Tidal Constraints on Planetary Habitability
Rory Barnes, Brian Jackson, Richard Greenberg, Sean N. Raymond, Rene Heller
aa r X i v : . [ a s t r o - ph . E P ] D ec **FULL TITLE**ASP Conference Series, Vol. **VOLUME**, c (cid:13) **YEAR OF PUBLICATION****NAMES OF EDITORS** Tidal Constraints on Planetary Habitability
Rory Barnes , , Brian Jackson , , Richard Greenberg , Sean N.Raymond , , and Ren´e Heller Abstract.
We review how tides may impact the habitability of terrestrial-like planets. If such planets form around low-mass stars, then planets in thecircumstellar habitable zone will be close enough to their host stars to experiencestrong tidal forces. We discuss 1) decay of semi-major axis, 2) circularization ofeccentric orbits, 3) evolution toward zero obliquity, 4) fixed rotation rates (notnecessarily synchronous), and 5) internal heating. We briefly describe theseeffects using the example of a 0.25 M ⊙ star with a 10 M ⊕ companion. Wesuggest that the concept of a habitable zone should be modified to include theeffects of tides. Exoplanet surfaces are probably the best places to look for life beyond the SolarSystem. Remote sensing of these bodies is still in its infancy, and the technologydoes not yet exist to measure the properties of terrestrial exoplanet atmospheresdirectly. Indeed, the scale and precision of the engineering required to do so isbreathtaking. Given these limitations, a reliable model of habitability is essentialin order to maximize the scientific return of future ground- and space-basedmissions with the capability to remotely detect exoplanet atmospheres.Here we review one often misunderstood issue: The effect of tides. If thedistance between a star and planet is small, < ∼ . Department of Astronomy, University of Washington, Seattle, WA, 98195-1580 Virtual Planetary Lab Planetary Systems Laboratory, Goddard Space Flight Center, Code 693, Greenbelt, MD20771 NASA Postdoctoral Program Fellow Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721 Center for Astrophysics and Space Astronomy, University of Colorado, UCB 389, BoulderCO 80309-0389 Hamburger Sternwarte, University of Hamburg, Gojenbergsweg 112, 21029 Hamburg, Ger-many e , the two models may diverge signif-icantly when e> ∼ . . Throughout this review the reader should remember thatthe presented magnitudes of tidal effects are model-dependent. For more onthese differences and the details of tidal models, the reader is referred to recentreviews by Ferraz-Mello et al. (2008) and Heller et al. (2009).We consider tidal effects in the habitable zone (HZ) model proposed byBarnes et al. (2008) which utilizes the 50% cloud cover HZ of Selsis et al. (2007),but assumes that the orbit averaged flux determines surface temperature (Williamsand Pollard 2002). We use the example of a 10 M ⊕ planet orbiting a 0.25 M ⊙ star. This choice is arbitrary, but we note that large terrestrial planets orbit-ing small stars will be preferentially discovered by current detection techniques.This chapter is organized as follows: In § § § § Orbital evolution due to tides should be considered for any potentially habit-able world. The asymmetry of the tidal bulge leads to torques which transferangular momentum between rotation and orbits, and the constant flexing ofthe planet’s figure between pericenter and apocenter dissipates energy insidethe planet. These two effects act to circularize and shrink most orbits. In theconstant-phase-lag model, the orbits of close-in exoplanets evolve in the followingway (Goldreich and Soter 1966; see also Jackson et al. dadt = − (cid:16) p GM ∗ R p m p Q ′ p e + 92 p G/M ∗ R ∗ m p Q ′∗ h e ]) a − / (1) dedt = − (cid:16) p GM ∗ R p m p Q ′ p + 22516 p G/M ∗ R ∗ m p Q ′∗ (cid:17) a − / e, (2)where a is semi-major axis, G is Newton’s gravitational constant, m p is the massof the planet, Q ′ p is the planet’s tidal dissipation function divided by two-thirdsits Love number, Q ′∗ is the star’s tidal dissipation function divided by two-thirds its Love number, R p is the planet’s radius, and R ∗ is the stellar radius.The Q ′ values represent the body’s response to tidal processes and combinesa myriad of internal properties, such as density, equation of state, etc. It is adifficult quantity to measure, so here we use the standard values of Q ′∗ = 10 and Q ′ p = 500 (Mathieu 1994; Mardling and Lin 2002; Jackson et al. a and e decay with time. As tides slowly change aplanet’s orbit, the planet may move