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 Planet Models
 Mass-Radius Contours (2013) Matlab Code (2008) Plots Tables Numerical Analytical

★Planets with both mass and radius measurements can be plotted on a mass-radius diagram. This diagram helps to classify different planets, like rocky planets, gas giants, mini-Neptunes, super-Earths, etc. The model composition curves on a mass-radius diagram help us to understand the interior structures of these planets.

(1.1) Mass-Radius Plot of Planets up to 30 Earth Masses: cite as:

(1) Lisa Kaltenegger’s review article to be published in The Astronomy and Astrophysics Review (http://link.springer.com/journal/159), 2017:

(2) “Mass-Radius Relation for Rocky Planets based on PREM”. Li Zeng, Dimitar D. Sasselov, and Stein B. Jacobsen. ApJ, 819, 127, 2016. (ADS link) (PDF)

Note for the plot:

1. The data of exoplanets are largely taken from the NASA Exoplanet Archive (http://exoplanetarchive.ipac.caltech.edu), with some recent updates done by hand.

2. Curves show models of different compositions, with solid indicating single composition (Fe, MgSiO3, i.e. rock, H2O) and dashed indicating Mg-silicate planets with different amounts of H2O or Fe added.

3. Planets are color-coded by their incident bolometric stellar flux (compared to the Earth) and equilibrium temperatures. Equiibrium tempratures and fluxes are calculated based on the following assumptions: (1) circular orbit (zero-eccentricity), (2) uniform surface temperaure of planet (complete heat redistribution, for those close-in planets tidally locked to their host stars, thus, one-hemisphere always facing the star, that hemisphere's temperature could be higher with incomplete heat redistrbution), (3) zero-albedo (A=0, for non-zero albedo, simply multiply the temperature by a factor of (1-A)^(1/4) to obtain the correction).

(1.2) Mass-Radius Plot of Planets up to 20 Earth Masses:

with better than 30% mass measurement accuracy cite as:

(1) “Kepler-21b: A rocky planet around a V = 8.25 magnitude star”. Mercedes Lopez-Morales, Raphaelle Haywood, Jeffrey Coughlin, Li Zeng, Lars Buchhave, et al. AJ, 152, 204, 2016. (ADS link) (PDF)

(2) “Mass-Radius Relation for Rocky Planets based on PREM”. Li Zeng, Dimitar D. Sasselov, and Stein B. Jacobsen. ApJ, 819, 127, 2016. (ADS link) (PDF)

Note for the plot:

1. The data of exoplanets are largely taken from the NASA Exoplanet Archive (http://exoplanetarchive.ipac.caltech.edu), with some recent updates done by hand.

2. Planets with masses measured by Transit Timing Variation (TTV) are shown as triangles, and planets with masses measured by ground-based Radial Velocity (RV) are shown as circles.

3. Curves show models of different compositions, with solid indicating single composition (Fe, MgSiO3, i.e. rock, H2O) and dashed indicating Mg-silicate planets with different amounts of H2O or Fe added.

4. Planets are color-coded by their incident bolometric stellar flux (compared to the Earth) and equilibrium temperatures. Equiibrium tempratures and fluxes are calculated based on the following assumptions: (1) circular orbit (zero-eccentricity), (2) uniform surface temperaure of planet (complete heat redistribution, for those close-in planets tidally locked to their host stars, thus, one-hemisphere always facing the star, that hemisphere's temperature could be higher with incomplete heat redistrbution), (3) zero-albedo (A=0, for non-zero albedo, simply multiply the temperature by a factor of (1-A)^(1/4) to obtain the correction).

