ON THE DIRECT IMAGING OF TIDALLY HEATED EXOMOONS

Reynolds et al. (1987) and Segatz et al. (1988) show that the average total luminosity of a moon due to tidal heating, L_tidal is given by

where G is the gravitational constant, μ is the moon's elastic rigidity, Q is the moon's dissipation function (or quality factor), e is the eccentricity of the moon's orbit, a is the semi-major axis of the moon's orbit, ρ is the density of the moon, R_s is the radius of the moon, and M_p is the mass of the planet it orbits. Equation (1) assumes zero obliquity.

It is useful to eliminate the explicit dependence on the planet's mass by parameterizing Equation (1) in terms of the Roche radius. Let the moon's semi-major axis be some multiple, β, of the Roche radius, a_R , such that

Then we can rewrite the tidal energy flux equation as

where we have grouped the terms that depend on the moon's physical properties and those that describe its orbit separately. Note that although β is grouped with the orbital terms, in addition to its linear dependence on the moon's semi-major axis, β is also more weakly dependent on the planet's mass and the moon's density.

Alternatively, we can write this equation in terms of R_s and M_s or ρ and M_s rather than R_s and ρ. Alternative forms of Equation (3) are

The scaling relation for Equation (3) relative to the luminosity of Earth (which is L_⊕ = 1.75 × 10²⁴ erg s⁻¹) is

Note that Equation (6) adopts Q = 36, μ = 10¹¹ dynes cm⁻² and Earth's radius and density as reference values. The reference values of β and e were then chosen to give L_⊕. Note that the first set of bracket terms in Equation (6) includes exomoon's physical parameters, and the second contains the orbital parameters. This scheme is also used in Equations (8) and (11) below.

Using conventions of stellar astrophysics, we can define the exomoon's effective temperature, T_s , from the luminosity via the Stefan–Boltzmann law

where σ is the Stefan–Boltzmann constant. Equation (7) can also be written as a scaling relation relative to a 279 K (the equilibrium temperature of Earth) exomoon:

This would give ∼60 K for Io, which would be the effective temperature of Io given no solar flux. Note that Equation (8) adopts the same reference values for μ, Q, ρ_s , R_s , β, and e as Equation (6) since the reference temperature, 279 K, corresponds to a blackbody the size of earth with the luminosity given by the reference luminosity in Equation (6). These are the Q and μ values for Io from the literature (Segatz et al. 1988; Peale et al. 1979). Equation (8) also assumes there is only tidal heating with no additional energy sources, such as stellar irradiation or interior radiogenic heat. This is a conservative assumption since additional heat sources only serve to make the exomoon more luminous and thus easier to detect. We maintain that neglecting the stellar irradiation term is a good approximation for cases of substantial tidal heating and at THEM–star separations (≳ 12 AU, and typically tens of AU) that are currently accessible to high-contrast instrumentation. For example, we can consider one of the most challenging direct imaging cases, an Earth-sized THEM heated to 300 K and at a 12 AU separation from its host star. For this case, the additional heating due to stellar irradiation is <1% of the tidal heating. However, should one want to calculate the temperature due to both tidal heating and stellar irradiation, it is given by

The scaling relations in Equation (8) are illustrated in Figure 1. In Figure 1, we adopt the Q and μ values for Io in the literature. However, although Q and μ are taken to be a constant for most of the curves shown in the figure, they are clearly not constant, even for the small number of objects in our solar system. Table 1 lists Q and μ for several moons in the solar system as well as the Earth. From Table 1 we see that Q and μ vary by two to three orders of magnitude and generally increase as the effective temperature of the moon increases. If these parameters increase as the tidal heating increases then one might expect a THEM's effective temperature to saturate for sufficiently strong tidal heating. It is likely that Q and μ will change particularly rapidly at phase transitions such as melting for the material that constitutes the bulk of a THEM. Certainly T_melt will be different for rocky verses icy moons and the THEM's temperature could plausibly equilibrate near these melting points where the Q and μ values might change very quickly by many orders of magnitude. We apply a function of the form

to two of the curves in both the right and left plots of Figure 1 as a "toy model" illustration of how the effective temperature might plateau at the melting temperature. The plateaus in Figure 1 given by Equation (10) are meant only to show that in general L_tidal = L_tidal(μ(L_tidal), Q(L_tidal)) and T_s = T_s (μ(T_s ), Q(T_s )) (i.e., the tidal heating depends on Q and μ which themselves depend on the tidal heating). Two values of T_melt are shown as horizontal dashed lines in the plot. The gray line is the melting temperature of water. The orange line is at 1200 K which is representative of the melting temperature of rock (the melting temperature of igneous rock is 800–1800 K; Emiliani 2007). The dotted red and black lines correspond to the solid red and black lines, respectively, but with Equation (10) applied. In short, it is quite possible that the extreme THEM temperatures shown if Figure 1 exist for some value of R_s , ρ_s , β, and e, however, it is difficult to estimate exactly what those values will be since the Q and μ dependence on tidal heating at these extreme temperatures is poorly understood and depends on the composition of the THEM.

