Theoretical Mass-radius Relation Research Articles

Mardigras: A Visualization Tool of Theoretical Mass–Radius Relations in the Context of Planetary Science

Over the past two decades, mass–radius relations have become a crucial tool for inferring the bulk composition of exoplanets using only their measured masses and radii. These relations, often referred to as isocomposition curves, are derived from interior structure models by calculating the theoretical radius as a function of mass for a given fixed planetary composition. Each mass–radius curve can be influenced by a variety of parameters, such as planetary composition, age, and equilibrium temperature. Navigating this parameter space can be cumbersome, particularly when models or their results are not open-source. To address this challenge, I have developed MAss–Radius DIaGRAm with Sliders, a visualization tool that enables simple, fast, and interactive exploration of the parameter space that governs mass–radius relations for any given model.

Machine learning techniques in studies of the interior structure of rocky exoplanets

Context. Earth-sized exoplanets have been discovered and characterized thanks to new developments in observational techniques, particularly those planets that may have a rocky composition that is comparable to terrestrial planets of the Solar System. Characterizing the interiors of rocky exoplanets is one of the main objectives in investigations of their habitability. Theoretical mass-radius relations are often used as a tool to constrain the internal structure of rocky exoplanets. But one mass-radius curve only represents a single interior structure and a great deal of computation time is required to obtain all possible interior structures that comply with the given mass and radius of a planet. Aims. We apply a machine-learning approach based on mixture density networks (MDNs) to investigate the interiors of rocky exoplanets. We aim to provide a well-trained MDN model to quickly and efficiently predict the interior structure of rocky exoplanets. Methods. We presented a training data set of rocky exoplanets with masses between 0.1 and 10 Earth masses based on three-layer interior models by assuming Earth-like compositions. This data set was then used to train the MDN model to predict the layer thicknesses and core properties of rocky exoplanets, where planetary mass, radius, and water content are inputs to the MDN. The performance of the trained MDN model was investigated in order to discern its predictive ability. Results. The MDN model is found to show good performance in predicting the layer thicknesses and core properties of rocky exoplanets through a comparison with the real solutions obtained by solving the interior models. We also applied the MDN model to the Earth and the super-Earth exoplanet LHS 1140b. The MDN predictions are in good agreement with the interior model solutions within the uncertainties of planetary mass and radius. More importantly, the MDN model takes a much shorter computational time compared to the cost of the interior model calculations, offering a convenient and powerful tool for quickly obtaining information on planetary interiors.

Internal water storage capacity of terrestrial planets and the effect of hydration on theM-Rrelation

Context.The discovery of low density exoplanets in the super-Earth mass regime suggests that ocean planets could be abundant in the galaxy. Understanding the chemical interactions between water and Mg-silicates or iron is essential for constraining the interiors of water-rich planets. Hydration effects have, however, been mostly neglected by the astrophysics community so far. As such effects are unlikely to have major impacts on theoretical mass-radius relations, this is justified as long as the measurement uncertainties are large. However, upcoming missions, such as the PLATO mission (scheduled launch 2026), are envisaged to reach a precision of up to ≈3 and ≈10% for radii and masses, respectively. As a result, we may soon enter an area in exoplanetary research where various physical and chemical effects such as hydration can no longer be ignored.Aims.Our goal is to construct interior models for planets that include reliable prescriptions for hydration of the cores and mantles. These models can be used to refine previous results for which hydration has been neglected and to guide future characterization of observed exoplanets.Methods.We have developed numerical tools to solve for the structure of multi-layered planets with variable boundary conditions and compositions. Here we consider three types of planets: dry interiors, hydrated interiors, and dry interiors plus surface ocean, where the ocean mass fraction corresponds to the mass fraction of the H2O equivalent in the hydrated case.Results.We find H and OH storage capacities in the hydrated planets equivalent to 0−6 wt% H2O corresponding to up to ≈800 km deep ocean layers. In the mass range 0.1 ≤M∕M⊕≤ 3, the effect of hydration on the total radius is found to be ≤2.5%, whereas the effect of separation into an isolated surface ocean is ≤5%. Furthermore, we find that our results are very sensitive to the bulk composition.

