Signature Redacted Author Department of Earth, Atmospheric, and Planetary Sciences May 8, 2009 Certified by Signature Redacted Linda T

Home , ELODIE spectrograph, Gliese 682, HD 181433, HD 40307 b, HD 47186

Mantle Thermal Evolution of Tidally-locked Super-Earths

by Sarah E. Gelman

Submitted to the Department of Earth, Atmospheric, and Planetary Sciences in Partial Fulfillment of the Requirements for the Degree of Bachelor of Science in Earth, Atmospheric, and Planetary Sciences at the Massachusetts Institute of Technology

The author hereby grants to M.I.T. permission to reproduce and distribute publicly paper and electronic copies of this thesis and to grant others the right to do so. Signature redacted Author Department of Earth, Atmospheric, and Planetary Sciences May 8, 2009 Certified by Signature redacted Linda T. Elkins-Tanton Signature redacted Thesis Supervisor Accepted by Samuel - .,'.- Bowring 2' Chair, Committee on Undergraduate Program

MASSACHUSESINSTITUTE OF TECHNOLOGY OCT 2 4 2017 The author hereby grants to MWT permission to reproduce and to distuibute publicly paper and electronic copies of this thesis document ip LIBRARIES whole or in part in any medium now known or hereafter created. ARCHIV. ABSTRACT

Mantle Thermal Evolution of Tidally-locked Super-Earths

by Sarah E. Gelman

Advisors: Linda T. Elkins-Tanton, Sara Seager

Most super-Earth (mass < 10MD) detection techniques are biased towards massive planets with close-in orbits. A planet's orbital decay timescale decreases with a lower semi-major axis, thereby providing a high probability of detecting exoplanets which are in tidal-lock with their star. We model the effect of fixed stellar flux on an Earth- like planet's mantle convection structure and evolution using an axisymmetric finite element fluid convection code, SSAXC. Three punctuating evolutionary steps have been identified. First, a sequence of three initial downwellings form at the antistellar point, the substellar point, and at the terminator. After approximately 250,000 years, lithospheric instabilities drip down into the mantle, inducing pervasive small scale convective cells. Finally, after 3.5 billion years of planetary evolution, a cold region

(~ 300'C) develops at the antistellar point, spanning the depth of the mantle, and inducing a pseudo-Mode 1 convective pattern. Though initial models have focused on an incoming stellar flux equivalent to that at the Earth, we also discuss the possibility of steady-state partial magma oceans in the substellar region induced by higher stellar fluxes. We propose that, though these tidally-locked planets may not be considered habitable in general, they may contain locally habitable regions due to their variety of long-lived extreme environments. Contents

Table of Contents 3

List of Figures 4

List of Tables 5

1 Introduction to Super-Earths 6 1.1 Detection Techniques for Extrasolar Planets ...... 6 1.2 Survey of Earth-Sized Exoplanet Discoveries ...... 10 1.3 Survey of Super-Earth Models ...... 11 1.4 The Significance of Synchronous Rotation ...... 12

2 Numerical Methods 15 2.1 Introduction to SSAXC ...... 15 2.2 Derivation of Boundary Conditions ...... 16 2.3 Input Parameters and Simplifications ...... 18

3 Results 20 3.1 Conduction and Radiation Only ...... 20 3.2 Implementation of Convection ...... 21 3.3 Mode 1 Convection and Long-Term Evolution ...... 28

4 Discussion: Partial Magma Oceans 31

5 Conclusions 35

Bibliography 37

A SSAXC Modifications 42

B Magma Pond Stability Code 50

3 List of Figures

1.1 Illustration of Transit Light Curve Method ...... 7 1.2 HD 209458 Light Curve ...... 8 1.3 51 Pegasi b Radial Velocity ...... 9 1.4 OGLE-2005-BLG-390 Gravitational Microlensing ...... 10 1.5 Synchronous Rotation Illustration ...... 13

2.1 SSAXC Grid Setup ...... 16

3.1 Initial Conductive Results ...... 21 3.2 4-D Early Convective Results ...... 22 3.3 Highlight of Initial Convective Downwellings ...... 23 3.4 Downwellings and Small Scale Convection ...... 24 3.5 Hemispheric Dichotomy from Lithospheric Drips ...... 26 3.6 Stability of Surface Liquid Water ...... 27 3.7 4-D Long-term Convective Results ...... 29

4.1 Magma Pond Stability Timescales ...... 33

5.1 Image of Gliese 581 ...... 36

4 List of Tables

1.1 Detected Super-Earths ...... 11

2.1 SSAXC Input Parameters ...... 19

4.1 Magma Pond Stability Parameters ...... 32

5 1 Introduction to Super-Earths

To date, 347 extrasolar planets have been discovered, most of them large, Jupiter-mass gaseous bodies orbiting close to their star. Although the detection of these planets is in- triguing and challenges traditional views on planetary evolution and solar system formation, it is the by-product of a larger search for more Earth-like, rocky worlds beyond our solar system. Detection techniques preferentially discover these larger gas giant planets; however, as these techniques are continually refined, the minimum limit of detectability is closing in on Earth-mass planets. Indeed, only since 2005 have we been able to detect planets which may be small enough to be considered predominantly rocky (Rivera et al., 2005; Valencia et al., 2007a). Since then, the number of identified potentially rocky planets is growing, and with new space satellite missions underway (for example, Kepler, launched March 6, 2009), this number can be expected to continue its upward trend.

1.1 Detection Techniques for Extrasolar Planets

There will be three main ways to detect Earth-mass exoplanets in the near future: (1) transit light curve analysis, (2) radial velocity variations, and (3) gravitational microlensing events (e.g., Griest and Safizadeh, 1998; Woolf and Angel, 1998; Marcy and Butler, 1998; Santos, 2008). These techniques provide complementary information, and each has its own sensitivities and limitations. In the transit light curve method, the photon flux of a star is observed and plotted against time. A planet is detected when the intensity of the star drops off as an orbiting planet passes directly between the star and the observer on Earth (Fig. 1.1). This orbital transit will not be observed for every planet because of the random orientation of stellar

6 7

Figure 1.1 Schematic explanation of an exoplanet transit light curve. From the point of view of an observer on Earth, the exoplanet (shown in blue) transits in front of the star (shown in yellow) as it orbits. In this particular cartoon, the planet is moving from left to right, but in reality stellar disk orientations are random and so this is not necessarily the case for any given exoplanet. The corresponding fitted light curve data of the transit is shown just below. The black line represents the flux of photons an observer records as the transit proceeds. The overall intensity from the star drops as the planet transits, and is restored afterwards. Figure simplified from Brown et al. (2001). disks. This technique will thus favor planets whose probability of a transit observable from Earth is high: planets close to their star, very large in radius, and low in albedo. Since the transit light curve method is dependent on the absolute intensity of light from a star, atmospheric effects can severely distort a ground-based telescope's data quality. Space-based observations are not necessary to detect a planet, but do provide a level of precision which greatly increases our ability to constrain planetary and orbital characteristics, especially with smaller-sized objects. Of the 347 exoplanets, 59 were detected with the transit light curve method. An example of a light curve obtained by Brown et al. (2001) using the Hubble Space Telescope is shown in Fig. 1.2. Measuring the variability in the radial velocity of a star provides a higher level of precision than the transit light curve method in detecting extrasolar planets. In any planetary system, 8

1.000

0.995-

0 0.990-

0.985

-0.10 -0.05 0.00 0.05 0.10 time from center of transit (days)

Figure 1.2 Example of transit light curve data obtained by Brown et al. (2001) of a large (Mp = 1.347 Rj) gaseous extrasolar planet orbiting HD 209458. Orbital period is 3.52474 days. Data was obtained from the Hubble Space Telescope over four transit events. both star and planet are orbiting a common center of mass. Thus, as a planet makes a complete revolution a slight wobble movement can be observed via a Doppler shift in the stellar spectra, assuming the orbital plane is oriented optimally. When a star moves towards the Earth in its orbit, we measure an increase in the frequency of the star's photons, or, in other words, a slight blueshift. When the star moves away from the Earth, we observe a redshift. Even though the radial velocity method is still biased towards the detection of large, close-in planets (since more massive bodies have greater gravitational potential), these measurements have been the most successful. So far 321 planets have been detected by radial velocity perturbations. An example, 51 Pegasi b (the first exoplanet ever detected), is shown in Fig. 1.3.