Zoom In Zoom Out Reset image size

Figure 1 still provides an understanding of the steep temperature dependence on R_s , ρ_s , β, and e (for instance, the temperature goes almost as the square of β). On the other hand, the temperature dependency on Q and μ is rather mild and goes only as the 1/4th power. Thus, these extreme temperatures possibly exist for reasonable values of R_s , ρ_s , β, and e, but likely for different (perhaps by orders of magnitude) values of Q and μ than were used for the computation in Figure 1.

Given T_s , the peak wavelength of a exomoon's spectral energy distribution (SED) would be

if it was emitting as an ideal blackbody. However, solar system objects with significant tidal heating typically do not have a uniform temperature surface emitting at each point as an ideal blackbody. Rather they display "hot spots" and even vulcanism, locations on the surface through which a larger, often much larger, than average part of the internal heating is being radiated away. This implies an SED that deviates significantly from a Planck form and which, in particular, emits more at shorter wavelengths than the uniform temperature blackbody approximation indicates. This is seen in the SED of Io, for example (Spencer et al. 2005). However, modeling of the complexities of heat transport in the interior of a THEM far exceeds the scope of this initial discussion of detectability, and thus we hereinafter adopt the simple blackbody model of THEM SEDs. In most respects, this is a conservative assumption in that it makes them less detectable than they would be expected to be with a more realistic SED. However, we note that this is not always true as the presence of hot spots will shift the SED to bluer wavelengths where the THEM will have to compete against an increased amount of star light. The blackbody SED assumption will be more accurate for an exomoon on which a thick atmosphere and/or oceans effectively redistribute the tidal heat emitted at hot spots on its surface.

Discussions of direct imaging of exoplanets are typically heavily focused on issues of contrast and associated instrumental inner working angles for the star–exoplanet separation on the plane of the sky. The associated considerations for THEMs are more complex since both contrast with the planet orbited by the exomoon and contrast with the stellar primary must be taken into account. Since the star–exoplanet separation does not affect the tidal luminosity of a given exomoon system at separations large enough to permit direct imaging, the contrast with the star may not (or may) be an important observational issue. However, since tidal heating is very sensitive to the moon–planet separation, it is not plausible that a significantly bright THEM can be resolved from the planet it orbits with existing or planned facilities. In other words, any emission from an exoplanet will dilute that from any THEM which orbits it. Furthermore, one can imagine a scenario where the THEM emits more light at certain wavelengths than the exoplanet and vice versa. Note the fact that current instrumentation is only capable of directly imaging THEMs at large separations which allow us to ignore nuances such as perturbations from the star on the satellite's orbit which could induce variations on the exomoon's eccentricity and tidal heating (Cassidy et al. 2009).

Tidally heated moons are easier to detect if L_s is large and L_p and L_* are small. Equation (3) gives the tidal luminosity of an exomoon. The contrast of the planet with the moon (L_p /L_s ) decreases for colder exoplanets () that are further away from their host star and for older exoplanets which have already cooled from their initial formation temperatures. The planet–moon contrast also decreases for an exoplanet with less surface area (). The contrast of the star relative to the moon (L_*/L_s ) decreases for less massive stars ( for M < 0.43 M_☉ and for M > 0.43 M_☉; Burrows et al. 2001 and Duric 2004). Finally, the contrast requirement will be relaxed for relatively nearby star systems due to the resulting larger angular separations on the sky.

In order to sustain tidal heating, a moon must preserve its orbital eccentricity. In some cases, the eccentricity is maintained by resonance with another moon. For example, Io, Europa, and Ganymede are in a 1:2:4 mean motion resonance that sustains the former's orbital eccentricity. Io is being heated by tidal forces 4.5 Gyr after the formation of the solar system, which suggests tidal heating can occur in old planetary systems. Hence, we are likely to find THEMs in systems where they are in orbital resonances with other exomoons; however, there is no reason to believe that such circumstances are uncommon. Both Jupiter and Saturn have close-in moons that participate in orbital resonances (Peale 1976), and there are theoretical reasons to expect such resonances to develop naturally during the formation process (Yoder 1979; Cassidy et al. 2009; Ogihara & Ida 2012). Furthermore, it is also possible for an exomoon to maintain an eccentric orbit via perturbations from other planets (Matija 2007) or by the star (Georgakarakos 2002, 2003).

