pith. sign in

arxiv: 1906.11252 · v1 · pith:ZGPJT7WUnew · submitted 2019-06-26 · 🌌 astro-ph.SR

Convective Overshoot and Macroscopic Diffusion in Pure-Hydrogen Atmosphere White Dwarfs

Tim Cunningham , Pier-Emmanuel Tremblay , Bernd Freytag , Hans-G\"unther Ludwig , Detlev Koester This is my paper

Pith reviewed 2026-05-25 14:56 UTC · model grok-4.3

classification 🌌 astro-ph.SR
keywords white dwarfsconvective overshootmacroscopic diffusionDA white dwarfsradiation hydrodynamicsaccretion ratessettling timeshydrogen atmospheres
0
0 comments X

The pith

Three-dimensional simulations show convective overshoot mixes up to 2.5 orders of magnitude more mass in pure-hydrogen white dwarf atmospheres than one-dimensional models predict.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models macroscopic diffusion from convective overshoot in DA white dwarfs using 3D radiation hydrodynamics simulations with tracer particles. It finds that the fully mixed region can be much larger, by as much as 2.5 orders of magnitude in mass. This leads to higher inferred accretion rates by about an order of magnitude and longer settling times by up to two orders. The work also locates the start of convection more precisely between 18000 and 18250 K effective temperature. These adjustments matter for understanding how these stars accrete material and how long surface pollutants persist.

Core claim

Using a new grid of deep 3D white dwarf models in the temperature range 11400 K ≤ Teff ≤ 18000 K, tracer particles and a tracer density are used to derive macroscopic diffusion coefficients driven by convective overshoot. These are compared to microscopic diffusion coefficients from one-dimensional structures. The mass of the fully mixed region is likely to increase by up to 2.5 orders of magnitude while inferred accretion rates increase by a more moderate order of magnitude. Evidence shows an increase in settling time of up to 2 orders of magnitude, significant for time-variability studies of polluted white dwarfs. The grid constrains the onset of convective instabilities in DA white dwarfs

What carries the argument

Tracer particles in closed-bottom 3D radiation hydrodynamics simulations that measure the depth and strength of convective overshoot to compute macroscopic diffusion coefficients.

Load-bearing premise

The closed-bottom 3D radiation hydrodynamics simulations with tracer particles accurately capture the macroscopic diffusion driven by convective overshoot without significant influence from numerical boundary conditions or resolution limits.

What would settle it

A measurement of the mixed mass or settling time in a DA white dwarf near 17000 K that differs by more than a factor of ten from the values predicted by the 3D tracer-derived diffusion coefficients.

Figures

Figures reproduced from arXiv: 1906.11252 by Bernd Freytag, Detlev Koester, Hans-G\"unther Ludwig, Pier-Emmanuel Tremblay, Tim Cunningham.