A third important detection technique is the analysis of gravitational microlensing events.

In this case, due to differing relative velocities of objects in the Milky Way, a relatively close star system occasionally will pass between the Earth and a very distant star. The light transmitted from the distant star is bent and magnified by the gravity of the moving star system. This change in the shape of the distant star's light path by the occulting star can also reveal a slight bending overprint by a companion occulting exoplanet. In order for a planet to be detected, it must firstly be massive enough to bend light significantly. Secondly, it must have a large enough semi-major axis for its bending effect to be differentiable from 9

100

E 0 ......

-60

-100

0 0.5 1

Figure 1.3 Radial velocity measurement of the first confirmed extrasolar planet, 51 Pegasi b (Mayor and Queloz, 1995). This planet orbits every 4.23 days and has a mass of 0.468 Mj; the measurement was performed on the ELODIE spectrograph of the Haute-Provence Observatory, France. that of its much more massive star. Though the discovery of exoplanets by analysis of gravitational microlensing events has had some success, follow-up analysis on any detected planet is virtually impossible due to the improbability and unpredictability of subsequent occultations by that particular system. Nevertheless, 8 planets have been detected so far by this method; an example of one such observation, OGLE-2005-BLG-390, is given in Fig. 1.4. Several other ways of detecting extrasolar planets have been proposed and attempted with varying degrees of success. In particular, astrometry, or the measurement of a star's orbital 'wobble' in the plane of the sky (e.g., Woolf and Angel, 1998; Marcy and Butler, 1998), has been proposed but is as yet unsuccessful. Another method, the use of transit timing to indirectly measure the gravitational effect of additional planets on a transiting planet (e.g., Holman and Murray, 2005), has detected a few candidates so far. Finally, direct imaging is only currently becoming possible, but has a distinct bias towards extremely large planets (several Jupiter masses) which are extremely far from their star (10s to 100s of AU). 10

3 - 3 -.. 1.6 -. - OGLE - 1.5--

2.5- 25 -Planetary 0 - - - 1.3 deviation

2 -1,000 0 9.5 10 10.

OGLE * Danish 1.5 . Robonet e Perth Canopus * MOA

1 -20 0 20 Days since 31 July 2005 UT

Figure 1.4 Example of a magnification light curve of OGLE-2005-BLG-390, which was detected through use of the gravitational microlensing technique. The occulting planet has a mass of 5.5 Me and a semi-major axis of 3 AU. The parent star is an M-dwarf, with a mass of 0.22 M®. Detection and fit by Beaulieu et al. (2006).

1.2 Survey of Earth-Sized Exoplanet Discoveries

The definition of a super-Earth is not clearly agreed upon in the scientific literature. Several workers have shown that at 10 Earth-masses (Me), bodies will be able to gravitationally capture gaseous H/He in their atmosphere, thus implying the accretion of a larger, lower- density, more H/He-rich planet (Mizuno, 1980; Stevenson, 1982; Rafikov, 2006; Alibert et

al., 2006). Therefore, this study will follow the convention after Valencia et al. (2006) of a

10 Me maximum for a planet to be considered terrestrial.

Of the 347 detected planets, twelve are less than 10 Me; ten of these were detected through radial velocity variations and two were through gravitational microlensing events

(Table 1.1). Except for the gravitational microlensing discoveries, all of these super-Earths

have a semi-major axis of less than 0.22 AU and a corresponding orbital period of less than

66.8 days. The early February 2009 announcement of the detection of a sub-2 Re (the radii

of other super-Earths have not yet been determined) around CoRoT 7 marks potentially

the first transiting Earth-sized body; it has an orbital period of just 0.85 days and a mass 11

Name Mass (Me) Semi-major Axis (AU) Period (days) Detection Method Year Gliese 581 e9 1.94003 0.03 3.14942 Rad. Vel. 2009 MOA-2007-BLG-192-L b 3.1783 0.62 Grav. Lens 2008 HD 40307 b 2 4.195356 0.047 4.3115 Rad. Vel. 2008 Gliese 581 c3,9 5.35861 0.07 12.9292 Rad. Vel. 2007 OGLE-05-390L b4 5.40311 2.1 3500 Grav. Lens. 2005 Gliese 876 d5 5.72094 0.02081 1.93776 Rad. Vel. 2005 HD 40307 c 2 6.865128 0.081 9.62 Rad. Vel. 2008 Gliese 581 d, 9 7.09079 0.22 66.8 Rad. Vel. 2007 HD 181433 b6 7.564354 0.08 9.3743 Rad. Vel. 2008 HD 285968 b7 8.422495 0.066 8.7836 Rad. Vel. 2007 HD 40307 d2 9.153504 0.134 20.46 Rad. Vel. 2008 HD 7924 b8 9.21707 0.057 5.3978 Rad. Vel. 2009

Table 1.1 Basic data review of all sub-10 Me exoplanets discovered to date. ['Bennett et al., 2008; 2Mayor et al., 2009; 3 Udry et al., 2007; 4 Beaulieu et al., 2006; 5Rivera et al., 2005; 6 Bouchy 7 8 et al., 2008; Forveille et al., 2009; Howard et al., 2009; 9Mayor et al., submitted] which has been constrained to less than 11 Me (Rouan et al., 2009; Bouchy et al., 2009). By comparison, Mercury has a semi-major axis of 0.39 AU and orbits every 88 days. In general, these super-Earths are far closer to their star and much larger than any terrestrial planets in our solar system. Even though these common characteristics are at least in part an artifact of the limitations of our detection techniques, it is critical to build a theoretical framework to understand the effects of a close-in orbit on a rocky planet. From this we can predict what these planets' surfaces are like-for instance, whether or not they are likely to have plate tectonics, liquid water, or convecting mantles-and ultimately, determine whether they are habitable as a guide to the search for life outside of our solar system.

1.3 Survey of Super-Earth Models

Throughout the last decade there has been ongoing work to model the internal structure, composition, and evolution of super-Earths. Based on the high variation observed in exoplanet density, many new types of planets have been explored: low-mass ocean planets by 12

Kuchner (2003) and Lager et al. (2004); high semi-major axis and thus small, frozen planets by Ehrenreich et al. (2006); silicate-rich, Earth-like planets by Valencia et al. (2006, 2007a, 2007b), Papuc and Davies (2008), and Zeng and Seager (2008); iron-rich Mercury-like planets by Valencia et al. (2006, 2007a); mini-Neptune planets by Grasset et al. (2009), and even coreless silicate planets by Elkins-Tanton and Seager (2008). Additionally, mass-radius relationships for various-sized planets have been derived, both for strictly super-Earths (Sotin et al., 2007) and across the wide spectrum from Earth-like to Jupiter-like bodies (Fortney et al., 2007; Seager et al., 2007). The inherent uncertainties involved in deducing the balance between atmospheric mass and icy or rocky mass have also been explicitly studied (e.g., Adams et al., 2008; Baraffe et al., 2008; Grasset et al., 2009). These workers have initiated a preliminary first-order exploration into the evolution and potential habitability of planets outside of our solar system. By building on these previous studies, new research will aim to understand these small- sized bodies in finer detail. The extremity of these close-in orbits provides several new problems in planetary evolution which do not apply at all in our solar system. For instance, several studies have tried to understand the effects of tidal heating on close-in rocky planets (Jackson et al., 2008a; Jackson et al., 2008b; Jackson et al., 2008c; Barnes et al., 2008) and work to constrain surface temperatures from observables such as star size, planet size, semi- major axis, eccentricity, and obliquity. The focus of this study is another first order problem in these planets' evolutions: the effect of intense stellar flux on a tidally-locked close-in rocky body.

