# A 5.3-minute-period pulsing white dwarf in a binary detected from radio to
X-rays

Ingrid Pelisoli  T. R. Marsh  David A. H. Buckley  I. Heywood  Stephen. B.
Potter  Axel Schwope  Jaco Brink  Annie Standke  P. A. Woudt  S. G. Parsons
M. J. Green  S. O. Kepler  James Munday  A. D. Romero  E. Breedt  A. J. Brown
V. S. Dhillon  M. J. Dyer  P. Kerry  S. P. Littlefair  D. I. Sahman  J. F.
Wild

White dwarf stars are the most common stellar fossils. When in binaries, they
make up the dominant form of compact object binary within the Galaxy and can
offer insight into different aspects of binary formation and evolution. One of
the most remarkable white dwarf binary systems identified to date is AR
Scorpii (henceforth AR Sco). AR Sco is composed of an M-dwarf star and a
rapidly-spinning white dwarf in a 3.56-hour orbit. It shows pulsed emission
with a period of 1.97 minutes over a broad range of wavelengths, which led to
it being known as a white dwarf pulsar. Both the pulse mechanism and the
evolutionary origin of AR Sco provide challenges to theoretical models. Here
we report the discovery of the first sibling of AR Sco, J191213.72$-$441045.1
(henceforth J1912$-$4410), which harbours a white dwarf in a 4.03-hour orbit
with an M-dwarf and exhibits pulsed emission with a period of 5.30 minutes.
This discovery establishes binary white dwarf pulsars as a class and provides
support for proposed formation models for white dwarf pulsars.

The white dwarf pulsar AR Sco is detected over a broad range of wavelengths,
from radio [1] to X-rays [2]. The spin-down of its rapidly-rotating white
dwarf provides enough energy to power the pulses [3], but the exact driving
mechanism is not fully understood. Unlike in neutron star pulsars, where no
companion is required, binarity seems to play an important role in AR Sco’s
pulses. The observed periodicity of 1.97 min is consistent with a reprocessed
frequency, the beat frequency between the 1.95 min spin period of the white
dwarf and the 3.56-hour orbital period. This suggests that interaction between
the white dwarf and the M-dwarf is behind the pulse mechanism. Proposed models
for the origin of emission include the surface or coronal loops of the M-dwarf
companion [4], the magnetosphere of the white dwarf [5, 6], close to the
surface of the white dwarf [7], or through an associated bow shock [8].

One of the main challenges to explain AR Sco is to reconcile the present fast
spin-down rate with the rapid spin of the white dwarf. The observed spin
period requires previous spin-up via mass accretion. That is because non-
interacting main sequence stars slow down their rotation as they age [9],
resulting in rotation periods of the order of days for their white dwarf
remnants[10, 11]. Only white dwarfs in cataclysmic variables have rotation
periods comparable to AR Sco, which is explained by angular momentum gain via
mass accretion from the companion[12]. However, whereas the spin-down rate of
AR Sco suggests that a strong magnetic field of 50–100 MG provides the
synchronising torque [4], the rapid spin can only be achieved with typical
mass transfer rates via Roche-Lobe overflow if the magnetic field is much
smaller ($\sim 1$ MG)[13]. With the strong magnetic field, a very large mass
transfer rate of up to $\dot{M}\sim
10^{-4}\,\mathrm{M}_{\odot}\,\mathrm{yr}^{-1}$ would be required to achieve a
1.95 min spin[14]. This rate is well beyond the ordinary M-dwarf companion
seen in AR Sco, and is $10^{5}$ times greater than the rates typical of
similar binaries[15]. Accretion via diamagnetic blobs could allow for lower
rates[16], but the origin of the strong magnetic field in AR Sco would remain
unexplained.

A solution to this conundrum has recently been put forward [17]. In the
proposed model, the white dwarf in AR Sco was originally non-magnetic,
allowing for straightforward accretion-driven spin-up. When crystallisation
started to occur in the core of the cooling white dwarf, strong density
stratification combined with convection created the conditions for a dynamo,
generating the magnetic field [18, 19]. With a strong enough field, the rapid
transfer of spin angular momentum into the orbit causes the binary to detach
and mass transfer to briefly cease, leading to a rare system such as AR Sco.
After a few millions of years, the system comes into contact again due to
reduced magnetic braking and gravitational radiation, giving origin to a
rapidly rotating accreting magnetic white dwarf. As well as addressing the
issue of forming a system like AR Sco, the proposed model also provides a
solution to a long-lasting problem in the field of white dwarf binaries: the
discrepancy between the fraction of magnetic white dwarfs in detached versus
accreting binaries. Strongly magnetic white dwarfs are nearly absent in
detached white dwarf binaries [20, 21], whereas more than one third of those
in accreting systems are magnetic [22]. The dynamo mechanism can naturally
explain that, as magnetic accreting white dwarfs are typically old and cool
enough to have been crystallised, and they have been spun up by accretion to
short periods such that the dynamo effect is intensified.

An important aspect of this promising rotation- and crystallisation-driven
dynamo model is that it suggests that binary white dwarf pulsars like AR Sco
are a possible stage in the evolution of accreting magnetic white dwarfs.
Though the timescales in the model cannot be precisely established, given the
existence at the time of only one object available to calibrate it, the
properties of AR Sco itself suggest that other binary white dwarf pulsars
should exist. First, considering its distance to Earth of only 117 pc, more
objects should be expected at larger volumes; second, its spin-down rate
suggests that the lifetime in such a stage is of millions of years[3]. This
model also predicts that the white dwarfs in AR Sco-like systems should be
cool enough to have crystallised, and that their companions should be close to
Roche-lobe filling. Finally, based on the observed population of accreting
systems, the model also predicts white dwarf pulsars to have orbital periods
in the range of three to five hours.

Although a few systems have been proposed as potential white dwarf pulsars[23,
24, 25], none of them have been confirmed, and as such AR Sco has remained
unique even after six years of its discovery. To address the lack of other
systems that could confirm binary white-dwarf pulsars as an evolutionary stage
and help to constrain the timescales and predictions of the rotation- and
crystallisation-driven dynamo model, we performed a targeted search for binary
white-dwarf pulsars. We searched for objects showing observational properties
similar to AR Sco, in particular non-thermal infrared colours, variability,
and location in the Gaia colour-magnitude diagram[26]. A few tens of
candidates were identified, over two thirds of which have already been
followed-up. Follow-up high-speed photometry with ULTRACAM[27] on the 3.58 m
New Technology Telescope (NTT) revealed that one of the candidates,
J1912$-$4410, shows strong pulses with a period of 5.3 min, during which the
$g$-band flux increases by up to a factor of four. Independently, J1912$-$4410
was detected as an X-ray source with eROSITA during its all-sky surveys [28]
and identified as a compact binary candidate due to its combined optical (as
seen by Gaia) and X-ray properties. These discoveries prompted further follow-
up.

We obtained ULTRACAM photometry on a total of five nights. We also obtained
photo-polarimetry on five nights with the High speed Photo-Polarimeter
(HIPPO[29]) mounted on the 1.9 m telescope at the South African Astronomical
Observatory (SAAO). Additionally, we retrieved archival photometry from the
Transiting Exoplanet Survey Satellite (TESS)[30], the Asteroid Terrestrial-
impact Last Alert System (ATLAS)[31], the Catalina Real-time Transient Survey
(CRTS)[32], and the All-Sky Automated Survey for Supernovae (ASAS-SN)[33].
Spectroscopy was firstly obtained with the Goodman spectrograph [34] at the
4.0 m Southern Astrophysical Research (SOAR) telescope and with the 1.0 m
telescope at SAAO, which allowed us to confirm that J1912$-$4410 showed
similar spectral characteristics to AR Sco: an optical spectrum displaying a
blue continuum added to the red spectrum of an M-type dwarf, with strong
Balmer and neutral helium lines in emission (see Extended Data Figure 1). Fast
frame-transfer spectroscopy was obtained with the 9.2 m Southern African Large
Telescope (SALT), and full orbital coverage was obtained with X-shooter[35] at
the 8.2 m Very Large Telescope (VLT). Further X-ray data were obtained with
XMM-Newton, after the detection with eROSITA which demonstrated the
feasibility of such observations. Radio imaging follow-up observations were
carried out using MeerKAT’s L-band receivers (856–1712 MHz). Further details
on the observations and data reduction are given in the Methods section.