(1.3) Mass-Radius Plot of Selected Rocky Planets up to 10 Earth Masses: cite as:

(1) “A Simple Analytical Model for Rocky Planet Interior”. Li Zeng, and Stein B. Jacobsen. ApJ, 837, 164, 2017. (ADS link) (PDF) (PDF)

(2) “Mass-Radius Relation for Rocky Planets based on PREM”. Li Zeng, Dimitar D. Sasselov, and Stein B. Jacobsen. ApJ, 819, 127, 2016. (ADS link) (PDF)

Note for the plot:

1. The data of exoplanets are largely taken from the NASA Exoplanet Archive (http://exoplanetarchive.ipac.caltech.edu), with some recent updates done by hand.

2. Curves show models of different compositions, with solid indicating single composition (Fe, MgSiO3, i.e. rock, H2O) and dashed indicating Mg-silicate planets with different amounts of H2O or Fe added. Rocky planets without volatile envelope likely lie in the shaded region within uncertainty, and those ones with volatile envelope may lie above.

3. Planets are color-coded by their incident bolometric stellar flux (compared to the Earth) and equilibrium temperatures. Equiibrium tempratures and fluxes are calculated based on the following assumptions: (1) circular orbit (zero-eccentricity), (2) uniform surface temperaure of planet (complete heat redistribution, for those close-in planets tidally locked to their host stars, thus, one-hemisphere always facing the star, that hemisphere's temperature could be higher with incomplete heat redistrbution), (3) zero-albedo (A=0, for non-zero albedo, simply multiply the temperature by a factor of (1-A)^(1/4) to obtain the correction).

(2) Table of Mass-Radius Curves representing various compositions: cite as: “Mass-Radius Relation for Rocky Planets based on PREM”. Li Zeng, Dimitar D. Sasselov, and Stein B. Jacobsen. ApJ, 819, 127, 2016. (ADS link) (PDF)

Note for the table:

1. first set of columns are for Fe/Mg-silicates two-layer planet, then followed by columns of Mg-silicates/H2O two-layer planet. (and in some cases, the calculation for cold H2/He planet, and the maximum collisional stripping curve.)

2. Regarding the Equation of States (EOS) used to calculate these curves: (1) Fe-alloy in the core is assumed to be liquid, just as the Earth's outer core, which dominates over the solid inner core. Solid inner core likely grows from the solidification of the liquid outer core over time, and more massive planets likely possess liquid core due to higher heat content. (2) Mg-silicates are assumed to be solid, just as the Earth's mantle. Both Fe and Mg-silicates are extrapolated from the Seismically-determined Density Profile of the Earth (Preliminary Reference Earth Model, known as PREM). (3) H2O is assumed to be in solid phase (Ice Ih, Ice III, Ice V, Ice VI (Chaplin's website), Ice VII (Frank et al. 2004 (PDF)), Ice X (French et al. 2009 (PDF))) along its melting curve. (4) Cold "zero-temperature" H2/He EOS is used for the calculation (Seager et al. 2007 (PDF)). (5) At extremely high pressure, modified Thomas-Fermi-Dirac EOS (Salpeter & Zapolsky 1967 (PDF), Zapolsky & Salpeter 1969 (PDF)) is assumed for all components.

3. The maximum collisional stripping curve is interpolated from Marcus et al. (2010) (PDF).

Models of Planet Interiors

(1) Numerical Model:

Manipulate Planet

Manipulate Planet numerically solves the interior structure of a planet based on the Equation of State (EOS) extrapolated from Earth's Seismic Density Profile (PREM). Require the free Wolfram CDF player. You might need to grant your browser's security permission to allow it to run. If it fails to load in your web browser, try open it in Firefox. (updated 2015/12)

To access the tool, please click the diagram below: cite as:

(1) Mass-Radius Relation for Rocky Planets based on PREM”. Li Zeng, Dimitar D. Sasselov, and Stein B. Jacobsen. ApJ, 819, 127, 2016. (ADS link) (PDF)

(2) “A Detailed Model Grid for Solid Planets from 0.1 through 100 Earth Masses”. Li Zeng and Dimitar D. Sasselov. PASP, 125, 227, 2013. (ADS link) (PDF)

Note for Manipulate Planet:

1. The latest version of Wolfram CDF Player is available free for download at www.wolfram.com/cdf-player/, which may require the input of your institution's name and email.