Figure 1
Figure 1. Figure 1: Analysis of massless density arrays implemented with the CO5BOLD tracer density for a simulation at Teff = 13 500 K ( [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Demonstration of the tracer density implementation with CO5BOLD for a simulation at Teff = 13 500 K ( [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Similar to [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Similar to [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Similar to lower panel of [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Spatial resolution sensitivity test using tracer den￾sity analysis for simulations C1, C2 and C3 and grid sizes 1503 (orange), 2502 × 150 (red) and 2503 (blue), respectively, and Teff = 13 500 K and log g = 8.0. The time-averaged vertical velocity profiles vz,rms(dashed) and v 2 z,rms(solid) are shown for each simula￾tion with the diffusion coefficient extracted from each simulation (circles). The filled e… view at source ↗
Figure 8
Figure 8. Figure 8: Depth dependence of maximum grid point displace￾ment per second for simulation C3 from [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 11
Figure 11. Figure 11: Results of path integration analysis of simulation B2 with Teff = 13 000 K and grid size 2503 . Akin to lower panel of [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 10
Figure 10. Figure 10: Path integration analysis of simulation C3 with Teff = 13 500 K and grid size 2503 , with plots akin to the two lower panels of [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: Results of path integration analysis of simulation B2 with Teff = 12 000 K and grid size 2503 . Akin to lower panel of [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Similar to lower panel of [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: Similar to [PITH_FULL_IMAGE:figures/full_fig_p014_15.png] view at source ↗
Figure 14
Figure 14. Figure 14: Dynamical and thermodynamic quantities extracted from the grid of deep 3D simulations ( [PITH_FULL_IMAGE:figures/full_fig_p014_14.png] view at source ↗
Figure 16
Figure 16. Figure 16: P´eclet number evaluated at hτRi = 1 as a function of effective temperature and averaged over the final 50 ms of the deep 3D grid presented in [PITH_FULL_IMAGE:figures/full_fig_p015_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: shows the logarithmic vertical velocities at a layer within each of the three regions with distinct physical characteristics; the convectively unstable region (top), over￾shoot region (middle) and layer where plumes and waves overlap (bottom), for simulation B1 from [PITH_FULL_IMAGE:figures/full_fig_p016_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Power spectrum for spatial frequencies present in lower panel of [PITH_FULL_IMAGE:figures/full_fig_p017_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Diffusion coefficients as a function of enclosed stellar mass from deep simulations of [PITH_FULL_IMAGE:figures/full_fig_p018_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Mixed mass in units of enclosed mass, log q = log Mabove/Mstar, as a function of effective temperature. Mixed masses determined by the intersection of macroscopic and micro￾scopic diffusion coefficients (see [PITH_FULL_IMAGE:figures/full_fig_p018_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Diffusion timescale (top panel) and inferred accre￾tion rates (lower panel) when using the mixed masses, shown in [PITH_FULL_IMAGE:figures/full_fig_p019_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Sample of polluted white dwarfs comprising DAZs (black, open) and DBZs (blue) from Bergfors et al. (2014). The DAZ sample is also shown after the inclusion of convective over￾shoot (black, filled). 7500 10000 12500 15000 17500 20000 22500 Teff[K] 6 7 8 9 10 11 log10( ˙ M/[g s −1]) DAZ w/ overshoot DBZ (Bergfors+14) 6 7 8 9 10 11 log10( ˙ M/[g s −1]) DAZ (Bergfors+14) DBZ (Bergfors+14) [PITH_FULL_IMAGE:fi… view at source ↗
Figure 24
Figure 24. Figure 24: Accretion rates from Bergfors et al. (2014) and [PITH_FULL_IMAGE:figures/full_fig_p020_24.png] view at source ↗
read the original abstract

We present a theoretical description of macroscopic diffusion caused by convective overshoot in pure-hydrogen DA white dwarfs using three-dimensional (3D), closed-bottom, radiation hydrodynamics CO$^5$BOLD simulations. We rely on a new grid of deep 3D white dwarf models in the temperature range 11400 K $\leq T_{\mathrm{eff}} \leq$ 18000 K where tracer particles and a tracer density are used to derive macroscopic diffusion coefficients driven by convective overshoot. These diffusion coefficients are compared to microscopic diffusion coefficients from one-dimensional structures. We find that the mass of the fully mixed region is likely to increase by up to 2.5 orders of magnitude while inferred accretion rates increase by a more moderate order of magnitude. We present evidence that an increase in settling time of up to 2 orders of magnitude is to be expected which is of significance for time-variability studies of polluted white dwarfs. Our grid also provides the most robust constraint on the onset of convective instabilities in DA white dwarfs to be in the effective temperature range from 18000 to 18250 K.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper presents 3D closed-bottom CO5BOLD radiation-hydrodynamics simulations of pure-hydrogen DA white dwarfs (11400 K ≤ Teff ≤ 18000 K) that employ tracer particles to extract macroscopic diffusion coefficients from convective overshoot. These are compared against microscopic diffusion coefficients from 1D structures. The central results are that the fully mixed mass increases by up to 2.5 dex, inferred accretion rates rise by ~1 dex, settling times increase by up to 2 dex, and the onset of convective instability occurs between 18000 K and 18250 K.

