A Detailed Model Grid for Solid Planets from 0.1 through 100 Earth Masses

LI ZENG AND DIMITAR SASSELOVAstronomy Department, Harvard University, Cambridge, MA 02138; lzeng@cfa.harvard.edu, dsasselov@cfa.harvard.edu

Received 2012 July 11; accepted 2013 January 20; published 2013 March 12

ABSTRACT. This article describes a new grid for the mass–radius relation of three-layer exoplanets within themass range of 0:1–100 M⊕. The three layers are: Fe (ϵ-phase of iron), MgSiO3 (including both the perovskite phase,post-perovskite phase, and its dissociation at ultrahigh pressures), and H2O (including Ices Ih, III, V, VI, VII, X, andthe superionic phase along the melting curve). We discuss the current state of knowledge about the equations of state(EOS) that influence these calculations and the improvements used in the new grid. For the two-layer model, wedemonstrate the utility of contours on the mass–radius diagrams. Given the mass and radius input, these contourscan be used to quickly determine the important physical properties of a planet including its p0 (central pressure),p1=p0 (core–mantle boundary pressure over central pressure), CMF (core mass fraction) or CRF (core radius frac-tion). For the three-layer model, a curve segment on the ternary diagram represents all possible relative mass pro-portions of the three layers for a given mass–radius input. These ternary diagrams are tabulated with the intent tomake comparison to observations easier. How the presence of Fe in the mantle affects the mass–radius relations isalso discussed in a separate section. A dynamic and interactive tool to characterize and illustrate the interior structureof exoplanets built upon models in this article is available online.

Online material: color figures, extended table

1. INTRODUCTION

The transit method of exoplanet discovery has produced asmall but well-constrained, sample of exoplanets that are unam-biguously solid in terms of interior bulk composition. We callsolid planets the ones that possess no H and He envelopesand/or atmospheres, i.e., their bulk radius is determined by ele-ments (and their mineral phases) heavier than H and He. Suchsolid exoplanets are Kepler-10b (Batalha et al. 2011), CoRoT-7b(Queloz et al. 2009; Hatzes et al. 2011), Kepler-36b (Carter et al.2012) as well as—most likely—Kepler-20b, e, f; Kepler-18b,and 55 Cnc e, in which the solid material could include high-pressure water ice (see references in § 4.2).

There is an increased interest in comparing observed param-eters to current models of interior planetary structure. The mod-els, and their use of approximations and EOS, have evolvedsince 2005 (Valencia et al. 2006; Fortney et al. 2007; Grassetet al. 2009; Seager et al. 2007; Wagner et al. 2011; Swift et al.2012), mainly because the more massive solid exoplanets(called super-Earths) have interior pressures that are far in ex-cess of Earth’s model, bringing about corresponding gaps in ourknowledge of mineral phases and their EOS (see a recent reviewby Sotin et al. [2010]). Here, we compute a new grid of modelsin order to aid current comparisons to observed exoplanets onthe mass–radius diagram. As in previous such grids, we assumethe main constituents inside the planets to be differentiated andmodel them as layers in one dimension.

The first part of this article aims to solve the two-layer exo-planet model. The two-layer model reveals the underlying phys-ics of planetary interior more intuitively, for which we considerthree scenarios: an Fe-MgSiO3, Fe-H2O, or MgSiO3-H2Oplanet.

The current observations generally measure the radius of anexoplanet through transits and the mass through Doppler shiftmeasurement of the host star. For each assumption of core andmantle compositions, given the mass and radius input, the two-layer exoplanet model can be solved uniquely. It is a uniquesolution of radial dependence of interior pressure and density.As a result, all the characteristic physical quantities, such as thepressure at the center (p0), the pressure at core–mantle boundary(p1), the core mass fraction (CMF), and the core radius fraction(CRF) naturally fall out from this model. These quantities canbe quickly determined by invoking the mass–radius contours.

The next part of this article compares some known exopla-nets to the mass–radius curves of six different two-layer exopla-net models: pure Fe, 50% Fe and 50% MgSiO3, pure MgSiO3,50% MgSiO3 and 50% H2O, 25% MgSiO3 and 75%H2O, andpure H2O. These percentages are in mass fractions. The data ofthese six curves are available in Table 1.

Up to now, a standard assumption has been that the planetinterior is fully differentiated into layers: all the Fe is in the coreand all the MgSiO3 is in the mantle. In § 4.3, we will change thisassumption and discuss how the presence of Fe in the mantleaffects the mass–radius relation.

000

PUBLICATIONS OF THE ASTRONOMICAL SOCIETY OF THE PACIFIC, 125:000–000, 2013 March© 2013. The Astronomical Society of the Pacific. All rights reserved. Printed in U.S.A.

TABLE1

DATA

FOR

THESI

XCHARACTERISTIC

MASS—RADIU

SCURVES

log 1

0ðp0Þ

Fe50

%Fe-50%

MgS

iO3

MgS

iO3∇

50%MgS

iO3-50%

H2O

25%MgS

iO3-75%H

2O

H2O

Massa

Radiusb

Mass

Radius

CRFc

p1=p0

dMass

Radius

Mass

Radius

CRF

p1=p0

Mass

Radius

CRF

p1=p0

Mass

Radius

09.6

.....

0.00

1496

0.09

947

0.00

177

0.12

10.68

920.29

310.00

623

0.20

290.00

8278

0.29

630.59

690.23

740.01

217

0.36

160.44

130.37

820.04

663

0.58

09.7

.....

0.00

2096

0.11

120.00

2481

0.13

540.68

880.29

250.00

8748

0.22

70.01

156

0.32

860.60

130.23

980.01

699

0.40

090.44

440.38

030.06

174

0.63

09.8

.....

0.00

2931

0.12

430.00

3472

0.15

130.68

820.29

180.01

227

0.25

40.01

615

0.36

470.60

510.24

130.02

351

0.44

320.44

750.38

50.08

208

0.68

5309

.9.....

0.00

409

0.13

870.00

4848

0.16

90.68

760.29

090.01

717

0.28

380.02

255

0.40

490.60

850.24

20.03

213

0.48

690.45

130.39

440.10

910.74

5510

.0.....

0.00

5694

0.15

460.00

6752

0.18

850.68

680.28

980.02

399

0.31

70.03

130.44

840.61

190.24

460.04

399

0.53

50.45

540.40

160.14

450.81

0210

.1.....

0.00

7904

0.17

220.00

938

0.21

010.68

580.28

860.03

343

0.35

360.04

314

0.49

440.61

650.24

950.06

029

0.58

80.45

920.40

730.19

040.87

910

.2.....

0.01

094

0.19

150.01

299

0.23

390.68

470.28

710.04

644

0.39

390.05

943

0.54

490.62

090.25

30.08

255

0.64

610.46

290.41

190.24

940.95

1510

.3.....

0.01

507

0.21

260.01

792

0.26

0.68

340.28

550.06

430.43

810.08

170.60

030.62

490.25

540.11

270.70

930.46

620.41

590.32

491.02

810

.4.....

0.02

070.23

560.02

464

0.28

860.68

190.28

360.08

866

0.48

650.11

20.66

070.62

850.25

710.15

330.77

770.46

930.41

970.42

061.10

710

.5.....

0.02

829

0.26

060.03

372

0.31

970.68

010.28

150.12

170.53

910.15

280.72

620.63

180.25

850.20

740.85

090.47

210.42

340.54

161.19

10.6

.....

0.03

850.28

760.04

595

0.35

350.67

820.27

920.16

610.59

60.20

740.79

660.63

470.25

950.27

890.92

890.47

470.42

70.69

381.27

610

.7.....

0.05

212

0.31

670.06

231

0.39

010.67

590.27

670.22

550.65

730.27

990.87

180.63

730.26

030.37

261.01

10.47

70.43

050.88

661.36

610

.8.....

0.07

021

0.34

80.08

408

0.42

960.67

350.27

40.30

420.72

280.37

540.95

150.63

960.26

10.49

461.09

80.47

920.43

391.13

21.46

110

.9.....

0.09

408

0.38

140.11

290.47

210.67

080.27

110.40

750.79

250.49

981.03

50.64

160.26

150.65

171.18

80.48

110.43

71.44

41.56

211

.0.....

0.12

540.41

70.15

080.51

770.66

790.26

820.54

20.86

610.66

071.12

30.64

340.26

180.85

291.28

20.48

270.43

971.84

11.66

911

.1.....

0.16

630.45

480.20

030.56

630.66

480.26

510.71

430.94

290.86

531.21

40.64

480.26

151.10

71.38

0.48

390.44

112.34

61.78

211

.2.....