Remnants and ejecta of thermonuclear electron-capture supernovae

The explosion mechanism of electron-capture supernovae (ECSNe) remains equivocal: it is not completely clear whether these events are implosions in which neutron stars are formed, or incomplete thermonuclear explosions that leave behind bound ONeFe white dwarf remnants. Furthermore, the frequency of occurrence of ECSNe is not known, though it has been estimated to be of the order of a few per cent of all core-collapse supernovae. We attempt to constrain the explosion mechanism (neutron-star-forming implosion or thermonuclear explosion) and the frequency of occurrence of ECSNe using nucleosynthesis simulations of the latter scenario, population synthesis, the solar abundance distribution, pre-solar meteoritic oxide grain isotopic ratio measurements and the white dwarf mass–radius relation. Tracer particles from the 3d hydrodynamic simulations were post-processed with a large nuclear reaction network in order to determine the complete compositional state of the bound ONeFe remnant and the ejecta, and population synthesis simulations were performed in order to estimate the ECSN rate with respect to the CCSN rate. The 3d deflagration simulations drastically overproduce the neutron-rich isotopes 48Ca, 50Ti, 54Cr , 60Fe and several of the Zn isotopes relative to their solar abundances. Using the solar abundance distribution as our constraint, we place an upper limit on the frequency of thermonuclear ECSNe as 1−3% the frequency at which core-collapse supernovae (FeCCSNe) occur. This is on par with or 1 dex lower than the estimates for ECSNe from single stars. The upper limit from the yields is also in relatively good agreement with the predictions from our population synthesis simulations. The 54Cr/52Cr and 50Ti/48Ti isotopic ratios in the ejecta are a near-perfect match with recent measurements of extreme pre-solar meteoritc oxide grains, and 53Cr/52Cr can also be matched if the ejecta condenses before mixing with the interstellar medium. The composition of the ejecta of our simulations implies that ECSNe, including accretion-induced collapse of oxygen-neon white dwarfs, could actually be partial thermonuclear explosions and not implosions that form neutron stars. There is still much work to do to improve the hydrodynamic simulations of such phenomena, but it is encouraging that our results are consistent with the predictions from stellar evolution modelling and population synthesis simulations, and can explain several key isotopic ratios in a sub-set of pre-solar oxide meteoritic grains. Theoretical mass–radius relations for the bound ONeFe WD remnants of these explosions are apparently consistent with several observational WD candidates. The composition of the remnants in our simulations can reproduce several, but not all, of the spectroscopically-determined elemental abundances from one such candidate WD.

A Comprehensive Spectroscopic and Photometric Analysis of DA and DB White Dwarfs from SDSS and Gaia

Abstract We present a detailed spectroscopic and photometric analysis of DA and DB white dwarfs drawn from the Sloan Digital Sky Survey with trigonometric parallax measurements available from the Gaia mission. The temperature and mass scales obtained from fits to ugriz photometry appear reasonable for both DA and DB stars, with almost identical mean masses of and 0.620 M ⊙, respectively. The comparison with similar results obtained from spectroscopy reveals several problems with our model spectra for both pure hydrogen and pure helium compositions. In particular, we find that the spectroscopic temperatures of DA stars exceed the photometric values by ∼10% above T eff ∼ 14,000 K, while for DB white dwarfs, we observe large differences between photometric and spectroscopic masses below T eff ∼ 16,000 K. We attribute these discrepancies to the inaccurate treatment of Stark and van der Waals broadening in our model spectra, respectively. Despite these problems, the mean masses derived from spectroscopy— and 0.625 M ⊙ for the DA and DB stars, respectively—agree extremely well with those obtained from photometry. Our analysis also reveals the presence of several unresolved double degenerate binaries, including DA+DA, DB+DB, DA+DB, and even DA+DC systems. Finally, we take advantage of the Gaia parallaxes to test the theoretical mass–radius relation for white dwarfs. We find that 65% of the white dwarfs are consistent within the 1σ confidence level with the predictions of the mass–radius relation, thus providing strong support to the theory of stellar degeneracy.

The white dwarf mass–radius relation and its dependence on the hydrogen envelope.