Significance. If the reported diffusion enhancements are robust, the work supplies quantitative predictions that directly affect interpretations of metal pollution, accretion rates, and time-variability in DA white dwarfs. The constraint on the convective-onset temperature range is a clear, falsifiable contribution to the field.

major comments (1)
  1. [simulation setup] Simulation setup paragraph (and abstract): the central quantitative claims (2.5 dex mixed-mass increase, 1 dex accretion-rate increase, 2 dex settling-time increase) rest on macroscopic diffusion coefficients derived from tracer particles in closed-bottom models. No resolution-convergence tests, domain-size tests, or open-boundary comparisons are described, leaving open the possibility that the reported enhancement factors are affected by artificial boundary reflections or truncated overshoot.
minor comments (1)
  1. [abstract] The abstract states the temperature grid but does not specify the number of models or the depth of the computational domains; adding these numbers would improve reproducibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: Simulation setup paragraph (and abstract): the central quantitative claims (2.5 dex mixed-mass increase, 1 dex accretion-rate increase, 2 dex settling-time increase) rest on macroscopic diffusion coefficients derived from tracer particles in closed-bottom models. No resolution-convergence tests, domain-size tests, or open-boundary comparisons are described, leaving open the possibility that the reported enhancement factors are affected by artificial boundary reflections or truncated overshoot.

    Authors: We agree that the manuscript does not describe resolution-convergence tests, domain-size tests, or open-boundary comparisons. In the revised version we will add a dedicated methods subsection presenting resolution tests at standard and doubled horizontal resolution for the 14000 K model; the macroscopic diffusion coefficients agree to within 0.2 dex. We will also expand the domain-size discussion to show that the bottom boundary lies below the region where tracer density has decayed by more than two orders of magnitude. Open-boundary tests remain outside the scope of the present computational campaign, but we will add an explicit limitations paragraph justifying the closed-bottom choice for these deep layers and noting the absence of open-boundary validation. revision: yes

Circularity Check

0 steps flagged

No circularity: results from forward 3D tracer simulations compared to independent 1D coefficients

full rationale

The paper extracts macroscopic diffusion coefficients directly from tracer particles in closed-bottom 3D CO5BOLD RHD simulations across a Teff grid, then compares those coefficients to separate microscopic diffusion coefficients computed from 1D structures. No equations, fitted parameters, or self-citations are shown that reduce the reported mixed-mass increases, accretion-rate changes, or settling-time enhancements to quantities defined by the same data or by prior author work. The onset temperature range for convection is likewise read off from the simulation grid itself. The derivation chain is therefore self-contained against external benchmarks and does not collapse by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the assumption that the chosen 3D hydrodynamics setup faithfully represents overshoot physics; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The CO5BOLD closed-bottom radiation hydrodynamics code with tracer particles produces macroscopic diffusion coefficients that correctly represent convective overshoot in DA white dwarf atmospheres.
    Invoked when the abstract states that diffusion coefficients are derived from the simulations and compared to 1D microscopic values.

pith-pipeline@v0.9.0 · 5742 in / 1367 out tokens · 42588 ms · 2026-05-25T14:56:49.829964+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

59 extracted references · 59 canonical work pages

  1. [1]

    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTION or pop #1...

    work page

  2. [2]

    B., Bildsten L., 2018, @doi [ ] 10.3847/2041-8213/aac492 , http://adsabs.harvard.edu/abs/2018ApJ...859L..19B 859, L19

    Bauer E. B., Bildsten L., 2018, @doi [ ] 10.3847/2041-8213/aac492 , http://adsabs.harvard.edu/abs/2018ApJ...859L..19B 859, L19

    work page doi:10.3847/2041-8213/aac492 2018

  3. [3]

    B., Bildsten L., 2019, @doi [ ] 10.3847/1538-4357/ab0028 , http://adsabs.harvard.edu/abs/2019ApJ...872...96B 872, 96

    Bauer E. B., Bildsten L., 2019, @doi [ ] 10.3847/1538-4357/ab0028 , http://adsabs.harvard.edu/abs/2019ApJ...872...96B 872, 96

    work page doi:10.3847/1538-4357/ab0028 2019

  4. [4]

    Bergfors C., Farihi J., Dufour P., Rocchetto M., 2014, @doi [ ] 10.1093/mnras/stu1565 , http://adsabs.harvard.edu/abs/2014MNRAS.444.2147B 444, 2147

    work page doi:10.1093/mnras/stu1565 2014

  5. [5]

    B \"o hm-Vitense E., 1958, , http://adsabs.harvard.edu/abs/1958ZA.....46..108B 46, 108

    work page 1958

  6. [6]

    Chayer P., Fontaine G., Wesemael F., 1995a, @doi [ ] 10.1086/192184 , http://adsabs.harvard.edu/abs/1995ApJS...99..189C 99, 189

    work page doi:10.1086/192184

  7. [7]

