	icy	Schubert et al. (2004), Hussmann & Spohn (2004),c Chen et al. (2014), Quick & Marsh (2015)
Enceladus	0.0395	1.8 × 10−5	1610	0.11	75	4.5	2.73 × 108	1.6 × 1010	1.63 × 1010	icy	Howett et al. (2011),c Chen et al. (2014)
Triton	0.21	0.00359	2060	0.78	38	4.5	7.12 × 1010	d6.6 × 1010	1.37 × 1011	icy	Gaeman et al. (2012),c Chen et al. (2014) and references therein
Callisto	0.378	0.018	1834	1.235	126	4.5	3.2 × 1011	3.3 × 109	3.2 × 1011	icy	Moore & Schubert (2003),c Chen et al. (2014) and (1)
Ganymede	0.413	0.025	1942	1.43	117	4.5	4.67 × 1011	1.1 × 1010	4.78 × 1011	icy	Moore & Schubert (2003), Schubert et al. (2004),c Chen et al. (2014) and (1)
Titan	0.404	0.0225	2575	1.35	94	4.5	4.11 × 1011	8.75 × 1010	4.99 × 1011	icy	c Chen et al. (2014), Schubert et al. (1986) and (1)
Mercury	0.38	0.055	5427	3.7	440	4.5	2.25 × 1012	1.4 × 109	2.25 × 1012	rocky	Peplowski et al. (2011), Makarov & Efroimsky (2014), Ogawa (2016),e
Early Mercury	0.38	0.055	5427	3.7	?	4	3.2 × 1012	...	3.2 × 1012	rocky	Hauck et al. (2004), Breuer et al. (2007)
Mars	0.533	0.107	3934	3.72	210	4.5	2.74 × 1012	1 × 109	2.74 × 1012	rocky	Parro et al. (2017), Manga et al. (2019)
The Moon	0.273	0.012	3344	1.62	215(equator) 104 (poles)	4.5	3 × 1011	...	3 × 1011	rocky	Siegler & Smrekar (2014), Paige & Siegler (2016), Williams et al. (2017)
Early Moon	0.273	0.012	3344	1.62	?	0.1	1.15 × 1012	1.05 × 1012 f	2.2 × 1012	rocky	Meyer et al. (2010), Frank et al. (2014)
Pluto	0.1868	2.18 × 10−3	1854	0.620	44	4.5	5.32 × 1010	...	5 × 1010	icy	Robuchon & Nimmo (2011), McKinnon et al. (2016)
Charon	0.095	2.7 × 10−4	1702	0.288	53	4.5	4.4 × 109	...	4.4 × 109	icy	Hussmann et al. (2006), Cook et al. (2007)
Ceres	0.074	1.5 × 10−4	2160	0.28	150	4.5	4.5 × 109	...	4.5 × 109	ice/rock hybrid	Castillo-Rogez et al. (2019)
Miranda	0.037	1.1 × 10−5	1200	0.079	60	4.5	8.5 × 107	...	8.5 × 107	icy	c Chen et al. (2014)
Ariel	0.091	2.3 × 10−4	1592	0.269	60	4.5	3.6 × 109	...	3.6 × 109	icy	c Chen et al. (2014)
Rhea	0.12	3.9 × 10−4	1233	0.264	99 (dayside) 53(nightside)	4.5	3.3 × 109	⋯	3.3 × 109	icy	c Chen et al. (2014)
Mimas	0.03	6.3 × 10−6	1148	0.064	64	4.5	4 × 107	2.9 × 108	3.3 × 108	icy	c Chen et al. (2014)
Dione	0.09	1.8 × 10−4	1480	0.232	87	4.5	2.46 × 109	...	2.46 × 109	icy	c Chen et al. (2014)
55 Cancri e	1.91	8.08	6400	21.7	1958	10.2	1.14 × 1014	4.28 × 1021	4.28 × 1021	rocky (MOW)	Demory et al. (2011, 2015, 2016)
Corot 7 b	1.55	5.7	7500	23	1756	1.32	2.17 × 1014	0 (e = 0)	2.17 × 1014	rocky (MOW)	Queloz et al. (2009), Barros et al. (2014), Stassun et al. (2017)
GJ 1132 b	1.13	1.66	6300	12.9	529	5	3.66 × 1013	TBD: e unknown	3.66 × 1013	rocky (EVP)	Berta-Thompson et al. (2015), Bonfils et al. (2018), Dittmann et al. (2017a)
HD 219134 b	1.6	4.74	6340	18	1015	11	6.42 × 1013	0 (e = 0)	6.42 × 1013	rocky (MOW)	Gillon et al. (2017a)
HD 219134c	1.5	4.36	6950	18.7	782	11	5.39 × 1013	2.17 × 1016	2.17 × 1016	Rocky (SI)	Gillon et al. (2017a)
g HD 219134 f	1.31	unknown	unknown	unknown	522	11	3.51 × 1013	1.42 × 1014	1.77 × 1014	unknown	Gillon et al. (2017a)
HD 3167 b	1.7	5.02	5600	17	1669	7.8	9.49 × 1013	0 (e = 0)	9.49 × 1013	rocky (MOW)	Christiansen et al. (2017), Livingston et al. (2018)
Kepler 10 b	1.45	4.6	8255	21	2130	10.6	4.9 × 1013	TBD: e unknown	4.91 × 1013	rocky (MOW)	Esteves et al. (2015)
Kepler 11 b	1.8	1.9	1720	5.7	unknown	8.5	1.07 × 1014	h1.12 × 1015	1.23 × 1015	cold ocean planet	Lissauer et al. (2013), Borsato et al. (2014)
Kepler 21 b	1.61	5.09	6720	19	2025	3.03	1.48 × 1014	2.6 × 1017	2.6 × 1017	rocky (MOW)	López-Morales et al. (2016)
Kepler 36 b	1.46	4.45	7810	20	978	6.8	6.6 × 1013	2 × 1014	2.7 × 1014	rocky (MOW)	Carter et al. (2012)
Kepler 60 b	1.68	4.2	4620	14	unknown	5.1	1.3 × 1014	9.4 × 1014	1.1 × 1015	ocean planet	Gozdziewski et al. (2016), Jontof-Hutter et al. (2016), Morton et al. (2016)
Kepler 60 c	1.9	3.85	3060	10.4	unknown	5.1	1.7 × 1014	3.81 × 1015	3.98 × 1015	cold ocean planet	Gozdziewski et al. (2016), Jontof-Hutter et al. (2016), Morton et al. (2016)
Kepler 60 d	1.99	4.16	2910	10.3	unknown	5.1	1.97 × 1014	3.03 × 1014	5 × 1014	cold ocean planet	Gozdziewski et al. (2016), Jontof-Hutter et al. (2016), Morton et al. (2016)
Kepler 62 c	0.54	unknown	unknown	unknown	578	7	3.16 × 1013	TBD: e unknown	3.16 × 1013	unknown	Borucki et al. (2013)
Kepler 68 c	0.96	2.04	unknown	21.7	unknown	6.3	1.94 × 1013	0 (e = 0)	1.94 × 1013	unknown	Berger et al. (2018), Mills et al. (2019)
Kepler 70 b	0.76	0.44	5528	7.5	unknown	unknown	TBD: system age unknown	TBD: e unknown	unknown	rocky	Charpinet et al. (2011)
Kepler 70 c	0.87	0.66	5521	8.5	unknown	unknown	TBD: system age unknown	TBD: e unknown	unknown	rocky	Charpinet et al. (2011)
Kepler 78 b	1.1	3.2	unknown	14	2250	0.75	9.57 × 1013	TBD: e unknown	9.57 × 1013	candidate rocky (MOW)	Pepe et al. (2013), Stassun et al. (2017)
Kepler 80 d	1.53	6.75	7040	28	720	2	1.55 × 1014	TBD: e unknown	1.55 × 1014	rocky (SI)	Muirhead et al. (2012), MacDonald et al. (2016)
Kepler 80 e	1.6	4.1	3750	16	628	2	1.8 × 1014	TBD: e unknown	1.8 × 1014	candi-date ocean planet	Muirhead et al. (2012), MacDonald et al. (2016)
Kepler 93 b	1.6	3.2	4292	12	1037	6.6	8.74 × 1013	0 (e = 0)	8.74 × 1013	candi-date ocean planet	Dressing et al. (2015), Stassun et al. (2017)
Kepler 97 b	1.5	3.5	5440	16	unknown	8.4	5.66 × 1013	TBD: e unknown	5.66 × 1013	rocky	Marcy et al. (2014)
Kepler 99 b	1.5	6.2	10900	28	unknown	1.5	1.66 × 1014	TBD: e unknown	1.66 × 1014	rocky	Marcy et al. (2014)
Kepler 100 b	1.3	7.3	14250	42.5	unknown	6.5	4.7 × 1013	TBD: e unknown	4.7 × 1013	rocky	Marcy et al. (2014)
Kepler 101 c	1.2	unknown	unknown	unknown	1412	5.9	4.3 × 1013	0 (e = 0)	4.3 × 1013	candi-date rocky (MOW)	Bonomo et al. (2014)
Kepler 102 d	1.2	3.8	13270	28	unknown	1.4	8.66 × 1013	TBD: e unknown	8.66 × 1013	rocky	Marcy et al. (2014)
Kepler 105 c	1.3	4.6	11200	27.3	997	3.5	6.85 × 1013	TBD: e unknown	6.85 × 1013	rocky (MOW)	Everett et al. (2015), Jontof-Hutter et al. (2016), Morton et al. (2016)
Kepler 114 c	1.6	2.8	4039	11.3	508	2.7	1.49 × 1014	TBD: e unknown	1.49 × 1014	candi-date ocean planet	Muirhead et al. (2012), Xie (2014), Morton et al. (2016)
Kepler 138 b	0.7	0.19	3020	3.8	unknown	4.7	9 × 1012	5.33 × 1011	9.5 × 1012	cold ocean planet	Morton et al. (2016), Almenara et al. (2018)
Kepler 138 c	1.7	5.2	6100	18.3	398	4.7	1.3 × 1014	3.2 × 1013	1.6 × 1014	candi-date ocean planet	Muirhead et al. (2012), Morton et al. (2016), Almenara et al. (2018)
Kepler 138 d	1.7	1.2	1360	4	335	4.7	1.3 × 1014	1.5 × 1013	1.4 × 1014	ocean planet	Muirhead et al. (2012), Morton et al. (2016), Almenara et al. (2018)
Kepler 186 b	1.07	unknown	unknown	unknown	579	4	3.6 × 1013	TBD: e unknown	3.6 × 1013	unknown	Muirhead et al. (2012), Quintana et al. (2014), Torres et al. (2015)
Kepler 186 c	1.25	unknown	unknown	unknown	470	4	5.7 × 1013	TBD: e unknown	5.7 × 1013	unknown	Muirhead et al. (2012), Quintana et al. (2014), Torres et al. (2015)
Kepler 186 d	1.4	unknown	unknown	unknown	384	4	8 × 1013	TBD: e unknown	8 × 1013	unknown	Muirhead et al. (2012), Quintana et al. (2014), Torres et al. (2015)
Kepler 186 e	1.27	unknown	unknown	unknown	unknown	4	6 × 1013	TBD: e unknown	6 × 1013	unknown	Quintana et al. (2014), Torres et al. (2015)
Kepler 186 f	1.2	unknown	unknown	unknown	unknown	4	4.7 × 1013	1.1 × 109	4.7 × 1013	unknown	Torres et al. (2015)
Kepler 406 b	1.4	6.4	11820	30	unknown	5.8	6.8 × 1013	TBD: e unknown	6.8 × 1013	rocky	Marcy et al. (2014)
Kepler 406 c	0.85	2.7	24390	37	unknown	5.8	1.4 × 1013	TBD: e unknown	1.4 × 1013	rocky	Marcy et al. (2014)
Kepler 414 b	1.7	3.5	3845	11.7	unknown	5.5	1.14 × 1014	TBD: e unknown	1.14 × 1014	ocean planet	Hadden & Lithwick (2014), Morton et al. (2016)
L 98-59 b	0.8	0.5	5365	7.6	unknown	1i	3.33 × 1013	5.72 × 1017	5.72 × 1017	rocky	Kostov et al. (2019)
L 98-59 c	1.35	2.4	5358	12.9	unknown	1i	1.6 × 1014	5.38 × 1017	5.38 × 1017	rocky	Kostov et al. (2019)
L 98-59 d	1.57	3.4	4826	13.5	unknown	1i	2.52 × 1014	1.68 × 1017	1.69 × 1017	candi-date ocean planet	Kostov et al. (2019)
LHS 1140 b	1.73	6.98	7500	23	235	5j	1.31 × 1014	k 6.15 × 1013	1.93 × 1014	cold ocean planet (SE)	Ment et al. (2019)
LHS 1140c	1.3	1.81	4700	10.8	438	5j	5.34 × 1013	k4.38 × 1018	4.38 × 1018	candi- date ocean planet	Ment et al. (2019)
Pi Mensae c	2.04	4.82	2970	11.3	1169.8	2.98	3 × 1014	0 (e = 0)	3 × 1014	rocky (MOW)	Huang et al. (2018)
Trappist-1 b	1.07	0.85	3400	7.3	400	8	2.3 × 1013	7.02 × 1017	7.02 × 1017	candi-date ocean planet	Gillon et al. (2017b), Wang et al. (2017)
Trappist-1 c	1.04	1.38	7630	12.6	342	8	2.12 × 1013	3.13 × 1016	3.13 × 1016	ocean planet	Wang et al. (2017)
Trappist-1 d	0.76	0.41	3950	7	288	8	8.29 × 1012	2.3 × 1013	3.13 × 1013	ocean planet	Gillon et al. (2017b), Wang et al. (2017)
Trappist-1 e	0.9	0.64	1710	7.7	251	8	1.39 × 1013	3.84 × 1013	5.23 × 1013	cold ocean planet (SE)	Gillon et al. (2017b), Wang et al. (2017)
Trappist-1 f	1.03	0.67	1740	6.2	219	8	2.05 × 1013	2.31 × 1013	4.36 × 1013	cold ocean planet (SE)	Gillon et al. (2017b), Wang et al. (2017)
Trappist-1 g	1.11	1.34	2180	10.7	199	8	2.57 × 1013	5.77 × 1011	2.63 × 1013	cold ocean planet (ST)	Wang et al. (2017)
Trappist-1 h	0.72	0.086	1270	1.6	167	8	6.99 × 1012	6.02 × 1012	1.3 × 1013	cold ocean planet (ST)	Wang et al. (2017)