The TESS photometry revealed a dominant frequency at $5.948136(13)$ cycles/d.
Radial velocities obtained for the M-dwarf from the X-shooter spectra show
that this frequency corresponds to the orbital period of the binary, of
$0.16811989(36)$ days (Fig. 1). The photometric orbital modulation is
asymmetric, which cannot be explained by reflection alone. This is likely due
to contribution from non-thermal emission, with the asymmetry arising either
due to more power being dissipated in the leading face of the M-dwarf, or due
to a misaligment between spin and orbit axis, leading to phase-dependent
dissipation rate [4]. The observed radial velocity amplitude depends on the
spectral line, reflecting the fact that the emission or absorption features
originate in different locations on the M-dwarf. We used the difference
between the amplitude measured the from NaI absorption lines (8183 and 8195
Å), inherent to the M-dwarf photosphere, and the Balmer emission lines, which
originate on the irradiated side facing the compact object, to constrain the
Roche geometry of the system (details in the Methods section). Additionally,
we imposed an upper mass limit equal to the Chandrasekhar mass for the compact
object, requiring it to be a white dwarf due to its spin period, which is more
than a factor of four longer than any confirmed neutron star pulsar[36]. With
this upper mass limit, a mass ratio of $q=M_{2}/M_{1}>0.3$111We use the
subscripts 1 and 2 to refer to parameters describing the white dwarf and the
M-dwarf, respectively. is needed for the origin of both emission and
absorption features to fit within the Roche lobe of the M-dwarf (see Extended
Data Figure 2). The observed centre-of-mass radial-velocity amplitude of the
M-dwarf can only be explained for this range of $q$ if the mass of the white
dwarf is $M_{1}>0.32\leavevmode\nobreak\ $M⊙. An upper limit to the M-dwarf
mass can be obtained by requiring it to fit within its Roche lobe, where it
would have a mean density determined by the orbital period[37]. We obtain
$M_{2}<0.42$ M⊙. The spectral type of the M-dwarf is M4.5$\pm$0.5 (see
Methods), suggesting a mass of $\approx 0.3$ M⊙, much below the Roche-lobe
filling upper limit. This implies that either the M-dwarf is not close to
Roche lobe filling, unlike AR Sco[38], or that the system is seen at low
inclination, resulting on a lower observed difference between the velocities
of NaI and the Balmer lines due to projection along the line of sight. In
fact, we find that our mass constraints imply a maximum inclination for the
system of $i=37^{\circ}$, favouring the latter interpretation (see Extended
Data Figure 2, panel b). Combining these constraints with the Gaia data
release 3 distance of $237\pm 5$ pc[39], which allows us to constrain the
radius of the M-dwarf from spectral fitting, we estimate the system masses to
be $M_{1}=1.2\pm 0.2$ M⊙ and $M_{2}=0.25\pm 0.05$ M⊙ (see Extended Data Figure
3), with the companion potentially filling over 90% of its Roche lobe. The
white dwarf temperature cannot be determined due to the lack of contribution
of the white dwarf in the optical wavelengths, but the upper limit derived
from the lack of visible features is consistent with at least the onset of
crystallisation (details in the Methods section).

The amplitude spectra computed from our fast-speed photometry reveal several
peaks separated by multiples of the orbital frequency (Fig. 2). We interpret
the dominant peak of 270.55038(7) cycles/day as the spin frequency of the
white dwarf (more details in the Methods section), corresponding to a spin
period of 319.34903(8) s. Our short baseline of under 5 months reveals no
detectable spin period change, which is unsurprising: if J1912$-$4410 shows a
similar spin-down rate to AR Sco, the expected change in the spin period over
our baseline would be of the order of $10^{-9}$ s, much smaller than the
precision in our derived spin period. The pulses are detected in all observed
frequencies from radio to X-rays, showing that, like AR Sco, J1912$-$4410
shows a broad spectral energy distribution (SED) that can be orders of
magnitudes brighter than the thermal emission from its component stars,
particularly at radio and X-ray wavelengths (Fig. 3). Integrating the SED
shown in Fig. 3 and using the Gaia distance, we find the bolometric luminosity
to be $\sim 10^{33}$ erg/s, well in excess of the total stellar luminosity of
$\sim 10^{31}$ erg/s. The excess is even higher than in AR Sco, potentially
suggesting a faster spin-down rate, or another source of energy such as
accretion (see below).

We interpret the mechanism behind the broad-band pulsed emission to be the
same as in AR Sco: synchrotron radiation. A synchrotron emission source locked
to the rotating frame of the white dwarf, which receives an injection of
electrons as the magnetic field of the white dwarf sweeps past the companion,
can reproduce the observed pulse profile, as illustrated in Fig. 4 (details in
the Methods section). The electrons are accelerated to relativistic speeds due
to magnetic reconnection, resulting in beamed synchrotron emission. The pulsed
emission show a complex shape that varies with orbital phase and, to some
extent, over time (see Extended Data Figure 4). At least two components can be
identified. First, a narrow component which is particularly dominant in radio,
where its full width at half maximum (FWHM) is less than 4 s. This narrow
pulse is also seen in the optical, over a much broader second component that
dominates both optical and X-rays (see middle panels in Fig. 2). The broader
component seen in all wavelengths is likely the result of synchrotron beaming
coming at each spin cycle when the magnetic field of the white dwarf sweeps
past the companion. The narrow pulses that dominate in radio are reminiscent
of neutron star pulsar variability and unlike the broad radio pulses observed
for AR Sco[1]. Perhaps the low inclination of J1912$-$4410 provides a better
view of a magnetic pole, which could be emitting in a similar manner to
neutron star pulsars, with very narrow pulses that are directly detected. The
phase lag between the two pulses suggests different origins in the system, or
at least distinct optical path lengths depending on energy.

At least one feature seen both in the optical and X-ray data does not
correspond to either the narrow or broad pulses (see Figure 5). We interpret
this feature as a flare. If the flare were due to chromospheric activity from
the M-dwarf, increased amplitude towards the ultraviolet would be expected
[40, 41]. That is not what we observe (see Extended Data Figure 5). Instead,
we propose that this is an accretion-induced flare, suggestive of mass
transfer and subsequent ejection between the M-dwarf and white dwarf (more
details in the Methods section). This is in contrast to AR Sco, which shows no
evidence whatsoever of ongoing accretion. Recent models for the evolution of
magnetic white dwarfs in close binaries [17] predict accretion to cause the
white dwarf spin-up, magnetic field generation, and subsequent detachment of
the white dwarf from its companion. The flares in J1912$-$4410 suggest that
not enough energy has been transferred from spin to orbit to fully detach the
system yet, which would suggest it is in a slightly earlier stage of this
formation scenario than AR Sco.

The discovery of J1912$-$4410 offers support for the rotation- and
crystallisation-driven dynamo model as the origin of magnetic cataclysmic
variables. It establishes binary white dwarf pulsars as a class, and provides
evidence for residual accretion as predicted by the models[17]. The M-dwarf
companion is estimated to be nearly filling its Roche lobe, in agreement with
the models and explaining any residual mass transfer. The upper limit on the
white dwarf temperature is consistent with the onset of crystallisation,
suggesting an emerging magnetic field that provides synchronising torque,
which will fully detach the system once enough energy is transferred from the
white dwarf spin to the orbit. The orbital period of the system is also in
agreement with current model predictions. Assuming AR Sco and J1912$-$4410 are
the only two binary white dwarf pulsars in the thin disk suggests a space
density of $0.15\pm 0.10$ kpc-3, taking into account the effective volume
given by the thin disk stellar density [42]. This should be regarded as a
lower limit, as ongoing searches might reveal other members of this class.

