THE STEPPENWOLF: A PROPOSAL FOR A HABITABLE PLANET IN INTERSTELLAR SPACE

As a planetary system forms, some planets or planetesimals, referred to as "rogue" planets, can enter hyperbolic orbits and be ejected from the system as a result of gravitational interactions with gas giant planets (Lissauer 1987). Furthermore, interaction with passing stars can eject planets from mature systems (Laughlin & Adams 2000). The ability of a rogue planet to support life is of interest as a sort of pathological example of planetary habitability, because such a planet could potentially represent a viable option for interstellar panspermia (Durand-Manterola 2010), and because such a planet could be the closest source of extrasolar life for exploration by humanity in the distant future. Since some sort of starting point is required to discuss the issue, a planet is often defined as habitable if it can sustain liquid water at its surface (Kasting et al. 1993). Stevenson (1999) argued that if a rogue planet had an extremely high-pressure hydrogen atmosphere, pressure broadening of far-infrared absorption by molecular hydrogen could support liquid water on the planet's surface as a result of the geothermal heat flux alone, making the planet potentially habitable. Debes & Sigurdsson (2007) showed that terrestrial planets can be ejected with moons and that the resulting tidal dissipation could increase the geothermal heat flux by up to two orders of magnitude for $\cal {O}$(10⁸ yr).

Subglacial liquid water oceans sustained by internal heat flux on icy bodies represent an alternative type of habitat. It is well known that subglacial oceans are possible on moons around giant planets and on trans-Neptunian objects in the solar system (Hussmann et al. 2006), as well as water-rich exoplanets in distant orbits (Ehrenreich et al. 2006; Fu et al. 2010). A possible terrestrial analog is Lake Vostok, a ≈125 m deep lake which is sustained by geothermal heat flux under ≈4 km of ice on Antarctica (Kapitsa et al. 1996). Laughlin & Adams (2000) have even argued that a terrestrial rogue planet, not attached to any star and receiving negligible energy at its surface, could sustain a subglacial liquid ocean if it had a thick enough ice layer. We wish to consider this point in more depth, including issues such as the potential for solid-state convection of ice, the potential effect of a thermally insulating frozen gas layer from outgassing of the mantle on an Earth-like rogue planet, the effects of melting point suppression due to contaminants, and observational prospects. By Earth-like, we mean specifically within an order of magnitude in mass and water complement, similar in composition of radionuclides in the mantle, and of similar age. A subglacial ocean on a rogue planet is interesting because it could serve as a habitat for life which could, for example, survive by exploiting chemical energy of rock that is continually exposed by an active mantle. We will refer to a rogue planet harboring a subglacial ocean as a Steppenwolf planet, since any life in this strange habitat would exist like a lone wolf wandering the galactic steppe.

We can imagine that the ice layer on top of an ocean on a Steppenwolf planet will grow until either it reaches a steady state with the ice bottom at the melting point, or all available water freezes. Geothermal heat from the interior of the Steppenwolf planet will be carried through the ice layer by conduction, and potentially by convection in the lower, warmer, and less viscous portion of the ice layer. Since convection transports heat much more efficiently than conduction, the steady-state ice thickness will be much larger if convection occurs, making it harder to maintain a subglacial ocean.

Here we will calculate steady-state ice thicknesses when there is conduction only and when there is convection in the lower portion of the ice, and make the conservative assumption that the thicker solution is valid. We must acknowledge, however, that it is very difficult to establish definitively whether convection would occur, and the resulting ice thickness if it were to occur, without detailed knowledge of conditions in and microscale composition of the ice (Barr & Showman 2009). More generally, we will make many simplifications, including considering the question within the framework of a one-dimensional (vertical) model, since our primary objective is to establish whether or not a Steppenwolf planet is feasible.

First we calculate the conductive steady-state thickness, H_cond. Above ≈10 K, the temperature dependence of the thermal conductivity of water ice is well approximated by k(T) = AT⁻¹, where T is the temperature in Kelvin and A = 651 W m⁻¹ (Petrenko & Whitworth 2002). Dimensional analysis shows that thermal steady state is reached in ∼10⁶ years, much shorter than the timescale of decay of the geothermal heat flux. Geothermal heat flux through the shell will be constant at steady state, since no heat is produced within the ice, as would occur by tidal heating of a frozen moon. Since the Steppenwolf planets we consider would be much larger and drier than the icy moons on which subglacial oceans are typically studied, we can assume that the ice thickness is much less than the planetary radius, yielding an exponential temperature profile through the ice and steady-state thickness, so that

$$\begin{equation} H_{\rm cond} = \frac{A}{F}\log \left(\frac{T_H}{T_0}\right), \end{equation} \tag{ 1 }$$

where T_H is the temperature at the ice–water interface (the melting temperature), T₀ is the temperature at the top of the ice, and F is the geothermal heat flux.