Notes. In the case of non-transiting planets detected solely by the radial velocity technique, where only planet mass in terms of m sin i is known, we have assumed that planet mass, M_P, is equal to the minimum mass. Planet density, ρ, has been calculated for planets with an unknown bulk density using the relation: $\rho =3{M}_{P}/4\pi {R}_{P}^{3}$. Similarly, surface gravity, g, has been calculated for planets for which surface gravity is unknown using $g={{GM}}_{P}/{R}_{P}^{2}$.

^aT_S corresponds to the mean surface temperature of the planets and moons in our solar system. ^bNegligible tidal heating rate. ^cWe have assumed that moons do not have a fully deformable interior (i.e., k₂ $\ne$ 1.5). ^dH_Tidal for Triton corresponds to obliquity tides in the subsurface ocean, which contribute more internal energy than solid body tides. See Chen et al. (2014). ^eH_Radiogenic for Mercury calculated using internal heating rate from Peplowski et al. (2011) and mantle mass from Ogawa (2016). ^fWe have utilized radiogenic heating rates at 0.1 Gyr from Frank et al. (2014), and have calculated the mass of the lunar mantle, assuming that the radius and density of the mantle are 1.4 × 10⁶ m and 3500 kg m⁻³, respectively. This returns a mantle mass equal to 4 × 10²² kg for the Moon. ^gSmallest possible value of R_p from Gillon et al. (2017a). ^hIn order to obtain a conservative tidal heating rate, we used the eccentricity value reported in Borsato et al. (2014) (e = 0.026) in Equation (1). ⁱMinimum stellar age from Kostov et al. (2019). ^jMinimum stellar age from Ment et al. (2019). ^kEccentricity is poorly constrained; maximum eccentricity values substituted into (1). H_Tidal is preliminary.

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All of the geological activity on the planets and moons in our solar system is driven by internal heating. On Earth, the Moon, and the terrestrial planets, geological processes are driven by radiogenic heating. However on the moons of the giant planets such as Io, Enceladus and Europa, geological processes are primarily driven by tidal heating (Cassen et al. 1979; Peale et al. 1979; Hurford et al. 2007; Meyer & Wisdom 2007; Nimmo et al. 2007; Roberts & Nimmo 2008; Tobie et al. 2008). Terrestrial exoplanets (Jackson et al. 2008; Henning & Hurford 2014; Driscoll & Barnes 2015; Barr et al. 2018; Makarov et al. 2018; Hurford et al. 2020) and exomoons (Scharf 2006; Cassidy et al. 2009; Oza et al. 2019) may also experience geological activity as a result of tidal heating from their primaries and due to the decay of radioactive elements in their interiors (Frank et al. 2014). We therefore assume that volcanic activity on terrestrial exoplanets is driven by both tidal and radiogenic heating.