Figure 1: J1912$-$4410’s photometry and radial velocities. Panel (a) shows the
radial velocities obtained from the Nai doublet lines (8183 and 8195 Å) phased
to the orbital ephemeris (Eq. 2). The red dashed line is a sinusoidal fit used
to obtain the radial velocity semi-amplitude of the M-dwarf. Uncertainties are
of the same order as the symbol size. Panel (b) shows the residuals when the
sinusoidal is subtracted (uncertainties are the same as in panel a). Panel (c)
shows the TESS data with one-sigma uncertainties folded on the orbital
ephemeris. The Fourier transform of the TESS data is shown in panel (d),
evidencing the orbital period (strongest peak) and its first harmonic. Figure
2: Radio, optical, and X-ray fluxes of J1912$-$4410. The panels on the left
show high-speed photometry obtained with MeerKAT (a), with ULTRACAM (b–d,
filters $i_{s}$, $g_{s}$, and $u_{s}$, respectively), and XMM-Newton (e) as a
function of orbital phase. For XMM-Newton, the full dataset was phase-
averaged, whereas for MeerKAT the data covering two orbits were folded with no
averaging. For ULTRACAM, we show unfolded data for one orbit. The individual
measurement uncertainties are shown in panels (a)-(e), but are sometimes
comparable to symbol size. The dashed vertical lines mark regions of orbital
phase that were selected for the spin-phase average pulses shown in the
central panels, (f) to (j). For XMM-Newton, the whole dataset was used. The
error bars show the uncertainty on the mean in each phase bin. The panels on
the right show the Fourier transform for each dataset, with the main frequency
combinations between spin ($\omega$) and orbit ($\Omega$) identified in panel
(m). For MeerKAT we used only data around orbital phase 0.5, where the pulses
are visible, to calculate the Fourier transform. For ULTRACAM, all data for
the five nights are included. We also plot the Fourier transform of the HIPPO
data, in which the peaks indicated by dashed lines are better resolved, in the
background of panels l–n.  Figure 3: Spectral energy distribution of
J1912$-$4410. The red and blue lines show model atmospheres assuming
parameters close to the constraints that we placed for the M star ($R_{2}=0.3$
R⊙, $T_{2}=3100$ K) and white dwarf ($\log\leavevmode\nobreak\ g=9.0$, and
$T_{1}=13000$ K) at the Gaia distance of $237$ pc. The grey polygon shows the
flux and frequency ranges observed in radio. The dashed line models the XMM-
Newton spectrum in X-rays, which combines a power law and absorption due to
cold interstellar matter. The coloured symbols show the other flux
measurements, with vertical bars spanning minimum to maximum values when more
than one measurement is available. From left to right, the flux values are W4,
W3, W2, W1, H, J, K, z, i, r, V, g, U, u, NUV, and FUV. Figure 4:
Photopolarimetry of J1912$-$4410. The four images on the left show the total
intensity (upper two images) and linear polarisation (lower two images). The
four images on the right show the model simulations of synchrotron emission
from one emission region located close to one magnetic pole of a spinning
white dwarf. The images show how the spin and beat (left and right columns,
respectively) pulse profiles change with orbital phase. The images represent
the averaged phase-folded data of all the photo-polarimetric observations. The
total and linear flux have been normalised. Figure 5: Simultaneous XMM-Newton
and ULTRACAM data. Panel (a) shows in black the XMM-Newton data taken
simultaneously with the ULTRACAM $g_{s}$ data shown in panel (b). The red
curve in panel (a) is the spin-phase average XMM-Newton data over the whole
observation. The grey shaded areas in panel (b) indicate integer spin cycles,
which coincide with the narrow pulses, whereas the dashed grey lines shown in
both panels mark the maxima of the broad X-ray pulses. The red arrow indicates
a feature that cannot be explained by either the narrow or the wide pulses,
which we interpert as a flare.

## Methods

### Observations and data reduction

Time-series photometry: J1912$-$4410 was observed with ULTRACAM mounted on the
3.58 m ESO NTT on five nights. ULTRACAM uses dichroic beam splitters that
allow simultaneous observations in three filters. We used so-called "super"
SDSS filters, whose cut-on/off wavelengths match those of the commonly used
Sloan Digital Sky Survey (SDSS) filters, but with a higher throughput [43].
For the first three nights, $u_{s},g_{s},i_{s}$ filters were installed, and
for the next two $u_{s},g_{s},r_{s}$ filters were in place. We used the same
exposure time for $g_{s},r_{s},i_{s}$ filters, but set it to a factor of 3 or
4 times longer for $u_{s}$. Frame transfer capabilities were used so that the
dead time between exposures was negligible. Supplementary Table 1 provides
information about each of our runs. We performed bias subtraction and flat
field correction, with skyflats taken during twilight, using the HiPERCAM data
reduction pipeline222https://github.com/HiPERCAM/hipercam. The pipeline was
also employed to carry out aperture photometry. We used a variable aperture
size set to scale with the seeing estimated from a point-spread function (PSF)
fit. The same constant comparison star, Gaia EDR3 6712712349514963072
($G=13.8$), was used in all runs. SkyMapper[44] data were used to calibrate
the photometry to an approximate flux scale.

List of suptables 1  Log of ULTRACAM observations of J1912$-$4410. Date | Start time (UT) | Duration (min) | Filter | Cadence (s)
---|---|---|---|---
|  |  | $u_{s}$ | 18.6
2022-04-28 | $08:49:29$ | 31 | $g_{s}$ | 6.2
|  |  | $i_{s}$ | 6.2
|  |  | $u_{s}$ | 32.0
2022-06-07 | $02:02:02$ | 276 | $g_{s}$ | 8.0
|  |  | $i_{s}$ | 8.0
|  |  | $u_{s}$ | 9.1
2022-06-08 | $05:21:30$ | 243 | $g_{s}$ | 3.0
|  |  | $i_{s}$ | 3.0
|  |  | $u_{s}$ | 12.1
2022-09-18⋆ | $03:20:35$ | 51 | $g_{s}$ | 4.0
|  |  | $r_{s}$ | 4.0
|  |  | $u_{s}$ | 8.5
2022-09-24 | $00:30:08$ | 237 | $g_{s}$ | 2.8
|  |  | $r_{s}$ | 2.8
⋆ Simultaneous with XMM-Newton. |  |  |  |

Photopolarimetry: Photopolarimetry was performed with HIPPO[29] on the 1.9-m
telescope of the South African Astronomical Observatory during the nights of
2022 June 29 and 30, 2022 July 3–5, as detailed in Supplementary Table 2.
Observations were through a clear filter ($3500-9000$ Å) defined by the
response of the two RCA31034A GaAs photomultiplier tubes. Observations of
polarised and non-polarised standard stars were made during the course of the
observation run in order to calculate the waveplate position angle offsets,
instrumental polarisation, and efficiency factors. The photometry is not
absolutely calibrated and is instead given as total counts minus the
background-sky counts. J1912$-$4410 was observed for a total of $\sim 32$
hours.

List of suptables 2  Log of HIPPO observations of J1912$-$4410\. The cadence was 5.0 s for all runs. Date | Start time (UT) | Duration (min)
---|---|---
2022-06-29 | 20:09:09 | 477.3
2022-06-30 | 20:27:09 | 460.8
2022-07-03 | 23:01:09 | 274.5
2022-07-04 | 19:53:34 | 252.1
2022-07-05 | 20:14:24 | 462.0

Spectroscopy: Two exploratory spectra to confirm J1912$-$4410’s similarity
with AR Sco were obtained with SOAR on 2022 August 28 during time obtained for
project SO2022A-016. The data were reduced using iraf’s noao package. Upon
confirmation of that J1912$-$4410’s had spectral features like those seen in
AR Sco, we obtained Director’s Discretionary Time (DDT) with X-shooter
(proposal 109.24EM) to cover one full orbital period. The observations were
carried out on 2022 July 24, between 03:30:06 and 08:00:20 UT. We used a 1
arcsec slit for the UVB arm (300-559.5 nm, $R=5400$), and 0.9 arcsec slits for
the VIS (559.5-1024 nm, $R=8900$) and NIR (1024-2480 nm, $R=5600$) arms. The
exposure time in the UVB arm was set to a fifth of the spin period (63.6 s) to
sample the spin variability, and we obtained 168 exposures. In the VIS arm,
the exposure was set to twice the spin period (636.4 s) to average out the
effect of the spin variability, as our main interest with the VIS was to
characterise the M-dwarf companion. 21 exposures were obtained. Finally, in
the NIR arm the exposure was equal to the spin period (318.2 s) and we
obtained 42 exposures. 2x2 binning was used to reduce the readout time, which
was of 28 s in UVB, 34 s in VIS and 8.2 s in NIR. Automatic flexure
compensation (AFC) exposures were obtained every 1.5 h. The X-shooter data
were reduced using the xsh_scired_slit_stare routine in the ESO Recipe
Execution Tool (EsoRex), and telluric line removal was performed with molecfit
[45, 46].

We also obtained medium-resolution (5.7Å) time resolved spectra over the
wavelength range 4060–7120 Å of J1912$-$4410 on 2022 June 26 using the Robert
Stobie Spectrograph (RSS) [47, 48, 49] on the Southern African Large Telescope
(SALT) [50]. Two observations of 3000 s were obtained during both the rising
(east) and setting (west) tracks, commencing respectively at 20:10:54 UTC and
02:37:53 UTC. Frame transfer mode was used, with 60 repeat 50 s exposures for
each observation, with no dead-time.

The spectra were reduced using the PySALT package [51], which includes bias
subtraction, flat-fielding, amplifier mosaicing, and a process to remove
cosmetic defects. The spectra were wavelength calibrated, background
subtracted, and extracted using standard IRAF333IRAF is distributed by the
National Optical Astronomy Observatory, which is operated by the Association
of Universities for Research in Astronomy (AURA) under cooperative agreement
with the National Science Foundation (NSF). procedures. We obtained a relative
flux calibration of all spectra using the spectrophotometric standard star
Feige 110. The frame-transfer observations were used to create trailed spectra
in order to investigate emission line variability.

X-rays: The original X-ray detection of J1912$-$4410 was made during eRASS1,
the first X-ray all sky survey with eROSITA on the Spektrum-Roentgen-Gamma
mission [28, 52]. The eRASS1 proto-catalogue internal to the
collaboration444The full catalogue will be published by Merloni et al., in
preparation., which was produced with processing version c947[53], lists the
detection of J1912$-$4410 at RA$=288.05807$, DEC$=-44.17913$. Source detection
was performed on an image using photons in the $0.2-2.3$ keV band. The source
was found at a rate of $0.35\pm 0.07$ s-1, which corresponds to a flux of
$(3.3\pm 0.7)\times 10^{-13}$ erg cm-2 s-1.

