Revisiting the mass-radius relation of super Earth with new ice EOS measurement

Chenliang Huang 1, Zachary M. Grande1, Dean Smith1, Jesse S. Smith2, John H. Boisvert1, Oliver Tschauner1,Jason H. Steffen1, Ashkan Salamat1

1University of Nevada, Las Vegas, 2Argonne National Laboratory

Revisiting the mass-radius relation of super Earth with new ice EOS measurement

Chenliang Huang 1, Zachary M. Grande1, Dean Smith1, Jesse S. Smith2, John H. Boisvert1, Oliver Tschauner1,Jason H. Steffen1, Ashkan Salamat1

1University of Nevada, Las Vegas, 2Argonne National Laboratory

Abstract

In a broadly adopted terrestrial planet model, Zeng et al. (2016) derivedthe mass-radius relation of fully differentiated, two-layer planets. As alarge analog of the icy satellites in the solar system, water/ice can be amajor component of planets. Hakim et al. (2018) showed that the mod-eling uncertainty due to equation of states (EOS) applied can dominateover the observational uncertainties for best observed super-Earths. Us-ing the laser heating and quenching techniques, we significantly improvethe measurement accuracy of the phase diagram and EOS of water iceunder pressures up to 88 GPa. We show that the phase transition froma newly identified ice-VIIt to ice-X occurs just above 30 GPa and thatthe bulk modulus of ice-X is much larger than previously believed.

In this work, we discuss the impact of these new measurements of iceon the planetary mass-radius relation and interior structure, esp. thosewith substantial water content, compared to the results presented inZeng et al. (2016). The higher bulk modulus of ice-X produces largerplanets for a given mass, thereby either reducing the atmospheric contri-bution to the volume of many exoplanets or limiting their water content.The change in planet mass-radius relation caused by the systematic dif-ferences between previous and new EOS measurements are comparableto observational uncertainties in planet sizes for several systems (Grandeet al., 2019).

Rapid-Melt Quench Technique

The main challenges, thus far, for measuring the true EOS of the differentphases of water ice have been due to the geometry constraints of thediamond anvil cell (DAC), which leads to deviatoric stress conditions.Furthermore, water ices often crystallize over large domain sizes thatlead to poor statistics when dealing with X-ray diffraction.

By using a CO2 laser to directly couple with the water ice inside theDAC, we have been able to use a rapid-melt quench technique to pro-duce small domains of ice that lead to a very significant improve on theaveraging statistics when using X-ray diffraction, as well as annealing thedeviatoric stress at the grain-grain boundaries of neighboring domains.In using this new and powerful technique we are also able to anneal thepressure marker (gold) to ensure a more reliable pressure determinationfor a given volume.

Fig. 1: X-ray diffraction pattern of the traditional cold compressionmethod (left) and the laser heating and quenching method (right).

New Ice EOS and Extrapolation

Using the laser heating and quenching techniques, we identify a new ice transitionphase ice-VIIt, proceeding the phase transition to ice-X just above 30 GPa. Atroom temperature that we performed the measurement, the bulk modulus of ice-Xis much larger than previously believed. We obtain the three-phase ice VII-VIIt-XEOS by fitting to the measured P − V data.

200 400 600 800 1000

Pressure (GPa)

2

3

4

5

6

7

dens

ity

(gcm−

3)

This work (E0S I)

Statistical 1-σ uncertainty

Zeng et al. 2016 (EOS II, Frank et al. 2004 melting + FMNR2009)

Frank et al. 2004, T = 300K (EOS III)

Fig. 2: Compare EOSs extrapolated to high pressure.

The black solid line shows the three-phase EOS suggested (thereafter EOS I)in this work. The EOS of ice-X is extrapolated to a pressure near the center ofsuper-Earth. The 1-σ statistical uncertainty of the EOS is shown as the grey bandenveloped by dash-dotted lines. Because the error bars between EOS parametersare correlated, the resulting mass-radius curve has a much smaller uncertaintythan what if they were uncorrelated. The red dashed line shows the ice EOSsused in Zeng et al. (2016) (thereafter EOS II) — EOS measured by Frank et al.(2004)(FFH2004) along the ice melting temperature for P <= 37.4 GPa andinterpolated densities at a series of discrete pressure and temperature points forP > 37.4 GPa simulated by French et al. (2009)(FMNR2009). The red solidcurve shows the third-order Vinet EOS using fitting parameters given by Franket al. (2004) based on their measurement at 300 K (thereafter EOS III). Byremoving the temperature dependence, it is a more direct comparison to our result.Our measured density of ice X at 88 GPa is ∼ 5% smaller than the density ofextrapolated EOS III. Such density difference increases when extrapolating bothEOS to higher pressure.