If all energy imparted to a planet by tidal dissipation as it orbits its host star is dispersed as heat in the planet's interior, then the amount of heat imparted to the planet by tidal sources, H_Tidal, is well-represented by the energy dissipation, $\dot{E}$, and can be approximated as:

$$\begin{eqnarray}&&{H}_{\mathrm{Tidal}}\sim \dot{E}=\displaystyle \frac{21}{2}\displaystyle \frac{{k}_{2}{\omega }^{5}{{R}_{P}}^{5}{e}^{2}}{{GQ}}\end{eqnarray} \tag{ 1 }$$

(Roberts & Nimmo 2008; Quick & Marsh 2015; Barr et al. 2018). In Equation (1), k₂ is the degree 2 love number, which describes how the planet responds to the tide raised on it by its primary. k₂ ranges from 0 for a completely rigid planet, to 1.5 for a planet that is entirely fluid and therefore has a significant tidal response to the gravitational tug of its star. The quantity $\omega \approx \ \tfrac{2\pi }{T}$ is the planet's orbital mean motion, with orbital period T. R_P represents planet radius, and e is orbital eccentricity. Here G is the gravitational constant, and Q is the quality factor, which represents the fraction of energy that is dissipated as heat within the planet per orbital cycle. Q can range from 1 to 1 × 10⁶ depending on a planet or moon's composition and internal structure (Goldreich & Soter 1966; Ojakangas & Stevenson 1989; Henning & Hurford 2014). A significant portion of tidal energy is dissipated as heat in the interiors of planets with high Q-values.

Frank et al. (2014) provide the radiogenic heating rate, per kg of mantle mass, (henceforth $\dot{h}$) in W kg⁻¹, for terrestrial exoplanets as a function of age, considering the radioactive isotopes ⁴⁰ K, ²³²Th, ²³⁵U,  and ²³⁸U (this data is provided publicly in Supplementary Data 2 of Frank et al. 2014).Assuming that each planet's mantle makes up 84% of its total volume, as is the case for Earth (Stacey & Davis 2008), the volume of each planet's mantle, V_mantle, can be expressed as:

$$\begin{eqnarray}&&{V}_{\mathrm{mantle}}=0.84{V}_{P}=0.84\times \displaystyle \frac{4}{3}\pi {R}_{P}^{3}.\end{eqnarray} \tag{ 2 }$$

If we assume that the density of each planet's mantle, ρ_mantle, is 4000 kg m⁻³, as is the case for Earth, (Stacey & Davis 2008), then the mass of each planet's mantle may be approximated by:

$$\begin{eqnarray}&&{M}_{\mathrm{mantle}}=0.84\times \displaystyle \frac{4}{3}\pi {R}_{P}^{3}{\rho }_{\mathrm{mantle}}.\end{eqnarray} \tag{ 3 }$$

Upon obtaining $\dot{h}$ from Frank et al. (2014) and assuming that the age of each exoplanet is identical to the average estimated age of its host star, the total radiogenic heating rate of each planet may be expressed as:

$$\begin{eqnarray}&&{H}_{\mathrm{Radiogenic}}=\dot{h}\,{M}_{\mathrm{mantle}}.\end{eqnarray} \tag{ 4 }$$

To demonstrate the utility of Equations (1) and (4), H_Radiogenic, H_Tidal, and H_Total, are listed along with physical and orbital parameters for 53 terrestrial exoplanets in Table 1. We considered planets with M_P ≤ 8M_⊕ and R_P ≤ 2R_⊕, which ensures an Earth-like, rather than a sub-Neptune-like, composition (Stevenson 1982; Borucki et al. 2011; Fabrycky et al. 2014). In addition, we have utilized the relationship: H_Total = H_Tidal + H_Radiogenic and assume that the planets do not have substantial bound atmospheres so that their average surface temperatures, T_S, may be taken to be equal to their effective temperatures, T_EFF. We have also substituted k₂ = 0.3, consistent with past studies of the terrestrial planets and icy moons in our solar system (Jackson et al. 2008; Kozai 1968; Quick & Marsh 2015; Hurford et al. 2020), and Q = 100, commensurate with past studies of tidal dissipation in our solar system's moons (e.g., Cassen et al. 1979; Peale et al. 1979; Chen et al. 2014; Quick & Marsh 2015) and the conservative end of studies that considered tidal dissipation in terrestrial exoplanets (Henning & Hurford 2014; Tamburo et al. 2018), into Equation (1). For comparison, Table 1 also includes H_Total, H_Radiogenic and H_Tidal, along with physical parameters for the planets and moons in our solar system. In most cases, internal heating rates and physical parameters for the bodies in our solar system were extracted from the literature (Table 1 Refs. column). In cases where internal heating rates were not documented in the literature, they were calculated using the formulae introduced above. As is the case for the exoplanets in our study, these calculations assumed that solar system bodies are not fully deformable so that values of H_Tidal that are reflective of bodies with homogenous interiors and non-zero rigidities were adopted. (e.g., see Table 3 of Chen et al. 2014). Physical and orbital parameters for each exoplanet were extracted from the NASA Exoplanet Archive and from the references listed in Table 1.

We note that with H_Total on the order of 10¹⁴ W (Table 1), Io is the most volcanically active body in our solar system (Moore 2003; Lopes et al. 2004; McEwen et al. 2004). Further, volcanic activity is prevalent on Earth, which has H_Total ∼ 4 × 10¹³ W (Table 1) (Turcotte 1995; Turcotte & Schubert 2002), primarily due to radiogenic sources (Schubert et al. 1997; Turcotte & Schubert 2002). Volcanic activity was also widespread on early Venus (Turcotte 1995; Basilevsky et al. 1997; Schubert et al. 1997), which may have had H_Total as large as 2 × 10¹⁴ W (Turcotte 1995). Jupiter's icy moon Europa, which has H_Total ∼ 1 TW (Chen et al. 2014; Quick & Marsh 2015), is also cryovolcanically and tectonically active (Fagents 2003; Kattenhorn & Hurford 2009; Prockter & Patterson 2009; Prockter et al. 2017; Quick et al. 2017b). Although explosive cryovolcanism on Europa may be periodic or transient in nature, and although this activity may display spatial variability (Roth et al. 2014a; Rhoden et al. 2015; Teolis et al. 2017b), the eruption of cold, geyser-like plumes has been inferred from multiple spectroscopic detections of water vapor and its constituent molecules emanating from the icy moon (Figure 1) (Roth et al. 2014b; Sparks et al. 2016, 2017; Paganini et al. 2019), as well as from in situ plasma wave and magnetic field observations by the Galileo spacecraft (Jia et al. 2018; Arnold et al. 2019, 2020). The Cassini-spacecraft directly observed geyser-like plumes erupting from the south pole of Saturn's moon Enceladus (Porco et al. 2006; Spencer et al. 2009) (Figure 1). Effusive cryovolcanism, in which slurries of cryogenic fluids quiescently erupt, has been observed on both Europa (Fagents 2003; Miyamoto et al. 2005; Parro et al. 2016; Prockter et al. 2017; Quick et al. 2017b) and on Neptune's moon Triton (Croft et al. 1995) (Figure 2). We have utilized total internal heating rates, and the associated geological activity on these planets and moons, as a baseline from which the expected rates of volcanic activity on terrestrial exoplanets can be inferred. Comparing the values of ${H}_{\mathrm{Total}}$ calculated for the planets in this study, with H_Total for the volcanically active planets and moons in our solar system (Table 1 and Figure 3) suggests that all of the exoplanets we considered are very volcanically active or very cryovolcanically active worlds. The close-in, high-eccentricity orbits of many of these planets ensure that they experience significant amounts of tidal heating from their host stars. In addition, they are large enough to hold onto heat from the decay of radiogenic elements in their interiors for significant periods of time. This retention of heat ensures that the planets in our study will be very geologically active. In addition to being volcanically active, many of these worlds are likely to have pervasive tectonic activity at their surfaces (see Hurford et al. 2020) as they attempt to rid themselves of copious amounts of internal heat.