0.21

930.49

490.26

470.61

80.66

160.26

210.92

71.02

1.11

71.30

50.64

550.25

981.42

1.47

70.48

430.44

2.98

51.90

111

.3.....

0.28

770.53

70.34

80.67

280.65

820.25

91.2

1.10

11.43

71.4

0.64

580.25

891.81

81.58

0.48

40.44

3.77

2.02

311

.4.....

0.37

540.58

140.45

50.73

070.65

470.25

591.54

51.18

61.84

21.49

90.64

580.25

822.32

11.68

80.48

340.44

034.73

52.14

711

.5.....

0.48

750.62

790.59

190.79

160.65

120.25

291.98

11.27

42.35

11.60

20.64

530.25

742.95

71.80

20.48

280.43

995.90

92.27

411

.6.....

0.62

980.67

650.76

590.85

540.64

760.25

2.52

51.36

52.98

81.70

80.64

450.25

653.75

21.92

0.48

220.43

917.32

52.40

111

.7.....

0.80

960.72

70.98

60.92

210.64

40.24

733.20

31.45

83.78

1.81

80.64

350.25

524.73

22.04

10.48

130.43

839.03

82.52

911

.8.....

1.03

60.77

961.26

10.99

070.64

070.24

514.04

31.55

44.76

31.93

30.64

230.25

365.93

62.16

40.48

030.43

7711

.11

2.66

11.9

.....

1.31

90.83

41.60

61.06

20.63

740.24

295.07

71.65

35.97

22.05

0.64

10.25

27.40

72.28

90.47

920.43

7313

.55

2.78

912

.0.....

1.67

10.89

022.03

61.13

60.63

420.24

076.29

71.74

97.39

22.16

50.63

940.24

889.12

72.41

10.47

780.43

4116

.42

2.91

512

.1.....

2.10

80.94

812.56

81.21

10.63

10.23

857.71

41.84

29.04

32.27

50.63

720.24

4911

.12

2.52

90.47

550.42

9919

.77

3.03

912

.2.....

2.64

81.00

73.22

61.28

90.62

780.23

639.42

31.93

511

.03

2.38

60.63

450.24

213

.52

2.64

70.47

270.42

7523

.68

3.16

12.3

.....

3.31

1.06

84.03

21.36

90.62

470.23

4211

.47

2.02

913

.42.49

80.63

170.23

9616

.37

2.76

50.46

950.42

6528

.21

3.27

812

.4.....

4.11

91.13

5.01

81.45

10.62

160.23

2113

.87

2.12

116

.18

2.60

80.62

850.23

7119

.72

2.88

20.46

70.42

4833

.49

3.39

312

.5.....

5.10

31.19

36.21

61.53

40.61

840.23

16.73

2.21

319

.48

2.71

70.62

520.23

5323

.68

2.99

80.46

460.42

3739

.62

3.50

612

.6.....

6.29

31.25

77.66

51.61

80.61

530.22

820

.12.30

423

.36

2.82

50.62

190.23

3928

.31

3.11

10.46

230.42

346

.72

3.61

612

.7.....

7.72

71.32

19.39

1.7

0.61

260.22

6724

.07

2.39

427

.94

2.93

30.61

90.23

2333

.71

3.22

20.46

0.42

2554

.92

3.72

412

.8.....

9.44

51.38

611

.45

1.78

30.61

0.22

5428

.68

2.48

333

.24

3.03

70.61

620.23

0639

.97

3.33

0.45

790.42

1464

.22

3.82

612

.9.....

11.49

1.45

113

.91

1.86

50.60

740.22

4133

.99

2.56

939

.33

3.13

80.61

340.22

8847

.15

3.43

40.45

560.41

9974

.79

3.92

413

.0.....

13.92

1.51

516

.81

1.94

70.60

480.22

2740

.05

2.65

46.26

3.23

40.61

050.22

6655

.31

3.53

40.45

310.41

7786

.85

4.01

713

.1.....

16.78

1.57

920

.21

2.02

70.60

240.22

1346

.88

2.72

854

.07

3.32

50.60

750.22

3864

.47

3.62

70.45

030.41

4810

0.3

4.10

413

.2.....

20.14

1.64

224

.22.10

60.59

990.21

9854

.49

2.79

962

.77

3.40

90.60

420.22

0774

.65

3.71

30.44

720.41

1411

5.3

4.18

313

.3.....

24.05

1.70

428

.85

2.18

30.59

730.21

8263

.08

2.86

672

.58

3.48

80.60

060.21

7886

.14

3.79

30.44

380.40

8713

1.9

4.25

613

.4.....

28.6

1.76

534

.23

2.25

80.59

470.21

6672

.75

2.92

883

.62

3.56

30.59

670.21

5299

.08

3.87

0.44

040.40

6415

0.3

4.32

213

.5.....

33.87

1.82

440

.45

2.33

10.59

210.21

5183

.59

2.98

695

.97

3.63

10.59

280.21

2911

3.6

3.94

0.43

70.40

4317

0.8

4.38

213

.6.....

39.94

1.88

147

.59

2.40

10.58

950.21

3695

.68

3.03

910

9.8

3.69

50.58

880.21

0612

9.7

4.00

60.43

380.40

2219

3.6

4.43

513

.7.....

46.92

1.93

755

.76

2.46

80.58

690.21

2310

9.1

3.08

812

5.1

3.75

40.58

470.20

8314

7.6

4.06

40.43

060.40

0121

8.7

4.48

313

.8.....

54.93

1.99

65.07

2.53

10.58

440.21

1412

43.13

214

23.80

70.58

070.20

6116

7.2

4.11

60.42

740.39

7924

6.6

4.52

513

.9.....

64.08

2.04

275

.65

2.59

0.58

20.21

0614

0.3

3.17

160.6

3.85

40.57

680.20

3818

8.8

4.16

20.42

420.39

5627

7.3

4.56

114

.0.....

74.51

2.09

187

.65

2.64

50.57

970.21

158.2

3.20

418

13.89

50.57

280.20

1521

2.5

4.20

20.42

090.39

3231

1.3

4.59

214

.1.....

86.37

2.13

810

1.2

2.69

70.57

750.20

9417

7.8

3.23

220

3.3

3.93

0.56

880.19

9223

8.4

4.23

60.41

750.39

0834

8.7

4.61

814

.2.....

99.8

2.18

311

6.5

2.74

50.57

550.20

8919

9.2

3.25

522

7.7

3.95

90.56

480.19

7126

6.8

4.26

50.41

420.38

8439

0.1

4.63

914

.3.....

115

2.22

613

3.8

2.78

90.57

350.20

8522

2.5

3.27

425

4.2

3.98

30.56

070.19

5129

7.9

4.28

80.41

090.38

6143

5.9

4.65

614

.4.....

132.1

2.26

615

3.1

2.82

90.57

160.20

8224

8.1

3.28

828

3.3

4.00

20.55

670.19

3233

2.1

4.30

70.40

760.38

4148

6.4

4.66

9

aMassin

Earth

mass(M

⊕¼

5:9742×10

24kg).

bRadiusin

Earth

radius

(R⊕¼

6:371

×10

6m).

cCRFstands

forcore

radius

fractio

n,theratio

oftheradius

ofthecore

over

thetotalradius

oftheplanet,in

thetwo-layermodel.

dp1=p0

stands

forcore–m

antle-boundarypressure

fractio

n,theratio

ofthepressure

atthecore–m

antle

boundary

(p1)

over

thepressure

atthecenter

oftheplanet

(p0),in

thetwo-layermodel.

000 ZENG & SASSELOV

2013 PASP, 125:000–000

The final part of this article calculates the three-layer differ-entiated exoplanet model. Given the mass and radius input, thesolution for the three-layer model is non-unique (degenerate),thus a curve on the ternary diagram is needed to representthe set of all solutions. This curve can be obtained by solvingdifferential equations with iterative methods. The ensemble ofsolutions is tabulated, from which users may interpolate to de-termine planet composition in three-layer model. A dynamicand interactive tool to characterize and illustrate the interiorstructure of exoplanets built upon ternary diagrams and othermodels in this article is available online.1

The methods described in this article can be used to rapidlycharacterize the interior structure of exoplanets.