1.4 The Significance of Synchronous Rotation

Synchronous rotation refers to an orbit in which one rotation of the body takes precisely the same amount of time as one revolution about the orbital center. This also means that, in the case of a planet, one day is equal to one year, and one hemisphere will always be star-facing, and the other will never be (Fig. 1.5). Tidal-lock occurs when a star's rotation and mass raise tides on a planet, decaying its orbit, and effectively despinning it. If the 13

Figure 1.5 Illustration of synchronous rotation. The planet is tidally-locked to its star, and so one hemisphere always is star-facing (patterned orange), while the other always faces away (patterned blue). time to synchronization, Ts, is less than the age of the star system, then the planet may be predicted to be in tidal-lock:

3, M)2 (a )6 Ts Q ( )w((11) where Q is the planet's tidal dissipation factor, Rp is the planet radius, G is the gravitational constant, Mp is the mass of the planet, w = Ispin - Worbi is the difference between the spin and orbital angular velocities, M* is the mass of the star, and a is semi-major axis (Goldreich and Soter, 1966; Guillot et al., 1996; Rasio et al., 1996).

This effect of synchronous rotation on a rocky planet will, to first order, mean one side of the surface is hotter than the other. How different these absolute temperatures are depends particularly on the intensity of the stellar flux arriving on the star-side surface and the presence of an atmosphere, which would serve to redistribute heat more evenly across the entire planet. In this study we aim to understand the first of these in isolation, and ignore the effects of a potential atmosphere in this preliminary study.

We formulate a numerical model to (a) test the tidal-lock boundary conditions on a simplified planet whose mantle does not convect, (b) test the tidal-lock boundary conditions on a 14 more realistic planet with convection, and (c) calculate regions of possible stable liquid water. From this we discuss the possibility of extreme temperatures which would permit localized magma oceans around the substellar point, and finally we briefly explore the implications for the habitability of tidally-locked Earth-like planets. 2 Numerical Methods

In this chapter we define the methodology used to understand the effect of thermal influx on a super-Earth in synchronous rotation about its star. We first derive boundary condition equations based on Stefan-Boltzmann radiative energy loss from the planetary surface, with stellar flux on the star-side. We then implement these boundary conditions in the surface elements in a fluid convection code, SSAXC.

2.1 Introduction to SSAXC

SSAXC is a spherical axisymmetric version of the two-dimensional finite element code Con- Man (King et al., 1990). ConMan models incompressible thermochemical convection by solving several nondimensional flow equations simultaneously, including the equation of mo- mentum, V 2 u = -Vp + Ra T Z, (2.1)

continuity, V u = 0, (2.2)

and energy, = U VT 2 + V T, (2.3) 9t where u is the dimensionless velocity, T is the dimensionless temperature, p is the dimensionless pressure, i is the unit vector in the vertical direction, and t is the dimensionless time. All material properties used in these convection calculations are combined into the

15 16

(UQ

'Jdr dV

Figure 2.1 Schematic of the finite element grid setup in SSAXC. The 'star-side' is the top half of the planet and the 'far-side' is the bottom; r-nodes are increasing radially while 0-nodes are increasing tangentially. thermal Rayleigh number, Ra, given by

Ra = gaATd, (2.4) IpL where g is the acceleration due to gravity, a is the coefficient of thermal expansion, AT is the temperature drop across the mantle, d is the depth of the mantle, r, is the thermal diffusivity, and M is the dynamic viscosity. The axisymmetric grid used by SSAXC is illustrated in Fig. 2.1. The r-direction is radial

and begins at an approximate core-mantle boundary (!Rp) and increases to the surface of

the planet. The 9-direction is tangential and begins at the substellar point (top of the figure)

and increases to the antistellar point (bottom of the figure). Thus, the halfway point along

the 9-axis is the boundary between star-side and far-side, or the terminator.

2.2 Derivation of Boundary Conditions

The tidal-lock condition necessitates a calculation of stellar intensity arriving at the planet

for each surface element. This intensity is strongest at the substellar point, and decays to 17 zero at 0 = ! (the terminator); the far-side will receive no stellar flux. Every element, however, radiates as a black-body at each timestep. The calculation of these radiative fluxes is based on the Stefan-Boltzmann equation: F = oT4 , where F is the radiative energy flux, o- is the Stefan-Boltzmann constant, and T is the temperature of the radiative body or element. From this flux we calculate temperature through AF = mCpAT, where m is the mass (or density times volume) and C, is the specific heat capacity for each element.

For the stellar temperature addition, Tin, on the star-side of the planet at each timestep, this flux must be multiplied by a term describing the ratio of energy radiated at the surface

2 of the star and that at the planet's distance: 4irR,47ra ' where R, is the star's radius and a is the planet's semi-major axis. This gives:

R 2 -T* dt Ti= , (2.5) a2pdx2 dz CP' where T* is the star's effective temperature, dt is the Courant timestep, p is the density of rock in the planet, and dx and dz reference the width and depth of an element, respectively

(see Fig. 2.1). Here, Tin has units of [mK ], which is nondimensionalized by

4 Tin[ND] = in dX Tscale I

R 2-T' dx2 dt cos 6 Tin[NDI a*202 p C, dz Tscae(.6 where K is thermal diffusivity, cos 6 describes the decay of stellar flux intensity along the planetary surface and Tscale is the reference scaling temperature for the planet. An analogous derivation for the planet's radiative flux out yields uT'dx2 T 3 dt To [ND= p scale p Cpdz r, where Tp is the temperature of each element at the beginning of that timestep's calculation. Equations 2.6 and 2.7 are added and subtracted (respectively) to each temperature calculation for the surface elements on the star-side; only equation 2.7 is subtracted from the temperature calculations for the far-side surface elements (see Appendix A). 18

2.3 Input Parameters and Simplifications

Numerical experiments were developed in two stages: (1) initial implementation of boundary conditions with only conductive and radiative heat transfer, and (2) addition of convective heat flow in the solid planet. The initial model box was 120 0-nodes by 120 r-nodes, though higher resolutions of 300 0-nodes by 100 r-nodes were necessary by the second stage. All of these preliminary models assumed an Earth-like planet (Rp = Re, Mp = Me, and a = 1 AU) and a Sun-like star (R. = R® and T, = TO), though these parameters may be varied for future analysis of super-Earths. All mantle temperatures are set to a potential temperature of 1000'C, and a conductive lithosphere within the top 5% of planetary radius. Since this study is focused in understanding the isolated long-term effects of synchronous rotation on Earth-like planets, we make several simplifying assumptions. First, internal compositional variations are assumed to be negligible, and therefore a uniform approximate dry peridotite mantle is used. Material input parameters and other constants throughout these numerical experiments are listed in Table 2.1. Due to the small semi-major axes of these low-mass exoplanets, we assume potential atmospheres would be evaporated by high temperatures and stellar bombardment, and are therefore negligible. The presence of an atmosphere on a super-Earth has not yet been observed, though several theoretical approaches have been proposed (e.g., Miller-Ricci et al., 2009; Seager and Deming, submitted). While atmospheres remain beyond detection, future work should consider the validity of this assumption, as an atmosphere would redistribute heat and dampen the effect of a strong, fixed stellar heating. Finally, previous work has shown that the effect of tides on a planet without a perfectly circular orbit is another potentially large source for internal heating (Jackson et al., 2008a; Jackson et al., 2008b; Jackson et al., 2008c; Barnes et al., 2008). Here we assume negligible deviations in both obliquity and eccentricity. Future work should compare the effect of tidal heating with fixed stellar flux to determine the total effect on planets with noncircular orbits at various semi-major axes of massive stars. 19

Symbol Parameter Value

8 a- Stefan-Boltzmann constant 5.67 x 10- m2.14.S Ra thermal Rayleigh number I x 1010

lo reference viscosity I x 1021 Pa - s

K 1x10-6 2 thermal diffusivity S

Cp specific heat capacity 1250 kg.K' p density 3300 kg planetary radius 6400 km

Tscale reference temperature 1300 K T* effective stellar temperature 6000 K R, stellar radius 7x10 8 km semi-major axis 1 AU

Table 2.1 List of constants and variables included in this SSAXC numerical model, with fixed values unless otherwise noted. 3 Results

The results of our numerical experiments are shown in Fig. 3.1 for the conductive and radiative first stage, and in Fig. 3.2 and Fig. 3.7 for the addition of convective mantle flow in the second stage. In both cases, as expected, the star-side of the planet becomes significantly warmer than the far-side. Within the convective model there are at least three punctuating steps in the planet's evolution: an initial downwelling sequence, a nearly-symmetric 'middle' step, and finally the development of a pseudo-Mode 1 convective pattern.

