A highly magnetised and rapidly rotating white dwarf as small as the Moon

White dwarfs represent the last stage of evolution of stars with mass less than about eight times that of the Sun and, like other stars, are often found in binaries. If the orbital period of the binary is short enough, energy losses from gravitational-wave radiation can shrink the orbit until the two white dwarfs come into contact and merge. Depending on the component masses, the merger can lead to a supernova of type Ia or result in a massive white dwarf. In the latter case, the white dwarf remnant is expected to be highly magnetised because of the strong magnetic dynamo that should arise during the merger, and be rapidly spinning from the conservation of the orbital angular momentum. Here we report observations of a white dwarf, ZTF J190132.9+145808.7, that exhibits these properties, but to an extreme: a rotation period of 6.94 minutes, a magnetic field ranging between 600 megagauss and 900 megagauss over its surface, and a stellar radius of about 2,100 km, slightly larger than the radius of the Moon. Such a small radius implies that the star's mass is close to the maximum white-dwarf mass, or Chandrasekhar mass. ZTF J190132.9+145808.7 is likely to be cooling through the Urca processes (neutrino emission from electron capture on sodium) because of the high densities reached in its core.

White dwarfs represent the last stage of evolution of stars with mass less than about eight times that of the Sun and, like other stars, are often found in binaries 1,2 . If the orbital period of the binary is short enough, energy losses from gravitational-wave radiation can shrink the orbit until the two white dwarfs come into contact and merge 3 . Depending on the component masses, the merger can lead to a supernova of type Ia or result in a massive white dwarf 4 .
In the latter case, the white dwarf remnant is expected to be highly magnetised 5,6 because of the strong magnetic dynamo that should arise during the merger, and be rapidly spinning from the conservation of the orbital angular momentum 7 . Here we report observations of a white dwarf, ZTF J190132.9+145808.7, that exhibits these properties, but to an extreme: a rotation period of 6.94 minutes, a magnetic field ranging between 600 megagauss and 900 megagauss over its surface, and a stellar radius of 2, 140 +160 −230 km, slightly larger than the radius of the Moon. Such a small radius implies that the star's mass is close to the maximum white-dwarf mass, or Chandrasekhar mass. ZTF J190132.9+145808.7 is likely to be cooling through the Urca processes (neutrino emission from electron capture on sodium) because of the high densities reached in its core.
Using the The Zwicky Transient Facility 8 (ZTF) we searched for short period objects that lie below the main white dwarf cooling sequence in the Gaia 9 colour-magnitude diagram (see Hale telescope, confirmed a period of 6.94 minutes (see Fig. 2). The period of ZTF J1901+1458 is unusually short for a white dwarf, as white dwarf rotational periods typically are upwards of 2 hours 11 . The period and ephemeris are listed in Table 1.
We undertook phase-resolved spectroscopy using the Low Resolution Imaging Spectrometer (LRIS) 12 on the 10-m W. M. Keck I Telescope. As can be seen from Fig. 3, the phase-averaged spectrum exhibits broad and shallow features that we identify as hydrogen absorption lines in a high magnetic field. The presence of a strong magnetic field results in splitting and proportional shifting of the zero-field energy levels, leading to line broadening. To identify the field strength, we considered all the allowed bound-bound hydrogen transitions (tabulated in 13 ) and, as shown in  Table 2.