2. METHOD

Spherical symmetry is assumed in all the models. The inte-rior of a planet is assumed to be fully differentiated into layers inthe first part of the paper. The two-layer model consists of a coreand a mantle. The three-layer model consists of a core, a mantle,and another layer on top of the mantle. The interior structure isdetermined by solving the following two differential equations:

dr

dm¼ 1

4πρr2; (1)

dp

dm¼ � Gm

4πr4: (2)

The two equations are similar to the ones in Zeng & Seager(2008). However, contrary to the common choice of radius r asthe independent variable, the interior mass m is chosen, whichis the total mass inside shell radius r, as the independent vari-able. So the solution is given as rðmÞ (interior radius r as adependent function of interior mass m), pðmÞ (pressure as adependent function of interior mass m), and ρðmÞ (densityas a dependent function of interior mass m).

The two differential equations are solved with the EOS of theconstituent materials as inputs:

ρ ¼ ρðp; T Þ: (3)

The EOS is a material property, which describes the materi-al’s density as a function of pressure and temperature. The EOScould be obtained both theoretically and experimentally. Theo-retically, the EOS could be calculated by quantum-mechanicalmolecular dynamics ab initio simulation such as the VASP(Vienna Ab initio Simulation Package) (Kresse & Hafner1993, 1994; Kresse & Furthmüller 1996; French et al. 2009).Experimentally, the EOS could be determined by high-pressurecompression experiment such as the DAC (diamond anvil cell)

experiment, or shock wave experiment like the implosion exper-iment by the Sandia Z-machine (Yu & Jacobsen 2011). The tem-perature effect on density is secondary compared to the pressureeffect (Valencia et al. 2006, 2007b). Therefore, we can safelyignore the temperature dependence of those higher density ma-terials (Fe and MgSiO3) for which the temperature effect isweaker, or we can implicitly include a preassumed pressure-temperature (p-T ) relation (for H2O it is the melting curve)so as to reduce the EOS to a simpler single-variable form:

ρ ¼ ρðpÞ: (4)

To solve the set of equations mentioned above, appropriateboundaries conditions are given as:

p0: the pressure at the center of the planetp1: the pressure at the first layer interface (the core–mantle

boundary)p2: the pressure at the second layer interface (only needed for

three-layer model)psurface: the pressure at the surface of the planet (set to

1 bar [105 Pa])

3. EOS OF Fe, MgSiO3 AND H2O

The three layers that we consider for the planet interior areFe, MgSiO3, and H2O. Their detailed EOS are described asfollows:

3.1. Fe

We model the core of a solid exoplanet after the Earth’s ironcore, except that in our model we ignore the presence of all otherelements such as nickel (Ni) and light elements such as sulfur(S) and oxygen (O) in the core. As pointed out by Valencia et al.(2010), above 100 GPa, the iron is mostly in the hexagonalclosed packed ϵ phase. Therefore, we use the Fe-EOS by Seageret al. (2007). Below 2:09 � 104 GPa, it is a Vinet (Vinet et al.1987, 1989) formula fit to the experimental data of ϵ-iron at p ≤330 GPa by Anderson et al. (2001). Above 2:09 � 104 GPa, itmakes smooth transition to the Thomas–Fermi–Dirac (TFD)EOS (Seager et al. 2007). A smooth transition is assumed be-cause there is no experimental data available in this ultrahighpressure regime.

The central pressure could reach 250 TPa (terapascals, i.e.,1012 Pa) in the most massive planet considered in this article.However, the EOS of Fe above ∼1 TPa is beyond the currentreach of experiment and thus largely unknown (Swift et al.2012). Therefore, our best approximation here is to extendthe currently available ϵ-iron EOS to higher pressures and con-nect it to the TFD EOS.

The EOS of Fe is shown in Figure 1 as the upper curve.1 See http://www.cfa.harvard.edu/~lzeng/.

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3.2. MgSiO3

We model the silicate layer of a solid exoplanet using theEarth’s mantle as a proxy. The FeO-free Earth’s mantle withMg=Si ¼ 1:07 would consist of mainly enstatite (MgSiO3) orits high-pressure polymorphs and, depending upon pressure,small amounts of either forsterite and its high-pressure poly-morphs (Mg2SiO4) or periclase (MgO) (e.g., Bina 2003).

The olivine polymorphs, as well as lower-pressure enstatiteandmajorite (MgSiO3with garnet structure), are not stable above27 GPa. At higher pressures, the system would consist ofMgSiO3-perovskite (pv) and periclase or their higher-pressurepolymorphs (Stixrude & Lithgow-Bertelloni 2011; Bina 2003).Given the high pressures at the H2O-silicate boundary usuallyexceeding 27 GPa, we can safely ignore olivine and lower-pressure pyroxene polymorphs. For the sake of simplicity, wealso ignore periclase, which would contribute only 7 at.% tothe silicatemantlemineralogical composition (Bina 2003). Thereare also small amounts of other elements such as aluminum (Al),calcium (Ca), and sodium (Na) present in Earth’s mantle (Sotinet al. 2007). For simplicity, we neglect them and thus the phases

containing them are not included in our model. We also do notconsider SiC because carbon-rich planets might form under veryrare circumstances, and are probably not common.

Some fraction of Fe can be incorporated into the minerals ofthe silicate mantle, which could then have the general formula asðMg; FeÞSiO3. For now, we simply assume all the Fe is in thecore and all the Mg is in the mantle in the form of MgSiO3-perovskite and/or its high-pressure polymorphs; thus the planetis fully differentiated. In a later section, we will discuss how theaddition of Fe to the mantle can affect mass–radius relation andcompare the differences between differentiated and undifferen-tiated as well as reduced and oxidized planets in § 4.3.

We first consider the perovskite (pv) and post-perovskite(ppv) phases of pure MgSiO3. MgSiO3 perovskite (pv) is be-lieved to be the major constituent of the Earth mantle. It makestransition into the post-perovskite (ppv) phase at roughly120 GPa and 2500 K (corresponding to a depth of 2600 kmin Earth) (Hirose 2010). The ppv phase was discovered experi-mentally in 2004 (Murakami et al. 2004) and was also theoreti-cally predicted in the same year (Oganov & Ono 2004). The ppvis about 1.5% denser than the pv phase (Caracas & Cohen 2008;Hirose 2010). This 1.5% density jump resulting from the pv-to-ppv phase transition can be clearly seen as the first density jumpof the MgSiO3 EOS curve shown in Figure 1. Both the MgSiO3

pv EOS and MgSiO3 ppv EOS are taken from Caracas & Cohen(2008). The transition pressure is determined to be 122 GPa forpure MgSiO3 according to Spera et al. (2006).

Beyond 0.90 TPa, MgSiO3 ppv undergoes a two-stage dis-sociation process predicted from the first-principle calculationsby Umemoto &Wentzcovitch (2011). MgSiO3 ppv first dissoci-ates into CsCl-type MgO and P21c-type MgSi2O5 at the pres-sure of 0.90 TPa, and later into CsCl-type MgO and Fe2 P-typeSiO2 at pressures higher than 2.10 TPa. The EOS of CsCl-typeMgO, P21c-type MgSi2O5, and Fe2P-type SiO2 are adoptedfrom Umemoto & Wentzcovitch (2011) and Wu et al. (2011).Therefore, there are two density jumps at the dissociation pres-sures of 0.90 and 2.10 TPa. The first one can be seen clearly inFigure 1. The second one cannot be seen in Figure 1 since it istoo small, but it surely exists.

Since Umemoto & Wentzcovitch’s (2011) EOS calculationapplies only up to 4.90 TPa, beyond 4.90 TPa, a modified ver-sion of the EOS by Seager et al. (2007) is used to smoothlyconnect to the TFD EOS. TFD EOS assumes electrons in aslowly varying potential with a density-dependent correlationenergy term that describes the interactions among electrons.It is therefore insensitive to any crystal structure or arrange-ments of atoms and it becomes asymptotically more accurateat higher pressure. Thus, the TFD EOS of MgSiO3 would beno different from the TFD EOS of MgO plus SiO2 as longas the types and numbers of atoms in the calculation are thesame. So it is safe to use the TFD EOS of MgSiO3 as an ap-proximation of the EOS of MgO and SiO2 mixture beyond4.90 TPa here.

FIG. 1.—p� ρ EOS of Fe (ϵ-phase of iron, red curve), MgSiO3 (perovskitephase, post-perovskite phase and its high-pressure derivatives, orange curve),and H2O (Ice Ih, Ice III, Ice V, Ice VI, Ice VII, Ice X, and superionic phasealong its melting curve, i.e., solid–liquid phase boundary) See the online editionof the PASP for a color version of this figure.

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Seager et al.’s (2007) EOS is calculated from the method ofSalpeter & Zapolsky (1967) above 1:35 � 104 GPa. Below1:35 � 104 GPa, it is a smooth connection to TFD EOS fromthe fourth-order Birch-Murbaghan Equation of State (BME)(see Birch 1947; Poirier 2000) fit to the parameters ofMgSiO3 pv obtained by ab initio lattice dynamics simulationof Karki et al. (2000).