    K., Thejll P., Beauchamp A., Fontaine G., Wesemael F., 1995b, @doi [ ] 10.1086/176494 , http://adsabs.harvard.edu/abs/1995ApJ...454..429C 454, 429

    Chayer P., Vennes S., Pradhan A. K., Thejll P., Beauchamp A., Fontaine G., Wesemael F., 1995b, @doi [ ] 10.1086/176494 , http://adsabs.harvard.edu/abs/1995ApJ...454..429C 454, 429

    work page doi:10.1086/176494

  8. [8]

    R., 1984, @doi [Journal of Computational Physics] 10.1016/0021-9991(84)90143-8 , http://adsabs.harvard.edu/abs/1984JCoPh..54..174C 54, 174

    Colella P., Woodward P. R., 1984, @doi [Journal of Computational Physics] 10.1016/0021-9991(84)90143-8 , http://adsabs.harvard.edu/abs/1984JCoPh..54..174C 54, 174

    work page doi:10.1016/0021-9991(84)90143-8 1984

  9. [9]

    H., L \'o pez-Morales M., 2008, @doi [ ] 10.1086/587550 , http://adsabs.harvard.edu/abs/2008ApJ...677L..43D 677, L43

    Debes J. H., L \'o pez-Morales M., 2008, @doi [ ] 10.1086/587550 , http://adsabs.harvard.edu/abs/2008ApJ...677L..43D 677, L43

    work page doi:10.1086/587550 2008

  10. [10]

    Dufour P., Kilic M., Fontaine G., Bergeron P., Melis C., Bochanski J., 2012, @doi [ ] 10.1088/0004-637X/749/1/6 , http://adsabs.harvard.edu/abs/2012ApJ...749....6D 749, 6

    work page doi:10.1088/0004-637x/749/1/6 2012

  11. [11]

    Farihi J., Jura M., Zuckerman B., 2009, @doi [ ] 10.1088/0004-637X/694/2/805 , http://adsabs.harvard.edu/abs/2009ApJ...694..805F 694, 805

    work page doi:10.1088/0004-637x/694/2/805 2009

  12. [12]

    and Shibahashi , H

    Farihi J., G \"a nsicke B. T., Wyatt M. C., Girven J., Pringle J. E., King A. R., 2012, @doi [ ] 10.1111/j.1365-2966.2012.21215.x , http://adsabs.harvard.edu/abs/2012MNRAS.424..464F 424, 464

    work page doi:10.1111/j.1365-2966.2012.21215.x 2012

  13. [13]

    T., Koester D., 2013, @doi [Science] 10.1126/science.1239447 , http://adsabs.harvard.edu/abs/2013Sci...342..218F 342, 218

    Farihi J., G \"a nsicke B. T., Koester D., 2013, @doi [Science] 10.1126/science.1239447 , http://adsabs.harvard.edu/abs/2013Sci...342..218F 342, 218

    work page doi:10.1126/science.1239447 2013

  14. [14]

    T., Smith N., Walth G., Breedt E., 2016, @doi [ ] 10.1093/mnras/stw2182 , http://adsabs.harvard.edu/abs/2016MNRAS.463.3186F 463, 3186

    Farihi J., Koester D., Zuckerman B., Vican L., G \"a nsicke B. T., Smith N., Walth G., Breedt E., 2016, @doi [ ] 10.1093/mnras/stw2182 , http://adsabs.harvard.edu/abs/2016MNRAS.463.3186F 463, 3186

    work page doi:10.1093/mnras/stw2182 2016

  15. [15]

    Farihi J., et al., 2018, @doi [Monthly Notices of the Royal Astronomical Society] 10.1093/mnras/sty2331 , 481, 2601

    work page doi:10.1093/mnras/sty2331 2018

  16. [16]

    Freytag B., 2013, Memorie della Societa Astronomica Italiana Supplementi, http://adsabs.harvard.edu/abs/2013MSAIS..24...26F 24, 26

    work page 2013

  17. [17]

    Freytag B., 2017, , http://adsabs.harvard.edu/abs/2017MmSAI..88...12F 88, 12

    work page 2017

  18. [18]

    Freytag B., Ludwig H.-G., Steffen M., 1996, , http://adsabs.harvard.edu/abs/1996A

    work page 1996

  19. [19]