We find that most of the spectral features are well characterised by a magnetic field strength of 800 million Gauss (MG, red horizontal line in Fig. 3), comparable to the field detected on the most magnetic white dwarfs known 15 . As the absorption features indicate an average field strength over the surface, the field at the magnetic pole is bound to be higher. From the phaseresolved spectra (Extended Data Fig. 4 and 5) we see that some of the features become narrower or broader depending on the phase, and the feature at ∼ 4600Å shifts in wavelength, which accounts for the dip at ∼ 4500Å in the co-added spectrum. This means that in some regions of the surface the magnetic field is as low as 600 MG. The explanation for the photometric variation, confirmed by the variations of absorption features with phase, is thus the combination of magnetic dichroism and rotation: the high magnetic field causes variations in the continuum opacities and in the surface temperature, and therefore, as the star rotates, we detect changes in flux as a function  Figure 1: Gaia colour-magnitude diagram. Gaia color-magnitude diagram for the white dwarfs that are within 100 pc from Earth and within the SDSS footprint 28 , where the x-axis depicts the difference between the Gaia BP and RP bands, and the y-axis the absolute magnitude in the Gaia G filter. Solid lines show theoretical cooling tracks for white dwarfs with masses between 0.6 M (top) and 1.28 M (bottom), equally spaced in mass; the atmosphere is assumed to be hydrogendominated 29 and the interior composition to be carbon-oxygen 30 for M < 1.1 M and oxygenneon 19 for M > 1.1 M . Coloured markers indicate white dwarfs for which a magnetic field was measured 15 . ZTF J1901+1458 is shown as a red star, and its location in the colour-magnitude diagram reveals its high mass. Vertical dashed lines indicate the location of the ZZ-Ceti instability strip 31 (the pre-white dwarf, or DOV, instability strip lies above the plot 32, 33 ) . Reddening corrections were applied only to ZTF J1901+1458; as the objects in the sample are close, reddening is expected to be small. 1σ error bars are smaller than the size of the coloured markers,and are omitted for the black background dots for clarity.  phase-folded at a period of 6.94 minutes in the g -band (a) and in the r -band (b). The flux has been normalised to the minimum of the lightcurve in each band. The amplitude of the photometric variation is higher in the g -band (about 3% peak-to-peak) than in the r -band (∼ 1.5%). Additionally, the two filters show a difference in phase: the red lags the green by about 51 s. The right panels show the similarly normalised ZTF discovery lightcurve in the ZTF g-band (c) and r−band (d). The error bars indicate 1σ errors. 5 of the field strength across the stellar surface. Depending on the magnetic field configuration, this dichroism can account for up to 10% photometric variation 14 , so it can easily account for the 3% amplitude observed in ZTF J1901+1458. ZTF J1901+1458's period could also be consistent with non-radial pulsations; however, its temperature and surface gravity place it far away from theoretical predictions for known instabilities (see Fig. 1), and its magnetic field may be strong enough to suppress gravity-mode pulsations (see the Method section for further discussion).
To determine the temperature and radius of the white dwarf, we obtained photometric measurements in the near UV using the UVOT (Ultraviolet/Optical Telescope) instrument 16 on the Neil Gehrels Swift Observatory 17 while the Pan-STARRS Survey 18 (PS1) and the Gaia mission provided optical photometry and parallax, respectively. We estimated the temperature, reddening and radius from the photometry, by comparing the observations with non-magnetic white dwarf atmosphere models. Because the white dwarf is very hot, the photometric constraints on the temperature are weak; however, the precise distance measurement from Gaia allows us to obtain a good estimate for the radius. We found the effective temperature, stellar radius and interstellar reddening to be T eff = 46, 000 +19,000 −8,000 K, R * = 2, 140 +160 −230 km and E(B − V ) = 0.044 +0.017 −0.015 , respectively (see the Methods section for more details). The radius is smaller than those measured for other white dwarfs and only slightly larger than that of the Moon. As explained below, the small radius also means that ZTF J1901+1458 maybe the most massive white dwarf yet discovered.
The mass can be inferred from the mass-radius relation which, as can be seen in  At the densities reached in the centre of ZTF J1901+1458, the nuclei of some elements may undergo electron capture (also called inverse beta-decay), removing electrons that contribute to the degeneracy pressure that keeps the star from collapsing. This lowers the maximum mass that can be sustained against gravity and reduces the equilibrium radius for a fixed mass (see Fig. 4, solid red curve). If ZTF J1901+1458 has an oxygen-neon internal composition (as is expected from its mass 7, 19 ), its central density is right at the threshold for electron capture on 23 Na, and its mass is within 2% of the highest possible mass for a white dwarf.
Neutrinos produced by inverse beta-decay escape and carry away energy, contributing to the cooling of the white dwarf. Based on the luminosity of the white dwarf 19   temperature in the core to be about 2 − 3 × 10 7 K. At such a high central temperature and density, the neutrino cooling of ZTF J1901+1458 will dominated by the "Urca" process 24, 25 acting on 23 Na.
This unusual neutrino cooling makes an age determination difficult. A recent work 26 studied the evolution of Urca-cooling white dwarfs; from the same models, we estimate the cooling age of ZTF J1901+1458 to be between 10 and 100 Myr.