3.1 Conduction and Radiation Only

For the conductive and radiative stage, Tscaie was set to 300'C, Ra to 0, and ro to 1x10 30 Pa -s, in order to make convective heat transfer impossible. The difference between the hottest surface temperatures at the substellar point (- 270'C) and the coolest surface temperatures at the antistellar point (- 90'C) produces a sharp gradient (< 300 of the planet circumference) near the terminator. The degree of focusing of this hot surface region is affected by the ratio of overall effective temperature of the planet (Tscaie) and intensity of stellar radiation (R,, T4, and a). This is implicit in the boundary conditions themselves (Equations 2.6, 2.7):

R T443 4 3 Tin[ND] OC and Tout[ND] oc T T (3.1) a2 Tscale p scale

In other words, as Tscaie increases, Tout[ND] will have a far greater effect on the temperature of any given surface element than Tin[ND}. Conversely, as Tscale decreases, the opposite is true and Tin[ND] dominates. The physical meaning of a higher Tscae is a greater difference between the average mantle temperature and surface temperatures. Thus, for cooler (and therefore older) planets, tidal-locking will produce a greater hemispheric dichotomy than

20 21

6000

0.9 4000

0.8 Z 0 2000

0.7 L. 0 :3 0 CL 0.6 r-5 3

-2000 - 0.5

-4000 0.4

-6000 0.3

2000 4000 6000 Distance from Axis of Symmetry [km]

Figure 3.1 The restultant thermal structure of the conductive and radiative first stage of the boundary condition implementation. Stellar flux arrives from the top of the figure. Color corresponds to nondimensionalized temperature, with 1 corresponding to 300'C. Notice that, as expected, the star-side is warmed to a higher temperature than the far-side. a planet with a hot interior. Similarly, greater stellar flux (R, and T. are high, while a is small) will also add to this dichotomy by increasing the effect of Tin[ND while leaving TUt[ND) unaffected at any given timestep.

3.2 Implementation of Convection

The second modeling stage, with the addition of convective heat flow, experiences at least 3 main evolutionary steps. Fig. 3.2 shows, in a sequence of six panels, the evolution of the first two of these steps. Before the development of full mantle convection, a series of three focused downwellings forms: a drip first at the antistellar point, then at the substellar 22

6000 1.0

r . / 0.9 z 4000 0 0.8 D 3 ~ ~ 0.7 TO 2000 N 0.67 C

Ii, I he 0 0.56

0.4 3 -2000 0.3 C -4000 0.2 'D 0.1 -6000 2000 4000 6000 Total Time Elapse: Distance from Axis of Symmetry [kin] -195 Million Years

Figure 3.2 Time sequence of the fully convective numerical model for the first -195 million years. Stellar flux arrives from the top of the figure. Color corresponds to nondimensionalized temperature, with 1 corresponding to 1000"C; black arrows correspond to direction and magnitude of mantle convective velocity. Note the development of a series of downwellings: first in the coldest region (antistellar point), then in the hottest region (substellar point), and finally at the terminator, before development of many convective cells throughout the mantle. 23

6000 1.0

0.9 4000 z 0.8 0

2000- 0.7 E 0 0.6 _ 0 X- 0.5 E - 0.4 CU -2000 @

4-.. 0.3

-4000 0.2

0.1 -6000 - 2000 4000 6000 Axis [km]

Figure 3.3 Highlight of initial convective downwellings, culminating with a drip at the terminator. This stage is very soon after the bottom left panel of Fig. 3.2. Stellar flux arrives from the top of the figure; colors correspond to nondimensionalized temperature, with the scale focused to highlight these initial drips. Left Panel: black lines are arrows whose magnitude and direction correspond to velocity of convecting mass within the mantle. Right Panel: black lines are temperature contours. 24

6000 6000 1.0

0.9 4000 4000 0.8 0 0.7 20002000 20000 0 E -/0.6 o$ -0 - 0 - -D x $0.5 0- E 0 3 0.4 S -2000- -2000- M 0.3

-4000 -400002

0.1

-6000 -6000 2000 4000 6000 2000 4000 6000 Axis [km]

Figure 3.4 Highlight of active downwellings and small scale, or half-mantle, convective patterns after -250,000 years. These drips, upwellings, and downwellings are not long-lived, and to a first- order, suggest a stagnant lid tectonic regime. Stellar flux arrives from the top of the figure; colors correspond to nondimensionalized temperature, with the scale focused to highlight these initial drips. Left Panel: black lines are arrows whose magnitude and direction correspond to velocity of convecting mass within the mantle. Right Panel: black lines are temperature contours. 25 point, and finally a concurrent series of three converging drips at the terminator (Fig. 3.3). Following this sequence, after approximately 250,000 years, many paired convection cells form as responses to drips from the cooler lithosphere (Fig. 3.4). Though our models so far provide only a small statistical sample, we found that generally more drips occur on the star-side than the far-side. This is illustrated in Fig. 3.5, where over 10 panels which begin at the the transition of the initial downwelling sequence into the pseudo-steady-state second step, 95 drips were counted on the star-side compared to only 77 on the far-side. Some of these cells span the depth of the mantle, while others seem to be small-scale convection; there seems to be no significant correlation, however, with the depth or width of a drip or plume with the hemispheric surface dichotomy. The lithosphere, which we define by a conductive geotherm, is significantly thinner at the substellar point compared to the antistellar regions. Even after the model has reached initial 'step 2' steady-state convection, the lithosphere on the star-side continues to slowly thin while the far-side thickens. An increase in stellar flux should amplify this lithospheric dichotomy and, since increasing the mass and radius of the planet should lower the overall lithospheric thickness (Valencia et al., 2006), this dichotomy may be exaggerated for hot, tidally-locked super-Earths than for our Earth-mass models. Outermost surface temperatures for the first 2.5 billion years are closely linked with radiative heating and cooling, while conductive heat flow becomes a dominant control on temperature just subsurface and within the lithosphere. After 1.68 billion years of evolution of an Earth-like model, we find that the surface node temperatures on the star-side vary from 390'C at the substellar point, and only drop below 100'C at the terminator. The star-side gradient is also more gradual near the substellar point, and decays away drastically at the terminator: temperature changes from 1200 to 47'C between two nodes at the terminator

(0 = 149 to 150; r = 100), while it takes 12 nodes to change from 3890 to 388'C at the substellar point (0 = 1 to 12; r = 100). In these outermost nodes, all far-side temperatures range from 0.5' to 47'C. At one node depth (32 km), temperatures become much hotter than at the surface nodes, and differences in temperature gradients along 6 become far less dramatic. The only regions which are less than 100'C closely surround the antistellar point 26

.00 0 doQ Q. 22

0 V-A

0 0 0

00C

Figure 3.5 Lithospheric drips through time, progressing from top left to bottom right, in the first ~1 billion years of convective evolution. Stellar flux arrives from the top of each panel; colors correspond to temperature, though actual scaling varies between panels to highlight plumes and drips. Each drip has been approximately boxed. Though the identification of a drip is subjective, over ten panels a statistically significant difference in the number of plumes between the star-side and the far-side suggest convection may be more vigorous in the warmer hemisphere. The total number of drips on the star-side was 95, while on the far-side only 77 were identified. 27

0.9 E a

0.7 between 0 and 1 00*C E 2- Dark Blue indicates temperatures E at depths up to 1 node (32 km).