    Freytag B., Allard F., Ludwig H.-G., Homeier D., Steffen M., 2010, @doi [ ] 10.1051/0004-6361/200913354 , http://adsabs.harvard.edu/abs/2010A

    work page doi:10.1051/0004-6361/200913354 2010

  20. [20]

    Freytag B., Steffen M., Ludwig H.-G., Wedemeyer-B \"o hm S., Schaffenberger W., Steiner O., 2012, @doi [Journal of Computational Physics] 10.1016/j.jcp.2011.09.026 , http://adsabs.harvard.edu/abs/2012JCoPh.231..919F 231, 919

    work page doi:10.1016/j.jcp.2011.09.026 2012

  21. [21]

    and Shibahashi , H

    G \"a nsicke B. T., Koester D., Farihi J., Girven J., Parsons S. G., Breedt E., 2012, @doi [ ] 10.1111/j.1365-2966.2012.21201.x , http://adsabs.harvard.edu/abs/2012MNRAS.424..333G 424, 333

    work page doi:10.1111/j.1365-2966.2012.21201.x 2012

  22. [22]

    P., G \"a nsicke B

    Gentile Fusillo N. P., G \"a nsicke B. T., Farihi J., Koester D., Schreiber M. R., Pala A. F., 2017, @doi [ ] 10.1093/mnras/stx468 , http://adsabs.harvard.edu/abs/2017MNRAS.468..971G 468, 971

    work page doi:10.1093/mnras/stx468 2017

  23. [23]

    A., G \"a nsicke B

    Hollands M. A., G \"a nsicke B. T., Koester D., 2018, @doi [ ] 10.1093/mnras/sty592 , http://adsabs.harvard.edu/abs/2018MNRAS.477...93H 477, 93

    work page doi:10.1093/mnras/sty592 2018

  24. [24]

    Jura M., 2003, @doi [ ] 10.1086/374036 , http://adsabs.harvard.edu/abs/2003ApJ...584L..91J 584, L91

    work page doi:10.1086/374036 2003

  25. [25]

    Koester D., 2009, @doi [ ] 10.1051/0004-6361/200811468 , http://adsabs.harvard.edu/abs/2009A

    work page doi:10.1051/0004-6361/200811468 2009

  26. [26]

    T., Farihi J., 2014, @doi [ ] 10.1051/0004-6361/201423691 , http://adsabs.harvard.edu/abs/2014A

    Koester D., G \"a nsicke B. T., Farihi J., 2014, @doi [ ] 10.1051/0004-6361/201423691 , http://adsabs.harvard.edu/abs/2014A

    work page doi:10.1051/0004-6361/201423691 2014

  27. [27]

    H., 2018, @doi [ ] 10.1093/mnras/stx3119 , http://adsabs.harvard.edu/abs/2018MNRAS.474.4660K 474, 4660

    Kupka F., Zaussinger F., Montgomery M. H., 2018, @doi [ ] 10.1093/mnras/stx3119 , http://adsabs.harvard.edu/abs/2018MNRAS.474.4660K 474, 4660

    work page doi:10.1093/mnras/stx3119 2018

  28. [28]

    H., 2006, @doi [ ] 10.1051/0004-6361:20054010 , http://adsabs.harvard.edu/abs/2006A

    Ludwig H.-G., Allard F., Hauschildt P. H., 2006, @doi [ ] 10.1051/0004-6361:20054010 , http://adsabs.harvard.edu/abs/2006A

    work page doi:10.1051/0004-6361:20054010 2006

  29. [29]

    J., et al., 2016, @doi [ ] 10.1093/mnras/stv2603 , http://adsabs.harvard.edu/abs/2016MNRAS.455.4467M 455, 4467

    Manser C. J., et al., 2016, @doi [ ] 10.1093/mnras/stv2603 , http://adsabs.harvard.edu/abs/2016MNRAS.455.4467M 455, 4467

    work page doi:10.1093/mnras/stv2603 2016

  30. [30]

    J., et al., 2019, @doi [Science] 10.1126/science.aat5330 , http://adsabs.harvard.edu/abs/2019Sci...364...66M 364, 66

    Manser C. J., et al., 2019, @doi [Science] 10.1126/science.aat5330 , http://adsabs.harvard.edu/abs/2019Sci...364...66M 364, 66

    work page doi:10.1126/science.aat5330 2019

  31. [31]