3. You might need to click "Enable Dynamics" at the upper right-hand corner when necessary, to allow the tool to display properly in your web browser.

5. When you click the Locator (Locator refers to the Target-shaped object which can be dragger around by your mouse to any desired location. It is where the calculation actually takes place.), please wait one second before dragging it around, to allow it enough time to respond. DO NOT release the click during the entire dragging process until the Locator is moved to the desired location in the mass-radius diagram.

6. This interactive tool is for research & education purpose only, all rights reserved.

7. Regarding the Equation of States (EOS) used for this model: (1) Fe-alloy in the core is assumed to be liquid, just as the Earth's outer core, which dominates over the solid inner core. Solid inner core likely grows from the solidification of the liquid outer core over time, and more massive planets likely possess liquid core due to higher heat content. (2) Mg-silicates are assumed to be solid, just as the Earth's mantle. Both Fe and Mg-silicates are extrapolated from the Seismically-determined Density Profile of the Earth (Preliminary Reference Earth Model, known as PREM). (3) H2O is assumed to be in solid phase (Ice Ih, Ice III, Ice V, Ice VI (Chaplin's website), Ice VII (Frank et al. 2004 (PDF)), Ice X (French et al. 2009 (PDF))) along its melting curve. (4) At extremely high pressure, modified Thomas-Fermi-Dirac EOS (Salpeter & Zapolsky 1967 (PDF), Zapolsky & Salpeter 1969 (PDF)) is assumed for all components.

(1) Mass and Radius uncertainties (δM+/-, δR+/-) of planets can now be plotted as as an uncertainty ellipse on the M-R diagram.

(2) p0 can now be entered in unit of GPa (10^9 Pascal).

(3) 11 equally log-spaced values of p0 range for each Locator {Mass,Radius} are available as 11 clickable buttons. Min and Max values are the two extreme values out of the 11, each corresponds to a particular two-layer model of the planet.

(4) The two dashed curves in the ternary diagram show uncertainty in composition due to mass and radius errors (δM+/-, δR+/-). The thick black curve in the ternary diagram show all the possible solutions of 3-layer model due to the intrisic degeneracy of the problem, not due to mass or radius uncertainties.

(2) Analytical Model:

★Alternatively, some analytical formulae can provide quick and approximate solutions for the interiors of rocky exoplanets:

The following gives the solutions of 4 planets as examples. For details, please refer to the paper attached. Numerical calculations based on PREM-extrapolated EOS (black) versus simple analytical models: Core (red, purple, and pink-area in between) and Mantle (green). Panel (1-4)a: Earth, Panel (1-4)b: GJ 1132b, Panel (1-4)c: Kepler-93b, Panel (1-4)d: Kepler-20b. Panel (1)a-d: Gravity Profiles (core is proportional to r and mantle is constant). Panel (2)a-d: Density Profiles (core is constant and mantle is inversely proportional to r). Panel (3)a-d: Pressure Profiles (core is parabolic in r and mantle is logarithmic in r). Panel (4)a-d: Temperature Profiles (best estimates shall lie in the green area (mantle) and pink area (core)). The solidus (where mixture starts to melt) and liquidus (where mixture completely melts) are plotted for comparison. The temperature profiles are calculated based on the scheme from (Stixrude 2014 (PDF)).

The following table gives the parameters of planets above: cite as:

(1) “A Simple Analytical Model for Rocky Planet Interior”. Li Zeng, and Stein B. Jacobsen. ApJ, 837, 164, 2017. (ADS link) (PDF) (PDF)

(2) “Variational Principle for Planetary Interiors”. Li Zeng, and Stein B. Jacobsen. ApJ, 829, 18, 2016. (ADS link) (PDF)

H2O-rich Planets

★Study of the Phase Diagram of H2O together with the discovery of some Kepler planets suggest the possible existence of H2O-rich planets. With thermal evolution, some of these planets could have super-ionic phase of H2O in their interior, leading to the possibility of magnetic field.