The mass-radius relation in Fig. 4 was calculated assuming that the core composition is homogeneous -a good assumption since ZTF J1901+1458 is less than 100 Myr old. However, over a few hundred million years, the heaviest elements, including Na, will gradually sink to the centre 27 . If the star lies at the small end of the radius constraint and if at least sixty percent of the 23 Na manages to sink and undergo beta decay before the core crystallises and sedimentation stops, electron-capture on 24 Mg would ensue. The star would shrink and the internal pressure would no longer be able to support the star, as the maximum allowed mass for the new composition would be lower than the mass of the white dwarf (see the Methods section for a more detailed discussion).
The star would therefore collapse and heat up, leading to the onset of electron capture onto Ne and to the ignition of oxygen nuclear burning. The white dwarf would then undergo a disruptive thermonuclear supernova or implode to form a neutron star 25 .
The probability of collapse is highly uncertain, as it depends on the timescales of sedimentation and crystallisation, both unconstrained at these high masses. However, the possibility of this novel formation channel for neutron stars is intriguing. If, upon collapse, no angular momentum is lost and magnetic flux is conserved, the newly born neutron star, with a magnetic field strength of ∼ 2 × 10 13 G and a spin period of ∼ 15 ms, would resemble a young pulsar. Owing to the gradual collapse, the neutron star would likely not receive a strong natal velocity kick. We would therefore expect such a neutron star population to be more concentrated to the Galactic plane. Furthermore, the proximity of ZTF J1901+1458 (41 parsecs) means it is not a rare type of object, and thus this formation channel would contribute appreciably to the total neutron star population. ZTF is currently discovering large numbers of similarly massive and rapidly variable white dwarfs. This enlarged sample will help us better understand the origin and fate of such objects. combined data from multiple filters by computing the median magnitude in each filter, and shifting the gand i-band so that their median magnitude matched the r-band data. We used a graphics processing unit (GPU) implementation of the conditional entropy period finding algorithm 36 . We cross-matched our candidates with the Gaia DR2 catalogue 37 and visually inspected the lightcurves of those objects that lie below the main white dwarf cooling track in the Gaia colour-magnitude diagram. ZTF J1901+1458 stood out because of the high-significance detection of its short period and its blue and faint location in the colour-magnitude diagram (see Fig. 1).
Magnetic Field At low magnetic field strengths, the effect of magnetic field on hydrogen transitions can be calculated as a perturbation to the zero-field electron wavefunction. The perturbation lifts the degeneracy in the m quantum number, and each transition is split into three Zeeman components, corresponding to a change in m of +1, 0 and −1. In the strong regime (above ∼ 100 MG), the magnetic and Coulomb terms are comparable, and the wavefunction does not have a spherical symmetry anymore. The perturbation method is not viable in this case, and the energies and oscillator strengths of hydrogen transitions in this regime were calculated using variational methods [38][39][40][41] .
An important characteristic of high-field transitions is that some transitions become 'stationary', i.e. they go through a minimum or a maximum in energy and, at those field strengths, appreciable changes in B only yield small changes in wavelength. The magnetic broadening at these transitions is considerably reduced, and therefore the absorption features are still detectable even after averaging the field over the surface of the white dwarf. We considered all the bound-bound transi- In particular, some of the features become narrower or broader at different times, indicating that the field is more or less uniform over the surface at different rotation phases. Additionally, the feature at ∼ 4600Å(line 7) shifts in wavelength, going as low as ∼ 4500Å, which indicates that the magnetic field is as low as 600 MG on some regions on the surface of the white dwarf. This is also confirmed by the contamination of line 10 by line 11.  The best fit is shown in Extended Data Fig. 1, while the left panel of Extended Data Fig. 3 illustrates the corner plots. In the fit, we assumed the nominal value of the Gaia parallax for the dis- Since ZTF J1901+1458 is so compact, general relativistic corrections are important and therefore, contrary to H&S, we did not integrate the Newtonian hydrostatic equilibrium equation, but rather the Tolman-Oppenheimer-Volkoff (TOV) equation 48,49 . The results are shown in Fig. 4 for several compositions. Our calculations were performed assuming the zero-temperature limit because the temperature of the white dwarf is insufficient to change its structure significantly. For the composition of the carbon-burning ash, we use the results from Camisassa et al. 19 . For the density thresholds for the electron capture onto magnesium and neon 50 , we use the values of 4×10 9 g cm −3 and 9 × 10 9 g cm −3 respectively. For sodium 25 we use 1.7 × 10 9 g cm −3 . We estimate the temperature at a density of 1.7 × 10 9 g cm −3 to be about 3 × 10 7 K; therefore, the neutrino cooling ZTF J1901+1458, unlike for all other known white dwarfs, is cooling dominated by the Urca process on 23 Na 24, 25 which makes an age determination difficult. A recent work 26 simulated the evolution of Urca-cooling white dwarfs with the Modules for Experiments in Stellar Astrophysics code (MESA) [51][52][53][54][55] , showing that the Urca processes are the main cooling mechanism in the core of stars like ZTF J1901+1458 when they are younger than about 30 Myr. From the same models, we can estimate the cooling age of the white dwarf to be between 10 and 100 Myr.