out (through the inner edge) of the habitablezone (HZ). This possibility was considered in Barnes et al. (2008), who showed Figure 1. Contours of equilibrium rotation period in days for a 10 M ⊕ planet orbiting a 0.25 M ⊙ star. The gray region is the HZ from Barnes etal. (2008). for some example cases the time for a planet to pass through the inner edge ofthe HZ. Such sterilizing evolution is most likely to occur for planets with initiallylarge eccentricity near the inner edge of the HZ of low mass stars ( < ∼ . ⊙ ).Even if a planet does not leave the HZ, the circularization of its orbit can requirebillions of years, potentially affecting the climatic evolution of the planet. Planetary rotation rates may be modified by tidal interactions. Although planetsmay form with a wide range of rotation rates Ω, tidal forces may fix Ω such thatno net exchange of rotational and orbital angular momenta occurs during oneorbital period. The planet is then said to be “tidally locked,” and the rotationrate is “pseudo-synchronous” or in equilibrium. The equilibrium rotation ratein the constant-phase-lag model isΩ eq = n (1 + 192 e ) , (3)where n is the mean motion (Goldreich 1966). Note that planets only rotatesynchronously (one side always facing the star) if e = 0 (the constant-time-lagmodel makes the same prediction). Therefore, the threat to habitability mayhave been overstated in the past, as independently pointed out by several recent Figure 2. Time in years for a 10 M ⊕ planet orbiting a 0.25 M ⊙ star toevolve from an obliquity ψ = 23 ◦ .5 to 5 ◦ . The gray region is the HZ fromBarnes et al. (2008). investigations (Barnes et al. et al. et al. ⊕ planetorbiting a 0.25 M ⊙ star as a function of a and e . Tidal effects tend to drive obliquities to zero or π . The constant-time-lag modelof Levrard et al. (2007) found a planet’s obliquity ψ changes asd ψ d t = sin ( ψ ) K p C p Ω n (cid:18) cos ( ψ ) ǫ Ω n − ǫ (cid:19) (4)where ǫ = 1 + 3 e + e (1 − e ) / , (5) K p = 32 k , p GM R p τ p n M s M p ! (cid:18) R p a (cid:19) , (6) C p = r , p M p R , (7)and ǫ = 1 + e + e + e (1 − e ) . (8) Figure 3. Tidal heating fluxes for a 10 M ⊕ planet orbiting a 0.25 M ⊙ star.Contour labels are in W m − . The dashed contours represent the boundariesof the tidal habitable zone (Jackson et al. et al. et al. (2008). In the preceding equations r g , p (= 0.5) is the planet’s radius of gyration (ameasure of the distribution of matter inside a body), Ω is the initial rotationfrequency, and τ p is the “tidal time lag”, which in this constant-time-lag modelreplaces Q ′ p . We assumed Q ′ p = 500 for the planet at its initial orbital configura-tion and set τ p = 1 / ( nQ ′ p ), i.e. initially the planet responds in the same way asin a constant-phase-lag model. In the course of the orbital evolution, τ p was thenfixed while n and Q p evolved in a self-consistent system of coupled differentialequations. In Fig. 2 we show the time for a planet with an initial obliquity of23 ◦ .5 to reach 5 ◦ , a value which may preclude habitability (F. Selsis, personalcommunication). However, obliquities may easily be modified by other planetsin the system (Atobe et al. etal. As a body on an eccentric orbit is continually reshaped due to the varyinggravitational field, friction heats the interior. This “tidal heating” is quantifiedin the constant-phase-lag model as H = 634 ( GM ∗ ) / M ∗ R p Q ′ p a − / e (9)(Peale et al. h = H/ πR p , through the planetary surface. Jackson et al. (2008c; see alsoBarnes et al. h ≥ − (the value for Io [McEwen et al. h ≤ .
04 W m − (the limit for plate tectonics [Williams etal. et al. (2009b) suggested that theselimits represent a “tidal habitable zone”. In Fig. 3 contours of tidal heating areshown for a 10 M ⊕ planet orbiting a 0.25 M ⊙ star. The tidal habitable zone isthe region between the dashed curves. Note that a and e evolve as prescribedby Eqs. (1 – 2), and hence the heating fluxes evolve with time as well. Acknowledgments.
RB and SNR acknowledge funding from NASA As-trobiology Institute’s Virtual Planetary Laboratory lead team, supported byNASA under Cooperative Agreement No. NNH05ZDA001C. RG acknowledgessupport from NASA’s Planetary Geology and Geophysics program, grant No.NNG05GH65G. BJ is funded by an NPP administered by ORAU. RH is sup-ported by a Ph.D. scholarship of the DFG Graduiertenkolleg 1351 “ExtrasolarPlanets and their Host Stars”.
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