'-4W 'T C - 1I1 -M 0.2 2OW0

'01- . - 0 -- , A A - - Axis [km]

0 2000 4000

Figure 3.6 Model temperature conditions after 1.68 billion years of evolution, with an inset of the region surrounding the antistellar point. Stellar flux arrives from the top of the figure, and colors correspond to nondimensionalized temperatures, where 1 = 1000*C. Dark blue highlights the region where T < 100*C at one node of depth (32 km, r = 99). Surface (r = 100) temperatures everywhere on the star-side are above 100'C, while they are below everywhere on the far-side. 28

(0 = 233 to 300; r = 99); at depths greater than this second node (r < 99), temperatures never drop below 100'C (Fig. 3.6). Obviously the stability of liquid water depends on pressure conditions in addition to temperatures, but without an atmosphere, surface pressures will be too low for liquid water. However, here we show that at some shallow subsurface depths during the first few billion years of planetary evolution, water may be stable in reservoirs or in highly permeable rock units, even without the assumption of atmospheric surface pressures.

3.3 Mode 1 Convection and Long-Term Evolution

As described in the previous section, the dichotomy in lithospheric thickness increases slowly for the first - 2.5 billion years of planetary evolution (approximately 30,000 SSAXC timesteps); however, in the final step of the convecting models, the far-side lithosphere sub- sequently grows around the substellar point over the course of the next 8 to 10 billion years, producing a very cold reservoir in the far-side of the planet (Fig. 3.7). By the time the planet is -3.5 billion years old, this cold reservoir extends to the core-mantle boundary in the substellar region, and the bulk mantle adiabatic temperature noticeably decreases by

~ 100'C. This cold region (- 300'C) then spreads laterally along the core-mantle boundary back toward the star-side of the planet before reaching an apparent steady-state at 12.8 billion years. The extension of the cold, lower mantle region halts slightly to the star-side of the terminator. Meanwhile, the bulk mantle adiabatic temperature continues to cool, nearing 800'C by the end of these experiments. To a large extent, these mantle features are not expressed in temperature changes along the surface of the planet. While the internal structure in the first steps of convective models show a lack of the hypothesized Mode 1 whole-mantle convective pattern, we begin to see it develop in this final step. We predicted that the intense fixed thermal flux would cause a dichotomy that penetrated deep into the mantle, and caused a local upwelling of hot material at the substellar point, and a paired downwelling at the coolest, antistellar point. After this extreme antistellar cold region develops, in general, hot material does rise at the substellar point, and 29

6000 1.0

0.9 z 40000 0.8 2-

2000 0.7 _ E0.6 0

- 0.5 a. 0.4 3 -2000 ~- 0.3 r+

-4000 0.2 * 0.1 -6000 2000 4000 6000 Total Time Elapse: Distance from Axis of Symmetry [ki] -2.5 to -12.8 Billion Years

Figure 3.7 Time sequence of the fully convective numerical model. The first panel (top left) shows the mantle thermal structure at -2.5 billion years of planetary evolution, and each subsequent panel steps -1.7 billion years further. The final panel (bottom right) shows the mantle thermal structure at -12.8 billion years. Color corresponds to nondimensionalized temperature, with 1 correlating to 1000*C; black arrows correspond to direction and magnitude of mantle convective velocity. Note the extreme cooling near the antistellar point, which expands back towards the terminator upon reaching the core-mantle boundary, along with the overall cooling of the mantle. Our models end with a pseudo-Mode 1 convective pattern in apparent steady-state. 30 flows laterally towards the far-side. Locally, however, this is frequently disrupted by continued lithospheric drips causing small-scale convective cells. In this sense, we interpret the final evolutionary stage of mantle convection to be an imperfect, or pseudo-Mode 1 pattern. We expect that the mantle will continue to cool, eventually stalling convection, though in these Earth-like models this might take longer than the lifetime of the star. The timescale of the development of these various steps in planetary evolution will clearly depend on stellar flux. Future work will raise the temperature input on the star-side of an Earth-like planet to better constrain these timescales at lower semi-major axes, and with different stellar intensities. We expect that higher stellar flux will amplify these effects, and possibly induce Mode 1 convection earlier in the planet's history. Moreover, here we have neglected the effect of stellar and orbital evolution, which undoubtedly will change the temperature inputs over the course of a single planet's evolutionary history. In other words, given the 'faint young sun' theory, it seems likely that planets will enter the mature stages of mantle evolution sooner, since they will cool faster earlier in their history due to a weaker star. As the star becomes more intense, planets will receive greater flux, which would dampen the the cooling effect in the mantle and may extend the lifetime of Mode 1 convection. 4 Discussion: Partial Magma Oceans

Given the full range of observed exoplanets so far, here we consider the possibility and characteristics of localized molten surface regions around the substellar point of Earth-like tidally-locked exoplanets. Though our preliminary models do not yet consider the large stellar flux which many detected super-Earths realistically receive, our results do indicate a possibility that temperatures may rise above the solidus of peridotite along shallow substellar regions. Localized magma oceans have been studied previously in order to understand the effects of large impactors early in terrestrial planet evolution (e.g., Tonks and Melosh, 1993; Reese et al., 2004; Reese and Solomatov, 2006; Reese et al., 2007; Watters et al., 2009). A significant difference between the dynamics of impact-induced transient molten regions and stellar flux- induced magma ponds is long-term evolution and stability. In the former, crystallization timescales depend on the depth, extent, composition, and, in general, the cooling efficiency of the region, while in the latter, the same is true but, given enough stellar heating, large- scale magma ponds may persist indefinitely. If the arriving stellar flux delivers more energy than is lost by the convecting magma pond, we assume that the feature will be stable over long timescales.

According to Kraichnan (1962), the heat lost from the top of a cooling magma pond, Fm, in a high Prandtl number convective regime is

1/ Fm 0.089Ra 3 (4.1)

We raise Equation 2.4 (the definition of Ra) to a power of ', dimensionalize this value to appropriate units for flux, , and multiply this by the appropriate experimentally-derived

31 32

Symbol Parameter Value

k thermal conductivity 4 W

p silicate liquid density 3300 k

a coefficient of thermal expansion 2x 10-5 1

silicate liquid dynamic viscosity 10-12 Pa - s thermal diffusivity 1x10-6S

T, potential temperature 14000 C 0 TS surface temperature 1200 C

Table 4.1 List of parameters used to calculate long term localized magma pond stability. efficiency coefficient from Equation 4.1 to yield

4 3 Fm = 0.089k(Tp - T8 ) / (ACT )1/3 (.2 where k is thermal conductivity, p, is the silicate liquid density, a is the coefficient of thermal expansion, ql is the silicate liquid dynamic viscosity, K is the thermal diffusivity, Tp is the potential temperature, and T, is the surface temperature (Reese and Solomatov, 2006). For these approximate calculations, we assume a sufficiently high surface temperature, T,~

12000 C, such that peridotite would be molten. From Zaranek and Parmentier (2004), we assume a change in viscosity by a factor of 10 between the uppermost fluid region not participating in convection and the lower convecting material. A mafic silicate melt at

0 1200 C at the surface corresponds to a viscosity of - 10-1.; once we decrease this by an order of magnitude, we expect the viscosity of the convecting fluid to be - 10-1.2, which correlates to a potential temperature, TP, of ~ 1400'C (Shaw, 1972). A summary of the parameters and values used in these calculations is given in Table 4.1.

To a first order, if the flux released by the surface remains less than or equal to the flux received from the star, F, = -T,', then the magma ocean should be stable over long timescales. A simple calculation of maximum semi-major axes for the stability of an exposed molten surface around various stars which have detected super-Earths is shown in Fig. 4.1.

