# Mapping the accretion disc of the short period eclipsing binary SDSS
J0926+3624

Wagner Schlindwein1& Raymundo Baptista1
1Departamento de Física, Universidade Federal de Santa Catarina, Campus
Trindade, 88040-900 Florianópolis, SC, Brazil

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We report the analysis of time-series of optical photometry of SDSS J0926+3624
collected with the Liverpool Robotic Telescope between 2012 February and March
while the object was in quiescence. We combined our median eclipse timing with
those in the literature to revise the ephemeris and confirm that the binary
period is increasing at a rate $\dot{P}=(3.2\pm 0.4)\times 10^{-13}\,s/s$. The
light curves show no evidence of either the orbital hump produced by a bright
spot at disc rim or of superhumps; the average out-of-eclipse brightness level
is consistently lower than previously reported. The eclipse map from the
average light curve shows a hot white dwarf surrounded by a faint, cool
accretion disc plus enhanced emission along the gas stream trajectory beyond
the impact point at the outer disc rim, suggesting the occurrence of gas
stream overflow/penetration at that epoch. We estimate a disc mass input rate
of $\dot{M}=(9\pm 1)\times 10^{-12}\,M_{\odot}\,yr^{-1}$, more than an order
of magnitude lower than that expected from binary evolution with conservative
mass transfer.

###### keywords:

stars: binaries: close – eclipses – stars: individual: SDSS J0926+3624 –
novae, cataclysmic variables – accretion, accretion discs

††pubyear: 2018††pagerange: Mapping the accretion disc of the short period
eclipsing binary SDSS J0926+3624–Mapping the accretion disc of the short
period eclipsing binary SDSS J0926+3624

##  1 Introduction

AM Canum Venaticorum (AM CVn) stars are ultracompact binaries with orbital
periods ranging from 5.4 to 65 min and optical spectra dominated by helium
lines (see e.g., Nelemans, 2005; Ramsay et al., 2007; Roelofs et al., 2010).
They consist of a white dwarf (the primary) accreting matter via an accretion
disc from a significantly less massive, hydrogen-deficient donor star (the
secondary). In order to fills its Roche lobe and sustain the mass transfer
process, the donor star needs to be at least partially degenerate. AM CVn
stars offer new insights into the formation and evolution of binary star
systems, with the short periods implying at least one common envelope phase in
the history of the binary, and the chemical composition suggesting helium
white dwarfs or cataclysmic variables (CVs) with evolved secondaries as
possible progenitors (Nelemans et al., 2001, 2010, see also Marsh et al.,
2011; Kulkarni & van Kerkwijk, 2010 for the recent identification of a
possible AM CVn progenitor).

SDSS J0926+3624 (hereafter J0926) was the first eclipsing AM CVn star and is
also one of the shortest period eclipsing binary known (Anderson et al.,
2005). Its light curve displays deep ($\sim 2$ mag) eclipses every 28.3 min,
which lasts for $\sim 2$ min, as well as $\sim 2$ mag amplitude outbursts
recurring on timescales of $\sim 100-200$ d (Copperwheat et al., 2011,
hereafter C11). Superhumps were seen in its light curves several days after
the end of an outburst (C11; Szypryt et al., 2014, hereafter Z14); these are
believed to result from the tidal interaction between the mass-donor star and
an elliptical precessing disc (e.g., Whitehurst, 1988; Hirose & Osaki, 1990).
C11 modeled the eclipse light curves to estimate masses and radii of both
stars, a mass ratio $q=M_{2}/M_{1}=0.041\pm 0.002$ and an inclination
$i=82\aas@@fstack{\circ}6\pm 0\aas@@fstack{\circ}3$, a white dwarf temperature
of $T_{wd}=17000\,K$ and a corresponding distance estimate of $(465\pm 5)$ pc.
Z14 found that the orbital period of J0926 is increasing at a rate
$\dot{P}=(3.07\pm 0.56)\times 10^{-13}\,s/s$ and, as a consequence, inferred a
conservative mass transfer rate $\dot{M}\simeq 1.8\times
10^{-10}\,M_{\odot}\,yr^{-1}$. This is in agreement with the $\dot{M}=(1.4\pm
0.3)\times 10^{-10}\,M_{\odot}\,yr^{-1}$ predicted by Deloye et al. (2007)
based on their stellar evolutionary calculations (corrected for the $q=0.041$
value of C11).