Decay of radioactive elements in Earth's interior and primordial heat remaining from Earth's formation lead to an average geothermal heat flux emanating from Earth's surface of F_⊕ = 0.087 W m⁻² (Pollack et al. 1993). This heat flux decays with time such that Earth's geothermal heat flux may have been roughly twice its present value 3 Gyr ago (Turcotte 1980). In order to consider Steppenwolf planets of different sizes, we use the radius–mass scaling R∝M^v for super-Earths with v = 0.27 (Valencia et al. 2006). Heuristically, this yields a geothermal heat flux that scales as (M/M_⊕)^{1 − 2v}, or roughly as the square root of the mass.

The pressure at the bottom of the ice layer is ≈9 MPa for each kilometer of ice, scaling with mass as (M/M_⊕)^{1 − 2v}. The melting point of pure ice is 250–270 K at pressures less than 620 MPa (Choukroun & Grasset 2007), although contaminants to pure ice such as chloride salts or ammonia could suppress the melting point by ≈50–100 K (Kargel 1991, 1992; Fortes & Choukroun 2010; Grasset & Sotin 1996). The steady-state thickness in the conductive regime is only logarithmically sensitive to T_H, so we will take T_H = 260 K in the estimates here, except that we will consider the eutectic point of an ammonia-water ice mixture at 176 K to demonstrate the impact of contaminants. In steady state, the temperature at the surface of a Steppenwolf planet (T_s), i.e., the top of the ice or frozen gas layer, will be set by a balance between thermal emission and geothermal heat flux, F = σT⁴_s, where σ is the Stefan–Boltzmann constant. Astrophysical radiation backgrounds (Mathis et al. 1983; Dole et al. 2006) are negligible.

Any gas present in the atmosphere at planetary ejection or outgassed subsequently by geological processes will tend to freeze into a low-thermal-conductivity blanket that could allow T₀ to exceed T_s. A blanket formed by freezing Earth's current atmosphere would only increase T₀ by about 4 K (Laughlin & Adams 2000). Isostatic adjustment (Fowler 1990), however, will tend to cause some continents or islands to rise above the layers of water and ice on an Earth-like Steppenwolf planet with an active mantle, allowing volcanoes to continuously emit gasses that can freeze onto the water-ice surface. Here, we will consider carbon dioxide because it is likely to be outgassed in significant quantities from an Earth-like planet and it supports a stable layer with relatively high base temperature relative to other common gases. To find an upper bound on the impact of a frozen gas layer, we will assume that the layer is Rayleigh–Taylor stable with respect to the underlying water ice, so it remains as a surface blanket.

To find the temperature at the carbon-dioxide layer base, we note that the thermal conductivity of carbon dioxide again scales as T⁻¹ (here, in a more limited regime), but with constant of proportionality A ≈ 100 W m⁻¹ (Sumarokov et al. 2003). We find that the maximum temperature supported is robustly ≈220 K for Earth-like Steppenwolf planets and is determined by the weak temperature dependence of the melting curve (Giordano et al. 2006). Setting T₀ = 220 K reduces the required steady-state water-ice thickness by an order of magnitude.

At Earth mass, the temperature at the bottom of a layer of solid CO₂ reaches the melting temperature of CO₂ for a layer thickness of ≈2 km, or ≈3 × 10⁶ kg m⁻². Venus' atmosphere has a partial pressure of CO₂ of ≈90 bar (≈10⁶ kg m⁻²), which is roughly the vapor pressure of carbonate rocks at Venus' surface temperature, implying that there may be more carbon locked in rock in equilibrium with the atmosphere (Pierrehumbert 2010). The store of carbon in carbonate rocks in Earth's interior is uncertain, but the continental crust is estimated to contain the equivalent of ≈7 × 10⁵ kg m⁻² CO₂ and the mantle may contain 2–4 times this amount (Zhang & Zindler 1993). Therefore, it appears reasonable to assume that a Steppenwolf planet could have a sufficient complement of CO₂ to significantly elevate T₀.