Seager et al.’s EOS is slightly modified to avoid any artificialdensity jump when connected with Umemoto &Wentzcovitch’sEOS at 4.90 TPa. At 4.90 TPa, the ratio of the density ρ be-tween Umemoto &Wentzcovitch’s EOS and Seager et al.’s EOSis 1.04437. This ratio is multiplied to the original Seager et al.’sEOS density ρ to produce the actual EOS used in our calcula-tion for p > 4:90 TPa. A smooth transition is assumed becauseno experimental data is available in this ultrahigh-pressure re-gime. This assumption does not affect our low or medium massplanet models, since only the most massive planets in our modelcould reach this ultrahigh pressure in their MgSiO3 part.

The EOS of MgSiO3 is shown in Figure 1 as the mid-dle curve.

3.3. H2O

The top layer of a planet could consist of various phases ofH2O. Since H2O has a complex phase diagram, and it alsohas a stronger temperature dependence, the temperature effectcannot be ignored. Instead, we follow the solid phases alongthe melting curve (solid–liquid phase boundary on the p-T plotbyChaplin [2012]). Along themelting curve, theH2Oundergoesseveral phase transitions. Initially, it is Ice Ih at low pressure, thensubsequently transforms into Ice III, Ice V, Ice VI, Ice VII, Ice X,and superionic phase (Chaplin 2012; Choukroun & Grasset2007; Dunaeva et al. 2010; French et al. 2009).

3.3.1. Chaplin’s EOS

The solid form of water has very complex phases in thelow-pressure and low-temperature regime. These phases arewell determined by experiments. Here we adopt the Chaplin’sEOS for Ice Ih, Ice III, Ice V, and Ice VI below 2.216 GPa (seeChaplin 2012; Choukroun & Grasset 2007; Dunaeva et al.2010). Along the melting curve (the solid–liquid boundary onthe p-T diagram), the solid form of water first exists as IceIh (hexagonal ice) from ambient pressure up to 209.5 MPa(Choukroun & Grasset 2007). At the triple point of 209.5 MPaand 251.15 K (Choukroun & Grasset 2007; Robinson et al.1996), it transforms into Ice III (Ice-three), whose unit cellforms tetragonal crystals. Ice III exists up to 355.0 MPa andtransforms into a higher-pressure form Ice V (Ice-five) at thetriple point of 355.0 MPa and 256.43 K (Choukroun & Grasset2007). Ice V’s unit cell forms monoclinic crystals. At the triplepoint of 618.4 MPa and 272.73 K (Choukroun & Grasset 2007),Ice V transforms into yet another higher-pressure form Ice VI(Ice-six). Ice VI’s unit cell forms tetragonal crystals. A single

molecule in Ice VI crystal is bonded to four other water mole-cules. Then at the triple point of 2.216 GPa and 355 K (Daucik& Dooley 2011), Ice VI transforms into Ice VII (Ice-seven). IceVII has a cubic crystal structure. Ice VII eventually transformsinto Ice X (Ice-ten) at the triple point of 47 GPa and 1000 K(Goncharov et al. 2005). In Ice X, the protons are equally spacedand bonded between the oxygen atoms, where the oxygen atomsare in a body-centered cubic lattice (Hirsch & Holzapfel 1984).The EOS of Ice X and Ice VII are very similar. For Ice VII(above 2.216 GPa), we switch to Frank, Fei, & Hu’s EOS (Franket al. [2004], hereafter FFH2004).

3.3.2. FFH2004’s EOS

We adopt FFH2004’s EOS of Ice VII for 2:216 GPa ≤p ≤ 37:4 GPa. This EOS is obtained using the Mao-Bell typediamond anvil cell with an external Mo-wire resistance heater.Gold and water are put into the sample chamber and com-pressed. The diffraction pattern of both H2O and gold are mea-sured by the energy-dispersive X-ray diffraction (EDXD)technique at the Brookhaven National Synchrotron LightSource. The gold here is used as an internal pressure calibrant.The disappearance of the diffraction pattern of the crystal IceVII is used as the indicator for the solid–liquid boundary (melt-ing curve). The melting curve for Ice VII is determined accu-rately from 3 to 60 GPa and fit to a Simon equation. The molardensity ( ρ in mol cm�3) of Ice VII as a function of pressure (pin GPa) is given by the following formula in FFH2004:

ρ ðmol cm�3Þ ¼ 0:0805þ 0:0229 � ð1� exp�0:0743�pÞþ 0:1573 � ð1� exp�0:0061�pÞ: (5)

We use equation (5) to calculate ρ from 2.216 GPa up to37.4 GPa. The upper limit 37.4 GPa is determined by the inter-section between FFH2004’s EOS and French et al.’s EOS(French et al. [2009], hereafter FMNR2009).

3.3.3. FMNR2009’s EOS

FMNR2009’s EOS is used for Ice VII, Ice X, and superionicphase of H2O for 37:4 GPa ≤ p ≤ 8:893 TPa. This EOS is de-termined by quantum-molecular dynamics simulations usingthe Vienna Ab Initio Simulation Package (VASP). The simula-tion is based on finite temperature density-functional theory(DFT) for the electronic structure and treating the ions as clas-sical particles. Most of French et al.’s simulations consider54 H2O molecules in a canonical ensemble, with the standardVASP PAW potentials, the 900 eV plane-wave cutoff, and theevaluation of the electronic states at the Γ point considered, forthe three independent variables: temperature (T ), volume (V ),and particle number (N). The simulation results are the thermalEOS pðT; V ;NÞ, and the caloric EOS UðT; V ;NÞ. The data aretabulated in FMNR2009.

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In order to approximate density ρ along the melting curve,from Table V in FMNR2009, we take one data point from IceVII phase at 1000 K and 2:5 g cm�3, four data points from Ice Xphase at 2000 K and 3:25 g cm�3 to 4:00 g cm�3, and the restof the data points from the superionic phase at 4000 K and5:00 g cm�3 up to 15 g cm�3. Since the temperature effecton density becomes smaller towards higher pressure, all the iso-thermal pressure–density curves converge on the isentropicpressure–density curves as well as the pressure–density curvealong the melting curve.

The FMNR2009’s EOS has been confirmed experimentallyby Thomas Mattson et al. at the Sandia National Laboratories.At 8.893 TPa, FMNR2009’s EOS is switched to the TFD EOSin Seager et al. (2007) (hereafter SKHMM2007).

3.3.4. SKHMM2007’s EOS

At ultrahigh pressure, the effect of electron–electron interac-tion can be safely ignored and electrons can be treated as a gasof noninteracting particles that obey the Pauli exclusion princi-ple subject to the Coulomb field of the nuclei. Assuming theCoulomb potential is spatially slowly varying throughout theelectron gas that the electronic wave functions can be approxi-mated locally as plane waves, the so-called TFD solution couldbe derived so that the Pauli exclusion pressure balances out theCoulomb forces (Eliezer et al. 2002; Macfarlane 1984).

In SKHMM2007, a modified TFD by Salpeter & Zapolsky(1967) is used. It is modified in the sense that we have added ina density-dependent correlation energy term which charac-terizes electron interaction effects.

Here, Seager et al.’s EOS is slightly modified to connect tothe FMNR2009’s EOS. At 8.893 TPa, the ratio of the density ρbetween the FMNR2009’s EOS and Seager et al.’s EOS is1.04464. This ratio is multiplied to the original Seager et al.’sEOS density ρ to produce the actual EOS used in our calcula-tion for p > 8:893 TPa. Only the most massive planets in ourmodel could reach this pressure in the H2O layer, so this choiceof EOS for p > 8:893 TPa has small effect on the overall mass–radius relation to be discussed in the next section.

The EOS of H2O is shown in Figure 1 as the lower curve.

4. RESULTS

4.1. Mass–Radius Contours

Given mass and radius input, various sets of mass–radiuscontours can be used to quickly determine the characteristic in-terior structure quantities of a two-layer planet including its p0(central pressure), p1=p0 (ratio of core–mantle boundary pres-sure over central pressure), CMF (core mass fraction), and CRF(core radius fraction).

The two-layer model is uniquely solved and represented as apoint on the mass–radius diagram given any pair of two param-eters from the following list: M (mass), R (radius), p0, p1=p0,CMF, CRF, etc. The contours of constant M or R are trivial on

the mass–radius diagram, which are simply vertical or horizon-tal lines. The contours of constant p0, p1=p0, CMF, or CRF aremore useful.