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In an effort to further categorize the type of volcanic activity on the rocky exoplanets in our study, we will employ the naming conventions presented in Henning et al. (2018) to describe supervolcanic worlds (Table 3). For the purposes of this study, a supervolcanic world is any exoplanet with a total internal heating rate that is comparable to Earth's, where volcanic activity manifests as numerous explosive eruptions and/or lava flows at the surface, or as intense internal convection or magmatism. We consider Super-Ios to be rocky exoplanets where the amount of volcanic activity induced by internal heating is expected to exceed, or be comparable to, the amount of volcanic activity on Jupiter's moon Io. This activity may manifest as numerous explosive or effusive eruptions at a planet's surface. Regardless of their surface temperatures, H_Total for Super-Ios will be ≥∼10¹⁴ W. We envision that the surfaces of Super-Ios may resemble that of Io, which is dotted with more than 150 active volcanic centers (Figures 1(c) and (d)) (Lopes et al. 2004). Planets that are covered by global magma oceans or that have at least one hemisphere that is covered by a magma ocean, will be denoted as Magma Ocean Worlds (Henning et al. 2018). Depending upon their SiO₂ content, melting temperatures for volcanic rocks range from ∼973 to 1473 K, (Parfitt & Wilson 2008). Thus, T_s = 973 K can be considered to be a reasonable lower limit for the effective temperature of planets with molten surfaces. We therefore assume that any rocky planet in Table 1 with T_s ≥ 973 K will have a completely molten surface and could therefore be a magma ocean world.

Henning et al. (2018) also mention Exo-Venus Planets, of which only a subset is expected to have high levels of volcanic activity. We therefore consider Exo-Venus planets to be planets that generally have low levels of volcanic activity that manifest as occasional explosive eruptions and lava flows. Exo-Venus planets may also have internal convection and magmatism. Venus' surface temperature is 700 K and its H_Total = 3 × 10¹³ W (Table 1). Exo-Venus planets may therefore have similar surface temperatures and similar total internal heating rates, on the order of a few 10¹³ W. Henning et al. (2018) also mention lava worlds, which they describe as planets with extensive lava lakes at their surfaces. As there are no planets or moons in our solar system that are considered to be lava worlds, it is not possible to specify ranges of T_S and H_Total for these worlds. For this reason, we do not consider lava worlds as an appropriate designation for the planets in our study and have not labeled any planets as such in Table 1. We have chosen to apply only the terms Super-Io, Magma ocean world and Exo-Venus planet to further characterize the rocky supervolcanic worlds in our study. Our categorizations for supervolcanic planets should be revisited should the characteristics of lava worlds, with respect to minimum surface temperature and characteristic internal heating rates, be further elucidated in the future.

3.1. Rocky Exoplanet Activity

3.2. Extrasolar Ocean Worlds

Several of the terrestrial planets considered in our study may be more specifically described as ocean worlds or ocean planets. Ocean planets are a class of low-density, terrestrial exoplanet with substantial water layers that may be common throughout the galaxy (Kuchner 2003; Léger et al. 2004; Ehrenreich & Cassan 2007; Sotin et al. 2007; Fu et al. 2010). These planets may exist in one of a variety of climactic states, including, ice-free, partially ice covered, and completely frozen (Budyko 1969; Sellers 1969; Tajika 2008). In the context of this study, planets that have conditions which are favorable to the maintenance of liquid water on their surfaces or in their interiors are assumed to be ocean planets. Conditions which may be favorable to the maintenance of liquid water include T_EFF < 373 K, or, in the case of high-gravity planets with potentially high surface pressures, i.e., surface pressures between 10⁶ and 10⁷ Pa (e.g., Kepler 138 c), T_EFF below the critical point (i.e., T_EFF < 647 K), and bulk densities below 5000 kg m⁻³. The effective temperatures (Table 1) and proposed internal structures of Trappist-1 c & d (Barr et al. 2018) suggest that they could be ocean planets with surfaces that are covered in liquid water. This could be especially true of Trappist-1d (Dobos et al. 2019), as its low density is suggestive of a substantial fraction of volatiles (Table 1). We hasten to add, however, that owing to the extreme XUV emission of Trappist-1, it is possible that Trappist-1b-d may be currently experiencing a runaway greenhouse and/or that these planets have lost many oceans' worth of water (Bolmont et al. 2017; Bourrier et al. 2017b). This type of extreme water loss could be an important signature that aids in the detectability and characterization of ocean planets. Nonetheless, H_Total for Trappist-1 c & d are 3 × 10¹⁶ W and 3 × 10¹³ W, respectively (Table 1). This indicates that they are likely to have extreme volcanism at their surfaces, or on their ocean floors, with Trappist-1 c being more volcanically active than Corot-7b (Figure 3), in agreement with the results of previous analyses (Dobos et al. 2019).

We refer to exoplanets with ice-covered surfaces that overlie internal oceans (Kuchner 2003; Ehrenreich et al. 2006; Ehrenreich & Cassan 2007; Tajika 2008; Yang et al. 2017) as cold ocean planets. The internal structures of cold ocean planets may resemble the internal structures of our solar system's icy moons (Figure 4) (Ehrenreich et al. 2006; Sotin et al. 2007; Vance et al. 2007, 2015; Fu et al. 2010; Henning & Hurford 2014; Noack et al. 2016; Luger et al. 2017; Barr et al. 2018), and they may exhibit similar geological activity at their surfaces, including ice tectonics (Fu et al. 2010; Levi et al. 2014) and cryovolcanism (Levi et al. 2013; Quick et al. 2017a; Barr et al. 2018; Quick & Roberge 2018). In the absence of an atmosphere, Earth's effective temperature of 255 K (Sagan & Mullen 1972) allows for the maintenance of liquid water at the surface. We therefore assume that cold ocean planets have T_EFF < 255 K. Several of the planets plotted in Figure 3 have T_EFF < 255 K, and/or densities that are ≤3500 kg m⁻³ (Table 1), commensurate with density values of our solar system's icy moons (Jacobson et al. 1992; Anderson et al. 1997; Hussmann et al. 2006). Based on their surface temperatures and bulk densities, we have designated these exoplanets as cold ocean planets.

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The effective temperatures, densities, and total internal heating rates of Kepler 138-b & d, Trappist-1e, f, g & h, Kepler 60-c & d and Kepler 11 b suggest that these planets could be cold ocean planets that have persistent cryovolcanic activity at their surfaces (Table 1 and Figure 3). Although LHS 1140 b's high-density is indicative of an iron-rich terrestrial planet (Dittmann et al. 2017b; Ment et al. 2019), its 235 K surface temperature (Ment et al. 2019) suggests that any water on its surface would be in a frozen state. Further, its estimated total internal heating rate of almost 2 × 10¹⁴ W may be high enough for it to maintain an internal ocean and cryovolcanic eruptions at its surface. While cryovolcanic activity on Kepler 138-b and Trappist 1-g & h may be more subdued than what has been observed on Enceladus and Europa (Figure 1), their positions in Figure 3 suggest that they may undergo periodic resurfacing by cryolava flows and/or that they may experience periodic outgassing of volatiles as is the case on planet Venus (e.g., see Bondarenko et al. 2010; Smrekar et al. 2010; Shalygin et al. 2015). Thus, cryolava flows on the surfaces of these planets may be similar to what has been observed in the walled plains units of Neptune's moon Triton and in certain regions of Europa (Figures 2(a) and (b)), albeit more widespread. Explosive cryovolcanism, similar to what has been observed on Enceladus and Europa (Figure 1), may occur at comparable magnitudes on Kepler 138 d, Trappist-1f and LHS 1140 b.

According to Figure 3, rates of cryovolcanism on Trappist-1e and Kepler 60 d may be similar to rates of volcanism on Jupiter's moon Io, while Kepler 11 b and 60 c are likely to exhibit extreme cryovolcanism. Internal heating rates for these planets, on the order of 10¹⁵ W (Table 1 and Figure 3), suggest that the rates of cryovolcanism at their surfaces could exceed the rate of silicate volcanism that occurs on Corot-7b by an order of magnitude. Based on their internal heating rates and surface area to volume ratios compared to the planets and moons in our solar system (Figure 1), all of the aforementioned exoplanets may contain extensive reservoirs of liquid water, possibly in the form of global oceans beneath layers of surface ice.

We have further characterized cold ocean planets as Super-Europa/Super-Enceladus planets and Super-Tritons (Table 3). We envision that the total internal heating rates of Super-Europa/Super-Enceladus planets are large enough for them to have explosive cryovolcanic eruptions in the form of geyser-like plumes (Figures 1(a) and (b)), effusive cryovolcanism that manifests as cryolava flows on their surfaces (Figures 2(a) and (b)), and internal convection and cryomagmatism. Conversely, cryovolcanism on Super-Tritons will be dominated by effusive cryovolcanism in the form of cryolava flows, although these planets may also have intense internal convection and cryomagmatism. Both types of cold ocean planets are likely to have internal oceans. We conclude that LHS 1140 b, Trappist-1e, and Trappist-1f may be Super-Europa/Super-Enceladus planets, while Trappist-1g and Trappist-1h may be Super-Tritons (Tables 1 and 3). As is the case for Europa (Crawford & Stevenson 1988; Fagents 2003) and Enceladus (Manga & Wang 2007; Běhounková et al. 2015), cryovolcanic eruptions on these worlds may provide a pathway by which ocean water, or the contents of discrete water pockets within their ice shells, reaches the surface. It must be noted that in the case of Trappist-1f, g, & h, H_Radiogenic is greater than, or equal to, H_Tidal (Table 1). Hence, even if these planets are too far away from their host star to experience substantial tidal heating (Dobos et al. 2019), they are still likely to be geologically active and able to maintain subsurface oceans.