For our solar system, in order for a localized magma ocean to be stable on an Earth-like 33 1

II 5 -

Assuming an Earth-like planet

The Sun 4 CoRoT 7 HD 40307 Gliese 581 Gliese 876 3 - I- - F. Fm

I I 0.013 0.04 0.067 0.094 0.12

Semi-major Axis [AU]

Figure 4.1 A simple calculation comparing stellar flux received by a planet against the energy flux that would be radiated out through a magma pond. Here, the ratio of these fluxes is on the vertical, and semi-major axis on the horizontal. This calculation assumes an Earth-like test planet, and calculates flux ratios for the Sun (green), CoRoT 7 (cyan), HD 40307 (blue), Gliese 581 (black), and Gliese 876 (red). To a first order, when the flux ratio is above or equal to 1, the magma pond is stable over long time scales. We find that the intersection points for each star system are: 0.123 AU for the Sun, 0.098 AU for CoRoT 7, 0.0658 for HD 40307, 0.017 AU for Gliese 581, and 0.015 AU for Gliese 876. Of known planets, only CoRoT 7 b and HD 40307 b are capable of stable magma oceans, based on this calculation. 34 planet, the semi-major axis must be less than -0.12 AU. The only known planets which, based on this calculation, are able to sustain a magma ocean for long timescales from fixed stellar flux are CoRoT 7 b (a = 0.017 AU) and HD 40307 b (a = 0.047 AU). Though several known exoplanets actually have significantly smaller semi-major axes than these two planets, those other planets orbit smaller stars (for example, Gliese 581 e is around an M-star) which do not deliver enough stellar flux, even for close-in planets, to match the efficiency of heat loss from a magma ocean. A planet with a localized magma pond will become asymmetric based on the differences in density between molten and solid silicates. Since density decreases in liquids, the solid mantle underlying the magma pond will isostatically rebound under the lighter load, causing uplifting and the magma to 'spill out' of the pond and eventually crystallize forming lava levees. Without any way of recycling this material, the initially fertile mantle will become depleted, which will effectively raise the melting temperature of the remaining mantle. 5 Conclusions

We use a numerical model to study the isolated effect of synchronous rotation on the thermal structure and evolution of the mantle in Earth-like exoplanets. We find that there are three punctuating steps in mantle evolution. Beginning each run, we observe an initial series of three downwellings: first at the antistellar point, then at the substellar point, and lastly at the terminator. This evolves after -250,000 years into pervasive small-scale convection as a result of short-lived lithospheric dripping. Finally, a cold antistellar region spanning the entire depth of the mantle develops, fueling a pseudo-Mode 1 convective pattern. Surface temperatures do not exhibit any noticeable expressions of the structure and evolution of the mantle underneath. In these initial Earth-like, Sun-like models, star-side temperatures generally remain around 600'C while far-side temperatures approach 00C, with a continuum between the two extremes. Given the wide range of super-Earths already detected, however, some of the closest-in planets may be so hot on the star-side that their surface becomes a partially molten localized magma pond over long time-scales. On the other hand, far-side temperatures may plummet below the freezing and condensing point of any volatiles, and serve as a glaciated region which becomes an unrecoverable sink for atmospheric gases. The Habitable Zone refers to a circumstellar region in which expected planetary surface temperatures would allow for the presence and stability of liquid water (e.g., Kasting et al., 1993). In this study, we show that extreme end-member environments, which are stable over long timescales, are likely to exist on the surface of tidally-locked planets. This implies that these bodies cannot be wholly habitable without the presence of an atmosphere. In less extreme planets, the asymmetric temperature profile may instigate locally habitable regions in which water may either pool on the surface, or exist in permeable rock reservoirs just subsurface. It is unclear from this work whether these planets, even if they are Earth-sized, 35 36

Figure 5.1 Above is a false color visual image centered on Gliese 581, located in the constellation Libra. This M-dwarf is known to harbor at least 3 tidally-locked super-Earths (see Table 1.1), including Gliese 581 d, which is in the Habitable Zone. Image credit to DSS, Skyview, AAO, ROE, CalTech, and STScI. will be tectonically active, but given substantial stellar heating they may be expected to be volcanically and potentially hydrothermally active. Future work on super-Earths will center around characterizing and detecting atmospheric compositional and temperature asymmetry, and coupling this to surface and subsurface conditions. While preliminary work has been done on the effect of tidal-locking on both magnetospheres (Grie3meier et al., 2005) and atmospheres (Joshi, 2003) individually, a fully coupled model is important in identifying where localized habitable regions may be found, given varying semi-major axes and stellar masses. Though future space missions promise an increasing catalog of close-in, terrestrial bodies, the Gliese 581 system (Fig. 5.1), with its three synchronously rotating super-Earths, provides a unique case study for testing the results of this and other studies, and developing our understanding of these exotic and exciting new extrasolar planets. In the absence of ground truth observations of exoplanet thermal profiles, this study provides tantalizing insight into the potential extreme conditions on the surface of worlds outside of our solar system. Bibliography

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Included here is the SSAXC file f-tmres.f, which contains the edits to the boundary conditions of the surface elements for synchronous rotation. It totals five pages, with this study's edits beginning on the fourth page. Following this, sample input and geometry decks are included for convecting runs.

42 43

subroutine fLtRes( dmhu tcon ien t & shl ,shdt ,shdr ,tdot & det ,tl ,vl xl & tdl ,uq ,vq ,ttq & trq ,tq ,tdq eval & lcblk ,x ,v ,idiag & tlhs ,trhs ,idt ,lmt & tbc ,mat ,diff ) c------c c This routine calculates the R.H.S. temperature Residual c and solves for and predicts the new temperature. c This is a ConMan routine modified to solve for c spherical axisymmetric polar coordinates. c Recall theta => x1 and r => x2. c c S.D.K. 9/20/91 c (modifications from L.H.K. 4/3/89). c ------c implicit double precision (a-h,o-z) double precision sigma,kappa,density,spht,ellength,stelint double precision latitude,planetradius,znodes,Tscale,xnodes double precision Tin,Tout,dz,semiaxis,steltemp,stelradius real nprint(7) c c include 'common.h' c C dimension dmhu(*) tcon(6,*) ien(numel,*) & t(*) shl(4,*) shdt(numel,4,*), & shdr(numel,4,*) tdot(*) det(numel,*) & tl(lvec,*) vl(lvec,*) xl(lvec,*) & tdl(lvec,*) uq(lvec,*) vq(lvec,*) & ttq(lvec,*) eval(numel,*), lcblk(2,*) & x(2,*) , v(2,*) , idiag(*) & tlhs(*) , trhs(*) , idt(*) & lmt(numel,*) , tbc(*) , mat(*) & tq(lvec,*) , trq(lvec,*) , tdq(lvec,* , & diff(*) C common /templ / el-rhs(lvec,4),blkmhu(lvec) , pg(lvec,4,5) & conv(lvec) ,tmass(lvec) , adiff(lvec) & unorm(lvec) ,eta(lvec) , xse(lvec) & blkdif(lvec) ,uxse(lvec) , ueta(lvec) & r(lvec,5) ,theta(lvec,5), cotte(lvec,5), & rdet(lvec,5) c c write(6,*)'f-tmres t beginning------c write(6,*) (t(ii),ii=1,1S)

if (npass .gt. 1 ) then flag = one else flag = zero end if c do 100 i = 1 , nEGnp trhs(i) = zero 100 continue c nelvec = numel c c.... loop over the element blocks c do 1000 iblk = 1 , nelblk c c.... set up the parameters c nel = lcblk(2,iblk) nenl = Icblk(2,iblk) nvec = lcblk(1,iblk+1) - iel c c..... localize the temperature ,tdot , velocity c c$dir no-recurrence do 150 iv = 1 , nvec ivel - iv + iel - 1 44