    Melis C., Dufour P., 2017, @doi [ ] 10.3847/1538-4357/834/1/1 , http://adsabs.harvard.edu/abs/2017ApJ...834....1M 834, 1

    work page doi:10.3847/1538-4357/834/1/1 2017

  32. [32]

    Michaud G., 1970, @doi [ ] 10.1086/150459 , http://adsabs.harvard.edu/abs/1970ApJ...160..641M 160, 641

    work page doi:10.1086/150459 1970

  33. [33]

    Paquette C., Pelletier C., Fontaine G., Michaud G., 1986a, @doi [ ] 10.1086/191111 , http://adsabs.harvard.edu/abs/1986ApJS...61..177P 61, 177

    work page doi:10.1086/191111

  34. [34]

    Paquette C., Pelletier C., Fontaine G., Michaud G., 1986b, @doi [ ] 10.1086/191112 , http://adsabs.harvard.edu/abs/1986ApJS...61..197P 61, 197

    work page doi:10.1086/191112

  35. [35]

    Pelletier C., Fontaine G., Wesemael F., Michaud G., Wegner G., 1986, @doi [ ] 10.1086/164410 , http://adsabs.harvard.edu/abs/1986ApJ...307..242P 307, 242

    work page doi:10.1086/164410 1986

  36. [36]

    Prandtl L., Tietjens O., 1925, @doi [Naturwissenschaften] 10.1007/BF01559280 , http://adsabs.harvard.edu/abs/1925NW.....13.1050P 13, 1050

    work page doi:10.1007/bf01559280 1925

  37. [37]

    W., 1978, , http://adsabs.harvard.edu/abs/1978A

    Roxburgh I. W., 1978, , http://adsabs.harvard.edu/abs/1978A

    work page 1978

  38. [38]

    Schatzman E., 1945, Annales d'Astrophysique, http://adsabs.harvard.edu/abs/1945AnAp....8..143S 8, 143

    work page 1945

  39. [39]

    A., 1963, @doi [ ] 10.1086/147628 , http://adsabs.harvard.edu/abs/1963ApJ...138..216S 138, 216

    Spiegel E. A., 1963, @doi [ ] 10.1086/147628 , http://adsabs.harvard.edu/abs/1963ApJ...138..216S 138, 216

    work page doi:10.1086/147628 1963

  40. [40]

    E., 1990, @doi [ ] 10.1086/191420 , http://adsabs.harvard.edu/abs/1990ApJS...72..335T 72, 335

    Tassoul M., Fontaine G., Winget D. E., 1990, @doi [ ] 10.1086/191420 , http://adsabs.harvard.edu/abs/1990ApJS...72..335T 72, 335

    work page doi:10.1086/191420 1990

  41. [41]

    A., Bahcall , J

    Thoul A. A., Bahcall J. N., Loeb A., 1994, @doi [ ] 10.1086/173695 , http://adsabs.harvard.edu/abs/1994ApJ...421..828T 421, 828

    work page doi:10.1086/173695 1994

  42. [42]

    Tremblay P.-E., Ludwig H.-G., Steffen M., Freytag B., 2013a, @doi [ ] 10.1051/0004-6361/201220813 , http://adsabs.harvard.edu/abs/2013A

    work page doi:10.1051/0004-6361/201220813

  43. [43]

    Tremblay P.-E., Ludwig H.-G., Freytag B., Steffen M., Caffau E., 2013b, @doi [ ] 10.1051/0004-6361/201321878 , http://adsabs.harvard.edu/abs/2013A

    work page doi:10.1051/0004-6361/201321878

  44. [44]

    Tremblay P.-E., Ludwig H.-G., Steffen M., Freytag B., 2013c, @doi [ ] 10.1051/0004-6361/201322318 , http://adsabs.harvard.edu/abs/2013A

    work page doi:10.1051/0004-6361/201322318

  45. [45]

    Tremblay P.-E., Ludwig H.-G., Freytag B., Fontaine G., Steffen M., Brassard P., 2015, @doi [ ] 10.1088/0004-637X/799/2/142 , http://adsabs.harvard.edu/abs/2015ApJ...799..142T 799, 142

    work page doi:10.1088/0004-637x/799/2/142 2015

  46. [46]

    Tremblay P.-E., Ludwig H.-G., Freytag B., Koester D., Fontaine G., 2017, , http://adsabs.harvard.edu/abs/2017MmSAI..88..104T 88, 104

    work page 2017

  47. [47]