The densities and/or effective temperatures of Trappist-1b, Kepler 93 b, Kepler 138 c, Kepler 114 c, Kepler 80 e, LHS 1140c, Pi Mensae c, and L98-59 d are such that they could also be ocean planets or cold ocean planets. If surface pressures on these planets are high enough, liquid water could be maintained at their surfaces at the critical point, or they could contain high pressure ices. Although these planets have been plotted as ocean planets in Figure 3, we label them as candidate ocean planets in Table 1. With the exception of Kepler 114 c, for which we were unable to obtain H_Total due to the unknown age of its host star and its unknown eccentricity, estimated internal heating rates for all of these planets suggest that they are all likely to be (cryo) volcanically active (Figure 3). Notably, Pi Mensae c and L98-59 d likely receive enough internal heating from radiogenic sources alone to be as volcanically active as Io (Table 1 and Figure 3), while LHS 1140c may receive enough internal heating from radiogenic sources alone to be as geologically active as Earth. Ment et al. (2019) report an eccentricity less than 0.31 for LHS 1140c. Using 0.31 as an upper bound for eccentricity in Equation (1) results in tidal and total internal heating rates for this planet that are several orders of magnitude greater than the tidal and total internal heating rates of Corot-7 b (Table 1 and Figure 3). Such high internal heating rates would suggest that LHS 1140 c is a magma ocean world instead of a candidate ocean world. However, owing to uncertainties in e (Ment et al. 2019) the estimated H_Total for this planet represents a maximum value that may change once e is better constrained. Bearing this in mind, we contend that LHS 1140c could be an ocean world that contains exotic forms of ice on its surface or within its interior. Due to its high T_EFF and low density, Pi Mensae c could also contain exotic forms of ice.

Europa and Ganymede's mantles comprise 69% and 40% of their total volumes, respectively. This information can be utilized to determine H_Radiogenic for ocean planets when their internal structures are similar to the internal structures of these icy moons (Table 2). For these cases, the procedures outlined in Equations (2)–(4) can be applied if Equation (2) is replaced by V_mantle = 0.69V_P and V_mantle = 0.4V_P for planets with mantle volumes that are similar to Europa and Ganymede, respectively (see Appendix). It is clear from Table 2 that assuming mantle volumes similar to Europa and Ganymede returns radiogenic heating rates and total internal heating rates that are generally of the same order of magnitude as when an Earth-like mantle volume was applied in Section 2. As before, H_Total obtained from these calculations suggest that all of the exoplanets considered here are likely to be cryovolcanically active. Thus, the results discussed in Sections 3.1 and 3.2 remain unchanged.

Table 2.  Exoplanet Heating Rates for Various Internal Structures [1 W = 10⁷ erg s⁻¹]

World	RP (${R}_{\oplus })$	MP (${M}_{\oplus })$	HRadiogenic for Europa-like Structure (W)	HRadiogenic for Ganymede-like Structure (W)	HTidal (W)	HTotal for Europa-like Internal Structure (W)	HTotal for Ganymede-like Internal Structure (W)	Type
aHD 219134 f	1.31	unknown	2.52 × 1013	1.46 × 1013	1.42 × 1014	1.67 × 1014	1.57 × 1014	unknown
Kepler 11 b	1.8	1.9	7.7 × 1013	4.5 × 1013	b1.12 × 1015	1.2 × 1015	1.16 × 1015	cold ocean planet
Kepler 60 b	1.68	4.2	8.53 × 1013	4.94 × 1013	9.4 × 1014	1.03 × 1015	9.89 × 1014	ocean planet
Kepler 60 c	1.9	3.85	1.23 × 1014	7.15 × 1013	3.81 × 1015	3.93 × 1015	3.88 × 1015	cold ocean planet
Kepler 60 d	1.99	4.16	1.42 × 1014	8.22 × 1013	3.03 × 1014	4.45 × 1014	3.85 × 1014	cold ocean planet
Kepler 62 c	0.54	unknown	2.33 × 1012	1.35 × 1012	TBD: e unknown	2.33 × 1012	1.35 × 1012	unknown
Kepler 68 c	0.96	2.04	1.4 × 1013	8.1 × 1012	0 (e = 0)	1.4 × 1013	8.1 × 1012	unknown
Kepler 80 e	1.6	4.1	1.36 × 1014	7.89 × 1013	TBD: e unknown	1.36 × 1014	7.89 × 1013	candidate ocean planet
Kepler 93 b	1.6	3.2	6.28 × 1013	3.64 × 1013	0 (e = 0)	6.28 × 1013	3.64 × 1013	candidate ocean planet
Kepler 114 c	1.6	2.8	1.13 × 1014	6.53 × 1013	TBD: e unknown	1.13 × 1014	6.53 × 1013	candidate ocean planet
Kepler 138 b	0.7	0.19	6.5 × 1012	3.77 × 1012	5.33 × 1011	7.04 × 1012	4.3 × 1012	cold ocean planet
Kepler 138 c	1.7	5.2	9.32 × 1013	5.4 × 1013	3.2 × 1013	1.25 × 1014	8.6 × 1013	candidate ocean planet
Kepler 138 d	1.7	1.2	9.32 × 1013	5.4 × 1013	1.5 × 1013	1.08 × 1014	6.9 × 1013	ocean planet
Kepler 186 b	1.07	unknown	2.58 × 1013	1.5 × 1013	TBD: e unknown	2.58 × 1013	1.5 × 1013	unknown
Kepler 186 c	1.25	unknown	4.12 × 1013	2.39 × 1013	TBD: e unknown	4.12 × 1013	2.39 × 1013	unknown
Kepler 186 d	1.4	unknown	5.79 × 1013	3.36 × 1013	TBD: e unknown	5.79 × 1013	3.36 × 1013	unknown
Kepler 186 e	1.27	unknown	4.32 × 1013	2.51 × 1013	TBD: e unknown	4.32 × 1013	2.51 × 1013	unknown
Kepler 186 f	1.2	unknown	3.65 × 1013	2.11 × 1013	1.1 × 109	3.65 × 1013	2.11 × 1013	unknown
Kepler 414 b	1.7	3.5	8.43 × 1013	4.89 × 1013	TBD: e unknown	8.43 × 1013	4.89 × 1013	ocean planet
L 98-59 d	1.57	3.4	1.81 × 1014	1.05 × 1014	1.68 × 1017	1.68 × 1017	1.68 × 1017	candidate ocean planet
LHS 1140 b	1.73	6.98	9.43 × 1013	5.47 × 1013	c 6.15 × 1013	1.56 × 1014	1.16 × 1014	cold ocean planet
LHS 1140c	1.3	1.81	4 × 1013	2.32 × 1013	c4.38 × 1018	4.38 × 1018	4.38 × 1018	candidate ocean planet
Trappist-1 b	1.07	0.85	1.68 × 1013	9.72 × 1012	7.02 × 1017	7.02 × 1017	7.02 × 1017	candidate ocean planet
Trappist-1 c	1.04	1.38	1.54 × 1013	8.92 × 1012	3.13 × 1016	3.13 × 1016	3.13 × 1016	ocean planet
Trappist-1 d	0.76	0.41	6.01 × 1012	3.48 × 1012	2.3 × 1013	2.9 × 1013	2.65 × 1013	ocean planet
Trappist-1 e	0.9	0.64	9.97 × 1012	5.78 × 1012	3.84 × 1013	4.84 × 1013	4.42 × 1013	cold ocean planet
Trappist-1 f	1.03	0.67	1.49 × 1013	8.67 × 1012	2.31 × 1013	3.8 × 1013	3.18 × 1013	cold ocean planet
Trappist-1 g	1.11	1.34	1.87 × 1013	1.08 × 1013	5.77 × 1011	1.93 × 1013	1.14 × 1013	cold ocean planet
Trappist-1 h	0.72	0.086	5.11 × 1012	2.96 × 1012	3.03 × 1012	8.14 × 1012	5.99 × 1012	cold ocean planet

Notes. Heating rates for Ocean Planets and planets of unknown type (see Table 1) considering internal structures similar to Europa (V_mantle = 0.69 V_P) and Ganymede (V_mantle = 0.4V_P) instead of Earth.

^aSmallest possible value of R_p from Gillon et al. (2017a). ^bIn order to obtain a conservative tidal heating rate, we have used the eccentricity value reported in Borsato et al. (2014) (e = 0.026) in Equation (1). ^cEccentricity is poorly constrained; maximum eccentricity values substituted into (1). H_Tidal is preliminary.