Since the viscosity of ice depends strongly on temperature (Barr & Showman 2009), if ice convection were to occur on a Steppenwolf planet, it would occur only in the lower, warmer ice regions and would be capped by a "stagnant" conducting lid (Solomatov 1995). We calculate the steady-state ice thickness when convection occurs following Hussmann et al. (2006), who assume a Newtonian rheology. We outline the solution here, but the reader should consult Hussmann et al. (2006) for more detail. We assume that convection occurs below temperature T_c (at higher temperature) and determine T_c by assuming that the viscosity is reduced by a factor γ = 10 over the convecting region, where the viscosity is given by the relation, η(T) = η₀exp [l(T_m/T − 1)], where η₀ = 10¹³ Pa s, T_m is the melting temperature, and l = 25. Assuming that T_m = T_H, we can solve for T_c. A Nusselt–Rayleigh number scaling (Nu = aRa^β) yields the thickness of the convecting region between temperature T_c and T_H,

$$\begin{equation} H_{\rm conv}^{1-3\beta } = \frac{a k (T_H-T_c)}{F } \left[ \frac{g \alpha \rho (T_H-T_c)}{\kappa \eta (\bar{T})} \right]^{\beta }, \end{equation} \tag{ 2 }$$

where we evaluate the viscosity at the mean temperature of the convecting layer, $\bar{T} = \frac{1}{2}(T_c+T_H)$, κ = 1.47 × 10⁻⁶ m² s⁻¹, α = 1.56 × 10⁻⁴ K⁻¹, ρ = 917 kg m⁻³, k = 3.3 W m⁻¹ K⁻¹, a = 0.12, and β = 0.3. Taking k to be constant in the convecting layer is a reasonable assumption given that the temperature is nearly constant within it. We add the thickness of the convective layer given by Equation (2) to the thickness of the stagnant lid, $H_{\rm lid} = \frac{A}{F}\log \left(\frac{T_c}{T_0}\right)$, to find the total ice thickness when there is convection. When the total thickness with convection exceeds the conductive thickness given by Equation (1), we use it for the ice thickness. This corresponds to a critical Rayleigh number of roughly 1000 within the convecting layer.

Given that the ice composition on a Steppenwolf planet is unknown and the ice material properties under appropriate conditions are poorly constrained, our convective calculation should be viewed as a rough estimate. For example, following Hussmann et al. (2006) and Fu et al. (2010), we have assumed Newtonian ice flow, which may or may not be realistic (Barr & Showman 2009). Creep mechanisms with stress-dependent viscosity, however, should yield results roughly similar to Newtonian flow (Kirk & Stevenson 1987). Another source of uncertainty is the appropriate ice grain size, the typical size of individual components of polycrystalline ice. In general, larger grain sizes correspond to higher viscosities, making convection less likely. The value of, η₀ that we have used, 10¹³ Pa s corresponds to an ice grain size of ≈0.1 mm (Barr & Showman 2009). For reference, the typical ice grain size on terrestrial ice sheets is 0.1–10 mm (Barr & Showman 2009) and Fu et al. (2010) assume an ice grain size of 0.2 mm for their calculations of convection on icy extrasolar planets. Competing estimates of ice grain size on Europa, however, place it at either 0.02–0.06 mm or >40 mm (Barr & Showman 2009). Although the appropriate ice grain size on a Steppenwolf planet is a source of uncertainty, we have used a relatively small value, which is conservative in that it makes convection more likely to occur. Finally, the constant β can take different values between 0.25–0.33 depending on the geometry and boundary conditions (Hussmann et al. 2006). We have used a fairly high value of β, which leads to a conservatively strong scaling of H_conv with M.

A Steppenwolf planet with mean ocean depth greater than the steady-state ice thickness, accounting for the ice-water density difference, will have an ocean under its ice layer. Figure 1 shows the required ice thickness to achieve liquid water in several scenarios. It is expected that the abundance of water on planetary surfaces varies greatly (Raymond et al. 2007), but we can consider a simple scaling to understand how the conditions for liquid water scale with planet mass and water complement. At fixed planetary water mass fraction and fraction of water at the surface (rather than in the mantle), the depth of the ocean scales approximately as the typical depth at Earth mass (D*) times (M/M_⊕)^{1 − 2v}. Figure 1 shows contours for planets with water complements that are 0.1, 1, 10, and 100 times that of Earth. Combining this scaling with Equations (1) and (2), we find that if a Steppenwolf planet is similar to Earth in water mass fraction (D* = D_⊕ ≈ 4 km), radionuclide composition, age, and has no frozen CO₂ layer, it must be ≈3.5 times more massive than Earth to sustain a subglacial liquid ocean. Contaminants which suppress the melting point have little effect on the conductive ice thickness, but significantly reduce the ability of the ice to convect. If a Steppenwolf planet has ten times more water (D* = 10D_⊕) than Earth or if it has a thick frozen CO₂ layer which reaches the maximum temperature of ≈220 K at its base, the planet must be only ≈0.3 times Earth's mass to have a liquid ocean.