We present a study of the dependence of the mass-radius relation for DA white dwarf stars on the hydrogen envelope mass and the impact on the value of log g, and thus the determination of the stellar mass. We employ a set of full evolutionary carbon-oxygen core white dwarf sequences with white dwarf mass between 0.493 and 1.05 Msun. Computations of the pre-white dwarf evolution uncovers an intrinsic dependence of the maximum mass of the hydrogen envelope with stellar mass, i.e., it decreases when the total mass increases. We find that a reduction of the hydrogen envelope mass can lead to a reduction in the radius of the model of up to ~12%. This translates directly into an increase in log g for a fixed stellar mass, that can reach up to 0.11 dex, mainly overestimating the determinations of stellar mass from atmospheric parameters. Finally, we find a good agreement between the results from the theoretical mass-radius relation and observations from white dwarfs in binary systems. In particular, we find a thin hydrogen mass of MH ~ 2 10^-8 Msun, for 40 Eridani B, in agreement with previous determinations. For Sirius B, the spectroscopic mass is 4.3% lower than the dynamical mass. However, the values of mass and radius from gravitational redshift observations are compatible with the theoretical mass-radius relation for a thick hydrogen envelope of MH = 2 10^-6 Msun.

Astrophysical Implications of a New Dynamical Mass for the Nearby White Dwarf 40 Eridani B

Abstract The bright, nearby DA-type white dwarf (WD) 40 Eridani B is orbited by the M dwarf 40 Eri C, allowing determination of the WD’s mass. Until recently, however, the mass depended on orbital elements determined four decades ago, and that mass was so low that it created several astrophysical puzzles. Using new astrometric measurements, the binary-star group at the U.S. Naval Observatory has revised the dynamical mass upward, to 0.573 ± 0.018 M ☉. In this paper, we use model-atmosphere analysis to update other parameters of the WD, including effective temperature, surface gravity, radius, and luminosity. We then compare these results with WD interior models. Within the observational uncertainties, theoretical cooling tracks for CO-core WDs of its measured mass are consistent with the position of 40 Eri B in the H-R diagram; equivalently, the theoretical mass–radius relation (MRR) is consistent with the star’s location in the mass–radius plane. This consistency is, however, achieved only if we assume a “thin” outer hydrogen layer, with q H = M H/M WD ≃ 10−10. We discuss other evidence that a significant fraction of DA WDs have such thin H layers, in spite of the expectation from canonical stellar-evolution theory of “thick” H layers with q H ≃ 10−4. The cooling age of 40 Eri B is ∼122 Myr, and its total age is ∼1.8 Gyr. We present the MRRs for 40 Eri B and three other nearby WDs in visual binaries with precise mass determinations, and show that the agreement of current theory with observations is excellent in all cases.

Measurements of Physical Parameters of White Dwarfs: A Test of the Mass–Radius Relation

Abstract We present a detailed spectroscopic and photometric analysis of 219 DA and DB white dwarfs for which trigonometric parallax measurements are available. Our aim is to compare the physical parameters derived from the spectroscopic and photometric techniques, and then to test the theoretical mass–radius relation for white dwarfs using these results. The agreement between spectroscopic and photometric parameters is found to be excellent, especially for effective temperatures, showing that our model atmospheres and fitting procedures provide an accurate, internally consistent analysis. The values of surface gravity and solid angle obtained, respectively, from spectroscopy and photometry, are combined with parallax measurements in various ways to study the validity of the mass–radius relation from an empirical point of view. After a thorough examination of our results, we find that 73% and 92% of the white dwarfs are consistent within 1σ and 2σ confidence levels, respectively, with the predictions of the mass–radius relation, thus providing strong support to the theory of stellar degeneracy. Our analysis also allows us to identify 15 stars that are better interpreted in terms of unresolved double degenerate binaries. Atmospheric parameters for both components in these binary systems are obtained using a novel approach. We further identify a few white dwarfs that are possibly composed of an iron core rather than a carbon/oxygen core, since they are consistent with Fe-core evolutionary models.

The white dwarf mass-radius relation with Gaia, Hubble and FUSE

AbstractWhite dwarfs are becoming useful tools for many areas of astronomy. They can be used as accurate chronometers over Gyr timescales. They are also clues to the history of star formation in our galaxy. Many of these studies require accurate estimates of the mass of the white dwarf. The theoretical mass-radius relation is often invoked to provide these mass estimates. While the theoretical mass-radius relation is well developed, observational tests of this relation show a much larger scatter in the results than expected. High precision observational tests to confirm this relation are required. Gaia is providing distance measurements which will remove one of the main source of uncertainty affecting most previous observations. We combine Gaia distances with spectra from the Hubble and FUSE satelites to make precise tests of the white dwarf mass-radius relation.