Here we report additional high-speed optical photometry of J0926, used to
confirm its orbital period change and to disentangle its optical light sources
with the aid of eclipse mapping techniques. The observations are described in
Sect. 2; data analysis and results are reported in Sect. 3 and summarized in
Sect. 4.

##  2 Observations and data reduction

Time series of high-speed CCD photometry of J0926 in the $V+R$ passband were
obtained with the RISE camera ($512\times 512$ pixels, $1."07$ pixel-1 in the
$2\times 2$ binning mode) attached to the 2.0 m Liverpool Robotic Telescope
(LRT), in the Canary Islands, during 2012 February-March while the object was
in quiescence and near the end of a 4.6 yr long period without recorded
outbursts. The CCD camera is operated in frame-transfer mode, with negligible
dead time between exposures.

The observations are summarized in Table 1. The fourth column lists the range
of eclipse number (E), the fifth column lists the seeing range, and the last
column gives an estimate of the quality of the observations. Each run consists
of a 2 h long sequence of 442 exposures at a time resolution of 16 s covering
4 orbits of the binary; the data set comprises a total of 20 eclipse light
curves. All light curves were obtained with the same set of instrument and
telescope, which ensures a high degree of uniformity to the data set. Poor
weather and seeing conditions forced us to discard about half of the data on
Mar 21.

Table 1: Journal of Observations Date | UT | E 111With respect to the ephemeris of eq. (1). | Seeing | Sky 222Sky: (A) Photometric; (B) good; (C) poor (large transparency variations, thin clouds,or both).
---|---|---|---|---
(2012) | Start | End | (cycle) | (arcsec) |
Feb 23 | 22:15 | 00:13 | 111131 - 111135 | 1.0 - 1.9 | A
Mar 18 | 22:49 | 00:47 | 112353 - 112357 | 1.3 - 2.5 | B
Mar 21 | 21:36 | 23:34 | 112503 - 112507 | 1.2 - 3.0 | C
Mar 23 | 21:17 | 23:15 | 112604 - 112608 | 1.1 - 1.8 | A
Mar 25 | 22:41 | 00:38 | 112709 - 112713 | 1.1 - 1.7 | A

As part of the observatory pipeline, data from the LRT are distributed to the
user already corrected from the instrumental effects of dark current, bias and
flat-field. Aperture photometry was used to extract fluxes for a reference
star, the variable, and for a set of nearby comparison stars with scripts
using the apphot/IRAF333IRAF is distributed by the National Optical Astronomy
Observatories, which are operated by the Association of Universities for
Research in Astronomy, Inc., under cooperative agreement with the National
Science Foundation. routines in order to derive light curves of differential
photometry. In order to flux calibrate the light curves, we used the aperture
photometry of the comparison field star LP 260-21, which has calibrated fluxes
in the $B,\ V,\ R,\ J,\ H$ and $K_{s}$ passbands in the SIMBAD database
(Wenger et al., 2000). We then performed synthetic photometry with the
synphot/IRAF package to find the best-fit match of the Bruzual-Persson-Gunn-
Stryker atlas of spectrophotometric stars (Strecker et al., 1979; Gunn &
Stryker, 1983) to the SIMBAD observed fluxes of LP 260-21, and convolved the
best-fit M4 V spectrum with the response of the RISE camera $V+R$ passband to
find a calibrated flux of $0.78\pm 0.08$ mJy for this star. The calibrated
flux of the reference star was then used to transform the light curves of the
variable and of the comparison stars from magnitude difference to absolute
flux. This absolute flux calibration has an estimated uncertainty of about 10
per cent. On the other hand, analysis of the relative flux of the comparison
stars in all data sets indicates that the internal error of the photometry is
less than 3 per cent. The error in the photometry of the variable is derived
from the photon-count noise and is transformed to flux units using the same
relation applied to the data. The individual light curves have typical signal-
to-noise ratios of $S/N\simeq 10$. Exposure times were transformed from
Universal Time (UTC) to Terrestrial Dynamical Time (TDT) and to the
Baricentric reference frame (BJDD), according to the code of Eastman et al.
(2010), to account for the light travel time effect.