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We expect that detection of reflected sunlight in the optical wavebands and IR follow-up present the only viable observational choice in the near term. For a single-visit limiting magnitude r ≈ 24.7 of LSST (Jones et al. 2009; and comparable r = 24 in the nearer-term Pan-STARRS; Jewitt 2003) and albedo of 0.5, the limiting distance out to which an object can be detected with reflected sunlight is ≈830(r/R_⊕)^1/2 AU. The Palomar survey of ∼12, 000 deg² to magnitude 21.3 (Schwamb et al. 2009) has discovered no such objects outside of the smaller trans-Neptunians such as Sedna.

The baseline requirement to identify a Steppenwolf planet is a detection of thermal emission in the far-IR. The flux at the Wien maximum is

$$\begin{eqnarray} S_{\rm max} & = & (108 {\rm \,mJy}) \cdot \left(\frac{T_{\rm s}}{\rm 1 K} \right)^3 \left(\frac{R}{R_\oplus } \right)^2 \left(\frac{d}{\rm 1 AU} \right)^{-2}, \end{eqnarray} \tag{ 3 }$$

where d is the Earth–object separation. At 10 M_⊕, T_s = 46 K so that λ_max = 110 μm. Here, the Herschel PACS instrument reaches a 40 beams/source confusion limit at a flux of ≈2 mJy (Berta et al. (2010), suggesting a limiting distance of ≈4000 AU; PACS reaches 10 mJy at 5σ in 1 hr (Poglitsch et al. 2010)). Higher resolution is required to progress to lower flux limits. At 200 μm, the planned 25 m Cornell-Caltech Atacama Telescope (CCAT) would reach the source confusion limit⁴ at 0.36 mJy.

Photometric microlensing has also been proposed as a method to detect rogue planets throughout the galaxy (Han et al. 2004). If each stellar system ejects one M_⊕ planet, a survey like the Galactic Exoplanet Survey Telescope could anticipate ∼20 detections of rogue planets (Bennett & Rhie 2002; Bennett et al. 2010). Typical distances to these objects would exceed the capabilities of follow-up that could elucidate their nature. Free-floating super-Jupiters have been discovered (Bihain et al. 2009), but these represent a different class of objects.

A Steppenwolf planet's lifetime will be limited by the decay of the geothermal heat flux, which is determined by the half-life of its stock of radioisotopes (⁴⁰K, ²³⁸U, ²³²Th) and by the decay of its heat of formation. As these decay times are ∼1–5 Gyr, its lifetime is comparable to planets in the traditional habitable zone of main-sequence stars (Kasting et al. 1993).

If a Steppenwolf planet harbors life, it could have originated in a more benign era before ejection from the host star. Alternatively, after ejection, life could originate around hydrothermal vents, which are a proposed location for the origin of life on Earth (Baross & Hoffman 1985). If life can originate and survive on a Steppenwolf planet, it must be truly ubiquitous in the universe.

We have shown that an Earth-like rogue planet drifting through interstellar space could harbor a subglacial liquid ocean despite its low emission temperature, and so might be considered habitable. Such an object could be detected and followed up using current technology if it passed within $\cal {O}$(1000 AU) of Earth.

D.S.A. was supported by a TC Chamberlin Fellowship of the University of Chicago and by the Canadian Institute for Advanced Research. E.R.S. acknowledges support by NSF Physics Frontier Center grant PHY-0114422 to the Kavli Institute of Cosmological Physics. We thank F. Adams, F. Ciesla, N. Cowan, R. Fu, C. Hirata, P. Kelly, N. Murray, S. Padin, L. Page, R. Pierrehumbert, F. Richter, D. Valencia, and the anonymous reviewer for conversations or comments on early versions of this Letter, and Amory Lovins for posing the question of what Earth's temperature would be if there were no Sun.