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★Planets with both mass and radius measurements can be plotted on a massradius diagram. This diagram helps to classify different planets, like rocky planets, gas giants, miniNeptunes, superEarths, etc. The model composition curves on a massradius diagram help us to understand the interior structures of these planets.
(1.1) MassRadius Plot of Planets up to 30 Earth Masses: Download Plot in Postscript: Click Here. (updated 2016/10) 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) “MassRadius 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 Mgsilicate planets with different amounts of H2O or Fe added. 3. Planets are colorcoded 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 (zeroeccentricity), (2) uniform surface temperaure of planet (complete heat redistribution, for those closein planets tidally locked to their host stars, thus, onehemisphere always facing the star, that hemisphere's temperature could be higher with incomplete heat redistrbution), (3) zeroalbedo (A=0, for nonzero albedo, simply multiply the temperature by a factor of (1A)^(1/4) to obtain the correction).
(1.2) MassRadius Plot of Planets up to 20 Earth Masses: with better than 30% mass measurement accuracy
Download Plot in Postscript: Click Here. (updated 2016/10) cite as: (1) “Kepler21b: A rocky planet around a V = 8.25 magnitude star”. Mercedes LopezMorales, Raphaelle Haywood, Jeffrey Coughlin, Li Zeng, Lars Buchhave, et al. AJ, 152, 204, 2016. (ADS link) (PDF) (2) “MassRadius 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 groundbased 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 Mgsilicate planets with different amounts of H2O or Fe added. 4. Planets are colorcoded 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 (zeroeccentricity), (2) uniform surface temperaure of planet (complete heat redistribution, for those closein planets tidally locked to their host stars, thus, onehemisphere always facing the star, that hemisphere's temperature could be higher with incomplete heat redistrbution), (3) zeroalbedo (A=0, for nonzero albedo, simply multiply the temperature by a factor of (1A)^(1/4) to obtain the correction).
(1.3) MassRadius Plot of Selected Rocky Planets up to 10 Earth Masses:
Download Plot in Postscript: Click Here. (updated 2016/10) 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) “MassRadius 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 Mgsilicate 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 colorcoded 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 (zeroeccentricity), (2) uniform surface temperaure of planet (complete heat redistribution, for those closein planets tidally locked to their host stars, thus, onehemisphere always facing the star, that hemisphere's temperature could be higher with incomplete heat redistrbution), (3) zeroalbedo (A=0, for nonzero albedo, simply multiply the temperature by a factor of (1A)^(1/4) to obtain the correction).
(2) Table of MassRadius Curves representing various compositions: Download Table: Click Here. (updated 2015/12) Download Table with finer grid: Click Here. (updated 2016/10) Download Table with even finer grid: Click Here. (updated 2016/10) cite as: “MassRadius 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/Mgsilicates twolayer planet, then followed by columns of Mgsilicates/H2O twolayer 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) Fealloy 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) Mgsilicates are assumed to be solid, just as the Earth's mantle. Both Fe and Mgsilicates are extrapolated from the Seismicallydetermined 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 "zerotemperature" H2/He EOS is used for the calculation (Seager et al. 2007 (PDF)). (5) At extremely high pressure, modified ThomasFermiDirac 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).
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) “MassRadius 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/cdfplayer/, which may require the input of your institution's name and email. 2. If Manipulate Planet fails to load in your web browser, try open it in Firefox. 3. You might need to click "Enable Dynamics" at the upper righthand corner when necessary, to allow the tool to display properly in your web browser. 4. Please be patient, the tool sometimes might take a while (up to ~30 seconds) to initialize and load in your web browser. If it runs too slow, try relaunch your web browser or restart your computer. 5. When you click the Locator (Locator refers to the Targetshaped 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 massradius 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) Fealloy 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) Mgsilicates are assumed to be solid, just as the Earth's mantle. Both Fe and Mgsilicates are extrapolated from the Seismicallydetermined 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 ThomasFermiDirac EOS (Salpeter & Zapolsky 1967 (PDF), Zapolsky & Salpeter 1969 (PDF)) is assumed for all components. 8. Any questions or comments are appreciated, please contact astrozeng@gmail.com
Major updates (2015/12): (1) Mass and Radius uncertainties (δM+/, δR+/) of planets can now be plotted as as an uncertainty ellipse on the MR diagram. (2) p0 can now be entered in unit of GPa (10^9 Pascal). (3) 11 equally logspaced 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 twolayer 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 3layer model due to the intrisic degeneracy of the problem, not due to mass or radius uncertainties.