Detailed follow-up was obtained with XMM-Newton following a DDT request. XMM-
Newton observed the field of J1912$-$4410 on 2022 Septemper 17/18 for a total
of 43 ks. For approximately 50 min, ULTRACAM observations were carried out
simultaneously with XMM-Newton. The EPIC X-ray cameras were operated in full
frame mode, and the optical monitor was used in fast imaging mode (time
resolution 0.5 s) with the UVM2 filter (effective wavelength 231 nm). The
original data were reduced with the latest version of the XMM-Newton SAS (SAS
20.0) using the most recent calibration files. Light curves and spectra were
produced using the signal in concentric annuli around the source position to
correct for background contamination. The mean rate in EPIC-pn was 0.0825 s-1
($0.2-10$ keV).

The mean X-ray spectrum was analysed with XSPEC (version 12.12.0) [54]. An
emission model as the sum of a power law (power law index $2.14\pm 0.11$) and
a thermal component ($kT=1.24\pm 0.11$ keV) modified with absorption due to
cold interstellar matter (N${}_{\rm H}=(5\pm 2)\times 10^{20}$ cm-2) reflected
the data satisfactorily. The best-fit model parameters yielded the model shown
in Fig. 3. A full description of the X-ray data analysis including data from
all eROSITA surveys and the XMM-Newton data is presented separately [55].

Radio: An exploratory observation of J1912$-$4410 was made with MeerKAT’s
L-band receivers (856–1712 MHz) for one hour on 22 June 2022 (block ID
1655920939). Following the detection of pulses, a longer observation was made
on 26 June 2022 (block ID 1656263834). The total on-target time was 7.55
hours, with the correlator configured to deliver 2 s integrations, which
defines the minimum imaging timescale. Instrumental bandpass and delay
corrections were derived from scans of the primary calibrator PKS B1934$-$638,
and time dependent complex gains were derived from scans of the secondary
calibrator J1830$-$3602\. The casa package [56] was used for reference
calibration and flagging of the calibrator scans. These gain corrections were
then applied to the target field, which was flagged using the tricolour
software [57]. The target data were imaged with wsclean [58], and self-
calibrated using cubical kenyon2018. The data reduction scripts[59] that
perform the processing up to this point are available online555oxkat, v0.3,
https://github.com/IanHeywood/oxkat, and contain detailed lists of all the
parameters used.

Following self-calibration and a final round of deconvolution, the resulting
set of spectral clean components (excluding the position of J1912$-$4410
itself) were inverted into a set of model sky visibilities and subtracted from
the data. This residual visibility set was then imaged using wsclean,
producing an image for every 2 s timeslot in the observation (13,552 snapshot
images in total). We experimented with deriving an in-band spectral index
using the strongest pulses, but found considerable scatter between pulses,
with values ranging from -4.4 to -0.8 for different pulses and uncertainties
reaching 40%. The median suggests a steep negative spectral index of
$\approx-3$, but higher signal-to-noise ratio than currently available is
required to obtain robust and conclusive results.

Archival data: J1912$-$4410 was observed by TESS during sectors 13 and 27,
with a cadence of 30 min and 10 min, respectively. We downloaded postcards and
performed aperture photometry using eleanor in a custom
script666https://github.com/ipelisoli/eleanor-LS. We also downloaded
photometry taken with the cyan ($c$) and orange ($o$) filters from the ATLAS
archive777https://fallingstar-data.com/forcedphot/. Data were also available
in CRTS and ASAS-SN, but due to the target’s relative faintness we could not
identify any of the periodic variability seen in other data in CRTS or ASAS-SN
data, which we therefore employed no further.

### The companion subtype and the orbital ephemeris

To determine the spectral subtype of the companion, we used the VIS arm of the
X-shooter spectra. To minimise contribution from the white dwarf, both direct
and due to its irradiation on the M-star, we started by deriving radial
velocities for the H$\alpha$ emission line by fitting it with a Gaussian
profile. H$\alpha$ should trace the irradiated face of the companion (as later
confirmed, see the section “Doppler tomography and the origin of emission"),
hence by fitting the velocities with a sinusoidal we were able to estimate the
point at which the companion is at its closest approach to Earth, when
contribution from the white dwarf is minimised. The spectrum closest to
inferior conjunction was then employed for spectral subtype determination. We
fitted the observed spectrum using M-dwarf templates obtained with the same
X-shooter configuration [60] as our observations. The template fluxes were
scaled by a factor $\alpha$ and combined with a smooth continuum to account
from any extra flux in addition to the M-star. This was parameterised by
$\exp(a_{1}+a_{2}\lambda)$ to ensure positivity[3]. To focus on the M-dwarf
contribution as well as avoid noisy regions, we only fit the spectrum between
6800 and 9250 Å. Additionally, we masked the CaII emission triplet lines
around 8500 Å. Similar values of $\chi^{2}$ were obtained for the M4, M4.5,
and M5 templates, hence we concluded the companion to be of type M$4.5\pm
0.5$.

Once the spectral type was determined, we proceeded to estimate the radial
velocity of the M-dwarf by cross-correlating the NaI 8200 Å doublet absorption
lines, which should more closely trace the centre of mass than the emission
lines, with a spectral template. We found the M4.0 template to provide a
better fit in this region, and therefore used it in the cross-correlation. We
normalised the region within 8140–8240 Å by the continuum using a first order
polynomial, and then subtracted a first-order spline fit to the normalised
continuum. The same procedure was applied to the 21 VIS spectra and the
template, which were then binned to the same velocity scale and cross-
correlated. The obtained radial velocities are listed in Supplementary Table
3.

List of suptables 3 Radial velocities obtained for J1912$-$4410. Radial velocities and one-sigma uncertainties obtained for the M-dwarf in J1912$-$4410 using the NaI 8200 Å doublet absorption lines. BJD(TDB) | RV (km/s) |  $\sigma_{\mathrm{RV}}$ (km/s)
---|---|---
2459784.65911 | 153.2 | 3.6
2459784.66687 | 195.8 | 3.5
2459784.67463 | 232.4 | 3.6
2459784.68259 | 261.5 | 4.0
2459784.69042 | 262.9 | 4.1
2459784.69818 | 252.2 | 4.5
2459784.70593 | 223.3 | 4.4
2459784.72079 | 125.6 | 5.0
2459784.72855 | 70.8 | 4.6
2459784.73632 | 11.4 | 4.6
2459784.74427 | -61.7 | 4.6
2459784.75208 | -112.2 | 4.5
2459784.75984 | -150.5 | 3.9
2459784.76760 | -172.5 | 4.0
2459784.78221 | -162.3 | 3.9
2459784.78997 | -136.0 | 4.7
2459784.79771 | -87.2 | 5.0
2459784.80566 | -27.0 | 5.2
2459784.81350 | 36.8 | 4.8
2459784.82127 | 99.7 | 4.4
2459784.82901 | 152.7 | 4.8

To determine the orbital ephemeris, the obtained radial velocities were
combined with an orbital period measurement from TESS data, whose time span
and continuous coverage is ideal for precisely determining the orbital
frequency. The Fourier transform of the TESS data showed a strong peak near
5.95 cycles/d, with the first harmonic also clearly visible (see Fig. 1). We
interpreted this as the orbital frequency, which is in agreement with the
observed radial velocity variability and within the orbital period range
predicted for white dwarf pulsars[17].

Reflection alone cannot explain the observed photometric orbital modulation,
which is asymmetric and shows amplitude higher than the sub-percent that would
be expected for a reflection effect for the system’s estimated parameters at
low inclination (which we find to be the case for J1912$-$4410, as discussed
in the next section). The larger amplitude can be explained by contribution
from non-thermal emission when the rotational energy of the white dwarf is
dissipated by interaction of its magnetic field with the M dwarf’s. The
asymmetry, also observed in AR Sco, has two proposed explanations [4]. The
first is that the power dissipated by interaction between the star’s magnetic
fields is greater on the leading face of the M-dwarf, where shock occurs, than
on its trailing face. The other possibility is that the spin axis of the white
dwarf is misaligned with the orbit, which would cause the dissipation rate in
the M dwarf to vary with orbital phase. A consequence of the latter is that
precession of the spin axis would make the orbital phase of the maximum to
drift, which has not been observed for AR Sco[61], but cannot be ruled out
given that the precession period can be of up to several hundreds of years.

