A measurement of the equation of state of carbon envelopes of white dwarfs


White dwarfs represent the final state of evolution for most stars1,2,3. Certain classes of white dwarfs pulsate4,5, leading to observable brightness variations, and analysis of these variations with theoretical stellar models probes their internal structure. Modelling of these pulsating stars provides stringent tests of white dwarf models and a detailed picture of the outcome of the late stages of stellar evolution6. However, the high-energy-density states that exist in white dwarfs are extremely difficult to reach and to measure in the laboratory, so theoretical predictions are largely untested at these conditions. Here we report measurements of the relationship between pressure and density along the principal shock Hugoniot (equations describing the state of the sample material before and after the passage of the shock derived from conservation laws) of hydrocarbon to within five per cent. The observed maximum compressibility is consistent with theoretical models that include detailed electronic structure. This is relevant for the equation of state of matter at pressures ranging from 100 million to 450 million atmospheres, where the understanding of white dwarf physics is sensitive to the equation of state and where models differ considerably. The measurements test these equation-of-state relations that are used in the modelling of white dwarfs and inertial confinement fusion experiments7,8, and we predict an increase in compressibility due to ionization of the inner-core orbitals of carbon. We also find that a detailed treatment of the electronic structure and the electron degeneracy pressure is required to capture the measured shape of the pressure–density evolution for hydrocarbon before peak compression. Our results illuminate the equation of state of the white dwarf envelope (the region surrounding the stellar core that contains partially ionized and partially degenerate non-ideal plasmas), which is a weak link in the constitutive physics informing the structure and evolution of white dwarf stars9.

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Fig. 1: Experimental configuration.
Fig. 2: Opacity of shock-compressed C9(H)10 at 9 keV.
Fig. 3: C9(H)10 shock Hugoniot measurements.
Fig. 4: Regime of white dwarf stars accessed by measurements.

Data availability

Source data are provided with this paper. Additional data are available upon request.

Code availability

Owing to its complexity, the data analysis algorithm that supports the findings of this study is available from D.C.S. upon reasonable request and will be outlined in more detail in a supporting publication.


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This work was performed under the auspices of the US Department of Energy under contract number DE-AC52-07NA27344 and contract number 89233218CNA000001. This work has been supported by Laboratory Directed Research and Development (LDRD) award number 13-ERD-073, and also by the University of California, Office of the President, Lab Fee Grant LFR-17-449059, by the Department of Energy, National Nuclear Security Administration, award DE-NA0003842, and by the Department of Energy, Office of Science, Office of Fusion Energy Sciences, awards DE-SC0018298, DE-SC0019269 and FWP 100182. We also acknowledge the NIF Discovery Science programme for providing access to the facility. This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favouring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.

Author information




A.L.K. wrote the manuscript and performed the design calculations. D.C.S. developed the analysis method and analysed the experimental data. B.B., B.M., M.J.M., M.E.M., J.N. and N.K. provided input and feedback to improve the data analysis method. T.D. fielded the experiments and follow-on supporting experiments together with B.B., D.K. and A.L. L.X.B., J.L.D., S.H., P.A.S., A.A.C. and H.D.W. provided theoretical calculations of equation of state models included in this paper and/or provided theoretical models for understanding the data. G.F. and D.S. performed calculations of white dwarf stars and worked closely with S.H.G. and A.L.K. to understand the impact of this work for white dwarfs. W.R.J., J.A.G. and J.N. provided calculations of theoretical opacities. W.S., F.E. and A.N. fabricated and characterized the targets for these experiments. D.C.S., S.H.G, R.W.F., P.N., B.A.R. and G.W.C. proposed the experiments and/or aided in obtaining experimental beamtime at NIF. All co-authors provided input on interpretation of the data and results and/or on their impact for white dwarf modelling.

Corresponding author

Correspondence to Andrea L. Kritcher.

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The authors declare no competing interests.

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Peer review information Nature thanks Stephanie Hansen and Don Winget for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 Schematic of the radiographic analysis method.

Normalized transmission intensity of an X-ray radiograph versus changes in compression and opacity. A given transmission corresponds to a range of possible compressions and opacities (black dashed curves). Density at the shock front is further constrained through knowledge of the initial material density ahead of the shock front and known mass contained within a region on the radiograph corresponding to a Ge marker layer in the sample. This further constrains the opacity and is included in the analysis as the shock traverses the sample. An example shock trajectory is denoted by a red dashed curve.

Extended Data Fig. 2 Comparison of two C9(H)10 shock Hugoniot measurements.

Measured pressure versus mass density (ρ) normalized to the initial density (ρ0) along the shock Hugoniot. Green, data from shot N130103-009-999 fielded at 24 K; red (from Fig. 3), shot N130701-002-999 fielded at ambient temperature. Also plotted are the theoretical Hugoniots for AA-DFT and AA-TFD at 24 K (ρ0 = 1.136 g cm−3), indicating that the initial density conditions are predicted to access approximately the same Hugoniot states at high pressure.

Extended Data Fig. 3 Sensitivity of the theoretical Hugoniot to fluorine.

Calculations show insensitivity of the theoretical Hugoniot (AA-DFT33,34,35) to fluorobenzene solvent (C6H5F) for concentrations up to 20%, corresponding to 1% atomic fraction of fluorine (green curve). Concentrations of 0.5% F (red curve) and 0% F (blue curve) are also shown, but not visibly distinguishable.

Extended Data Fig. 4 Simulations of the shock-front Hugoniot.

Extracted shock-front compressions and pressures from radiation hydrodynamic simulations38 of the experimental platform (red points). The theoretical shock Hugoniot33,34,35 input to the simulations is also shown with ±2% deviation in compression from the input Hugoniot (black curves).

Extended Data Fig. 5 Extended C9(H)10 shock Hugoniot measurements.

Measured pressure versus mass density (ρ) normalized to the initial density (ρ0) along the shock Hugoniot from this work (red and purple curves and shaded region). The purple curve corresponds to the extended dataset that may be impacted by radiative shock-front preheat. Also plotted are previous experimental data and theoretical modelling of the Hugoniot (see Fig. 3).

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Kritcher, A.L., Swift, D.C., Döppner, T. et al. A measurement of the equation of state of carbon envelopes of white dwarfs. Nature 584, 51–54 (2020). https://doi.org/10.1038/s41586-020-2535-y

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