The possibility of water-rich planets in our galaxy emerge from the discoveries and mass-radius measurements of some exoplanets. They could consist of more than 50 percent water by weight, compared to a tiny fraction of one percent for the Earth. However, it is not necessarily liquid water as on the Earth's surface. My research suggests that water in the interiors of these planets under high pressure and temperature could undergo various phase transitions, including the super-ionic form of water (wikipedia link), which has properties of both a solid and a liquid, where the oxygen atoms still sit in the crystal lattice while the hydrogen ions are mobile to conduct electricity. This form of water could then support global magnetic fields on these planets, just as the liquid outer core of the Earth supports the Earth’s magnetic field.

The following shows some examples. For details, please refer to the paper attached. cite as: “The Effect of Temperature Evolution on the Interior Structure of H2O-rich Planets”. Li Zeng and Dimitar D. Sasselov. ApJ, 784, 96, 2014. (ADS link) (PDF)

H2O EOS include Ice Ih, Ice III, Ice V, Ice VI (Chaplin's website), Ice VII (Frank et al. 2004 (PDF)), Ice X (French et al. 2009 (PDF)), Molecular Fluid, Ionic Fluid, Plasma, and Super-ionic (Cavazzoni et al. 1999 (PDF), Redmer et al. 2011 (PDF), French & Redmer 2015 (PDF), French et al. 2016 (PDF), Hernandez & Caracas 2016 (PDF)).

★Mass-radius contours are theoretical contours of pressure, pressure ratio, core mass fraction (CMF), and core radius fraction (CRF) of planets with two distinctive layers. They illustrate the relation between the interior structure and the mass-radius of a planet, thus can be used for interpolation and solving the inverse problem.

The following gives some examples. For details, please refer to the paper attached. Fe-MgSiO3 planet:

 Fe-MgSiO3 p0-p1/p0 table Fe-MgSiO3 p0-CMF table Fe-MgSiO3 p0-CRF table

MgSiO3-H2O planet:

 MgSiO3-H2O p0-p1/p0 table MgSiO3-H2O p0-CMF table MgSiO3-H2O p0-CRF table

Fe-H2O planet:

 Fe-H2O p0-p1/p0 table Fe-H2O p0-CMF table Fe-H2O p0-CRF table

cite as: “A Detailed Model Grid for Solid Planets from 0.1 through 100 Earth Masses”. Li Zeng and Dimitar D. Sasselov. PASP, 125, 227, 2013. (ADS link) (PDF)

Note for Mass-Radius contours of 2-layer planet:

1. x-axis is in unit of Earth Masses in logarithmic scale. y-axis is in unit of Earth Radii in linear scale. 1st row: Fe-MgSiO3 planet. 2nd row: MgSiO3-H2O planet. 3rd row: Fe-H2O planet. 1st column: contour mesh of p1/p0 with p0. 2nd column: contour mesh of CMF with p0. 3rd column: contour mesh of CRF with p0.

2. Given mass and radius input, various sets of mass-radius contours can be used to quickly determine the characteristic interior parameters of a 2-layer planet, such as its p0 (central pressure, i.e., pressure at the center of the planet), p1/p0 (ratio of core-mantle boundary pressure over central pressure), CMF (core mass fraction), and CRF (core radius fraction).

3. Given a unique set of mass and radius, the solution of a 2-layer model is unique. It is represented as a point on the mass-radius diagram. This problem has two degrees of freedon, thus, given any pair of two parameters from the following list: M (mass), R (radius), p0, p1/p0, CMF, CRF, and so on, the solution is unique. The contours of constant M or R are trivial on the mass-radius diagram. They are simply vertical or horizontal lines. The contours of constant p0, p1/p0, CMF, or CRF are generally curves, but can sometimes be approximated as straight lines in certain regions.