The system’s orbital period and its uncertainty were determined via
bootstrapping, using a Fourier model with two sine terms (one on the
fundamental frequency, and one on the first harmonic) to fit the TESS data. We
obtained $P_{\mathrm{orb}}=0.16811989(36)$ days, corresponding to
$\Omega=5.948136(13)$ cycles/d. To determine the reference epoch (phase 0) of
the ephemeris, defined here as the inferior conjunction of the M-dwarf, we
fitted the radial velocities with

$\displaystyle V_{R}$ $\displaystyle=$
$\displaystyle\gamma+K_{2}\sin[2\pi(t-T_{0})\Omega],$ (1)

where $\gamma$ is the systemic velocity, $K_{2}$ the radial velocity semi-
amplitude of the M-dwarf, $t$ the mid-exposure times of each spectrum yielding
a measurement, $T_{0}$ the reference time of inferior conjunction, and
$\Omega$ is the orbital frequency, which was fixed at the value obtained from
the TESS data. Using bootstrapping to determine uncertainties, we obtained
$\gamma=46.2\pm 1.0$ km/s, $K_{2}=220.9\pm 1.1$ km/s, and
$T_{0}=2459784.98308(19)$, thus the orbital ephemeris of J1912$-$4410 is:

$\displaystyle BJD(TDB)$ $\displaystyle=$ $\displaystyle
2459784.98308(19)+0.16811989(36)E,$ (2)

where $E$ is an integral cycle number, and BJD is the barycentric Julian date
(in TDB scale).

### Constraining the stellar masses

From the value of $K_{2}$ and the orbital period, we can use Kepler’s third
law to calculate the mass function:

$\displaystyle f_{M}$ $\displaystyle=$
$\displaystyle\frac{M_{1}^{3}\sin^{3}i}{(M_{1}+M_{2})^{2}}\;=\;\frac{P_{\mathrm{orb}}K_{2}^{3}}{2\pi
G}\;=\;(0.1879\pm 0.0027)\leavevmode\nobreak\ M_{{}_{\odot}}.$ (3)

This equation, in which $i$ is the orbital inclination, sets a lower limit to
the mass of the unseen compact object, met for $M_{2}=0$ and $i=90^{\circ}$.
This only stands if our assumption that the obtained radial velocities indeed
trace the centre of mass of the M-dwarf is correct. As we detected no
systematic deviation from a sinusoidal in the residuals (see Fig. 1), as would
be expected if irradiation caused weakening of the absorption lines on the
side of the cool star facing the compact companion, our assumption seems to be
correct.

An independent constraint on the system’s masses can be obtained from the
difference between the semi-amplitudes derived from the emission lines
(tracing the irradiated face) compared to the absorption lines (tracing the
centre of mass), shown in Supplementary Table 4. The simple assumption that
all measurements have to be within the M-dwarf’s Roche-lobe results in
$q>0.1$, as illustrated in panel (a) of Extended Data Fig. 2. Requiring that
star 1 is a white dwarf can provide a tighter constraint. Given the spin
period of 5.3 min, too long for a neutron star but completely consistent with
white dwarfs in magnetic cataclysmic variables[62], we consider this to be a
fair assumption. As shown in panel (b) of Extended Data Fig. 2, this results
in $q>0.3$. This can be combined with Equation 3 to obtain a tighter
constraint on $M_{1}$. Equation 3 implies

$\displaystyle M_{1}=f_{M}(1+q)^{2}/\sin^{3}i.$ (4)

Given the lower limit for $q$ and upper limit for $\sin^{3}i$ at
$i=90^{\circ}$, we obtain $M_{1}>0.32$ M${}_{{}_{\odot}}$ and $M_{2}>0.095$
M${}_{{}_{\odot}}$.

List of suptables 4 Radial velocity semi-amplitudes for different lines Line | Semi-amplitude (km/s)
---|---
H$\alpha$ | $177\pm 5$
H$\beta$ | $155.5\pm 1.1$
H$\gamma$ | $157.7\pm 1.0$
NaI doublet | $220.9\pm 1.1$

We can also obtain an upper limit on the M-dwarf mass. For a Roche-lobe
filling star, there is a tight relationship between orbital period and mean
density $\rho$ [37]:

$\displaystyle\rho[\mathrm{g/cm}^{3}]$ $\displaystyle=$
$\displaystyle(0.43/P_{\mathrm{orb}}[\mathrm{days}])^{2}.$ (5)

Because the mean density of a main-sequence star decreases with increasing
mass, this corresponds to an upper limit. Assuming a semi-empirical mass-
radius relationship[63], which takes the M-dwarf inflation problem[64] into
account, results in $M_{2}<0.42$ M${}_{{}_{\odot}}$. Combining the limit given
by the M-dwarf filling its Roche lobe with the observed $K_{2}$ values also
implies a maximum inclination at which star 1 is a white dwarf. Higher
inclinations would require a more compact M-dwarf to explain the observed
$K_{2}$ difference, which in turn requires the compact object to have a
higher-mass so that the M-dwarf still fills the Roche lobe. The maximum system
inclination is $i<37^{\circ}$, as demonstrated in Extended Data Fig. 2.

Equation 4 can also be interpreted as a relationship between $M_{1}$, $M_{2}$,
and inclination. With the limits derived above, useful constraints can be
obtained. This is shown in Extended Data Figure 3. The upper limit on
inclination requires the white dwarf mass to be above $\approx 1.0$
M${}_{{}_{\odot}}$. To further constrain the masses, we can rely on an
estimate for $M_{2}$ obtained from an estimate of the stellar radius. The
distance to J1912$-$4410 is well constrained by the Gaia parallax to $d=237\pm
5$ pc. Hence, when fitting the observed spectrum to models, the previously
mentioned scaling factor $\alpha$, which corresponds to $(R_{2}/d)^{2}$,
provides a radius estimate. Combining this with a mass-radius relationship
gives $M_{2}$. We fitted NextGen models [65] to the M-dwarf spectra via
$\chi^{2}$ minimisation using the same wavelength region as employed to obtain
the M-dwarf subtype. We kept the $\log\leavevmode\nobreak\ g$ fixed at 5.5
rather than free, given that the effects of gravity and rotation on the line
widths cannot easily be disentangled. We obtained $T_{2}=3\,100\pm 100$ K and
$R_{2}=0.23\pm 0.02$ R⊙. Using the same mass-radius relationship as above
results in $M_{2}=0.26\pm 0.02$ M⊙. The quoted uncertainties are statistical
only. For $R_{2}$ and $M_{2}$, the M-dwarf inflation problem suggests that the
uncertainty is of the order of 15%. The irradiation of the M-dwarf by the
white dwarf, which can also cause inflation, leads to similar uncertainty[66].
The derived $M_{2}$ value implies a white dwarf mass of $\approx 1.2$
M${}_{{}_{\odot}}$. To take into account the large systematic uncertainties in
$M_{2}$, we adopt the mass values of $M_{1}=1.2\pm 0.2$ M${}_{{}_{\odot}}$ and
$M_{2}=0.25\pm 0.05$ M${}_{{}_{\odot}}$ quoted in the main text. These masses
imply $q\sim 0.21$, and hence a Roche-lobe radius of $\sim 0.25$ R⊙[37] for
the M-dwarf, close to our estimated radius, suggesting that the M-dwarf is
near Roche-lobe filling, as inferred for AR Sco[38] and as predicted for a
white dwarf pulsar configuration[17].

### The spin and beat frequencies

The ground-based fast photometry obtained with ULTRACAM and HIPPO was used to
determine the spin frequency of J1912$-$4410\. This identification is not as
straightforward as the orbital period. Fourier transform of the ULTRACAM and
HIPPO data showed three main peaks separated by multiples of the orbital
frequency. We also analysed light curves derived from the continua and
emission lines of observed spectra (see Supplementary Figure 1), which show in
turn only one resolved peak. The lines originate in the irradiated face of the
companion, hence their variability would likely reflect the reprocessed or
beat frequency. The resolution of the spectral Fourier transforms is, however,
not high enough to separate spin and beat frequencies. We ultimately relied on
the modelling of the photopolarimetry (see Section "Photopolarimetry and
modelling of the emission") to determine whether the dominant frequency is the
spin or beat, concluding that the data can be better explained by the models
if the former is the dominant frequency. Therefore we interpret the two main
peaks as $\omega$ and $\omega-2\Omega$, with $\omega+\Omega$ also being
marginally detected. The absence of the beat frequency is somewhat puzzling,
although 2($\omega-\Omega$) is detected (Supplementary Figure 2). This could
be a consequence of low inclination, such that reprocessed radiation from only
one pole is detected.

To determine the spin ephemeris, we measured pulse arrival times from both the
HIPPO and the ULTRACAM data. We first estimated $\omega$ and the time of
maxima $T_{0}^{\omega}$ values by fitting a cosine function to the ULTRACAM
$g_{s}$ data, after subtracting the orbital modulation. This trial ephemeris
was used to estimate the approximate pulse arrival times and cycle numbers
expected from each dataset. To refine this estimate, we cross-correlated the
data around each expected peak with a Gaussian function, and found the time of
maxima by locating the maxima of the cross-correlation function using the
Newton-Raphson method.