##  3 Data analysis and results

###  3.1 Light curves and revised ephemeris

The individual light curves were phase-folded according to the linear
ephemeris of C11,

$T_{\mathrm{mid}}(BJDD)=2\,453\,796.445\,5191(5)+0.019\,661\,272\,89(2)\times
E\,\,,$ (1)

where $T_{\mathrm{mid}}$ is the primary mid-eclipse time and $E$ is the binary
cycle. At first, we grouped the data sets to obtain separate average light
curves for each of the observing nights. However, since the average light
curves of all nights show the same morphology, brightness level, eclipse shape
and depth, we combined all data to obtain a single average light curve with
increased S/N.

We used a phase-folded, concatenated light curve to obtain a single mid-
eclipse time from the whole data set. The concatenated light curve was median
filtered with a boxcar of phase width 0.015 and fitted with a model eclipse
light curve of the white dwarf (see Sect. 3.3) to find a white dwarf (WD) mid-
eclipse phase of $\phi_{0}=+0.0009\pm 0.0005$. The measured eclipse time is
delayed by 1.5 s with respect to Eq. 1, in very good agreement with the
results of Z14. We adopted the median eclipse cycle of our median observing
date, used Eq. 1 to compute its predicted mid-eclipse time and added the
measured $\phi_{0}$ value to obtain an equivalent observed mid-eclipse timing
of $T_{\mathrm{mid}}(E=112\,505)=\mathrm{BJDD}\;2\,456\,008.437\,044(10)$.

We evaluated the influence of the relatively long 16 s exposure time of the
individual light curves (with respect to the $\sim 120$ s total width of the
eclipse) on our ability (i) to precisely measure mid-eclipse timings and (ii)
to recover the eclipse shape (for the purpose of eclipse mapping) with the
following experiment. We simulated the eclipse of the surface brightness
distribution in the upper panel of Fig. 4 at a time resolution of 0.5 s to
construct a base light curve, to which a phase offset of $\phi_{0}=+0.0009$
was added. We then created two sets of 20 light curves from the base light
curve (by binning the data at a time resolution of 16 s with random starting
times), one noise-free data set and one data set with 10 per cent gaussian
noise added to each data point ($S/N=10$). We repeated the above procedure to
estimate a mid-eclipse timing from the noisy, simulated data set and were able
to recover the added phase offset with the same precision of the real data,
namely, $\phi_{0}=+0.0009\pm 0.0005$. Furthermore, the comparison of the
noise-free data set with the base light curve indicates that the long exposure
time of the observations lead to systematic, small differences in eclipse
shape at the 1 per cent level, which are negligible in comparison with the 10
per cent noise affecting the real data. This shows that the uncertainties in
the mid-eclipse timing and in the eclipse shape are largely dominated by the
noise in the light curves, with negligible contribution from the relatively
long exposure time of the observations.

In order to derive a revised ephemeris for J0926, we combined our timing with
those in the literature. We measured the Observed-minus-Calculated (O-C)
values of C11 from their Fig. 12; we computed median (O-C) values for each of
their 6 nights of observations and used their ephemeris with a representative
cycle for each night to derive equivalent BJDD eclipse times. The same
procedure was used to derive an equivalent, median eclipse time from Fig. 4 of
Z14. We excluded the timings of their first two nights of observations from
this analysis as the object was in outburst and the eclipses were much
shallower than usual, being systematically later than those of the last night.
444Possibly the consequence of the significant out-of-eclipse modulation
reminiscent of superhumps, particularly in the 2012 Dec 8 data. The results
are listed in Table 2. Table 3 presents the parameters of the best-fit linear
and quadratic ephemerides to these timings with their 1-$\sigma$ formal errors
quoted, together with the root-mean-square value of the residuals, $\sigma$,
and the reduced $\chi^{2}_{\nu}$, where $\nu$ is the number of degrees of
freedom. These fits assume equal errors of $2.2\times 10^{-6}\,d$ to the data
points, which ensures a unity $\chi^{2}_{\nu}$ for the quadratic ephemeris.