4. Within any pair of two parameters, by fixing one of them and continuously varying the other, the point on the mass-radius diagram will move to form a curve. This is how the contour curves are formed. Multiple values of the fixed parameter will give multiple parallel curves, forming one set of contours. The other set of contours can be obtained by switching the fixed parameter with the varying parameter.
The two sets of contours criss-cross each other to form a mesh, which is a natural coordinate system (see the plot above) of this pair of parameters, superimposed onto the existing Cartesian (M,R) coordinates of the mass-radius diagram. This mesh can be used to solve for the two parameters given the mass and radius input, and vice versa. Thus, it is useful for solving the inverse problem.

5. The data for the mass-radius contours with a finer grid are given in xls files. Each file contains two Excel sheets, one for mass and one for radius, both in Earth units. The rows of the table correspond to Log10[ p0 (Pa) ] from 8 to 14.4 with stepsize 0.025. The column of the table corresponds to one of the three parameters: p1/p0, CMF, or CRF from 0 to 1 in stepsize of 0.025.

6. The data can be used to translate a certain probability distribution in the mass-radius phase space, into the probability distribution of the interior structure parameter phase space, such as the probability distribution in p0-p1/p0 phase space, p0-CMF phase space, or p0-CRF phase space.

7. Regarding the Equation of States (EOS) used for this model: (1) Core is assumed to be solid pure ε-Fe (hexagonal close-packed phase of iron stable only at very high pressure). Due to impurities and partially molten state, the actual densities of planetary cores are likely lower. Thus, it might have over-estimated the densities of the core. (2) Mg-silicates are assumed to be solid MgSiO3 (perovskite, post-perovskite, and its higher-pressure derivatives). This is the major composition of Earth's lower mantle, however, the Earth's upper mantle (composed of various phases of Mg2SiO4) is less dense. Thus, it might have over-estimated the densities of the mantle. Considering all these, this model might have under-estimated the CMF by ~0.1 in some cases. (3) H2O is assumed to be in solid phase (Ice Ih, Ice III, Ice V, Ice VI (Chaplin's website), Ice VII (Frank et al. 2004 (PDF)), Ice X (French et al. 2009 (PDF))) along its melting curve. (4) At extremely high pressure, modified Thomas-Fermi-Dirac EOS (Salpeter & Zapolsky 1967 (PDF), Zapolsky & Salpeter 1969 (PDF)) is assumed for all components.

★Matlab codes for the interior structure of exoplanets built with Prof. Sara Seager. Assumptions: 3-layer planet: an iron core, a silicate mantle and a water crust. Temperature dependence of the Equation of State (EOS) is neglected for simplicity. cite as: “A Computational Tool to Interpret the Bulk Composition of Solid Exoplanets based on Mass and Radius Measurements”. Li Zeng and Sara Seager. PASP, 120, 983, 2008. (ADS link) (PDF)

Two matlab codes to interpret the bulk composition of solid exoplanets based on their mass and radius measurements are available for download below.

Code instructions:

Frist of all, download the code zip files and decompress them.

Please put the decompressed files into your Matlab work directory so that the Matlab can recognize them. Use addpath command if necessary.

Based in Matlab if using this computer code please cite Li Zeng & Prof. Sara Seager.

ExoterDE(M, Munc, R, Runc)

Take longer to run, but more accurate. This code can deal with zero uncertainty in planet mass and radius. It plots 1-σ, 2-σ, and 3-σ contours of given mass, radius, and uncertainties inputs.

example: for Mass=10 Earth Mass , Mass uncertainty=0.5 Earth Mass;

Type ExoterDE(10, 0.5, 2, 0.1); in your Matlab command window. Then hit Enter.

ExoterDB(M, Munc, R, Runc)

faster to run based on interpolation of data grid. This code plots a colormap showing the possible proportions of iron, silicate and water for a continuous range of σ from 0 up to 3.

example: for Mass=10 Earth Mass , Mass uncertainty=0.5 Earth Mass;