This procedure of estimating times of maxima given trial ephemeris and
subsequently fitting the obtained values was repeated until the trial and
fitted ephemeris showed no significant change. The $\sigma$ width of the
Gaussian was fixed at a value of 15 s, which we found to yield a good balance
between identified pulses and fit residuals (i.e. a large number of pulses was
obtained, and the identifications were good enough that the residuals to the
fit were not increased. Increasing the value of $\sigma$ resulted in lower
residuals, but fewer identified pulses, whereas decreasing the value increased
the number of identified pulses, but also increased residuals).

Next we fitted the obtained cycle numbers and time of maxima assuming linear
ephemeris. The residuals of the fit are modulated with orbital phase, as
illustrated in Supplementary Figure 3. The semi-amplitude of the modulation is
of the order of 15 s, significantly in excess of the $\lesssim 1.5$ s that
could be explained by the difference in light travel time throughout the
orbit. This behaviour likely arises due to the varying contribution of beat,
spin and their harmonics to the pulse shape throughout the orbit, as seen in
AR Sco [67, 68, 38], and suggests that the line-of-sight geometry plays a role
in the detected emission. To minimise the effect of this in the ephemeris
determination, we fitted the orbital modulation with a Fourier series (one
sine and cosine component) and subtracted the modulation from the derived
times. We additionally excluded from the fit any measurements with a residual
larger than 0.25 cycles.

We initially fitted each dataset (HIPPO and each of the ULTRACAM filters)
independently to probe for dependence of the pulse arrival times with
wavelength (as seen for AR Sco [68, 38]). We found the ephemeris to be
consistent between datasets, and hence fitted all data together. Following the
described procedure, we obtained the following spin ephemeris:

$\displaystyle BJD(TDB)$ $\displaystyle=$ $\displaystyle
2459772.142522(24)+0.0036961693(10)E,$ (6)

where $E$ is an integral cycle number. The best fit and uncertainties were
determined via bootstrapping.

Additionally, we have probed for the occurrence of spin-up or spin-down, as
observed for AR Sco[3, 67, 68, 38], by fitting a quadratic ephemeris and
performing an $F$-test to determine whether the addition of a quadratic term
significantly improved the fit. We have found that for none of the our
datasets a quadratic fit represented a significant improvement, which is
unsurprising given our short baseline. We attempted to increase our baseline
by deriving a pulse arrival measurement from the ATLAS $c$ data, which shows a
hint of the spin period in its Fourier transform. Our approach was to subtract
the orbital modulation modelled by a Fourier series, and fit the residuals
with a cosine with period fixed to the spin period, but we found that the
large uncertainties resulted in a poor fit with an amplitude consistent with
zero. Therefore, a constraint on the spin period change could not be obtained
with the current data.

### The possible occurrence of flares

The optical observations of J1912$-$4410 often showed hints of flaring, with
the amplitude of the possible flares even dominating over the orbital
modulation in a few occasions (see Supplementary Figure 4). We studied the
possibility of flares in more detail using the simultaneous XMM-Newton and
ULTRACAM data, as shown in Figure 5. The simultaneous data covers orbital
phases 0.05 to 0.26, where typically the optical pulses are weak and less
sharp. Yet, several narrow features consistent with the location of spin
maxima are seen (in particular at spin cycles 18532–18535). The X-ray pulses,
in contrast, are displaced in phase compared to narrow pulses, but coincide
with wider features in the optical data. Our interpretation is that the short-
lived features seen in the ULTRACAM data originate in the magnetosphere of the
white dwarf, thus repeating on the spin period. As seen in Fig. 2, they are
associated with the narrow pulses seen in radio data. The broader features
seen in both ULTRACAM and XMM-Newton show an offset compared to the narrow
maxima (as also seen in Fig. 2), suggesting they originate in another region
in the system, possibly near the M-dwarf companion where the strongest
emission is observed.

Associating the narrow features with the magnetosphere of white dwarf and the
broader pulses with emission elsewhere, causing a phase delay, still leaves at
least one unidentified feature around cycle 18534.5. The amplitude of this
feature shows no apparent colour dependency (Extended Data Figure 5), unlike
M-dwarf flares. We suggest that these are flares induced by accretion along
the white dwarf’s magnetic field lines. If this interpretation is correct, it
would imply that, unlike AR Sco, J1912$-$4410 is not completely detached yet,
thus being in an earlier stage of evolution according to the model of
Schreiber et al. 2021[17]. Continuous monitoring of this source to probe the
occurrence of flares will allow testing this hypothesis.

### Constraining the white dwarf temperature

The temperature of the white dwarf is a crucial parameter, especially in the
context of the theoretical model proposed to explain the origin of binary
white dwarf pulsars [17]. In this model, the generation of the white dwarf
magnetic field, which eventually becomes strong enough to connect with the
M-dwarf magnetic field and provide a synchronising torque that explains the
fast spin-down of systems like AR Sco, has been attributed to crystallisation-
and rotation-driven dynamo. Therefore, the white dwarf must have cooled down
enough for crystallisation to progress in the core. For the range of masses
derived, crystallisation will start below temperatures of $35\,000-14\,000$ K
for a carbon-oxygen core white dwarf, depending both on the white dwarf mass
and on the thickness of its atmosphere [69].

To constrain the white dwarf temperature, we first considered the observed
GALEX magnitudes, where the white dwarf completely dominates over the
companion. We calculated the absolute magnitude given the Gaia distance and
taking extinction at J1912$-$4410’s distance and sky direction [70] into
account, and compared to synthetic magnitudes for cooling models [71, 69]. For
a mass of 1.0 M⊙, the observed magnitudes are consistent with a maximum
temperature of $\sim 15\,000$ K, above which the white dwarf flux would exceed
the observed flux. For 1.2 M⊙, the upper limit is $18\,000$ K. Another
constrain can be obtained from the lack of any visible absorption lines from
the white dwarf. To derive an upper limit based on that, our approach was to
subtract a scaled white dwarf model from the observed spectrum and identify
when that introduced a slope near the emission lines, suggesting that the
white dwarf absorption line would be visible at that temperature. The models
were scaled taking into account J1912$-$4410’s distance and a white dwarf
radius interpolated from cooling models for each temperature[71, 69], assuming
a mass of 1.0 M⊙ for $\log\leavevmode\nobreak\ g=8.5$ and 1.2 M⊙ for
$\log\leavevmode\nobreak\ g=9.0$. We note that radius changes are minimal and
such that the $\log\leavevmode\nobreak\ g$ change is negligible for fixed a
mass. To numerically determine when there is a significant slope, we fitted
the continuum near the H$\gamma$ line, where the spectrum is close to flat,
after subtracting the scaled model. We fit both using a constant and a first
order polynomial (indicating a slope), and perform an $F$-test to determine
when the slope becomes significant. This suggested $T_{1}<13\,000$ K, above
which the wings of the absorption lines of the white dwarf would be detectable
(Supplementary Figure 5). For a mass of 1.2 M⊙, the upper limit is less
restrictive than the SED (23 000 K).

For 1.2 M⊙, crystallisation starts around 20 000 K for a carbon-oxygen
core[69], and around 24 000 K for a oxygen-neon core[72], therefore the
derived upper limit suggestions crystallisation already started. The 1.0 M⊙
case is less conclusive: crystallisation starts around our derived upper limit
[69]. Hence the derive maximum temperature cannot rule out that the core is
not significantly crystallised, but is consistent with it being at least
starting to crystallise. Since the 1.0 M⊙ case is the less conclusive in terms
of crystallisation, it is the one we have focused on and illustrated in the
main text.

### Doppler tomography and the origin of emission

The X-shooter spectra cover a full orbital period, allowing us to employ the
Doppler tomography technique [73], which maps the observed line profiles at
different orbital phases into velocity space. This allows the emission
distribution in the binary to be mapped. We computed Doppler maps for
H$\beta$, H$\alpha$, and the CaII triplet around 8500 Å. The trailed line
profiles for these lines can be seen in Supplementary Figure 6. For H$\alpha$
(panel b), in additional to the sinusoidal profile due to orbital motion,
satellite lines extending to higher velocities can be seen, similarly to what
has been observed for AR Sco [74]. Frame-transfer spectroscopy also showed
that H$\alpha$ is seen to pulsate with a period closer to the spin period (see
Supplementary Figure 7). Corresponding Doppler maps are shown in Supplementary
Figure 8. It can be seen that the emission originates mainly at the irradiated
face of the M-dwarf, with a somewhat extended region towards the white dwarf
for H$\alpha$ and H$\beta$. H$\alpha$ additionally shows signs of prominences
around -200 and 200 km/s, a pattern also seen in the Doppler maps of AR Sco
[74, 38]. Though these prominences could be attributed to accumulated material
in these regions, possibly due to the magnetic field altering the Roche
geometry and placing stable Lagrange points at these locations [75], an
alternative is that the H$\alpha$ line profile has components that are not
kinematic in origin, but result from variations in optical depth or other
changes in the radiative transfer in the M-dwarf photosphere that change the
line profile independently of velocity. This suggestion is motivated by the
fact that H$\alpha$ is seen to show atypical line profiles even in some
detached binaries of M-dwarfs with white dwarf companions [76].