Table 2: Median mid-eclipse times of J0926. E | BJDD | (O-C) | Reference
---|---|---|---
(cycle) | (2450000+) | (s) |
7 | 3796.583142 | $-0.10$ | C11
55 | 3797.526887 | $+0.24$ | C11
105 | 3798.509946 | $-0.15$ | C11
52750 | 4833.577663 | $+0.22$ | C11
52803 | 4834.619706 | $-0.15$ | C11
52857 | 4835.681416 | $-0.05$ | C11
112505 | 6008.437044 | $-0.05$ | This work
125963 | 6273.038464 | $+0.05$ | Z14
Table 3: Ephemerides for mid-eclipse times of J0926. Linear ephemeris: |
---|---
$\mathrm{BJDD}=T_{0}+P_{0}\,E$ |
$T_{0}=2\,453\,796.445\,510(\pm 1)\,d$ | $P_{0}=0.019\,661\,273\,12(\pm 2)\,d$
$\chi^{2}_{\nu_{1}}=9.56\,,\nu_{1}=6$ | $\sigma_{1}=6.23\times 10^{-6}\,d$
Quadratic ephemeris: |
$\mathrm{BJDD}=T_{0}+P_{0}\,E+c\,E^{2}$ |
$T_{0}=2\,453\,796.445\,514(\pm 1)\,d$ | $P_{0}=0.019\,661\,272\,75(\pm 5)\,d$
$c=(+3.18\pm 0.44)\times 10^{-15}\,d$ | $\sigma_{2}=1.85\times 10^{-6}\,d$
$\chi^{2}_{\nu_{2}}=1.00\,,\nu_{2}=5$ |

The top panel of Fig. 1 shows the (O-C) diagram with respect to the best-fit
linear ephemeris and the lower panel shows the residuals with respect to the
best-fit quadratic ephemeris. The quadratic fit provides a much better
description of the data than the linear fit, underscoring the findings of Z14.
The significance of adding an additional term to the linear ephemeris was
estimated with the F-test (Pringle, 1975). The quadratic ephemeris has a
statistical significance of 99.9 per cent with $F(1,6)=51.7$. The orbital
period of J0926 seems to be increasing at a rate $\dot{P}=(3.24\pm 0.43)\times
10^{-13}\,s/s$. The inferred $\dot{P}$ value is consistent with that found by
Z14 within the uncertainties. The observed period increase could possibly be
the consequence of binary evolution through angular momentum loss caused by
gravitational radiation (Z14). However, given that every eclipsing CV with a
well-sampled (O-C) diagram covering at least a decade of observations shows
cyclical period changes (e.g., Borges et al., 2008), the observed period
increase could alternatively be part of a cyclical period modulation of
amplitude several seconds on a decade long timescale. Given the precision with
which eclipses can be measured in this binary, a further decade of (regular)
observations will suffice to clarify these possibilities.

Figure 1: Top: (O-C) diagram of J0926 with respect to the best-fit linear
ephemeris of Table 3. The dashed line depicts the linear ephemeris of C11 and
the solid curve shows the best-fit quadratic ephemeris. Bottom: The residuals
with respect to the best-fit quadratic ephemeris of Table 3. The dashed lines
indicate the 2-$\sigma$ limits on the ephemeris, with the covariances taken
into account.