CEMP stars: possible hosts to carbon planets in the early Universe

We explore the possibility of planet formation in the carbon-rich protoplanetary disks of carbon-enhanced metal-poor (CEMP) stars, possible relics of the early Universe. The chemically anomalous abundance patterns ([C/Fe] $\geq$ 0.7) in this subset of low-mass stars suggest pollution by primordial core-collapsing supernovae (SNe) ejecta that are particularly rich in carbon dust grains. By comparing the dust-settling timescale in the protoplanetary disks of CEMP stars to the expected disk lifetime (assuming dissipation via photoevaporation), we determine the maximum distance $r_{max}$ from the host CEMP star at which carbon-rich planetesimal formation is possible, as a function of the host star's [C/H] abundance. We then use our linear relation between $r_{max}$ and [C/H], along with the theoretical mass-radius relation derived for a solid, pure carbon planet, to characterize potential planetary transits across host CEMP stars. Given that the related transits are detectable with current and upcoming space-based transit surveys, we suggest initiating an observational program to search for carbon planets around CEMP stars in hopes of shedding light on the question of how early planetary systems may have formed after the Big Bang.

Rotating neutron stars with exotic cores: masses, radii, stability

A set of theoretical mass-radius relations for rigidly rotating neutron stars with exotic cores, obtained in various theories of dense matter, is reviewed. Two basic observational constraints are used: the largest measured rotation frequency (716 Hz) and the maximum measured mass ($2\;M_\odot$). Present status of measuring the radii of neutron stars is described. The theory of rigidly rotating stars in general relativity is reviewed and limitations of the slow rotation approximation are pointed out. Mass-radius relations for rotating neutron stars with hyperon and quark cores are illustrated using several models. Problems related to the non-uniqueness of the crust-core matching are mentioned. Limits on rigid rotation resulting from the mass-shedding instability and the instability with respect to the axisymmetric perturbations are summarized. The problem of instabilities and of the back-bending phenomenon are discussed in detail. Metastability and instability of a neutron star core in the case of a first-order phase transition, both between pure phases, and into a mixed-phase state, are reviewed. The case of two disjoint families (branches) of rotating neutron stars is discussed and generic features of neutron-star families and of core-quakes triggered by the instabilities are considered.

Constraining the density dependence of the nuclear symmetry energy from an x-ray bursting neutron star

Neutrons stars lighter than the Sun are basically composed of nuclear matter of density up to around twice normal nuclear density. In our recent analyses, we showed that possible simultaneous observations of masses and radii of such neutron stars could constrain $\eta\equiv(K_0L^2)^{1/3}$, a combination of the incompressibility of symmetric nuclear matter $K_0$ and the density derivative of the nuclear symmetry energy $L$ that characterizes the theoretical mass-radius relation. In this paper, we focus on the mass-radius constraint of the X-ray burster 4U 1724-307 given by Suleimanov et al. (2011). We therefrom obtain the constraint that $\eta$ should be larger than around 130 MeV, which in turn leads to $L$ larger than around 110, 98, 89, and 78 MeV for $K_0=180$, 230, 280, and 360 MeV. Such a constraint on $L$ is more or less consistent with that obtained from the frequencies of quasi-periodic oscillations in giant flares observed in soft-gamma repeaters.

An accurate mass and radius measurement for an ultracool white dwarf

Studies of cool white dwarfs in the solar neighbourhood have placed a limit on the age of the Galactic disk of 8-9 billion years. However, determining their cooling ages requires the knowledge of their effective temperatures, masses, radii, and atmospheric composition. So far, these parameters could only be inferred for a small number of ultracool white dwarfs for which an accurate distance is known, by fitting their spectral energy distributions (SEDs) in conjunction with a theoretical mass-radius relation. However, the mass-radius relation remains largely untested, and the derived cooling ages are hence model-dependent. Here we report direct measurements of the mass and radius of an ultracool white dwarf in the double-lined eclipsing binary SDSS J013851.54-001621.6. We find M(WD)=0.529+/-0.010Msol and R(WD)=0.0131+/-0.0003Rsol. Our measurements are consistent with the mass-radius relation and we determine a robust cooling age of 9.5 billion years for the 3570K white dwarf. We find that the mass and radius of the low mass companion star, M(sec)=0.132+/-0.003Msol and R(sec)=0.165+/-0.001Rsol, are in agreement with evolutionary models. We also find evidence that this >9.5 Gyr old M5 star is still active, far beyond the activity lifetime for a star of its spectral type. This is likely caused by the high tidally-enforced rotation rate of the star. The companion star is close to filling its Roche lobe and the system will evolve into a cataclysmic variable in only 70 Myr. Our direct measurements demonstrate that this system can be used to calibrate ultracool white dwarf atmospheric models.