xl(iv,1) = x(1, ien(ivel,1) ) xl(iv,2) = x(2, ien(ivel,1) ) xl(iv,3) = x(1, ien(ivel,2) ) xl(iv,4) = x(2, ien(ivel,2) ) xl(iv,5) = x(1, ien(ivel,3) ) xl(iv,6) = x(2, ien(ivel,3) ) xl(iv,7) = x(1, ien(ivel,4) ) xl(iv,8) = x(Z, ien(ivel,4) ) C vl(iv,1) = v(1, ien(ivel,1) ) vl(iv,2) = v(2, ien(ivel,1) ) vl(iv,3) = v(Z, ien(ivel,Z) ) vl(iv,4) = v(2, ien(ivel,2) ) vl(iv,5) - v(1, ien(ivel,3) ) vl(iv,6) = v(1, ien(ivel,3) ) vl(iv,7) = v(1, ien(ivel,4) ) vl(iv,8) = v(2, ien(ivel,4)) C c---DEBUG c if (iv eq. 1) c & write (6,*) 'f-tmres v = vl(iv,2) c---DEBUG tl(iv,1) = t( lmt(ivel,2) ) tl(iv,2) = t( lmt(ivel,2) ) tl(iv,3) = t( Imt~ivel,3)) tl(iv,4) = t( lmt(ivel,4) ) 150 continue if( npass .gt. 1) then do 175 iv = 1, nvec ivel = iv + iel - 1 tdl(iv,1) = tdot( lmt(ivel,1) ) tdl(iv,2) = tdot( lmt(ivel,2) ) tdl(iv,3) = tdot( lmt(ivel,3) ) tdl(iv,4) = tdot( lmt(ivel,4) ) 175 continue end if C c.. form the the values of the temperature and velocity at integration points C do 400 intp - 1, 5 do 200 iv = 1 , nvec ivel = iv + iel - 1 theta(iv,intp)= shl(1,intp) * xl(iv,1) + shl(2,intp) * xl(iv,3) & + shl(3,intp) * xl(iv,5) + shl(4,intp) * xl(iv,7) C r(iv,intp) = shl(1,intp) * xl(iv,2) + shl(2,intp) * xl(iv,4) & + shl(3,intp) * xl(iv,6) + shl(4,intp) * xl(iv,8) C rdet(iv,intp) = r(iv,intp) * r(iv,intp) * sin(theta(iv,intp)) & * det(ivel,intp) C cotte(iv,intp)=one/(tan(theta(iv,intp))) C uq(iv,intp) = vl(iv,1) * shl(1,intp) + vl(iv,3) * shl(2,intp) & + vl(iv,5) * shl(3,intp) + vl(iv,7) * shl(4,intp) C vq(iv,intp) = vl(iv,2) * shl(1,intp) + vl(iv,4) * shl(2,intp) & + vl(iv,6) * shl(3,intp) + vl(iv,8) * shl(4,intp) C trq(iv,intp) = tl(iv,1)*shdr(ivel,1,intp) & + tl(iv,2)*shdr(ivel,2,intp) & + tl(iv,3)*shdr(ivel,3,intp) & + tl(iv,4)*shdr(ivel,4,intp) C ttq(iv,intp) = tl(iv,1)*shdt(ivel,1,intp)/r(iv,intp) & + tl(iv,2)*shdt(ivel,2,intp)/r(iv,intp) & + tl(iv,3)*shdt(ivel,3,intp)/r(iv,intp) & + tl(iv,4)*shdt(ivel,4,intp)/r(iv,intp) C 200 continue C if ( npass .gt. 1 ) then do 300 iv = 1, nvec tdq(iv,intp) = tdl(iv,1)*shl(1,intp) + tdl(iv,Z)*shl(2,intp) & + tdl(iv,3)*shl(3,intp) + tdl(iv,4)*shl(4,intp) 300 continue end if c c ...end loop over integration points 45

c 400 continue c c... for PG shape functions (see Brooks Thesis) c c$dir no-recurrence do 500 iv = 1 , nvec ivel - iv + iel - 1 uxse(iv) = abs(uq(iv,5)*eval(ivel,1) + vq(iv,5)*eval(ivel,2) ) & * eval(ivel,5) ueta(iv) = abs(uq(iv,5)*eval(ivel,3) + vq(iv,5)*eval(ivel,4) ) & * eval(ivel,6) c blkmhu~iv) = dmhu(mat(ivel)) blkdif(iv) = diff(mat(ivel)) c if( uxse(iv) .gt. two*blkdif(iv)) then xse(iv) = one - two*blkdif(iv)/uxse(iv) else xse(iv) = zero end if c if( ueta(iv) .gt. two*blkdif(iv)) then eta(iv) one - two*blkdif(iv)/ueta(iv) else eta(iv) = zero end if c unorm(iv) = uq(iv,5) * uq(iv,5) + vq(iv,5) vq(iv,5) if( unorm(iv) .gt. 0.000001 ) then adiff(iv) = (uxse(iv) * xse(iv) + ueta(iv) eta(iv))/ & (two*unorm(iv)) else adiff(iv) = zero end if c adiff(iv) = zero c 500 continue c c loop over integration points c do 650 intp = 1 , 4 c do 600 iv = 1 , nvec ivel - iel + iv - 1 C c..... form pg shape function's for intergration point intp c pg(iv,1,intp) = shl(1,intp) + & adiff(iv)*(uq(iv,intp)*shdt(ivel,1,intp) & +vq(iv,intp)*shdr(ivel,1,intp)) c pg(iv,2,intp) = shl(2,intp) + & adiff(iv)*(uq(iv,intp)*shdt(ivel,2,intp) & +vq(iv,intp)*shdr(ivel,2,intp)) c pg(iv,3,intp) = shl(3,intp) + & adiff(iv)*(uq(iv,intp)*shdt(ive,3,intp) & +vq(iv,intp)*shdr(ivel,3,intp)) c pg(iv,4,intp) = shl(4,intp) + & adiff(iv)*(uq(iv,intp)*shdt(ivel,4,intp) & +vq(iv,intp)*shdr(ivel,4,intp)) c el-rhs(iv,intp) . zero c 600 continue 650 continue c c.... form Residue term at intergation point INTP c c$dir no-recurrence do 750 intp = 1, 4 c c loop over elements in a block c do 700 iv = 1 ,nvec ivel = iv + iel - 1 46

conv(iv) = uq(iv,intp)*ttq(iv,intp) + vq(iv,intp)*trq(iv,intp) tmass(iv) = tdq(iv,intp) * flag + conv(iv) - blkmhu(iv) c el-rhs(iv,1) = el-rhs(iv,1) & -rdet(iv,intp)*( tmass(iv) * pg(iv,1,intp) + blkdif(iv) * & (shdr(ivel,1,intp)*trq(iv,intp) & + shdt(ivel,1,intp)*ttq(iv,intp)/r(iv,intp) )) c el-rhs(iv,2) = el-rhs(iv,2) & -rdet(iv,intp)*( tmass(iv) * pg(iv,2,intp) + blkdif(iv) * & (shdr(ivel,2,intp)*trq(iv,intp) & + shdt(ivel,2,intp)*ttq(iv,intp)/r(iv,intp) )) C el-rhs(iv,3) = el-rhs(iv,3) & -rdet(iv,intp)*( tmass(iv) * pg(iv,3,intp) + blkdif(iv) & (shdr(ivel,3,intp)*trq(iv,intp) & + shdt(ivel,3,intp)*ttq(iv,intp)/r(iv,intp) )) c el-rhs(iv,4) = el-rhs(iv,4) & -rdet(iv,intp)*( tmass(iv) * pg(iv,4,intp) + blkdif(iv) * & (shdr(ivel,4,intp)*trq(iv,intp) & + shdt(ivel,4,intp)*ttq(iv,intp)/r(iv,intp) )) C 700 continue c 750 continue c c..... assemble this block's element residual C c$dir no-recurrence do 800 iv - 1 , nvec ivel = iv + iel - 1 trhs(lmt(ivel,1)) = trhs(lmt(ivel,1)) + el-rhs(iv,1) trhs(lmt(ivel,2)) = trhs(lmt(ivel,2)) + el-rhs(iv,2) trhs(lmt(ivel,3)) = trhs(lmt(ivel,3)) + el-rhs(iv,3) trhs(lmt(ivel,4)) = trhs(lmt(ivel,4)) + el-rhs(iv,4) 800 continue c c.... end loop over element blocks c 1000 continue c c .... adjust group assembled residual (trhs) for boundray conditions c.... (i.e replace with boundray value) C do 1200 n = 1 , nEGnp if(idt(n) eq. 0 ) then trhs(n) = zero end if 1200 continue c c.... correct for new temperature c if (npass .eq. 1) then do 1300 n = 1 , nEGnp tdot(n) - tlhs(n) * trhs(n) t(n) = t(n) + alfadt * tdot(n) 1300 continue else do 1305 n = 1 , nEGnp tdot(n) = tdot(n) + tlhs(n) * trhs(n) t(n) = t(n) + alfadt * tlhs(n) * trhs(n) 1305 continue end if