### Photopolarimetry and modelling of the emission

An example of a single night’s (29 June 2022) time-series photo-polarimetry
data is shown in Supplementary Figure 9. The orbital modulation and the short
period pulses are clearly seen in the photometry. The circular polarisation
appears consistent with zero percent. The linear polarisation averages around
4 percent, although some single data points show larger values, up to 12
percent (see Supplementary Figure 10, which zooms into the region where the
linear polarisation pulses can be seen). The linear polarisation data is of
insufficient signal-to-noise to be binned with enough time resolution to show
the photometric pulses. Therefore, we subjected the entire series of photo-
polarimetry to a Fourier analysis, for which the amplitude spectrum is
presented in Supplementary Figure 11. The amplitude spectra, for both the
photometry and the linearly polarised flux, display the most prominent peaks
at the orbital frequency ($\Omega$), the spin frequency ($\omega$) and twice
the beat frequency ($2(\omega-\Omega)$). There were no significant frequencies
in the amplitude spectrum of the circular polarisation.

To increase the signal-to-noise, we phase-binned and folded the data on the
spin and beat frequencies as a function of orbital phase (Figure 4, left-hand
panels). These so-called dynamic pulse profiles show how the beat and spin
variations are modulated on the orbital period. Immediately obvious is that
both the photometric and linear pulses (spin and beat) evolve in amplitude
over the orbital cycle, peaking at $\sim 0.4$ in orbital phase. The spin and
beat pulses have a diagonal structure as they appear to drift later and
earlier, respectively, as a function of orbital phase, thus indicating that
the polarised emission has both spin and beat components. In addition, there
is a fainter photometric pulse during orbital phases $\sim 0.6-1.1$. We
performed the same exercise assuming the dominant frequency to be that of the
beat, rather than the spin, which resulted in smeared features (see
Supplementary Figure 12). This significant increase in smearing when the beat
is assumed to be dominant indicates that the data have not been folded on the
correct spin and beat frequencies. This was our main motivation behind the
interpretation of spin as the dominant frequency. In addition, the position
angle of linear polarisation seems to be better defined when assuming the
dominant peak to be the spin, though this is marginal at the moment and will
require further follow-up to be confirmed (see Supplementary Figure 13).

The observed dynamic pulse profile is remarkably identical (in morphology and
orbital phasing) to the main pulse in AR Sco (see fig. 2 of Potter & Buckley
2018[6]). On closer inspection, there may be an indication that the linearly
polarised spin and beat pulses are diagonally split, in the same manner as the
main pulse in AR Sco, given that there seems to be a valley of lower intensity
between peaks of higher intensity at each end of the profile. Future
observations with higher signal-to-noise ratio will confirm whether this is a
real feature.

Given the observed photopolarimetric similarities between J1912$-$4410 and AR
Sco, we adapted the simple synchrotron model previously used in the literature
to explain AR Sco’s emission[6]. The model photopolarimetric emission is shown
as dynamic pulse profiles in the right-hand panels of Figure 4. The model
assumes a synchrotron emission source is locked in the white dwarf rotating
frame, which receives a further injection of electrons as the white dwarf’s
magnetic field sweeps past the secondary star on the beat frequency. The
magnetic field of the rotating white dwarf accelerates the electrons to
relativistic speeds, resulting in beamed synchrotron emission. We used the AR
Sco model, but with a smaller inclination of 37∘ and with a single emission
region instead of two. As can be seen from the right-hand panels of Figure 4,
the model visually reproduces the observed dynamic pulse profile quite well,
in particular the orbital phasing of the pulses and their morphology. The
absence of a second emission region and a less significant splitting of the
linear pulses in J1912$-$4410 compared to AR Sco is simply explained as an
inclination effect, i.e. J1912$-$4410 is a lower inclination version of AR
Sco.

## Data availability

The TESS data used in this work are public and can be accessed via the Barbara
A. Mikulski Archive for Space Telescopes (https://mast.stsci.edu/). Other data
will be become public after the proprietary time expires, but can be made
available upon reasonable request to the corresponding author.

## Code availability

Any of the custom data analysis scripts used in this work can be made
available upon reasonable request to the corresponding author.

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Correspondence and requests for materials should be addressed to Dr Ingrid
Pelisoli (ingrid.pelisoli@warwick.ac.uk).

## Acknowledgements

IP and TRM acknowledge funding by the UK’s Science and Technology Facilities
Council (STFC), grant ST/T000406/1. IP also acknowledges funding from a
Warwick Astrophysics prize post-doctoral fellowship, made possible thanks to a
generous philanthropic donation. IP was additionally supported in part by the
National Science Foundation under Grant No. NSF PHY-1748958, and thanks the
organisers of the KITP Program "White Dwarfs as Probes of the Evolution of
Planets, Stars, the Milky Way and the Expanding Universe". SGP acknowledges
the support of a STFC Ernest Rutherford Fellowship. VSD and ULTRACAM are
funded by the UK’s Science and Technology Facilities Council (STFC), grant
ST/V000853/1.

This research made extensive use of Astropy (http://www.astropy.org) a
community-developed core Python package for Astronomy [77, 78]

This paper includes data collected by the TESS mission. Funding for the TESS
mission is provided by the NASA Explorer Program.

Based in part on observations obtained at the Southern Astrophysical Research
(SOAR) telescope, which is a joint project of the Ministério da Ciência,
Tecnologia e Inovações do Brasil (MCTI/LNA), the US National Science
Foundation’s NOIRLab, the University of North Carolina at Chapel Hill (UNC),
and Michigan State University (MSU).

Based on observations collected at the European Organisation for Astronomical
Research in the Southern Hemisphere under ESO programmes 109.24EM and
109.234F.

The SALT observations were obtained under the SALT Large Science Programme on
transients (2021-2-LSP-001; PI: DAHB). Polish participation in SALT is funded
by grant No. MEiN nr 2021/WK/01. DAHB and SBP acknowledge researh support by
the National Research Foundation.

This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

The MeerKAT telescope is operated by the South African Radio Astronomy
Observatory (SARAO), which is a facility of the National Research Foundation,
an agency of the Department of Science and Innovation. The authors thank SARAO
for the award of the MeerKAT Director’s Discretionary Time.

Based on observations obtained with XMM-Newton, an ESA science mission with
instruments and contributions directly funded by ESA Member States and NASA.

This work is based on data from eROSITA, the soft X-ray instrument aboard SRG,
a joint Russian-German science mission supported by the Russian Space Agency
(Roskosmos), in the interests of the Russian Academy of Sciences represented
by its Space Research Institute (IKI), and the Deutsches Zentrum für Luft- und
Raumfahrt (DLR). The SRG spacecraft was built by Lavochkin Association (NPOL)
and its subcontractors, and is operated by NPOL with support from the Max
Planck Institute for Extraterrestrial Physics (MPE). The development and
construction of the eROSITA X-ray instrument was led by MPE, with
contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU
Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz
Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and
Astrophysics of the University of Tübingen, with the support of DLR and the
Max Planck Society. The Argelander Institute for Astronomy of the University
of Bonn and the Ludwig Maximilians Universität Munich also participated in the
science preparation for eROSITA. The eROSITA data shown here were processed
using the eSASS/NRTA software system developed by the German eROSITA
consortium.

Support of the Deutsche Forschungsgemeinschaft (DFG) under grant number
536/37-1 is gratefully acknowledged.

## Author contributions

All authors contributed to the work presented in this paper. IP wrote the
manuscript and lead the follow-up and analysis of the system, with significant
input from TRM. DAHB carried out follow-up observations at SAAO and
contributed to analysis of the optical data. IH carried out reduction and
analysis of the radio data. SBP carried out follow-up observations at SAAO and
the analysis and modelling of polarimetric data. A. Schwope and A. Standke
obtained and analysed the X-ray data. PAW obtained the radio data. SGP and MJG
contributed to initial identification and analysis of the system. SOK, JM, and
ADR contributed to observational follow-up. EB, AJB, VSD, MJD, PK, SPL, DIS,
and JFW contributed to the maintenance of operations of ULTRACAM.

## Competing interests statement

The authors declare no competing interests.