★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 PREMextrapolated EOS (black) versus simple analytical models: Core (red, purple, and pinkarea in between) and Mantle (green). Panel (14)a: Earth, Panel (14)b: GJ 1132b, Panel (14)c: Kepler93b, Panel (14)d: Kepler20b. Panel (1)ad: Gravity Profiles (core is proportional to r and mantle is constant). Panel (2)ad: Density Profiles (core is constant and mantle is inversely proportional to r). Panel (3)ad: Pressure Profiles (core is parabolic in r and mantle is logarithmic in r). Panel (4)ad: 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)). Download Plot in Postscript: Click Here. (update 2016/10) 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)
★Study of the Phase Diagram of H2O together with the discovery of some Kepler planets suggest the possible existence of H2Orich planets. With thermal evolution, some of these planets could have superionic phase of H2O in their interior, leading to the possibility of magnetic field. The possibility of waterrich planets in our galaxy emerge from the discoveries and massradius 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 superionic 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. Download Plot in PDF: Click Here. (updated 2014/03) cite as: “The Effect of Temperature Evolution on the Interior Structure of H2Orich 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 Superionic (Cavazzoni et al. 1999 (PDF), Redmer et al. 2011 (PDF), French & Redmer 2015 (PDF), French et al. 2016 (PDF), Hernandez & Caracas 2016 (PDF)).
★Massradius 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 massradius 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. Download Data of Contours in Excel: FeMgSiO3 planet: MgSiO3H2O planet: FeH2O planet: 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 MassRadius contours of 2layer planet: 1. xaxis is in unit of Earth Masses in logarithmic scale. yaxis is in unit of Earth Radii in linear scale. 1st row: FeMgSiO3 planet. 2nd row: MgSiO3H2O planet. 3rd row: FeH2O 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 massradius contours can be used to quickly determine the characteristic interior parameters of a 2layer planet, such as its p0 (central pressure, i.e., pressure at the center of the planet), p1/p0 (ratio of coremantle 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 2layer model is unique. It is represented as a point on the massradius 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 massradius 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 massradius 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. 5. The data for the massradius 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 massradius phase space, into the probability distribution of the interior structure parameter phase space, such as the probability distribution in p0p1/p0 phase space, p0CMF phase space, or p0CRF phase space. 7. Regarding the Equation of States (EOS) used for this model: (1) Core is assumed to be solid pure εFe (hexagonal closepacked 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 overestimated the densities of the core. (2) Mgsilicates are assumed to be solid MgSiO3 (perovskite, postperovskite, and its higherpressure 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 overestimated the densities of the mantle. Considering all these, this model might have underestimated 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 ThomasFermiDirac EOS (Salpeter & Zapolsky 1967 (PDF), Zapolsky & Salpeter 1969 (PDF)) is assumed for all components.
Matlab Exoplanet Code for Download (2008) ★Matlab codes for the interior structure of exoplanets built with Prof. Sara Seager. Assumptions: 3layer 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.
Take longer to run, but more accurate. This code can deal with zero uncertainty in planet mass and radius. It plots 1σ, 2σ, and 3σ contours of given mass, radius, and uncertainties inputs. example: for Mass=10 Earth Mass , Mass uncertainty=0.5 Earth Mass; Radius= 2 Earth Radius, Radius uncertainty= 0.1 Earth Radius: Type ExoterDE(10, 0.5, 2, 0.1); in your Matlab command window. Then hit Enter.
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; Radius= 2 Earth Radius, Radius uncertainty= 0.1 Earth Radius: Type ExoterDB(10, 0.5, 2, 0.1); in your Matlab command window. Then hit Enter.
Regarding the Equation of States (EOS) used for this model: This model adopts the EOS from Seager et al. 2007 (PDF). (1) Core is assumed to be solid pure εFe (hexagonal closepacked 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 overestimated the densities of the core. (2) Mgsilicates are assumed to be solid MgSiO3 (perovskite, however, it does not include phase transtion of perovskite to postperovskite, and its further dissociation at higher pressure.). 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 overestimated the densities of the mantle, especially for small masses. Considering all these, this model might have underestimated the CMF by ~0.1 in some cases. (3) H2O is assumed to be in cold solid phase (Ice VII, VIII, X).(4) At extremely high pressure, modified ThomasFermiDirac EOS (Salpeter & Zapolsky 1967 (PDF), Zapolsky & Salpeter 1969 (PDF)) is assumed for all components.