Within a pair of parameters, fixing one and continuouslyvarying the other, the point on the mass–radius diagram movesto form a curve. Multiple values of the fixed parameter give mul-tiple parallel curves forming a set of contours. The other set ofcontours can be obtained by exchanging the fixed parameter forthe varying parameter. The two sets of contours crisscross eachother to form a mesh, which is a natural coordinate system (seeFig. 2) of this pair of parameters, superimposed onto the exist-ing Cartesian ðM;RÞ coordinates of the mass–radius diagram.This mesh can be used to determine the two parameters giventhe mass and radius input or vice versa.

4.1.1. Fe-MgSiO3 Planet

As an example, the mesh of p0 with p1=p0 for an Fe-coreMgSiO3-mantle planet is illustrated in the first subplot (upperleft) of Figure 2. The mesh is formed by p0-contours and p1=p0-contours crisscrossing each other. The more vertical set ofcurves represents the p0-contours. The ratio of adjacent p0-contours is 100:1 (see Table 1). The more horizontal set of curvesrepresents the p1=p0-contours. From bottom up, the p1=p0values vary from 0 to 1 with step size 0.1.

Given a pair of p0 and p1 input, users may interpolatefrom the mesh to find the mass and radius. On the other hand,given the mass and radius of a planet, users may also interpolatefrom the mesh to find the corresponding p0 and p1 of a two-layer, Fe-core MgSiO3-mantle planet.

Similarly, the contour mesh of p0 with CMF for the Fe-MgSiO3 planet is shown as the middle panel in the top rowof Figure 2. As a reference point, for a pure-Fe planet withp0 ¼ 1011 Pa, M ¼ 0:1254 M⊕, R ¼ 0:417 R⊕.

The contour mesh of p0 with CRF for the Fe-MgSiO3 planetis shown as the right panel of the top row of Figure 2.

4.1.2. MgSiO3-H2O Planet

For two-layer MgSiO3-H2O planet, the three diagrams (p0contours pair with p1=p0 contours, CMF contours, or CRF con-tours) are the subplots of the middle row of Figure 2.

As a reference point, for a pure-MgSiO3 planet withp0 ¼ 1010:5 Pa, M ¼ 0:122 M⊕, R ¼ 0:5396 R⊕.

4.1.3. Fe-H2O Planet

For two-layer Fe-H2O planet, the three diagrams (p0 con-tours pair with p1=p0 contours, CMF contours, or CRF con-tours) are the subplots of the bottom row of Figure 2.

As a reference point, for a pure-Fe planet with p0 ¼ 1011 Pa,M ¼ 0:1254 M⊕, R ¼ 0:417 R⊕.

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Fe H2O planetp0 contour stepsize: 0.1 in log10 p0

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.1 0.2 0.5 1 2 5 10 20 50M Earth Mass

MgSiO3 H2O planetp0 contour stepsize: 0.1 in log10 p0CMF contour stepsize: 0.1 from 1 to 0

Fe MgSiO3 planetp0 contour stepsize: 0.1 in log10 p0CMF contour stepsize: 0.1 from 1 to 0

.1 0.2 0.5 1 2 5 10 20 50

Fe H2O planetp0 contour stepsize: 0.1 in log10 p0CRF contour stepsize: 0.1 from 1 to 0

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FIG. 2.—Mass–radius contours of two-layer planet. Top: Fe-MgSiO3 planet.Middle row: MgSiO3-H2O planet. Bottom: Fe-H2O planet. Left: contour mesh of p1=p0with p0.Middle column: contour mesh of CMF with p0. Right: contour mesh of CRF with p0. To find out what p0 value each p0-contour corresponds to, please refer toTable 1. See the online edition of the PASP for a color version of this figure.

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4.2. Mass–Radius Curves

For observers’ interest, six characteristic mass–radius curvesare plotted (Fig. 3) and tabulated (Table 1), representing the pure-Fe planet, half-Fe half-MgSiO3 planet, pureMgSiO3 planet, half-MgSiO3 half-H2O planet, 75%H2O-25%MgSiO3 planet, andpureH2Oplanet. These fractions aremass fractions. Figure 3 alsoshows some recently discovered exoplanets within the relevantmass–radius regime for comparison. These planets includeKepler-10b (Batalha et al. 2011), Kepler-11b (Lissauer et al.2011), Kepler-11f (Lissauer et al. 2011), Kepler-18b (Cochranet al. 2011), Kepler-36b (Carter et al. 2012), and Kepler-20b, c, d (Gautier et al. 2012). They also include Kepler-20e(R ¼ 0:868þ0:074

�0:096 R⊕ [Fressin et al. 2012], the mass range is de-termined by pure-silicate mass–radius curve and the maximumcollisional stripping curve [Marcus et al. 2010]), Kepler-20f (R ¼ 1:034þ0:100

�0:127 R⊕ [Fressin et al. 2012], the mass rangeis determined by 75% water-ice and 25% silicate mass–radiuscurves and the maximum collisional stripping curve [Marcuset al. 2010]), Kepler-21b (R ¼ 1:64� 0:04 R⊕ [Howell et al.

2012]; the upper limit for mass is 10:4 M⊕: the 2- σ upper limitpreferred in the paper; the lower limit is 4 M⊕, which is inbetween the “Earth” and “50%H2O-50%MgSiO3” modelcurves—the planet is very hot and is unlikely to have muchwater content, if any at all.), Kepler-22b (R ¼ 2:38� 0:13 R⊕[Borucki et al. 2012]; the 1- σ upper limit for mass is 36 M⊕ foran eccentric orbit, or 27 M⊕, for circular orbit), CoRoT-7b(M ¼ 7:42� 1:21 M⊕, R ¼ 1:58� 0:1 R⊕ [Hatzes et al.2011; Leger et al. 2009; Queloz et al. 2009]), 55 Cancri e(M ¼ 8:63� 0:35 M⊕, R ¼ 2:00� 0:14 R⊕ [Winn et al.2011]), and GJ 1214b (Charbonneau et al. 2009).

4.3. Levels of Planet Differentiation: The Effect of FePartitioning Between Mantle and Core

All models of planets discussed so far assume that all Fe is inthe core, while all Mg, Si and O are in the mantle, i.e., that aplanet is fully differentiated. However, we know that in terres-trial planets some Fe is incorporated into the mantle. There aretwo separate processes which affect the Fe content of the man-tle: (1) mechanical segregation of Fe-rich metal from the mantleto the core, and (2) different redox conditions resulting in a dif-ferent Fe/Mg ratio within the mantle, which in turn affects therelative size of the core and mantle. In this section we show theeffects of (1) undifferentiated versus fully-differentiated, and(2) versus oxidized planetary structure on the mass–radius re-lation for a planet with the same Fe/Si and Mg/Si ratios.

For simplicity, herewe ignore the H2O and gaseous content ofthe planet and only consider the planet as made of Fe, Mg, Si, O.To facilitate comparison between different cases, we fix theglobal atomic ratios of Fe=Mg ¼ 1 and Mg=Si ¼ 1; these fitwell within the range of local stellar abundances (Grassetet al. 2009).

In particular, we consider a double-layer planet with a coreand a mantle in two scenarios. One follows the incomplete me-chanical separation of the Fe-rich metal during planet formation,and results in addition of Fe to the mantle as metal particles. Itdoes not change EOS of the silicate components, but requiresadding an Fe-EOS to the mantle mixture. Thus, the planet gen-erally consists of a Fe metal core and a partially differentiatedmantle consisting of the mixture of metallic Fe and MgSiO3

silicates. While the distribution of metallic Fe may have a radialgradient, for simplicity we assume that it is uniformly distrib-uted in the silicate mantle. Within scenario 1, we calculate threecases to represent different levels of differentiation:

Case 1: complete differentiation—Metallic Fe core andMgSiO3 silicate mantle. For Fe=Mg ¼ 1, CMF ¼ 0:3574.

Case 2: partial differentiation—Half the Fe forms a smallermetallic Fe core, with the other half of metal being mixed withMgSiO3 silicates in the mantle. CMF ¼ 0:1787.

Case 3: no differentiation—All the metallic Fe is mixed withMgSiO3 in the mantle. CMF ¼ 0 (no core).

The other scenario assumes different redox conditions,resulting in different Fe/Mg ratios in mantle minerals, and

FIG. 3.—Currently known transiting exoplanets are shown with their mea-sured mass and radius with observation uncertainties. Earth and Venus areshown for comparison. The curves are calculated for planets composed of pureFe, 50% Fe-50% MgSiO3, pure MgSiO3, 50% H2O-50% MgSiO3, 75% H2O-25%MgSiO3, and pure H2O. The red dashed curve is the maximum collisionalstripping curve calculated by Marcus et al. (2010). See the online edition of thePASP for a color version of this figure.