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Similar to the case for volcanism on rocky exoplanets, water vapor and other volatiles that are lofted into the exospheres of cold ocean planets during cryovolcanic eruptions could be detected in transmission spectra using high-resolution ground-based telescopes. The presence of copious amounts of water vapor and its constituent atoms in the exospheres of low-mass exoplanets with low effective temperatures and densities could therefore serve to confirm their status as cryovolcanically-active cold ocean planets.

We note that utilizing Equation (1) to determine H_Total for Io and Enceladus where Jupiter and Saturn serve as the primaries from which tidal heating is sourced, returns total internal heating rates of 2 × 10¹³ W and 3.2. × 10⁸ W, respectively, which are 1–2 orders of magnitude less than their actual values of 1 × 10¹⁴ W, and 1.6 × 10¹⁰ W, respectively (Veeder et al. 1994; Spencer et al. 2000; Schubert et al. 2004; Howett et al. 2011). This discrepancy suggests that substituting k₂ = 0.3 and Q = 100 in Equation (1) returns conservative values for exoplanet tidal heating rates, and by extension, for the magnitude of geological activity that may be occurring at the surfaces of the planets we have considered. In order to ensure maximum accuracy when constraining the amount of volcanic activity on the exoplanets listed in Table 1, specific k₂ and Q values must be determined for each planet. This would require a new modeling approach that would allow us to constrain the relative amounts of ice, metal, and rock likely to be present in each of the exoplanets considered. This approach was employed in Barr et al. (2018) and could be utilized to confirm the terrestrial status of the largest planets in our study, including, Kepler 60 c & d, LHS 1140 b, Kepler 138 c, and Pi Mensae c. All of these planets have radii greater than 1.6R_⊕, which could be indicative of a sub-Neptune-like composition (Rogers 2015; Jin & Mordasini 2018). Such an approach, which would also include an in-depth analysis of the internal structures of the cold ocean planets identified in this study, will be the focus of a forthcoming manuscript. Indeed, we have assumed that all of the cold ocean planets in our study are Super-Europa or Super-Enceladus planets, consisting of thin external ice shells overlying oceans that are in direct contact with a rocky mantle. Such worlds may also have a small iron core (Figure 4). However, the cold ocean planets listed in Tables 1 and 3 could also be Super-Ganymedes. In that case, these worlds would have oceans that are sandwiched between very thick (≥approximately 100 km thick) external ice shells and sub-ocean layers of high-pressure ice, beneath which rocky mantles and iron cores lie (see Anderson et al. 1996; Vance et al. 2014, 2015). Owing to this internal configuration in which oceans would not be in direct contact with silicate mantles, cryovolcanism on Super-Ganymedes would occur sporadically and would likely manifest as effusive cryovolcanism in the form of occasional cryolava flows at their surfaces (Kay & Head 1999; Schenk et al. 2001; Showman et al. 2004) Both Europa-like and Ganymede-like internal structures are permissible for cold ocean planets (Luger et al. 2017; Barr et al. 2018) and should be explored further in the future.

Table 3.  Supervolcanic Exoplanet Categories [1 W = 10⁷ erg s⁻¹] This Table includes Descriptions and Examples of each Type of Supervolcanic World for which an Analog Exists in our Solar System

Note. Magma Ocean Worlds (e.g., 55 Cancri e, Corot-7b) have not been included here, as no analogs for these types of planets exist in our solar system. See Figures 1 and 2 for detailed descriptions of the ways in which volcanism/cryovolcanism may manifest on these worlds.

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We have not considered additional exoplanet heating sources, such as those that might be contributed by stellar radiation (see Barr et al. 2018), or dynamical planet–planet or planet-moon interactions within exoplanetary systems in our analyses. However, contributions from stellar radiation may be significant (Guenther & Kislyakova 2020). We have also neglected to consider cases in which the dynamical histories and internal structures of planetary bodies could preclude substantial internal heating. This is the case for Saturn's moon Mimas. Although Mimas is known to be a geologically dead world (Rhoden et al. 2017; Kirchoff et al. 2018), utilization of Equation (1) returns a tidal heating rate that suggests that the small moon should be more geologically active than Enceladus. Neveu & Rhoden (2019) suggest that the absence of current geological activity on Mimas may be due to a loss of radiogenic heating early in its evolution. The authors assert that loss of radiogenic heating precluded Mimas from having a dissipative interior. However, such considerations for Earth-like exoplanets are beyond the scope of this work.

Additionally, as the planets in our study have been categorized as Super-Ios, Magma Ocean Worlds, Exo-Venus Planets, Super-Europa/Super-Enceladus planets and Super-Tritons based on their surface temperatures and total internal heating rates, we have made no attempt to further categorize planets for which T_s is unknown, planets of "unknown" type (i.e., planets for which there is not enough information to determine whether they are rocky planets, ocean planets or cold ocean planets in Table 1) or ocean planets, for which it is unknown whether the bulk of their volcanic activity would occur on their ocean floors, or at their surfaces.

Finally, our analysis has not included the effects of induction heating, which could serve as an additional heat source for driving volcanic activity on close-in exoplanets (Garraffo et al. 2017; Kislyakova et al. 2018; Guenther & Kislyakova 2020). Recent studies focusing on induction heating of the Trappist-1 planets have revealed that the amount of heating imparted by this process would be at least an order of magnitude less than the amount of heating that these planets would experience due to tidal interactions with their host stars (Garraffo et al. 2017; Kislyakova et al. 2017). In the case of Trappist-1b, the amount of heat imparted by tidal interactions, 7 × 10¹⁷ W (Table 1), is greater than that experienced due to induction heating, 5 × 10¹² W (Kislyakova et al. 2017), by 5 orders of magnitude. This is in agreement with past work which suggested that tidal heating surpasses induction heating in the Trappist-1 system (Luger et al. 2017). Guenther & Kislyakova (2020) explored prospects for volcanism on HD3167 b assuming that volcanic activity would be driven by induction heating or melting of the surface due to intense stellar radiation. The amount of induction heating these authors estimate for HD3167 b ranges from 10¹² to 10¹⁵ W, with the latter value being representative of the estimated amount of heating imparted by induction when the planet was young. Induction heating rates of 1.8 × 10¹²–4.5 × 10¹³ W, corresponding to stellar dipole fields on the order of 10⁻⁴ T (1–5 G), are much less than the ∼10¹⁴ W of radiogenic heating that this planet experiences today (Table 1). Moreover, while induction heating may have played a significant role in driving volcanic activity on HD 3167 b when HD 3167 was young (Guenther & Kislyakova 2020), the system's average estimated age of 7.8 Gyr (Christiansen et al. 2017) suggests that induction heating is much less of a driver for volcanic activity on this planet than radiogenic heating is today. Our results suggest that even without accounting for additional heating from its host star, HD 3167 b can be classified as a supervolcanic planet owing to its present-day radiogenic heating rate alone (Table 1 and Figure 3). These results are in agreement with previous assessments by Guenther & Kislyakova (2020) that HD3167 b has a molten surface and is an ideal place to search for an exosphere. Indeed, HD3167 b's high surface temperature and H_Total suggest that this planet is a magma ocean world (Table 1). Since ${H}_{{\mathrm{Total}}_{\mathrm{HD}3167{\rm{b}}}}\sim {H}_{{\mathrm{Total}}_{\mathrm{Io}}}$, we argue that in addition to intense heating by its host star, that widespread magmatism induced by the planet's endogenic heating plays a significant role in creating HD 3167 b's molten surface.

It must be noted that variations in exoplanet eccentricity may affect the timing and strength of volcanism at the surfaces of the planets plotted in Figure 3. Recent studies have attempted to predict the timing and strength of volcanism at Io's surface based on variations in its eccentricity as it orbits Jupiter. Similar to the case of the geyser-like plumes on Enceladus (Figure 1(a)) (Hedman et al. 2013), observations of the Ionian volcano Loki Patera, which is responsible for ≥10% of Io's heat flow (Veeder et al. 1994), suggest that volcanic activity on the Jovian moon may vary with its orbital phase. Io experiences the greatest amount of tidal heating and volcanic activity at apojove, when its average eccentricity is the highest (de Kleer et al. 2019a, 2019b). In addition, periodicities in tidal forcing and eruption strength are likely to exist on Io, with recent eruptions occurring on 440–475 day periods (de Kleer et al. 2019a; Rathbun et al. 2019). If volcanism on tidally heated exoplanets operates similarly to volcanism on the tidally heated bodies in our solar system, this would suggest that extrasolar volcanism may also be periodic, and that tidally locked planets may experience the greatest amount of internal heating and elevated volcanism at apoapsis. The results of the aforementioned studies could then be used as a baseline to predict the time during their orbits in which volcanic eruptions are likely to be the strongest, and hence most detectable, on solid exoplanets. We also hasten to add that while explosive cryovolcanic activity on Enceladus is abundant and continuous (Spencer et al. 2009; Postberg et al. 2018a), spectroscopic detections using ground-based telescopes (Roth et al. 2014a, 2014b; Paganini et al. 2019), modeling (Fagents et al. 2000; Quick et al. 2013; Rhoden et al. 2015; Quick & Hedman 2020), and previous searches for plumes in the Galileo data set (Phillips et al. 2000), all suggest that explosive cryovolcanism on Europa may be small-scale and/or sporadic in nature. Thus, future observational searches for explosive cryovolcanic activity on cold ocean planets should be undertaken with the knowledge that activity on these bodies could be transient in nature and possibly difficult to detect. Accordingly, searches for variability at transiting exoplanets would be useful.