References

Frank, M. R., Fei, Y., & Hu, J. 2004, Geochimica et Cosmochimica Acta, 68, 2781

French, M., Mattsson, T. R., Nettelmann, N., & Redmer, R. 2009, Phys. Rev. B,79, 054107

Grande, Z. M., Huang, C., Smith, D., et al. 2019, submitted to Science Advances

Hakim, K., Rivoldini, A., Van Hoolst, T., et al. 2018, Icarus, 313, 61

Southworth, J. 2011, MNRAS, 417, 2166

Zeng, L., Sasselov, D. D., & Jacobsen, S. B. 2016, ApJ, 819, 127

Interior Density Map of Planet Model

0.3 1 3 100.0

0.4

0.6

0.8

0.9

1.0

r/Rp

PREM mantle

ice X

EOS

I

ice VIIt

ice V

II

ice VI

Liquid water

0.3 1 3 10

PREM mantle

ice VII

Low pressure ice

EOS

III

1

2

3

5

8

11

(g c

m3 )

0.3 1 3 10

PREM mantle

ice X

ice VII

Low pressure ice

EOS

II

M (M )Fig. 3: The planet interior density map for 50 wt% water/rocky planet.

We reproduce the two-layer planet modelin Zeng et al. (2016) by using the sameEOS of the iron core and rocky mental,while varying the EOS of the ice layer.

The y-axis in the figure indicates the frac-tion of the total planet radius. Black solidcurves show the phase boundary betweendifferent composition or phases. Therocky part of the planet in three panelsapply the same EOS, which is the directextrapolation from Earth’s seismic model:the Preliminary Reference Earth Model(PREM) following Zeng et al. (2016).

Three panels show planet models using theice EOS I, II, and III respectively. Theice in the pressure range that studied inthis work can occupy a substantial planetvolume for water-rich planets.

Planet Mass-Radius Relation

1 2 5 10Mass (M )

1.00

1.25

1.50

1.75

2.00

2.25

2.50

2.75

3.00

Radi

us (R

)

55 Cnc e

CoRoT-7 b

GJ 1214 b

GJ 892 b

GJ 892 c

GJ 3651 b

K2-2 b

K2-18 b

K2-141 b Kepler-10 b

Kepler-10 c Kepler-19 b

Kepler-307 b

Kepler-30 b

Kepler-36 b

WASP-47 e

Men c

TRAPPIST-1 bTRAPPIST-1 c

TRAPPIST-1 d

TRAPPIST-1 eTRAPPIST-1 f

TRAPPIST-1 g

TRAPPIST-1 h

EPIC 246471491 b

(TESS)

pure water

50 wt% rock/waterpure rock

pure iron

EOS IEOS IIEOS III

300 500 1200 2300Teq(K)

1 2 10Mass (M )

0.0

2.5

5.0

7.5

10.0

R/R

(%)

50 wt% rock/waterPure water

Fig. 4: Mass-radius curve of planet models compared to observational data.

Applying the ice EOS II, which is affected by planet interior temperature variations, dotted curves reproduce the planetmass-radius curves in Zeng et al. (2016). Replacing the ice EOS in the model by EOS I (this study) and EOS III, bothmeasured at room temperature, mass-radius curves are shown in solid and dashed curves respectively. Planets whose radiiand masses are measured to better than ∼ 10% and ∼ 20% respectively are plotted and color-coded by their surfacetemperatures. Source: NASA Exoplanet Archive, TEPCat (Southworth, 2011) and thereafter. The increased planet radiussuggested by the new EOS is larger than the radius uncertainty of many planets. Thus, the contribution of the atmosphereor of water content to the planet structure for planets such as π Mensae-c, Kepler-10c, and EPIC-246471491b may be lessthan previously inferred. Inset: The percent difference in radius between solid (EOS I) and dashed lines (EOS III).