In general, more massive planets and more massive moons are expected to allow longer lifetimes for the orbital resonances that sustain the tidal heating at fixed tidal heating luminosity.

There is evidence for strong tidal dissipation in Io and Jupiter (Lainey et al. 2009). To determine if such enormous amounts of tidal heating can be sustained over the lifetime of a planetary system, we can divide the orbital energy of the moon and rotational energy of its host planet by the THEM's luminosity. It turns out that the rotational energy of Jupiter is substantially larger than the orbital energy associated with Io, and the orbital energy term can be ignored for the Jovian system. A few percent of Jupiter's rotational energy could provide enough energy to maintain Io at ≈300 K for the age of the solar system. And for a larger planet or smaller THEM, temperatures of 1000 K could be sustained for billions of years. Io's orbital energy alone (excluding Jupiter's rotational energy) could sustain a 300 K Io for the order of 100 Myr.

In general the observed brightness and SED of a THEM is expected to be substantially variable for multiple reasons: most dramatically the exomoon may be eclipsed by the much larger and darker exoplanet it orbits, thus causing a sharp drop (and later increase) in the observed flux. The steep dependence of tidal heating on semi-major axis implies that the most luminous exomoons will have close-in orbits and thus particularly large eclipse probabilities. Even in the absence of eclipses, phase curve variations are expected since tidal heating typically produces moderate to extreme temperature variation across an object's surface. Moreover, even at a single location on the exomoon's tidally heated surface, the temperature may well fluctuate due to time varying transport of interior heat to the surface, e.g., vulcanism. In general short timescale (hours to days would be expected from solar system analogs) variability would be a signature of THEMs which would help distinguish them from the relatively steady emission expected from a cooling gas-giant exoplanet. The period of the orbit, T_orbit is just

And the fraction, F_ellipse of that time that the THEM spends in ellipse, τ_ellipse, assuming a circular, coplanar orbit (given in Heller 2012) is

where R_p is the radius of the planet. For Io the parameters in Equations (12) and (13) are, T_orbit = 1.77 days and F_ellipse = 5.4% respectively.

Because THEMs can be much smaller than objects that are kept warm by their own internal heat (such a brown dwarf or a giant planet) and much hotter than they would be due to stellar irradiation from their parent stars, some THEMs are expected to be easily and conclusively identified by their photometric properties (given that the distance to the system is known, which is very likely to be the case). In particular, rocky THEMs would have an SED significantly too blue for their maximum stellar irradiation temperature and a flux much too small to be consistent with a giant planet or brown dwarf. For example, an Earth sized object with a surface temperature of 800 K seen at a projected (and thus minimum) separation of 30 AU from a G-dwarf star would have an apparent magnitude and colors quite inconsistent with either a cooling brown dwarf or giant planet.

In this paper we have approximated THEM spectra as blackbodies. While this is perhaps a reasonable rough approximation, it is worth noting that deviations are expected from this simplistic model. The blackbody model gives us moons that will be hotter and smaller than their host exoplanets. One correction is that THEMs would likely have an excess emission at bluer wavelengths due to hot spots on their surfaces compared to that expected for a single effective temperature model (as is the case for Io, see Section 2.2). Additionally, absorption features could substantially modify their SEDs, depending on the THEM's surface composition. THEM spectral absorption features might be similar to those observed in the SEDs of lava on Earth (Oppenheimer et al. 1998; Ramsey & Fink 1999) or Io (McEwen et al. 1998; Schmitt et al. 1994; Geissler 2003) or even similar to the models of extremely hot, rocky exoplanets (Hu et al. 2012; Kaltenegger et al. 2010). Unfortunately, fully predictive modeling of THEM SEDs would be complex and very probably underdetermined since it would require an understanding of the temperatures and distribution of hot spots as well as assumptions about surface and atmosphere composition, pressure etc. Nevertheless, SEDs indicating surprisingly high temperatures (given the stellar irradiation) and small (relative to giant planet sizes) luminosities are still likely to be valuable and relatively reliable indicators of the presence of a THEM.