Flux calibrated and phase-folded light curves of J0926 and of a comparison
star are shown in Fig. 2. An offset of $\phi_{0}=-0.0009$ was applied to the
data to make the WD eclipse center coincide with phase zero. The average light
curve shows the deep, narrow and partial eclipse of the WD at disc center
superimposed on a broad, shallow eclipse of the disc plus incoming gas stream
displaced towards positive phases. There is no evidence of superhump or of an
orbital hump produced by anisotropic emission from a bright spot at disc rim.
The average out-of-eclipse flux level is $0.053\pm 0.005$ mJy, $\sim 20$ per
cent lower than that obtained by C11 after subtracting the superhump
contribution from their light curves. The lack of evidence of superhumps and
the clear differences in the light curve morphology with respect to the data
of C11 and Z14 is not surprising. According to the Catalina Real-Time
Transient Survey data (CRTS, Drake et al., 2009), J0926 had no recorded
outburst during the $\sim 4.6$ yr period between MJD 54550 and 56250 (Fig. 3).
Our data set frames this extended quiescent state. The last recorded outburst
occurred $\sim 1450\,d$ ($\sim 4\,yr$) before our observations. Both the 2006
and the 2009 observations of C11 occurred at an epoch where the binary showed
frequent outbursts, and the observations of Z14 occured right after the first
recorded outburst at the end of this long quiescent state.

Figure 2: Phase-folded light curves of J0926 (green crosses) and of a nearby
comparison star (blue crosses). The solid line is the median light curve at a
phase resolution of 0.005 cycle. Figure 3: J0926 observations made by the CRTS
for the interval from 2005 January 29 to 2013 November 3. Vertical dotted
lines mark the epoch of the observations of C11, Z14 and of this work. Our
observations frame the 4.6 yr long interval without recorded outbursts.

We end this section noting that the small inferred $\dot{P}$ value inplies a
change in eclipse center of about $1\,\mu s$ over the time span ($\sim 30$ d)
of our observations, several orders of magnitude lower than the uncertainty in
the measured median eclipse timing.

###  3.2 Search for periodicities

We searched for periodicities in the light curves by computing a Lomb-Scargle
periodogram (Press et al., 1992) for each separate night. We subtracted the
combined, average light curve of all nights from each individual light curve
to eliminate the DC component, and we masked data in the phase range
$(-0.1,+0.1)$ to avoid the eclipse. Amplitude levels in the power spectrum
were evaluated by injecting artificial signals of known amplitude in the time
series. The periodograms show no prominent signal at orbital
$(f_{\mathrm{orb}}=50.9\,d^{-1})$, superhump ($f_{\mathrm{sh}}=50.4\,d^{-1}$,
C11), or at any other frequency in the range $f<2000\,d^{-1}$. We find upper
limits of 2 and 3 percent for the semi-amplitude of any superhump or orbital
hump periodicity in the light curves, respectively for the nights of Feb 23,
Mar 23-25 and the nights of Feb 18-21 (the lower quality nights), in agreement
with the observed lower brightness level and the lack of evidence for out-of-
eclipse modulation in the light curves (Fig. 2).

###  3.3 Eclipse mapping

We applied eclipse mapping techniques (Horne, 1985; Baptista, 2016) to the
average light curve of J0926 to solve for a map of the surface brightness
distribution of its accretion disc and for the flux of an additional
uneclipsed component.

Our eclipse map is a flat Cartesian grid of $51\times 51$ pixels centered on
the primary star with side $2R_{L_{1}}$ (where $R_{L_{1}}$ is the distance
from the disc center to the inner Lagrangian point). The eclipse geometry is
defined by the mass ratio $q$ and the inclination $i$. We adopted the
parameters of C11, $R_{L_{1}}=(0.231\pm 0.009)\,R_{\odot}$, $q=0.041\pm 0.002$
and $i=82.6^{\circ}\pm 0.03^{\circ}$, which correspond to a WD eclipse phase
width of $\Delta\phi=0.022$. This combination of parameters ensures that the
WD is at the center of the map.