OBSERVATIONAL CONSTRAINTS ON THE DEGENERATE MASS-RADIUS RELATION

The white dwarf mass-radius relationship is fundamental to modern astrophysics. It is central to routine estimation of DA white dwarf masses derived from spectroscopic temperatures and gravities. It is also the basis for observational determinations of the white dwarf initial-final mass relation. Nevertheless, definitive and detailed observational confirmations of the mass-radius relation (MRR) remain elusive due to a lack of sufficiently accurate white dwarf masses and radii. Current best estimates of masses and radii allow only broad conclusions about the expected inverse relation between masses and radii in degenerate stars. In this paper we examine a restricted set of 12 DA white dwarf binary systems for which accurate (1) trigonometric parallaxes, (2) spectroscopic effective temperatures and gravities, and (3) gravitational redshifts are available. We consider these three independent constraints on mass and radius in comparison with an appropriate evolved MRR for each star. For the best-determined systems it is found that the DA white dwarfs conform to evolved theoretical MRRs at the 1-{\sigma} to 2-{\sigma} level. For the white dwarf 40 Eri B (WD0413-077) we find strong evidence for the existence of a "thin" hydrogen envelope. For other stars improved parallaxes will be necessary before meaningful comparisons are possible. For several systems current parallaxes approach the precision required for the state-of-the-art mass and radius determinations that will be obtained routinely from the Gaia mission. It is demonstrated here how these anticipated results can be used to firmly constrain details of theoretical mass-radius determinations.

High-resolution optical spectroscopy of Praesepe white dwarfs

We present the results of a high-resolution optical spectroscopic study of nine white dwarf candidate members of Praesepe undertaken with the VLT and Ultraviolet and Visual Echelle Spectrograph. We find, contrary to a number of previous studies, that WD0836+201 (LB390, EG59) and WD0837+199 (LB393, EG61) are magnetic and non-magnetic white dwarfs, respectively. Subsequently, we determine the radial velocities for the eight non-magnetic degenerates and provide compelling evidence that WD0837+185 is a radial velocity variable and possibly a double-degenerate system. We also find that our result for WD0837+218, in conjunction with its projected spatial location and position in initial mass–final mass space, argues it is more likely to be a field star than a cluster member. After eliminating these two white dwarfs, and WD0836+199 which has no clean Sloan Digital Sky Survey photometry, we use the remaining five stars to substantiate modern theoretical mass–radius relations for white dwarfs. In light of our new results, we re-examine the white dwarf members of Praesepe and use them to further constrain the initial mass–final mass relation (IFMR). We find a near-monotonic IFMR, which can still be adequately represented by simple linear function with only one outlier which may have formed from a blue straggler star.

Mass-radius relations for massive white dwarf stars

We present detailed theoretical mass-radius relations for massive white dwarf stars with oxygen-neon cores. This work is motivated by recent observational evidence about the existence of white dwarf stars with very high surface gravities. Our results are based on evolutionary calculations that take into account the chemical composition expected from the evolutionary history of massive white dwarf progenitors. We present theoretical mass-radius relations for stellar mass values ranging from 1.06 to with a step of and effective temperatures from 150 000 K to ≈5000 K. A novel aspect predicted by our calculations is that the mass-radius relation for the most massive white dwarfs exhibits a marked dependence on the neutrino luminosity. Extensive tabulations for massive white dwarfs, accessible from our web site, are presented as well.