Io has a history of extreme volatile loss, much of which includes the loss of volcanically produced SO₂, S, and O to its exosphere via interactions with solar energetic particles (McGrath et al. 2004 and references therein). Thus, comparisons of the magnitudes of atmospheric loss between Io and the rocky exoplanets we have considered could further confirm their categorization as Super-Ios, magma ocean worlds, or Exo-Venus planets, while simultaneously revealing the state of their atmospheres. We have therefore employed methods for observing an exo-Io (see Oza et al. 2019) to perform a basic atmospheric loss calculation for several of the rocky planets listed in Table 1, assuming that SO₂ is their primary atmospheric constituent and volcanic volatile, as on Io. This basic calculation allows us to estimate the magnitude of energy-limited escape induced by incoming XUV radiation, and the associated line of sight (LOS) column densities of SO₂ for these worlds (Table 4). Incident XUV fluxes for each planet, F_XUV, as a function of system age and host star spectral type, were obtained using the methods of Lammer et al. (2009), while the magnitude of energy-limited escape was calculated using Equation (9) of Oza et al. (2019), assuming an efficiency factor for SO₂, η_XUV, of 0.35, a mass fraction of SO₂, x_i, of ∼1, as is the case for Io's atmosphere (Lellouch et al. 1992, 1996), and assuming that R_a ≅ R_P so that U_s is representative of the gravitational binding energy of SO₂ at an adsorption altitude R_P (see Johnson et al. 2015). Equation (13) of Oza et al. (2019) was then employed to obtain the disk-averaged, line of sight column density of SO₂, $\langle N\rangle$, as a function of host star radius and the SO₂ photoionization time, τ_i, of each exoplanet. The results of these calculations represent the evaporating column density of SO₂ in transmission spectroscopy (Table 4).

Table 4.  Column Densities and Outgassing Rates for Rocky Supervolcanic Planets Assuming that their Atmospheres/exospheres are Dominated by SO₂ and/or that SO₂ is their Primary Volcanic Volatile, as is the case for Jupiter's Moon Io

Planet	Stellar Spectral Type	aP (au)	Line of Sight Column Density of SO2 (Evaporative Column Density) $\langle N\rangle$ (molecules m−2)	SO2 Mass Loss Rate (kg s−1)	Volcanic Column Density of SO2, NO (molecules m−2)	Tidally-driven Volcanic Outgassing Rate of SO2 (kg s−1)
Kepler 78 b	G	0.01	2.0 × 1016	2.9 × 108	a...	a...
55 Cancri e	G	0.015	3.2 × 1014	3.3 × 106	4.9 × 1025	3.0 × 1014
GJ 1132 b	M	0.0153	6.5 × 1015	3.2 × 106	a...	a...
Corot-7b	G	0.017	9.9 × 1015	6.1 × 107	b0	b0
Kepler 10 b	G	0.0172	1.8 × 1014	1.8 × 106	a...	a...
HD 3167 b	K	0.0186	2.6 × 1014	1.5 × 106	b0	b0
L 98-59 b	M	0.0233	3.0 × 1016	1.4 × 107	c...	4.0 × 1010
L 98-59 c	M	0.0324	3.0 × 1016	7.3 × 106	c...	3.7 × 1010
Kepler 80 d	K	0.0372	2.4 × 1015	2.1 × 106	a...	a...
HD 219134 b	K	0.03876	1.6 × 1014	1.6 × 105	b0	b0
Kepler 21 b	F	0.0427	1.5 × 1015	7.3 × 106	4.9 × 1021	1.8 × 1010
HD 219134c	K	0.0653	1.4 × 1014	5.2 × 104	3.0 × 1020	1.5 × 109
Pi Mensae c	G	0.06839	3.8 × 1015	2.6 × 106	b0	b0
Kepler 100 b	G	0.0727	9.6 × 1013	1.0 × 105	a...	a...
Kepler 105 c	G	0.0731	1.2 × 1015	4.7 × 105	a...	a...
Kepler 36 b	G	0.1153	1.7 × 1014	8.9 × 104	3.0 × 1018	1.4 × 107

Notes.

^aCannot be calculated because H_Tidal is unknown (see Table 1). ^bEquals 0 because H_Tidal = 0 (see Table 1). ^cUnknown because atmospheric scale height, H = kT_s/mg, cannot be calculated (see Table 1).

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The photoionization time of SO₂ on Io, ${\tau }_{{i}_{\mathrm{Io}}}$, is approximately 20 days (1.7 × 10⁶ s) (Kumar 1979). As the photon flux from the host star is proportional to 1/a_P², where a_P is the planet's semimajor axis, the photoionization time of SO₂ on each exoplanet may be obtained by scaling according to Io's SO₂ photoionization lifetime, i.e., ${\tau }_{i}={\tau }_{{i}_{\mathrm{Io}}}\times {\left({a}_{P}/{a}_{\mathrm{Io}}\right)}^{2}=1.7\times {10}^{6}\,{\rm{s}}\times {\left({a}_{P}/{a}_{\mathrm{Io}}\right)}^{2}$, where a_P for each exoplanet is listed in Table 4. Table 19.1 in McGrath et al. (2004) reports line of sight column densities of SO₂ at Io on the order of 10¹⁸–10²¹ molecules m⁻², while Lellouch et al. (2015) report column densities ranging from 3 × 10²⁰ to 1.5 × 10²¹ molecules m⁻² for SO₂ on Io. According to our calculations, line of sight column densities of SO₂ on the exoplanets listed in Table 4 would be ∼2 to 4 orders of magnitude less than the conservative end of estimates for Io's SO₂ column densities. While these calculations suggest that even in the absence of volcanism the extreme surface heating of close-in exoplanets may cause evaporation of their surfaces and outgassing, they also suggest that without abundant volcanism, all of the planets listed in Table 4 would have exospheres similar to Europa and Earth's moon rather than Io-like collisional atmospheres.

Constraining the expected tidally-driven volcanic outgassing rates and volcanic column densities, NO, for the rocky exoplanets in our study represents a novel approach to confirming their status as supervolcanic worlds. Thus we have also employed Equation (11) of Oza et al. (2019) to calculate the tidally-driven volcanic outgassing rates of several of the planets listed in Table 4, assuming that their atmospheres are dominated by SO₂. Thus their tidal efficiency, ${\eta }_{T}\,={H}_{\mathrm{Tidal}}/{H}_{{\mathrm{Tidal}}_{\mathrm{Io}}}$. In addition we have employed Equation (5) of Oza et al. (2019) to determine the expected volcanic column densities of these planets, obtaining their scale heights, H, using the relation: H  = kT_s/mg. Here k = 1.38 × 10⁻³ m² kg s⁻² K is Boltzmann's Constant, T_s is the exoplanet's surface temperature (Table 1), m = 1.06 × 10⁻²⁵ kg is the mass of one SO₂ molecule, and g is acceleration due to gravity (Table 1). H_Tidal for each exoplanet and ${H}_{{\mathrm{Tidal}}_{\mathrm{Io}}}$ are listed in Table 1. Based on the observations of Lellouch et al. (2015), Oza et al. (2019) concluded that the volcanic outgassing rate of SO₂ on Io is ∼6.9 × 10⁶ kg s⁻¹. Predicted tidally-driven volcanic outgassing rates for SO₂ on 55 Cancri e, L 98-59 b & c, Kepler 21 b, HD 219134c, and Kepler 36 b would be between 1 and 8 orders of magnitude greater than this (Table 4). In addition, in the presence of strong tidal heating, volcanic column densities for 55 Cancri e, Kepler 21 b, HD 219134 c, and Kepler 36 b would be on the order of 10¹⁸–10²⁵ molecules m⁻² (Table 4). These calculations suggest that these worlds experience very high levels of volcanic activity, due in large part to the tidal heating they experience because of their close-in, eccentric orbits. As is the case for Io, these exoplanets would also experience extreme loss of large amounts of volcanically-produced SO₂, S, and O which could lead to the production of Io-like collisional volcanic atmospheres. Even 55 Cancri e, which is categorized as a magma ocean world in Table 1, would have enough volcanic outgassing, because of its large tidal heating rate, to support a collisional atmosphere.