Our eclipse mapping code implements the scheme of double default functions,
$D_{+}\,D_{-}$, simultaneously steering the solution towards the most nearly
axi-symmetric map consistent with the data ($D_{+}$), and away from the criss-
crossed arcs along the edges of the shadow of the occulting, mass-donor star
($D_{-}$) (Spruit, 1994; Baptista et al., 2005; Baptista, 2016). It is
optimized to recover asymmetric structures in eclipse maps such as spiral arms
and enhanced gas stream emission. The positive default function is a polar
Gaussian with radial and azimuthal blur widths of $\Delta r=0.02\,R_{L_{1}}$
and $\Delta\theta=30^{\circ}$, respectively. The negative default function is
a Gaussian along the ingress/egress arcs of phase width $\Delta\phi=0.01$.

The input, average light curve was obtained by slicing the combined data in
phase bins of width 0.005 and by computing the median flux for each bin. The
median of the absolute deviations with respect to the median flux was taken as
the corresponding uncertainty at each bin (Fig. 4, upper panel). We also used
the eclipse geometry to model the contribution of the WD to the light curve,
assuming a DB white dwarf with $M_{WD}=0.85\,M_{\odot}$, $\log g=8.33$,
$T_{WD}=17000\,K$ and a distance of 465 pc (C11), taking into account the
effects of limb darkening (Gianninas et al., 2013). The WD model light curve
was subtracted from the average light curve and the resulting curve was input
to the eclipse mapping code in order to isolate the surface brightness of the
faint accretion disc plus gas stream emission (Fig. 4, lower panel). All
reconstructions reached a unity final reduced chi-square.

Figure 4: Upper left panel: Data (green crosses with error bars) and model
(black line) light curves. Lower left panel: WD-subtracted data (green crosses
with error bars) and model (black line) light curves. The red solid line
depicts the model WD eclipse light curve. Horizontal dashed lines indicate the
uneclipsed flux in each panel. Vertical dashed lines mark the ingress/egress
phases of the WD. Upper right panel: Surface brightness distribution of the
combined disc+stream+WD map in a logarithmic grayscale. Lower right panel: WD-
subtracted surface brightness distribution. Regions inside the green/red
contour lines are above the 2- and 3-$\sigma$ confidence levels, respectively.
Dashed lines depict the primary Roche lobe, a disc of radius
$0.65\,R_{L_{1}}$, and the ballistic stream trajectory. The horizontal bar
shows the logarithmic intensity grayscale; brighter regions are darker.

The statistical uncertainties in the eclipse maps are estimated with a Monte
Carlo procedure (see e.g. Rutten et al., 1992). For a given input data curve,
a set of $10^{2}$ artificial curves is generated in which the data points are
independently and randomly varied according to a Gaussian distribution with
standard deviation equal to the uncertainty at that point. The artificial
curves are fitted with the eclipse-mapping algorithm to produce a set of
randomized eclipse maps. These are combined to produce an average map and a
map of the residuals with respect to the average, which yields the statistical
uncertainty at each pixel. The uncertainties obtained with this procedure are
used to plot the confidence level contours in the eclipse maps of Fig. 4 and
to estimate the errors in the derived radial temperature distributions.

Data and model light curves and corresponding eclipse maps in a logarithmic
grayscale are shown in Fig. 4. The upper panel displays the eclipse map of the
full light curve. The brightness distribution is dominated by the contribution
from the WD at disc center. Aside of the deep, narrow eclipse of the WD, the
additional broad, shallow and asymmetric eclipse shape maps into two
asymmetric brightness sources, one running along the disc rim at $R_{d}\simeq
0.65\,R_{L_{1}}$ (signalling the presence of a weak bright spot), and the
other extending along the ballistic stream trajectory beyond the impact point
at disc rim (suggesting the presence of gas stream overflow/penetration). The
fact that this residual bright spot leads to no perceptible orbital hump in
the light curve suggests that at least the outer disc regions are optically
thin. The lower panel shows the WD-subtracted light curve and corresponding
eclipse map, revealing a faint disc at the center of the map as well as the
two above mentioned asymmetric sources. These three brightness sources are
statistically significant at the $2.5\sigma$ (bright spot) and 3-5 $\sigma$
(disc plus gas stream emission) confidence levels. We find small and non
significant uneclipsed components of $0.002\pm 0.001$ mJy for both the full
and the WD-subtracted light curves.