Hubble Space Telescope spectroscopy of the Balmer lines in Sirius B★

Sirius B is the nearest and brightest of all white dwarfs, but it is very difficult to observe at visible wavelengths due to the overwhelming scattered light contribution from Sirius A. However, from space we can take advantage of the superb spatial resolution of the Hubble Space Telescope to resolve the A and B components. Since the closest approach in 1993, the separation between the two stars has become increasingly favourable and we have recently been able to obtain a spectrum of the complete Balmer line series for Sirius B using HST?s Space Telescope Imaging Spectrograph (STIS). The quality of the STIS spectra greatly exceed that of previous ground-based spectra, and can be used to provide an important determination of the stellar temperature (Teff = 25193K) and gravity (log g = 8.556). In addition we have obtained a new, more accurate, gravitational red-shift of 80.42 +/- 4.83 km s-1 for Sirius B. Combining these results with the photometric data and the Hipparcos parallax we obtain new determinations of the stellar mass for comparison with the theoretical mass-radius relation. However, there are some disparities between the results obtained independently from log g and the gravitational redshift which may arise from flux losses in the narrow 50x0.2arcsec slit. Combining our measurements of Teff and log g with the Wood (1995) evolutionary mass-radius relation we get a best estimate for the white dwarf mass of 0.978 M. Within the overall uncertainties, this is in agreement with a mass of 1.02 M obtained by matching our new gravitational red-shift to the theoretical M/R relation.

Analysis of a Very Massive DA White Dwarf via the Trigonometric Parallax and Spectroscopic Methods

By two different methods, we show that LHS 4033 is an extremely massive white dwarf near its likely upper mass limit for destruction by unstable electron captures. From the accurate trigonometric parallax reported herein, the effective temperature (Teff = 10,900 K) and the stellar radius (R = 0.00368 R☉) are directly determined from the broadband spectral energy distribution—the parallax method. The effective temperature and surface gravity are also estimated independently from the simultaneous fitting of the observed Balmer line profiles with those predicted from pure-hydrogen model atmospheres—the spectroscopic method (Teff = 10,760 K, log g = 9.46). The mass of LHS 4033 is then inferred from theoretical mass-radius relations appropriate for white dwarfs. The parallax method yields a mass estimate of 1.310-1.330 M☉ for interior compositions ranging from pure magnesium to pure carbon, respectively, while the spectroscopic method yields an estimate of 1.318-1.335 M☉ for the same core compositions. This star is the most massive white dwarf for which a robust comparison of the two techniques has been made.

Measuring the Fundamental Parameters of Hot Hydrogen-Rich White Dwarfs

This review considers the observations of hot, hydrogen-rich white dwarf stars, with particular reference to measurements of temperature, surface gravity and composition. Spectroscopic data from a variety of wavelength ranges are required for this work and, in particular, the important contributions from optical, ultraviolet and extreme ultraviolet studies are discussed. Using the values of Teff and log g determined for an individual white dwarf, estimates of mass and radius might be derived from the theoretical mass-radius relation. The issue of the accuracy of the theoretical mass-radius calculations and the prospects for making empirical tests using observational data are outlined.

Confirmation of eclipses in KPD 0422+5421, a binary containing a white dwarf and a subdwarf B star

We report additional photometric CCD observations of KPD 0422+5421, a binary with an orbital period of 2.16 hours which contains a subdwarf B star (sdB) and a white dwarf. There are two main results of this work. First, the light curve of KPD 0422+5421 contains two distinct periodic signals, the 2.16 hour ellipsoidal modulation discovered by Koen, Orosz, & Wade (1998) and an additional modulation at 7.8 hours. This 7.8 hour modulation is clearly not sinusoidal: the rise time is about 0.25 in phase, whereas the decay time is 0.75 in phase. Its amplitude is roughly half of the amplitude of the ellipsoidal modulation. Second, after the 7.8 hour modulation is removed, the light curve folded on the orbital period clearly shows the signature of the transit of the white dwarf across the face of the sdB star and the signature of the occultation of the white dwarf by the sdB star. We used the Wilson-Devinney code to model the light curve to obtain the inclination, the mass ratio, and the Omega potentials, and a Monte Carlo code to compute confidence limits on interesting system parameters. We find component masses of M_sdB = 0.36 +/- 0.16 solar masses and M_WD = 0.47 +/- 0.16 solar masses (M_total = 0.86 +/- 0.35 solar masses, 68 per cent confidence limits). If we impose an additional constraint and require the computed mass and radius of the white dwarf to be consistent with a theoretical mass-radius relation, we find M_sdB = 0.511 +0.047 -0.050 solar masses and M_WD = 0.526 +0.033 -0.030 solar masses (68 per cent confidence limits). In this case the total mass of the system is less than 1.4 solar masses at the 99.99 per cent confidence level. We briefly discuss possible interpretations of the 7.8 hour modulation and the importance of KPD 0422+5421 as a member of a rare class of evolved binaries.