We note in passing that the data collected on Fomalhaut b prior to the recent detection of F435W emission (Currie et al. 2012) fits the general SED properties expected for a THEM. Specifically, aside from the F435W flux, Fomalhaut b is consistent with a THEM that is a modest fraction of the size of Io and has a surface temperature of ∼1600 K if (1) an absorption feature in H band is present and (2) surface hot spots lead to increased emission at bluer wavelengths (e.g., the F606W flux). However, the new F435W point in the object's SED curve would require the presence of unrealistically high temperature (∼6000 K) hot spots. Moreover, the optical colors of Fomalhaut b are a good match to those of its primary star, thus strongly suggesting a scattered stellar radiation explanation (Currie et al. 2012). It therefore appears quite unlikely that Fomalhaut b is a THEM. Nevertheless, it remains an interesting example of an object with some of the spectral characteristics expected for a THEM.

Before exploring the THEM discovery space it is necessary to understand constraints on exomoon temperatures that can be produced by tidal heating. The temperature power law dependencies are so strong that the simple scaling relations presented in the previous section yield temperatures of thousands of degrees for seemingly plausible hypothetical exomoon systems. However, there are other physical constraints that limit the effective temperatures that can be achieved by tidal heating. These constraints are discussed in Cassidy et al. (2009). The Cassidy et al. (2009) results suggest that mass loss from the satellite could erode THEMs on a Gyr timescale. We note that long-lived resonances in the solar system exist, and thus should exist in other systems as well.

The Cassidy et al. (2009) analysis does not yield any precise upper limit on the surface temperature of an exomoon; we therefore only present calculations of THEMs up to T_s = 1000 K in our following analysis. As context for this number note that the surface temperatures of known rocky bodies, such as Mercury and UCF-1.01 in the GJ 436 system, are of this order. The Sun facing side of Mercury reaches a temperature of 700 K, and UCF-1.01's surface is estimated to be at ∼860 K (Stevenson et al. 2012). Given that rocky planets exist at surface temperatures near 1000 K and that some rocks have even higher melting temperatures, it is plausible that rocky exomoons can survive at or above 1000 K.

The quantities Q and μ are poorly known even for most solar system objects. For the first-approximation models considered in Figure 1, we adopted the values for Io in the literature and then discussed where this assumption was valid. The following calculations for Figures 2 and 3 do not require that we assume a particular value of μ or Q. They only require that we choose the radius of a moon, and assume that there is some combination of the remaining parameters in Equation (8) (e.g., R_s , ρ_s , Q, and μ) that will give us the desired luminosity and hence temperature shown in the following plots. We will discuss the physical implications and viability of those hypothetical exomoon parameterizations.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figure 2 illustrates the detectability of THEMs with existing ground and space-based facilities. The sensitivity of two future ground based instruments is also shown for comparison. The exomoon curves are shown as blackbodies at a given temperature and radius. The curves shown for the three stars are from the Kurucz models (Kurucz 1970). Note that a 2 R_⊕ moon would have a mass of 5–20 M_⊕ whereas the expected theoretical upper limit is near 1 M_⊕ (Canup & Ward 2006; Ogihara & Ida 2012). Although the theoretical limits discussed in these two papers implies that the 2 R_⊕ THEMs shown in Figure 2 do not exist, it is possible that moons form via other processes not discussed in Canup & Ward (2006) and Ogihara & Ida (2012). For instance, the process that created the Earth's Moon produced a moon-to-planet mass ratio which is equivalent to a 20 M_⊕ moon orbiting a 5 M_J exoplanet. The Moon's formation is difficult to model (Canup & Asphaug 2001), and it is feasible that an analogous process occurs for gas giants that is capable of producing moons with >1 M_⊕.

The instrument sensitivities are shown for a 5σ, 1 hr integration detection limit. The VLT's NACO and Subaru's HiCIAO ground-based instruments are two of the most sensitive high-contrast imaging instruments currently on sky. The high contrast and adaptive optic (AO) systems currently on the infrared platform at the Subaru telescope, namely, HiCIAO (Hodapp et al. 2008) and AO188 (Hayano et al. 2010; Minowa et al. 2010), are currently capable of achieving 10⁻⁵ contrast at 0.2 arcsec angular separation from the host star. The implementation of the next generation of high contrast instrumentation, SCExAO (Martinache et al. 2011) and CHARIS (McElwain et al. 2012) will be able to do at least an order of magnitude better than that (10⁻⁶ contrast) at the same separation, and 10⁻⁷ contrast at a 2 arcsec separation. Based on Figure 2, the ability of these instruments to detect THEMs is likely to be limited by their sensitivity rather than their achievable contrast. It is possible future ground based instrumentation operating in J, H, and K bands (such as MICADO; Davies & Team 2010) will be contrast, rather than sensitivity, limited unless observing late-type stars. However, the high contrast exoplanet imagers, such as E-ELT's EPICS (Kasper et al. 2010), claim they will be able to achieve contrasts of 2 × 10⁻¹⁰ at 0.2 arcsec angular separations, which is again likely to make directly imaging THEMs a sensitivity and not a contrast problem.