... Hot Tidally-locked Super-Earths SSAXC Edits for this study: c... Adjust new temperature for boundary conditions c... Modified by SEG Summer/Fall 2008, Spring 2009. In file ftrEs.f c znodes = nelz +1 xnodes = nelx +1 (continued on next page) sigma = 0.0000000567 kappa = 0.000001 spht = 1000 density - 3000 planetradius = 6400000 ellength = (planetradius*3.14/xnodes) dz = (planetradius/2/znodes) stelint = 1300

I r r 47

stelradius = 700000 steltemp = 6000 semiaxis = 100000000 Tscale = 1000

do 1400 n = 1 , numnp if (mod(n,znodes) eq. 0) then if (n .le. (numnp/2)) then

latitude - (((n/znodes))*3.14159/xnodes)

t(n) = t(n) & -(sigma*(t(n)**(4))/spht/density & *dt*(ellength)/kappa*(Tscale**(3))) & + (stelint*cos(latitude)/density/spht & *dt*(ellength)/kappa/Tscale)

else

t(n) = t(n) & -(sigma*(t(n)**(4))/spht/density & *dt*(ellength)/kappa*(Tscale**(3)))

end if

else if (idt(n) .eq. 0 ) then t(n) = tbcCn) endif end if

1400 continue if (nwrap .gt. 0) then c$dir no-recurrence do 1500 n = 1, nwrap+1 iwrap = nEGnp - nwrap-1 + n t(iwrap) = t(n) tdot(iwrap) = tdot(n) 1500 continue end if return end SO+300000' S0+300000'I 00+3000S8'0 00+300000'1 00+300081'O 00+00000*02 00+300000'0 00+300000'0 0T+300000*I 00+300000'0 00+300000'I LO+300000'I 00+300000'I 0 0 S 0 Z T t7 t T096Z2 6STtT'E 0000050 000000*0 000000*0 0 0 0 00T 66662 66 00T 20662 2 0 0 0 00T 8966T 00T 00T T066Z T 0 0 0 0 0 0 0 0 1 00T 0000E 00T T 00T T066Z T 0 0 0 0 0 T T T OOOOE 0000E 1 1 T 00T 00T 1 1 1 T066Z T0662

0 1 1 OOOOE 10662 0 1 T 00T T T 0 00T OOOOE 00T T 0 00T 10662 1 0002 0002 0002 0002 90-300000'T V0+300000'T 00000S'0 2 0008 0 T 0 0 T T T 1 009 T T T 0 T Z 66 662 Z 2 0000E

>pap indui aldw es

8f7 49

Sample geometry deck

1 4 0.000000 0.500000 29901 1 3.141590 0.500000 30000 1 3.141590 1.000000 100 1 0.000000 1.000000 299 100 99 1 0 0 0 0 1 2 0 0 29901 0 0 0 299 100 0 0 100 2 0.000000 0 30000 0 0.000000 0 299 100 0 0 1 2 0 0 100 0 0 0 0 0 99 1 29901 2 0 0 30000 0 0 0 0 0 99 1 0 0 0 0 1 2 1.000000 29901 0 1.000000 299 100 0 0 100 2 0.000000 30000 0 0.000000 299 100 0 0 0 0 0 0 0 0 1 1 1 1 101 102 2 299 99 100 99 1 1 0 0 0 0 0 0 0 I " 4

B Magma Pond Stability Code

Included here is the Matlab m-file magmapondtester.m, in which the stability of localized magma oceans on the star-side of hot, tidally-locked super-Earths is calculated, as described in Chapter 4. It totals two pages, with the results being Fig. 4.1.

50 51

% Sarah E. Gelman % 19 April 2009 % magmapondtester.m % Simple comparitive calculation of stellar flux % vs. magma pond radiation to see if the magma pond % is stable for long time-scales. clear all;

% Set fixed parameters. Sunrad = 6.96*(10A8); Suntemp = 5778; GJ581rad = 0.38*Sunrad; GJ581temp = 3480; CoRoTrad = 0.95*Sunrad; CoRoTtemp = 5300; HD40307rad = 0.72*Sunrad; HD40307temp = 4977; GJ876rad = 0.36*Sunrad; GJ876temp = 3350; sigma = 5.67 * (10A(-8)); J/(mA2 sec KA4) k = 4; J/(m sec K) Tpot = 1673; K Tsurf = 1473; K Rhol = 3300; kg/(mA3) alpha = 2 * (10^(-5)); 1/K BigG = 6.67 * (10^(-11)); (mA3)/((secA2) kg) planmass = 6 * (10A24); kg planrad = 6400000; m viscosity = 0.0631; Pa sec = kg/(m sec) kappa = 10A(-6); (mA2)/sec a = 1*(10A9):5*(10A7):Z*(10A10);

% Calculate littleg (in case you wanna, you know, make real Super-Earths). littleg = BigG*planmass/(planradA2);

%% Calculate magma ocean flux (Reese & Solomatov, 2006/2007; % Solomatov 2000). Fmagma = 0.089*k*((Tpot-Tsurf)A(4/3))*(((Rhol*alpha*littleg)... /(viscosity*kappa))A(1/3));

%%%%%%%%%%%%%%% For the Sun %%%%%%%%%%%%%%%%%%%%%%%% % Calculate stellar flux (from Stefan-Boltzmann). Sunflux = ((SunradA2)*sigma*(SuntempA4));

% Calculate ratio. Sunratio = Sunflux./(a.A2)/Fmagma;

%%%%%%%%%%%%%%% For Gli ese 581 %%%%%%%%%%%%%%%%%%%%%%%% % Calculate stellar flux (from Stefan-Boltzmann). GJ581flux = ((GJ581radA2)*sigma*(GJ581tempA4));

% Calculate ratio. GJ581ratio = GJ581flux./(a.A2)/Fmagma;

%%%%%%%%%%%%%%% For CoRoT 7 %%%%%%%%%%%%%%%%%%%%%%%% % Calculate stellar flux (from Stefan-Boltzmann). CoRoTflux = ((CoRoTradA2)*sigma*(CoRoTtempA4));

% Calculate ratio. CoRoTratio = CoRoTflux./(a.A2)/Fmagma;

%%%%%%%%%%%%%%% For HD 40307 %%%%%%%%%%%%%%%%%%%%%%%% 52

% Calculate stellar flux (from Stefan-Boltzmann). HD40307flux = ((HD40307radA2)*sigma*(HD40307temp4));

% Calculate ratio. HD40307ratio = HD40307flux./(a.^2)/Fmagma;

%%%%%%%%%%%%%%% For Gliese 876 %%%%%%%%%%%%%%%%%%%%%%%% % Calculate stellar flux (from Stefan-Boltzmann). GJ876flux = ((GJ876radA2)*sigma*(GJ876tempA4));

% Calculate ratio. GJ876ratio = GJ876flux./(a.A2)/Fmagma;

%% Plot it. plot(a,Sunratio,'color','green','linewidth',2); hold on; plot(a,GJ581ratio,'color','black','linewidth',2); plot(a,CoRoTratio,'color','cyan','linewidth',2); plot(a,HD40307ratio,'color','blue','linewidth',2); plot(a,GJ876ratio,'color','red','linewidth',2); axis([(10^9) (2*(10^10)) 0 5]); hold off;

%Find where each intersects 1 (this is boundary for steady-state). diffi = 100; diff2 = 100; diff3 = 100; diff4 = 100; diff5 = 100; for i = 1:length(a) diff-newl = abs(Sunratio()-1); diff-new2 = abs(GJ581ratio(i)-1); diff-new3 = abs(CoRoTratio(i)-1); diff-new4 = abs(HD40307ratio(i)-1); diff-new5 = abs(GJ876ratio(i)-1); if diff-newl < diffi lowestindexi = i; diffi = diffnewl; end if diff-new2 < diff2 lowestindex2 = i; diff2 = diffnew2; end if diff.new3 < diff3 lowestindex3 = i; diff3 = diffnew3; end if diff.new4 < diff4 lowestindex4 = i; diff4 = diff-new4; end if diff.new5 < diff5 lowestindex5 = i; diff5 = diffnew5; end end a(lowestindexl) % 1.8400e+10 a(lowestindex2) % 2.5500e+09 a(lowestindex3) % 1.4700e+10 a(lowestindex4) % 9.8500e+09 a(lowestindex5) % 2.2500e+09