Sedimentation and collapse The solid red curve in Fig. 4 was calculated assuming that the core composition of the white dwarf is homogeneous and that all the sodium currently at densities above the threshold for electron capture has already undergone inverse beta decay, and similarly for magnesium and neon. As the sedimentation proceeds, much of the remaining 23 Na will sink to densities above the threshold and undergo beta-decay, reducing the number of electrons in the star and reducing its radius. This means that the solid red curve in Fig. 4 will be lowered, because the equilibrium radius for any given mass will be smaller, and therefore the red triangle (the maximum mass allowed) will sink. This process can only be stopped if the core crystallises before enough sodium can reach the centre, as crystallisation would de facto freeze the composition gradient.
If the star lies at the small end of the radius constraint and if at least sixty percent of the 23 Na manages to sink and decay before the core crystallises, electron-capture on 24 Mg would ensue and the radius of the white dwarf would shrink to about 1,550 km. The central density at this point would be 6 × 10 9 g cm −3 , still below the threshold for electron capture on neon, but the mass of the white dwarf would be above the maximum mass (the red triangle in Fig. 4 would have sunk below the current white dwarf mass). The internal pressure would be then insufficient to support the star, and the star would begin to collapse, heat up and start electron capture onto neon and nuclear burning of oxygen.
The possibility of collapse is highly uncertain as it depends on the timescales for crystallisation and for the sedimentation of sodium, both currently poorly constrained, especially for such an extremely massive white dwarf. Furthermore, the inferred temperature and surface gravity of ZTF J1901+1458 characterise it as an unlikely pulsator for known instability mechanisms. The temperature of ZTF J1901+1458, ∼50,000 K, is much higher than the predicted blue edge of the ZZ-Ceti instability strip, located at a temperature of about 12,500-14,000 K 31, 61 , and of the helium white dwarfs (DBV) instability strip, at about 30,000 K 62 (the detection of hydrogen also discourages the DBV interpretation).

Origin of Photometric Variability
The DOV or GW Vir instability strip includes similar and higher temperatures than what we found for ZTF J1901+1458, but both instability mechanisms involved, the κ − γ mechanism for carbon and oxygen and the −mechanism for hydrogen, are inefficient at such high surface gravities (see Follow-up work showed that magnetic fields greater than B ∼ 0.1 MG are sufficient to suppress g modes of typical periods in ZZ Ceti stars 68 , so magnetic suppression is a possibility for ZTF J1901+1458.
For these reasons, we believe the variability of ZTF J1901+1458 is most likely to be caused by rotation rather than pulsations, though we cannot rule out either mechanism. Follow-up observations can place more stringent limits on the presence of other non-harmonic periodicities which would be expected in the pulsation hypothesis. Finally, even if pulsations are the source of variabil-ity in ZTF J1901+1458, it would further enhance the extraordinary nature of this star by making it the most massive pulsating white dwarf and the only known magnetic pulsating white dwarf.
Data Availability Upon request, I.C. will provide the reduced photometric lightcurves and spectroscopic data, and available ZTF data for the object. The spectroscopic data and photometric lightcurves are also available in the GitHub repository https://github.com/ilac/ZTF-J1901-1458, while ZTF data is accessible in the ZTF database. The astrometric and photometric data are already in the public domain, and they are readily accessible in the Gaia and Pan-STARSS catalogues and in the Swift database.
Code availability We used the pyphot package (https://mfouesneau.github.io/docs/pyphot/) and the corner.py package 69 . The LRIS spectra were reduced using the Lpipe pipeline 70 . Upon request, I.C. will provide the code used to analyse the spectroscopic and photometric data.