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therefore requiring different EOS for the Fe2þ-bearing silicatesand oxides. More oxidized mantle means adding more Fe in theform of FeO to the MgSiO3 silicates to form ðMg; FeÞSiO3 sil-icates and (Mg,Fe)O magnesiowüstite (mv), thus reducing theamount of Fe in the core. The exact amounts of ðMg; FeÞSiO3

and (Mg,Fe)O in the mantle are determined by the followingmass balance equation:

xFeOþMgSiO3 → ðMg 11þx; Fe x

1þxÞSiO3 þ xðMg 1

1þx; Fe 1

1þxÞO;(6)

wherex denotes the relative amount of FeO added to the silicatemantle, x ¼ 0 being the most reduced state with no Fe in themantle, and x ¼ 1 being the most oxidized state with all Fe ex-isting as oxides in the mantle. This oxidization process con-serves the global Fe/Mg and Mg/Si ratios, but increases Ocontent and thus the O/Si ratio of the planet since Fe is addedto the mantle in the form of FeO. Because in stellar environ-ments O is excessively abundant relative to Mg, Si, and Fe(e.g., the solar elemental abundances [Asplund et al. 2009]),it is not a limiting factor in our models of oxidized planets.We calculate the following three cases to represent the full rangeof redox conditions:

Case 4: no oxidization of Fe—x ¼ 0. Metal Fe core andMgSiO3 silicate mantle. For Fe=Mg ¼ 1, O=Si ¼ 3,CMF ¼ 0:3574.

Case 5: partial oxidization of Fe—x ¼ 0:5. Half the Fe formssmaller metal core, the other half is added as FeO to the mantle.O=Si ¼ 3:5, CMF ¼ 0:1700.

Case 6: complete oxidization of Fe—x ¼ 1. All Fe is addedas FeO to the mantle, resulting in no metal core at all.O=Si ¼ 4, CMF ¼ 0.

Notice that Case 4 looks identical to Case 1. However, thesilicate EOS used to calculate Case 4 is different from Case 1 atultrahigh pressures (beyond 0.90 TPa). For Cases 4, 5, and 6,the ðMg; FeÞSiO3 EOS is adopted from Caracas & Cohen(2008) and Spera et al. (2006) which only consider perovskite(pv) and post-perovskite (ppv) phases without including furtherdissociation beyond 0.90 TPa, since the Fe2þ-bearing silicateEOS at ultrahigh pressures is hardly available. On the otherhand, comparison between Case 1 and Case 4 also showsthe uncertainty on mass–radius relation resulting from the dif-ferent choice of EOS (see Table 2). Fe2þ-bearing pv and ppvhave the general formula: ðMgy; Fe1�yÞSiO3, where y denotesthe relative atomic number fraction of Mg and Fe in the silicatemineral. The ðMgy; Fe1�yÞSiO3 silicate is therefore a binarycomponent equilibrium solid solution. It could either be pvor ppv or both depending on the pressure (Spera et al.2006). We can safely approximate the narrow pressure regionwhere pv and ppv co-exist as a single transition pressure frompv to ppv. This pressure is calculated as the arithmetic mean ofthe initial transition pressure of pv → pvþ ppv mixture and thefinal transition pressure of pvþ ppv mixture → ppv (Spera

et al. 2006). The pv EOS and ppv EOS are connected at thistransition pressure to form a consistent EOS for all pressures.

Addition of FeO to MgSiO3 results in the appearance of asecond phase, magnesiowüstite, in the mantle according toequation 6. The (Mg,Fe)O EOS for Cases 4, 5, and 6 is adoptedfrom Fei et al. (2007), which includes the electronic spin tran-sition of high-spin to low-spin in Fe2þ. For simplicity, we as-sume that pv/ppv and mw have the same Mg/Fe ratio.

Figure 4 shows fractional differences (η) in radius of Cases 2and 3 compared to Case 1 as well as Cases 5 and 6 compared toCase 4 (r0 is radius of the reference case, which is that of Case 1for Cases 2 and 3 and is that of Case 4 for Cases 5 and 6):

η ¼ r� r0r0

: (7)

Oxidization of Fe (partitioning Fe as Fe-oxides from the coreinto the mantle) makes the planet appear larger. The completeoxidization of Fe makes the radius 3% larger for small planetsaround 1 M⊕, then the difference decreases with increasingmass within the mass range of 1 to 20 M⊕. Undifferentiatedplanets (partitioning of metallic Fe from the core into the man-tle) appear smaller than fully differentiated planets. The com-pletely undifferentiated planet is practically indistinguishablein radius for small planets around 1 M⊕, then the differenceincreases to 1%-level around 20 M⊕. The mass, radius, CRFand p1=p0 data of Cases 1 through 6 are listed in Table 2.

4.4. Tabulating the Ternary Diagram

For the three-layer model of solid exoplanet, points of acurve segment on the ternary diagram represent all the solutionsfor a given mass–radius input. These ternary diagrams are tabu-lated (Table 3) with the intent to make comparison to observa-tions easier.

Usually, there are infinite combinations (solutions) of Fe,MgSiO3, and H2O mass fractions which can give the samemass–radius pair. All the combinations together form a curvesegment on the ternary diagram of Fe, MgSiO3, and H2O massfractions (Zeng & Seager 2008; Valencia et al. 2007a). Thiscurve segment can be approximated by three points on it:two endpoints where one or more out of the three layers areabsent and one point in between where all three layers are pres-ent to give the same mass and radius. The two endpoints corre-spond to the minimum central pressure (p0min) and maximumcentral pressure (p0max) allowed for the given mass–radius pair.The middle point is chosen to have the central pres-sure p0mid ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p0max � p0minp

.Table 3 contains eight columns:

1. Mass. The masses range from 0:1 to 100 M⊕ with 41points in total. The range between 0.1 and 1 M⊕ is equally di-vided into 10 sections in logarithmic scale. The range between 1and 10 M⊕ is equally divided into 20 sections in logarithmic

MODEL GRID FOR SOLID PLANETS 000

2013 PASP, 125:000–000

TABLE2

DATA

OFCASES1–6

log 1

0ðp0Þ

Case1(CMF¼

0:357

4)Case2(CMF¼

0:178

7)Case3

Case4(CMF¼

0:357

4)

Case5(CMF¼

0:1700)

Case6

Massa

Radiusb

CRFc

p1=p0d

Mass

Radius

CRF

p1=p0

Mass

Radius

Mass

Radius

CRF

p1=p0

Mass

Radius

CRF

p1=p0

Mass

Radius

9.6

......

0.00

2009

0.13

020.59

720.38

470.00

2395

0.13

810.47

360.56

340.00

4165

0.16

590.00

2009

0.13

020.59

720.38

470.00

2526

0.14

280.45

840.56

250.00

4921

0.18

049.7

......

0.00

2816

0.14

560.59

670.38

390.00

3359

0.15

450.47

310.56

260.00

5846

0.18

560.00

2816

0.14

560.59

670.38

390.00

3542

0.15

970.45

790.56

170.00

6905

0.20

189.8

......

0.00

3941

0.16

280.59

610.38

30.00

4704

0.17

270.47

260.56

160.00

8194

0.20

760.00

3941

0.16

280.59

610.38

30.00

4959

0.17

860.45

740.56

070.00

9673

0.22

569.9

......

0.00

5504

0.18

190.59

540.38

20.00

6573

0.19

290.47

190.56

040.01

146

0.23

20.00

5504

0.18

190.59

540.38

20.00

693

0.19

950.45

670.55

960.01

352

0.25

210

......

0.00

767

0.20

30.59

460.38

070.00

9165

0.21

540.47

110.55

910.01

60.25

890.00

767

0.20

30.59

460.38

070.00

9662

0.22

270.45

60.55

820.01

887

0.28

1210

.1.....

0.01

066

0.22

630.59

360.37

920.01

275

0.24

010.47

020.55

750.02

228

0.28

870.01

066

0.22

630.59

360.37

920.01

344

0.24

840.45

50.55

670.02

625

0.31

3510

.2.....

0.01

477

0.25

20.59

240.37

740.01

767

0.26

740.46

90.55

570.03

093

0.32

150.01

477

0.25

20.59

240.37

740.01

863

0.27

660.45

40.55

480.03

639

0.34

8910

.3.....