It must be reiterated that the above calculations are based on the assumptions that the rocky exoplanets in our study have atmospheres that are dominated by SO₂, and that SO₂ serves as the main volcanic volatile on these worlds. However, previous research has suggested that the atmosphere of 55 Cancri e is dominated by water vapor, N₂, CO₂, CO, or O₃ (Angelo & Hu 2017). Moreover, several volcanoes on Earth erupt carbon-rich lavas (Radebaugh et al. 2020), a phenomenon which could be pervasive on rocky exoplanets if extrasolar carbon planets are common throughout the galaxy (Kuchner & Seager 2005). Thus, comparing expected SO₂ loss rates on extrasolar worlds with those on Io may not represent the most reliable method to confirm the supervolcanic status of the putative Super-Ios, Magma Ocean Worlds, etc., in our study. While we note that with careful consideration, the estimates provided above can be reformulated for CO₂ or other volcanic volatiles, an alternative approach that involves constraining the depth to magma layers in rocky exoplanets based on their surface temperatures, total internal heating rates, and the temperatures at which igneous rocks melt, may represent a more reliable way to confirm their status as supervolcanic worlds and discriminate between Super-Ios, Magma Ocean Worlds, and Exo-Venus Planets. Similar methods have been employed to determine the depth to subsurface oceans and liquid layers within the icy moons and TNOs in our solar system (Ruiz 2003; Quick & Marsh 2015; Saxena et al. 2018) and will be the focus of a future effort for the rocky exoplanets surveyed here.

Dorn et al. (2018) suggested that stagnant-lid planets with masses greater than 3M_⊕ would not have significant outgassing, while Noack et al. (2017) found that on stagnant-lid planets, outgassing only occurs on planets with masses below 4–7 M_⊕. Furthermore, previous studies have suggested that the majority of volcanism on stagnant-lid planets occurs prior to 4.5 Gyr (Kite et al. 2009; Dorn et al. 2018). However, we find that even massive super-Earths, including 55 Cancri e, which is 8M_⊕, and ∼10 Gyr old, are likely to be volcanically active. We also find that all of the Trappist planets, which have an upper limit age of 8 Gyr (Luger et al. 2017), will be volcanically or cryovolcanically active. Comparing calculated internal heating rates of each exoplanet in our study to the estimated internal heating rates of the planets and moons in our solar system indicates that all 53 exoplanets we considered are likely to exhibit moderate to extreme rates of volcanism at their surfaces. We find that the majority of exoplanets considered in this study are likely to exhibit volcanic activity at levels that are at least on par with, if not greater than, the most volcanically active bodies in our solar system. The agreement of our results with past studies that considered the magnitude of geological activity on exoplanets (e.g., Jackson et al. 2008; Barnes et al. 2010; Demory et al. 2015, 2016; Dobos et al. 2019) illustrates that the tidal heating rates obtained using Equation (1), and the radiogenic heating rates extracted from Frank et al. (2014), can be reliably employed to place conservative estimates on the expected magnitude of internal heating, and by extension, volcanic activity, on solid exoplanets. Hence from the standpoint of comparative exoplanetology, in which the internal heating rates and geological activity of the planets and moons in our solar system are used as a baseline, our results regarding the expected magnitude of volcanic activity on terrestrial exoplanets are robust.

Additionally, out of the 53 exoplanets that were surveyed, 14 (∼26%) have effective temperatures and/or densities that are consistent with them being candidate ocean planets. 9 out of these 14 planets (∼64%) have effective temperatures and/or densities that are indicative of them being cold ocean planets. Hence, about 17% of all of the planets surveyed in this study may be cold ocean planets with internal structures that are similar to the moons of our solar system's giant planets (Figure 4), with similar geological activity at their surfaces. If conditions at the surfaces of candidate ocean planets Trappist-1b, Kepler 93 b, Kepler 138 c, Kepler 114 c, Kepler 80 e, LHS 1140c, Pi Mensae C and L98-59 d are such that they can maintain liquid water or high-pressure ices, then 21/53 or ∼40% of the planets surveyed here may be ocean planets. Of note is that these totals do not include cold exoplanets such as OGLE 2005-BLG-390-Lb (Beaulieu et al. 2006), MOA-2007-BLG-192 (Bennett et al. 2008), or OGLE 2016-BLG-1195-Lb (Shvartzvald et al. 2017) which were detected by gravitational microlensing. These worlds may be cold ocean planets in their own rights, with ice-covered surfaces and internal oceans (Ehrenreich et al. 2006; Ehrenreich & Cassan 2007; Bond et al. 2017; Bourrier et al. 2017b). Depending on their total internal heating rates, they could also be geologically active. Our results suggest that a significant number of exoplanets that have been previously classified as terrestrial planets may instead be extrasolar oceans worlds that contain significant amounts of water and may exhibit cryovolcanism at their surfaces.

In our solar system, cryovolcanism on the moons of the giant planets serves as an important process that transports liquid water, energy and organics between their interiors and surfaces (Fagents 2003; Manga & Wang 2007; Lopes et al. 2013; Postberg et al. 2018b). In some cases, the circulation of cryomagmatic fluids could create transient habitable niches in the interiors of ice-covered worlds (Ruiz et al. 2007). In addition, recent studies have suggested that even planets that are mostly ice-covered may have substantial amounts of unfrozen land near their equators (Paradise et al. 2019) or small, equatorial regions of salt-rich water (del Genio et al. 2019) where life could flourish. The significant number of candidate cold ocean planets listed in Table 1 suggests that it is important to consider the possibility of habitable environments on planets that exist beyond the snowline in extrasolar planetary systems. Cryovolcanic eruptions on cold ocean planets could be detected by next-generation telescopes as transient or periodic excesses in H₂O, O₂, and/or H in transit spectra that display spatial variability (Bourrier et al. 2017b; Quick et al. 2017a) or are localized to one hemisphere, as is the case for the cryovolcanically active moons in our solar system (Hurford et al. 2007; Hedman et al. 2013; Roth et al. 2014a, 2014b; Rhoden et al. 2015; Sparks et al. 2016, 2017; Teolis et al. 2017a, 2017b). If traces of cryovolcanic activity on exoplanets could be detected in transit spectra, this activity could be used as an indicator to determine which planets have substantial amounts of liquid water and internal energy, both of which are necessary ingredients for life.

We thank Dr. Orenthal Tucker for helpful discussions and an anonymous reviewer for helpful critiques that improved the quality of the manuscript.

According to Schubert et al. (2009), Europa's mantle likely comprises 81% of its mass. Assuming that the densities of ice and silicates inside of Jupiter's moon Ganymede are 950 kg m⁻³ and 3500 kg m⁻³, respectively, Equation (65) of Chen et al. (2014) reveals that the total mass of silicates within Ganymede is approximately 1 × 10²³ kg. If all of Ganymede's silicates are found within its mantle, then this suggests that Ganymede's mantle comprises 70% of its total mass. This information has been used to deduce that the mantles of Europa- and Ganymede-like planets comprise 69% and 40% of their total volumes, respectively, by employing the following procedure:

We know that ${V}_{\mathrm{mantle}}={M}_{\mathrm{mantle}}/{\rho }_{\mathrm{mantle}}$. If we assume that ρ_Mantle = 3500 kg m⁻³, we have for the case of Europa that:

V_mantle = 0.81M_Europa/3500 kg m⁻³ = (0.81 ∗ 4.8 × 10²² kg)/3500 kg m⁻³ = 1.1 × 10¹⁹ m³. We can then employ the relationship:

${V}_{\mathrm{mantle}}/{V}_{P}=\tfrac{1.1\times {10}^{19}\,{{\rm{m}}}^{3}}{(4/3)\pi {R}_{\mathrm{Europa}}^{3}}=\tfrac{1.1\times {10}^{19}\,{{\rm{m}}}^{3}}{(4/3)\pi {\left[1.56\ast {10}^{6}{\rm{m}}\right]}^{3}}$ to obtain V_mantle = 0.69V_P for planets with Europa-like internal structures.

Similar steps may be taken to obtain V_mantle = 0.4V_P for planets with Ganymede-like internal structures when we assume that the total mass of Ganymede's mantle is equivalent to the total mass of silicates in its interior so that M_mantle = 1 × 10²³ kg = 0.7M_Ganymede.

From here, equations similar in form to Equations (2)–(4) were used to determine H_Radiogenic in Table 2 for planets with internal structures that are more similar to our solar system's icy moons than to Earth.