The high contrast planet imagers (such as GPI, SPHERE, and CHARIS; Macintosh et al. 2008; Beuzit et al. 2008; Peters et al. 2012) coming on sky in the next 1–3 yr should be able to detect 800 K, one Earth-radius moons in the K band in systems with late-type M-dwarfs at a distance of a few parsecs. This same exomoon would have even more favorable contrast with its host star at λ = 4.5 μm with warm Spitzer. It is interesting to note that at these temperatures, the exomoon would likely be many orders of magnitude brighter than its host planet at most wavelengths, even if the planet were 10 × more massive than Jupiter, assuming a system with an age comparable to the solar system's.

Beyond K band, warm Spitzer is currently the most sensitive telescope. Spitzer's Infrared Array Camera (IRAC) could detect a 850 K, 1 R_⊕ moon at 5 pc. Figure 2 shows that THEM would be less than 10⁴ times dimmer than a late-type M-dwarf at these wavelengths. The next generation of ground based instrumentation with comparable spectral coverage (such are ELT's METIS; Kendrew et al. 2010) will be even more sensitive than Spitzer and have a much higher angular resolution (see METIS detection limits in Figure 2). The ELT's METIS and MACADO should be able to detect 600 K THEMs with radii of 0.5–1 R_⊕.

Future instruments, for example JWST's Mid-Infrared Instrument (MIRI), offer even more potential for direct imaging of exomoons. Figure 3 shows the discovery space for MIRI. Note that this discovery space should be similar to SPICA's with a slight correction for SPICA's marginally smaller aperture. Similar to Figure 2, the solid lines labeled with temperatures and radii correspond to exomoon blackbody curves with those parameters. Note that the exomoons shown here have temperatures similar to Earth's rather than the much hotter temperatures shown in Figure 2. Thus, it is plausible that some of the exomoons JWST is capable of detecting, could potentially be habitable, in the sense of having surface temperatures that would allow liquid water to be present. Some of these exomoons have comparable irradiance to the gas giants in our solar system. At ∼14 μm a 300 K, Earth-radius exomoon would be as luminous as Jupiter. However, if Jupiter were colder due to being less heated by its primary and/or being older, the Earth-like moon would be much brighter than the planet. The red bars are the 5σ detection limits for JWST-MIRI with a 10,000 s integration time (Glasse et al. 2010).

A 300 K, Earth-radius THEM is only 3 × 10⁵ times fainter at λ ∼ 14 μm than a Sun-like star but can be at a large distance from it's host star. For example, at 30 AU projected separation, it would be 15 arcsec from the star at a distance of 2 pc. At λ = 14 μm, λ/D = 0.44 arcsec for a 6.5 m telescope, which means this moon would be at 30λ/D. This far away from the star, the airy rings are ∼3 × 10⁵ times fainter than the core of the star and about the same intensity as the 300 K, Earth-radius moon, indicating that the detection should be possible. This example is not at the limit of JWST's sensitivity and inner working angle. The most challenging THEM detection JWST will be capable of making is a 300 K THEM as far as 4 pc from the sun. If there is such an Earth-sized 300 K moon orbiting αCen, MIRI will be able to detect it in eight of its nine spectral bands with better than 15σ signal to noise in a 10⁴ s integration. Thus, directly imaging a 300 K, Earth-radius moon that is tidally heated is potentially much easier than resolving an Earth-like exoplanet orbiting in the HZ of its primary.

SPICA is another future space telescope that is ideal for exomoon detection. The SPICA coronagraph is being designed to operate from 3.5 to 27 μm with a contrast of order 10⁻⁶ and a 3.3λ/D inner working angle (Enya et al. 2011). SPICA has a slightly higher noise floor than JWST, but should be able to achieve similar contrast. Thus SPICA's exomoon discovery space will be similar to JWST's, though JWST's larger aperture will give MIRI a modest advantage.

The next generation of ground based telescopes such as the Giant Magellan Telescope (Johns 2008), the Thirty Meter Telescope (Nelson & Sanders 2008), and the European Extremely Large telescope (E-ELT; McPherson et al. 2012) are best suited for detecting THEMs at closer exomoon–star separations and at slightly shorter wavelengths than SPICA and JWST (and are therefore plotted in Figure 2 which has shorter wavelengths plotted on the x-axis) and discussed in the previous section.