0.02

039

0.28

020.59

10.37

540.02

442

0.29

740.46

780.55

360.04

278

0.35

750.02

039

0.28

020.59

10.37

540.02

574

0.30

760.45

270.55

280.05

027

0.38

7810

.4.....

0.02

805

0.31

10.58

940.37

320.03

362

0.33

030.46

630.55

130.05

893

0.39

680.02

805

0.31

10.58

940.37

320.03

544

0.34

160.45

130.55

040.06

915

0.43

0110

.5.....

0.03

842

0.34

470.58

760.37

060.04

610.36

610.46

470.54

870.08

081

0.43

950.03

842

0.34

470.58

760.37

060.04

859

0.37

860.44

970.54

790.09

465

0.47

610

.6.....

0.05

239

0.38

140.58

560.36

790.06

292

0.40

50.46

280.54

590.11

020.48

580.05

239

0.38

140.58

560.36

790.06

631

0.41

890.44

80.54

510.12

890.52

5710

.7.....

0.07

111

0.42

110.58

330.36

490.08

548

0.44

720.46

080.54

290.14

960.53

560.07

111

0.42

110.58

330.36

490.09

006

0.46

240.44

60.54

220.17

440.57

8910

.8.....

0.09

605

0.46

40.58

090.36

170.11

550.49

270.45

870.53

980.20

170.58

880.09

605

0.46

40.58

090.36

170.12

170.50

940.44

390.53

920.23

450.6358

10.9

.....

0.12

910.51

010.57

820.35

840.15

530.54

160.45

640.53

660.27

020.64

560.12

910.51

010.57

820.35

840.16

360.55

990.44

170.53

610.31

130.6946

11......

0.17

250.55

950.57

530.35

50.20

770.59

390.45

40.53

340.35

960.70

560.17

250.55

950.57

530.35

50.21

870.61

390.43

940.53

30.40

80.75

4811

.1.....

0.22

940.61

230.57

230.35

150.27

620.64

950.45

150.53

020.47

440.76

840.22

940.61

230.57

230.35

150.29

030.67

090.43

710.53

020.53

150.8177

11.2

.....

0.30

330.66

840.56

920.34

80.36

520.70

850.44

890.52

710.61

780.83

230.30

330.66

840.56

920.34

80.38

330.73

130.43

470.52

750.69

050.88

411

.3.....

0.39

90.72

780.56

60.34

450.48

0.77

080.44

630.52

410.80

180.89

960.39

90.72

780.56

60.34

450.50

240.79

450.43

230.52

550.89

320.95

3611

.4.....

0.52

190.79

060.56

270.34

120.62

660.83

580.44

370.52

161.03

60.96

990.52

190.79

060.56

270.34

120.65

420.86

040.43

0.52

371.15

1.02

611

.5.....

0.67

90.85

650.55

940.33

790.81

180.90

310.44

120.51

991.33

11.04

30.67

90.85

650.55

940.33

790.84

780.92

940.42

780.52

191.47

21.10

111

.6.....

0.87

790.92

510.55

610.33

521.04

70.97

330.43

880.51

811.70

21.11

90.87

790.92

510.55

610.33

521.09

31.00

10.42

550.52

011.87

41.17

811

.7.....

1.12

80.99

60.55

30.33

31.34

31.04

60.43

650.51

632.16

61.19

71.12

80.99

60.55

30.33

31.40

21.07

60.42

330.51

842.37

31.25

811

.8.....

1.44

31.07

0.55

0.33

071.71

51.12

20.43

410.51

452.74

11.27

71.44

31.07

0.55

0.33

071.78

91.15

40.42

110.51

672.99

1.34

11.9

.....

1.83

71.14

60.54

70.32

842.17

91.2

0.43

180.51

273.45

31.36

1.83

71.14

60.54

70.32

842.27

21.23

30.41

90.51

523.75

1.42

312

......

2.32

71.22

50.54

410.32

622.75

51.28

10.42

950.51

14.30

31.44

22.32

71.22

50.54

410.32

622.87

11.31

60.41

680.51

374.68

1.50

912

.1.....

2.93

41.30

60.54

120.32

43.46

81.36

40.42

730.50

945.30

61.52

22.93

41.30

60.54

120.32

43.61

1.4

0.41

470.51

235.81

81.59

612

.2.....

3.68

21.39

0.53

830.32

194.34

41.44

80.42

50.50

776.52

21.60

33.68

21.39

0.53

830.32

194.51

91.48

60.41

260.51

097.20

31.68

412

.3.....

4.6

1.47

50.53

550.31

975.40

31.53

30.42

290.50

687.98

61.68

54.6

1.47

50.53

550.31

975.63

1.57

40.41

050.50

958.88

61.77

512

.4.....

5.72

1.56

20.53

270.31

766.67

11.61

70.42

080.50

689.72

41.76

65.72

1.56

20.53

270.31

766.98

31.66

40.40

850.50

8110

.93

1.86

612

.5.....

7.07

11.64

90.53

0.31

618.20

21.70

20.41

890.50

6711

.81.84

87.08

21.65

10.52

990.31

558.62

61.75

40.40

640.50

6513

.39

1.96

12.6

.....

8.68

61.73

60.52

750.31

5210

.04

1.78

70.41

70.50

6514

.27

1.92

98.72

91.74

0.52

70.31

3310

.61

1.84

70.40

430.50

4916

.37

2.05

412

.7.....

10.62

1.82

20.52

510.31

4312

.23

1.87

10.41

520.50

6417

.18

2.01

10.71

1.83

10.52

420.31

113

1.94

0.40

210.50

319

.96

2.15

12.8

.....

12.93

1.90

80.52

280.31

3414

.83

1.95

50.41

340.50

6320

.62.09

13.09

1.92

30.52

130.30

8615

.86

2.03

40.39

990.50

0924

.29

2.24

812

.9.....

15.67

1.99

30.52

060.31

2517

.92.03

80.41

160.50

5924

.57

2.16

815

.93

2.01

50.51

830.30

6119

.29

2.12

90.39

760.49

8529

.48

2.34

713

......

18.9

2.07

80.51

830.31

1621

.53

2.12

0.40

980.50

5429

.15

2.24

419

.32.10

80.51

530.30

3323

.36

2.22

40.39

520.49

5835

.73

2.44

813

.1.....

22.69

2.16

10.51

610.31

0425

.76

2.20

10.40

790.50

4934

.37

2.31

623

.28

2.2

0.51

210.30

0428

.19

2.32

0.39

270.49

2843

.22

2.55

113

.2.....

27.12

2.24

20.51

390.30

9230

.67

2.27

90.40

60.50

4440

.27

2.38

427

.97

2.29

30.50

890.29

7233

.92.41

70.39

010.48

9352

.21

2.65

613

.3.....

32.27

2.32

20.51

160.30

7836

.33

2.35

40.40

40.50

3946

.99

2.44

933

.46

2.38

50.50

540.29

3740

.62

2.51

30.38

730.48

5362

.98

2.76

213

.4.....

38.2

2.39

90.50

930.30

6642

.82.42

60.40

20.50

3854

.61

2.51

139

.88

2.47

70.50

180.29

48.52

2.61

0.38

440.48

175

.88

2.87

13.5

.....

45.05

2.47

20.50

690.30

5650

.18

2.49

40.40

010.50

4263

.25

2.56

947

.37

2.56

90.49

810.28

6257

.81

2.70

80.38

130.47

6591

.32

2.98

113

.6.....

52.85

2.54

20.50

460.30

4858

.56

2.55

70.39

820.50

4572

.97

2.62

456

.06

2.66

0.49

420.28

2168

.68

2.80

60.37

820.47

1410

9.8

3.09

313

.7.....

61.72

2.60

80.50

240.30

4468

.09

2.61

80.39

630.50

4983

.87

2.67

566

.14

2.75

20.49

010.27

7981

.38

2.90

50.37

50.46

613

1.9

3.20

813

.8.....

71.81

2.67

0.50

020.30

4278

.88

2.67

40.39

450.50

5596

.04

2.72

277

.81

2.84

20.48

590.27

3596

.23.00

40.37

160.46

0415

8.4

3.32

613

.9.....

83.26

2.72

80.49

820.30

4191

.07

2.72

70.39

280.50

6210

9.6

2.76

591

.29

2.93

30.48

160.26

8911

3.5

3.10

40.36

820.45

4419

03.44

614

......

96.2

2.78

10.49

620.30

4210

4.8

2.77

50.39

110.50

712

4.6

2.80

410

6.8

3.02

40.47

720.26

4213

3.6

3.20

40.36

470.44

8122

7.8

3.56

814

.1.....