List of extfigures 1 Optical spectrum of J1912$-$4410. The black line shows
the SOAR spectrum obtained for J1912$-$4410, which confirmed its spectral
characteristics to be similar to AR Sco. The grey line shows X-Shooter spectra
from the UVB and VIS arms obtained around the same orbital phase as the SOAR
spectrum (0.85). The flux calibration of the SOAR spectrum is poor towards the
blue due to reduced sensitivity. List of extfigures 2 Constraints from Roche
geometry. In panel (a) the star marks measurements from the NaII line (by our
definition the centre of mass of the M-dwarf). The cross, diamond, and circle
mark respectively H$\alpha$, H$\beta$, and H$\gamma$, which were fitted as
$V=\gamma-V_{X}\cos(2\pi\phi)$ $+V_{Y}\sin(2\pi\phi)$, where $\phi$ is the
orbital phase and $\gamma$, the systemic velocity, was kept fixed to the
previously determined value. H$\beta$, and H$\gamma$ give consistent
measurements, whereas H$\alpha$ seems to trail the leading face of the
M-dwarf. The red dashed line is the Roche lobe of the M-dwarf for $q=0.1$. The
black cross and blue triangle mark the centre of mass of the system and of the
white dwarf, respectively. The black line in panel (b) shows the expected
difference between the irradiated face and centre of mass radial velocity
semi-amplitudes as a function of $q$, assuming the M-dwarf fills its Roche
lobe. The right-hand $y$-axis shows the required inclination to explain the
detected $K$ difference. The observed value of $K$ difference sets a minimum
for $q$, which would happen if the system was seen at 90∘ inclination, as
indicated by the red dashed line. The area shaded in grey corresponds to
$M_{1}$ values consistent with a white dwarf for a Roche lobe-filling
companion, and implies the minimum $q$ value of 0.3. This minimum $q$
corresponds to a maximum inclination of 37∘ (blue dashed lines). List of
extfigures 3 Constraints from the binary mass function. The colour map shows
the required inclination to explain the observed $K_{2}$ for given values of
$M_{1}$ and $M_{2}$ shown in the x- and y-axis. The red line marks the maximum
inclination of 37∘, derived from Roche geometry, and the blue shaded area
indicates the $M_{2}$ mass derived from a mass-radius relationship. Given the
high systematic uncertainty on $M_{2}$, we adopt less strict constraints of
$M_{1}=1.2\pm 0.2$ M${}_{{}_{\odot}}$ and $M_{2}=0.25\pm 0.05$
M${}_{{}_{\odot}}$. List of extfigures 4 Pulse shape for different orbital
phases and nights. The thick red line shows all the X-ray data averaged and
folded on the spin ephemeris. The thin lines and symbols show ULTRACAM $g_{s}$
data averaged and folded on the same ephemeris, but considering data only
within the orbital phase ranges shown on the right of the plot. The green
dashed line shows data taken on 2022 June 07, the black symbols are data taken
on 2022 September 17 (simultaneously with the X-ray data), and the solid blue
line shows data for 2022 September 23. All data were normalised to the
strongest peak to facilitate comparison. As also seen in Fig. 2, the peak of
the X-ray pulses does not align with the bulk of optical pulses. However, it
does align with the optical peaks observed on 2022 September 17. This
difference cannot be attributed to uncertainty in the ephemeris, given the
agreement between data taken on nights before and after the X-ray
observations. Additional simultaneous data is needed to determine the cause of
misalignment, which could possibly be due to sporadic changes on pulse
profile.  List of extfigures 5 Flux and colour of the possible flare. Panel
(a) shows the flux in the $u_{s}$ (blue triangles), $g_{s}$ (green circles),
and $r_{s}$ (red crosses) bands in the region of the feature that we identify
as a flare (marked by the shaded grey area). Panels (b) and (c) show the
$u_{s}-g_{s}$ and $g_{s}-r_{s}$ colours. Unlike typical M-dwarf flares, there
is no evidence of flux increase towards the blue.

## Supplementary Information

List of suppfigures 1 Fourier transforms of data derived from spectroscopy.
Panels (a)-(c) show Fourier transforms of the continuum, H$\beta$ flux, and
H$\beta$ equivalent width derived from VLT/X-shooter spectra. Panels (d)-(f)
show the same but for H$\alpha$ and SALT spectra. For the continuum, flux was
integrated over the spectrum masking the line regions, whereas only the lines
themselves were integrated when obtaining the line flux. The solid line shows
the spin frequency, whereas the dashed line shows beat frequency
$\omega-\Omega$. List of suppfigures 2 Fourier transforms of HIPPO and
ULTRACAM $g_{s}$. The left panels show the Fourier transform of HIPPO data (a)
and ULTRACAM $g_{s}$ (b) around the spin frequency $\omega$. Other peaks also
visible are labelled. The beat frequency $\omega-\Omega$ (red dotted line) is
not detected. The right panels show the same Fourier transforms around the
first harmonic of the spin frequency, showing that the first harmonic of the
beat frequency is detected and even dominant over the spin harmonic. List of
suppfigures 3 Residuals from linear fit. Panel (a) shows the residuals as a
function of orbital phase after an initial linear fit to the pulse arrival
times derived from ULTRACAM and HIPPO data. There is modulation with orbital
phase, modelled by a Fourier series shown by the black dashed line. The red
dashed lines show the same model plus or minus 0.25 cycles; points below or
above the red lines were excluded from the final fit. Panel (b) shows the
residuals after the fit that resulted in the ephemeris of Equation 6. In panel
(b) the symbols are grey circles for HIPPO data, red triangles for ULTRACAM
$r_{s}$ and $i_{s}$ data, green squares for ULTRACAM $g_{s}$, and blue crosses
for ULTRACAM $u_{s}$. List of suppfigures 4 Optical light curve showing
possible flaring. Light curve of J1912$-$4410 taken with the Mookodi
instrument on the SAAO 1-m Lesedi telescope on 2022 August 21. There are
strong changes in amplitude, e.g. between BJD 2459813.8 and 2459813.85, which
are suggestive of flaring.  List of suppfigures 5 Constraining the temperature
of the white dwarf. Panel (a) shows in grey a spectrum obtained by co-adding
all UVB spectra taking around minimum flux. Model white dwarf spectra with
$\log g=8.50$ and different effective temperatures, indicated by the colour
bar, are shown. The models were scaled taking J1912$-$4410’s distance into
account and assuming a white dwarf radius from cooling models for $\log
g=8.50$. In panel (b), the model white dwarf spectra were subtracted from the
observed spectrum. For the correct white dwarf model, this should simply
remove the white dwarf contribution, and add no new features. From a certain
temperature, the strength of the lines in the model is such that the
subtraction visibly adds broad features to the spectrum. By numerically
determining when that happens, we placed an upper limit on the white dwarf
temperature of 13 000 K. List of suppfigures 6 Trailed spectra of
J1912$-$4410. Panel (a) shows spectra taken with X-shooter’s UVB arm, around
H$\beta$. Panels (b) and (c) show data from the VIS arm, around H$\alpha$ and
the CaII triplet, respectively. The left hand axis shows time in BJD, and the
right hand axis shows orbital phases for all panels. For H$\beta$, the
integration time was short enough to resolve the pulsations, which can be seen
in the continuum. List of suppfigures 7 Trailed H$\alpha$ spectra. Trail of
spectra obtained via frame-transfer spectroscopy with SALT. The pulsed
behaviour of H$\alpha$ can be clearly seen in the flux changes, in particular
towards the end of observations (top of the plot). List of suppfigures 8
Doppler tomography of J1912$-$4410. Doppler tomography of the H$\beta$ (panel
a), H$\alpha$ (panel b) and Ca II triplet (panel c) lines. The white cross
marks the position of the white dwarf, the red line indicates the Roche lobe
of the companion assuming $q=0.2$, and the red dot its centre of mass. The
colour bars show emission intensity after normalising and subtracting the
continuum. List of suppfigures 9 Example HIPPO photo-polarimetry of
J1912$-$4410 from 29 June 2022. Top to bottom show the photometry, circular
polarisation and linear polarisation, respectively, with one-sigma measurement
uncertainties for each datapoint. List of suppfigures 10 Zoomed in region of
Supplementary Figure 9 highlighting the linear polarisation pulses. Top to
bottom show the photometry, circular polarisation and linear polarisation,
respectively, with one-sigma measurement uncertainties for each datapoint..
List of suppfigures 11 Fourier transforms of HIPPO photometry and linear
polarimetry. Black is photometry, lightskyblue is linear polarised flux. List
of suppfigures 12 Photopolarimetry of J1912-4410 assuming beat to be the
dominant frequency. Panels show the same as in Figure 4, but assuming beat
rather than the spin as the dominant frequency, which results in considerable
smearing in the spin panel. The right column is the same as the second column
in Figure 4, because the same frequency is used for folding in both cases, but
with a different interpretation. List of suppfigures 13 Position angle of the
linear polarisation as a function of orbital and spin/beat phases. The left
panel shows the position angle when the dominant frequency is assumed to be
the spin, while the right shows the same when the beat is assumed to be
dominant. In the left panel, there is a hint of a consistent angle for orbital
phase $\approx 0.45$.