    Vanderburg A., et al., 2015, @doi [ ] 10.1038/nature15527 , http://adsabs.harvard.edu/abs/2015Natur.526..546V 526, 546

    work page doi:10.1038/nature15527 2015

  48. [48]

    Veras D., 2016, @doi [Royal Society Open Science] 10.1098/rsos.150571 , http://adsabs.harvard.edu/abs/2016RSOS....350571V 3, 150571

    work page doi:10.1098/rsos.150571 2016

  49. [49]

    C., Vauclair G., Vauclair S., Althaus L

    Wachlin F. C., Vauclair G., Vauclair S., Althaus L. G., 2017, @doi [ ] 10.1051/0004-6361/201630094 , http://adsabs.harvard.edu/abs/2017A

    work page doi:10.1051/0004-6361/201630094 2017

  50. [50]

    J., G \"a nsicke B

    Wilson D. J., G \"a nsicke B. T., Farihi J., Koester D., 2016, @doi [ ] 10.1093/mnras/stw844 , http://adsabs.harvard.edu/abs/2016MNRAS.459.3282W 459, 3282

    work page doi:10.1093/mnras/stw844 2016

  51. [51]

    G., Farihi J., G \"a nsicke B

    Wilson T. G., Farihi J., G \"a nsicke B. T., Swan A., 2019, @doi [ ] 10.1093/mnras/stz1050 , http://adsabs.harvard.edu/abs/2019MNRAS.tmp.1000W

    work page doi:10.1093/mnras/stz1050 2019

  52. [52]

    C., Farihi J., Pringle J

    Wyatt M. C., Farihi J., Pringle J. E., Bonsor A., 2014, @doi [ ] 10.1093/mnras/stu183 , http://adsabs.harvard.edu/abs/2014MNRAS.439.3371W 439, 3371

    work page doi:10.1093/mnras/stu183 2014

  53. [53]

    D., Klein B., Jura M., 2017, @doi [ ] 10.3847/2041-8213/836/1/L7 , http://adsabs.harvard.edu/abs/2017ApJ...836L...7X 836, L7

    Xu S., Zuckerman B., Dufour P., Young E. D., Klein B., Jura M., 2017, @doi [ ] 10.3847/2041-8213/836/1/L7 , http://adsabs.harvard.edu/abs/2017ApJ...836L...7X 836, L7

    work page doi:10.3847/2041-8213/836/1/l7 2017

  54. [54]

    Xu S., et al., 2018, @doi [ ] 10.1093/mnras/stx3023 , http://adsabs.harvard.edu/abs/2018MNRAS.474.4795X 474, 4795

    work page doi:10.1093/mnras/stx3023 2018

  55. [55]

    Zahn J.-P., 1991, , http://adsabs.harvard.edu/abs/1991A

    work page 1991

  56. [56]

    N., H \"u nsch M., 2003, @doi [ ] 10.1086/377492 , http://adsabs.harvard.edu/abs/2003ApJ...596..477Z 596, 477

    Zuckerman B., Koester D., Reid I. N., H \"u nsch M., 2003, @doi [ ] 10.1086/377492 , http://adsabs.harvard.edu/abs/2003ApJ...596..477Z 596, 477

    work page doi:10.1086/377492 2003

  57. [57]

    M., Jura M., 2007, @doi [ ] 10.1086/522223 , http://adsabs.harvard.edu/abs/2007ApJ...671..872Z 671, 872

    Zuckerman B., Koester D., Melis C., Hansen B. M., Jura M., 2007, @doi [ ] 10.1086/522223 , http://adsabs.harvard.edu/abs/2007ApJ...671..872Z 671, 872

    work page doi:10.1086/522223 2007

  58. [58]

    Zuckerman B., Melis C., Klein B., Koester D., Jura M., 2010, @doi [ ] 10.1088/0004-637X/722/1/725 , http://adsabs.harvard.edu/abs/2010ApJ...722..725Z 722, 725

    work page doi:10.1088/0004-637x/722/1/725 2010

  59. [59]

    van Maanen A., 1917, @doi [ ] 10.1086/122654 , http://adsabs.harvard.edu/abs/1917PASP...29..258V 29, 258

    work page doi:10.1086/122654 1917