110.8

2.83

10.49

440.30

4312

0.1

2.81

90.38

950.50

7914

1.3

2.83

912

4.7

3.11

40.47

270.25

9315

73.30

60.36

120.44

1527

33.69

414

.2.....

127.2

2.87

60.49

260.30

4513

7.3

2.85

90.38

80.50

915

9.6

2.87

145.2

3.20

50.46

80.25

4218

4.1

3.40

80.35

750.43

4532

7.1

3.82

314

.3.....

145.5

2.91

70.49

10.30

4915

6.4

2.89

40.38

650.51

0217

9.9

2.89

716

8.7

3.29

50.46

330.24

8921

5.5

3.51

10.35

380.42

7139

1.8

3.95

514

.4.....

166

2.95

40.48

940.30

5417

7.5

2.92

50.38

510.51

1520

2.3

2.92

195.5

3.38

60.45

840.24

3525

1.8

3.61

50.34

990.41

9546

9.1

4.09

1

aMassin

Earth

mass(M

⊕¼

5:9742

×102

4kg).

bRadiusin

Earth

radius

(R⊕¼

6:371×106

m).

cCRFstands

forcore

radius

fractio

n,theratio

oftheradius

ofthecore

over

thetotalradius

oftheplanet,in

thetwo-layermodel.

dp1=p0

stands

forcore–m

antle-boundarypressure

fractio

n,theratio

ofthepressure

atthecore–m

antle

boundary

(p1)

over

thepressure

atthecenter

oftheplanet

(p0),in

thetwo-layermodel.

000 ZENG & SASSELOV

2013 PASP, 125:000–000

scale. And the range between 10 and 100 M⊕ is equally dividedinto 10 sections in logarithmic scale.

2. Radius: For each mass M in Table 3 there are12 radius values, 11 of which are equally spaced within theallowed range: RFeðMÞ þ ðRH2OðMÞ �RFeðMÞÞ � i, wherei ¼ 0; 0:1; 0:2;…; 0:9; 1:0. The 12th radius value (RMgSiO3

ðMÞ)

is inserted into the list corresponding to the pure-MgSiO3 planetradius (see Table 1) for mass M. Here RFeðMÞ, RMgSiO3

ðMÞ,and RH2OðMÞ are the radii for planets with mass M composedof pure-Fe, pure-MgSiO3, and pure-H2O correspondingly.

Overall, there are 41 � 12 ¼ 492 different mass–radius pairsin Table 3. For each ðM;RÞ, three cases: p0min, p0mid, and p0max

are listed.3. Central pressure p0 (Pascal) in logarithmic base-10 scale.4. p1=p0, the ratio of p1 (the first boundary pressure, i.e., the

pressure at the Fe-MgSiO3 boundary) over p0.5. p2=p1, the ratio of p2 (the second boundary pressure, i.e.,

the pressure at the MgSiO3-H2O boundary) over p1.6. Fe mass fraction (the ratio of the Fe-layer mass over the

total mass of the planet).7. MgSiO3 mass fraction (the ratio of the MgSiO3-layer mass

over the total mass of the planet).8. H2Omass fraction (the ratio of the H2O-layer mass of over

the total mass of the planet).

The sixth, seventh and eighth columns always add up to one.A dynamic and interactive tool to characterize and illustrate

the interior structure of exoplanets built upon Table 3 and othermodels in this article is available online (see 1).

FIG. 4.—Fractional differences in radius resulting from Fe partitioning be-tween mantle and core. Case 1 and Case 4 (complete differentiation and no oxi-dization of Fe, black); Case 2 (partial [50%] differentiation: 50% metallic Femixed with the mantle, solid brown); Case 3 (no differentiation: all metallicFe mixed with the mantle, pink); Case 5 (partial [50%] oxidization of Fe, dashedbrown); Case 6 (complete oxidization of Fe, dashed pink). See the online editionof the PASP for a color version of this figure.

TABLE 3

TABLE FOR TERNARY DIAGRAM

MðM⊕Þ RðR⊕Þ log10ðp0Þ p1=p0 p2=p1 Fe MgSiO3 H2O

0.1 . . . . . 0.3888 10.9212 0 0 1 0 00.1 . . . . . 0.3888 10.9212 0 0 1 0 00.1 . . . . . 0.3888 10.9212 0 0 1 0 00.1 . . . . . 0.4226 10.9066 0.133 0 0.759 0.241 00.1 . . . . . 0.4226 10.9126 0.102 0.046 0.818 0.172 0.010.1 . . . . . 0.4226 10.9186 0.024 1 0.953 0 0.047...

. . . . . . . ... ..

. ... ..

. ... ..

. ...

100 4.102 13.0979 1 1 0 0 1

NOTE.—Table is shown in its entirety in the electronic edition of the PASP.A portion is shown here regarding its form and content.

FIG. 5.—The red, orange and purple points correspond tolog10ðp0 ðinPaÞÞ ¼ 11:4391, 11.5863, and 11.7336. The mass fractions are(1) FeMF ¼ 0:083,MgSiO3MF ¼ 0:917, H2OMF ¼ 0 (red); (2) FeMF ¼ 0:244,MgSiO3MF ¼ 0:711, H2OMF ¼ 0:046 (orange); (3) FeMF ¼ 0:728,MgSiO3MF ¼ 0, H2OMF ¼ 0:272 (purple). MF here stands for mass fraction.Theblue curve is theparabolic fit. The red dashedcurve is themaximumcollisionalstripping curve by Marcus et al. (2010). See the online edition of the PASP for acolor version of this figure.

MODEL GRID FOR SOLID PLANETS 000

2013 PASP, 125:000–000

4.5. Generate Curve Segment on Ternary Diagram UsingTable 3

One utility of Table 3 is to generate the curve segment on thethree-layer ternary diagram for a given mass–radius pair. As anexample, for M ¼ 1 M⊕ and R ¼ 1:0281 R⊕, the table pro-vides three p0s. For each p0, the mass fractions of Fe,MgSiO3, and H2O are given to determine a point on the ternarydiagram. Then, a parabolic fit (see Fig. 5) through the threepoints is a good approximation to the actual curve segment. Thisparabola may intersect the maximum collisional stripping curveby Marcus et al. (2010), indicating that the portion of parabolabeneath the intersection point may be ruled out by planet for-mation theory.

5. CONCLUSION

The two-layer and three-layer models for solid exoplanetscomposed of Fe, MgSiO3, and H2O are the focus of this article.The mass–radius contours (Fig. 2) are provided for the two-layer model, useful for readers to quickly calculate the interiorstructure of a solid exoplanet. The two-parameter contour meshmay also help one build physical insights into the solid exopla-net interior structure.

The complete three-layer mass–fraction ternary diagram istabulated (Table 3), useful for readers to interpolate and calcu-late all solutions as the mass fractions of the three layers for agiven mass–radius input. The details of the EOS of Fe, MgSiO3,and H2O and how they are calculated and used in this article arediscussed in § 3 and shown in Figure 1.

A dynamic and interactive tool to characterize and illustratethe interior structure of exoplanets built upon Table 3 and othermodels in this article is available online (see 1).

The effect of Fe partitioning between mantle and core onmass–radius relation is explored in § 4.3, and the result is shownin Figure 4 and Table 2.

With the ongoing Kepler Mission and many other exoplanetsearching projects, we hope this article could provide a handytool for observers to fast characterize the interior structure ofexoplanets already discovered or to be discovered, and furtherour understanding of those worlds beyond our own.

We acknowledge partial support for this work by NASA co-operative agreement NNX09AJ50A (Kepler Mission scienceteam).

We would like to thank Michail Petaev and Stein Jacobsenfor their valuable comments and suggestions. This research issupported by the National Nuclear Security Administration un-der the High Energy Density Laboratory Plasmas through DOEgrant # DE-FG52-09NA29549 to S. B. Jacobsen (PI) withHarvard University. This research is the authors’ views andnot those of the DOE.

Li Zeng would like to thank Professor Pingyuan Li, LiZeng’s grandfather, in the Department of Mathematics atChongqing University, for giving Li Zeng important spiritualsupport and guidance on research. The guidance includes re-search strategy and approach, methods of solving differentialequations and other numeric methods, etc.

Li Zeng would also like to give special thanks to MasterAnlin Wang. Master Wang is a Traditional Chinese Kung FuMaster and World Champion. He is also a practitioner and re-alizer of Traditional Chinese Philosophy of Tao Te Ching,which is the ancient oriental wisdom to study the relation be-tween the universe, nature and humanity. Valuable inspirationswere obtained through discussion of Tao Te Ching with MasterWang as well as Qigong cultivation with him.

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