We performed simulations in order to gauge the ability of our eclipse mapping
code to distinguish between a bright spot at disc rim and extended emission
along the ballistic stream trajectory. For this purpose, we created two
artificial brightness distributions, one with a WD at disc center plus a
bright spot at the intersection of the ballistic stream trajectory with a disc
of radius $R_{d}=0.485\,R_{L1}$ (C11), and the other with enhanced emission
along the ballistic stream trajectory well beyond the outer disc rim.
Grayscale maps of these distributions are shown in the uppermost panels of
Fig. 5. They were convolved with the eclipse geometry of J0926 to generate
synthetic light curves to which Gaussian noise was added in order to simulate
real data. Each of these light curves was then subjected to the eclipse
mapping code. The lower and middle panels of Fig. 5 show, respectively, the
synthetic data and model light curves, and corresponding eclipse maps for the
case of $S/N=30$, comparable to that of our light curves.

Figure 5: Reconstructions of asymmetric brighness sources with light curves of
$S/N=30$. The uppermost panel shows the grayscale maps of the models with a
bright spot (left) and with extended emission along the ballistic stream
trajectory (right). The notation is the same as that of Fig. 4.

Because of the small mass ratio of J0926, the projected shadow of its tiny
mass-donor star in the orbital plane is so narrow that it only starts covering
the bright spot when the WD has almost fully recovered from its eclipse (see
Fig. 3 of Marsh et al., 2007), thereby leading to two almost separate eclipses
(Fig. 5, left-hand panels). This eclipse shape is reminiscent of the light
curves of C11 and of the first two nights of observations of Z14, which show a
significant orbital hump intrepreted in terms of anisotropic emission from a
bright spot ‘painted’ at disc rim. On the other hand, the extended emission of
the ‘stream overflow’ distribution is eclipsed together with the WD at disc
center. Because of its extended distribution and asymmetric location, its
eclipse lasts longer and is displaced towards positive phases with respect to
that of the WD (Fig. 5, right-hand panels). The two artificial brightness
distributions lead to different eclipse shapes, which are easy to distinguish
even at relatively low S/N. As a consequence of the azimuthal blur of the
maximum entropy eclipse mapping method (see Baptista, 2016), the ‘bright spot’
is reconstructed as an arc extended by $\simeq 90^{\circ}\,(=3\,\Delta\theta)$
in azimuth at the correct radius, while the ‘stream overflow’ is smeared in
azimuth along the ballistic stream trajectory. These simulations show that the
data allow us to easily distinguish between a compact bright spot at disc rim
and an extended emission along the stream trajectory; they give us confidence
to argue that there is evidence of gas stream overflow/penetration in J0926 at
the epoch of our observations.

We assumed a distance of ($465\pm 5$) pc (C11) to convert the intensities of
the eclipse maps to blackbody brightness temperatures, $T_{b}$, which may then
be compared to the $T_{\mathrm{eff}}(R)\propto R^{-3/4}$ law of opaque,
steady-state disc models, as well as to allow a rough estimate of the disc
mass input rate $\dot{M}$. It is worth noting that $T_{b}$ may not be a proper
estimate of the disc & stream gas effective temperature, and that a relation
between these quantities can only be obtained by self-consistent modelling of
the atmosphere of these sources (Baptista et al., 1998). Additionally,
interstellar extinction effects were not taken into account in this procedure,
as the insterstellar extinction towards J0926 is expected to be negligible
(C11) due to its high galactic latitude ($+46^{\circ}$). We separated the
emission of the gas stream from that of the remaining disc by dividing the
eclipse map into two azimuthal ‘slices of pizza’ (e.g., Baptista et al.,
2000), one for $\theta=0^{\circ}-90^{\circ}$ 555$\theta$ is the azimuth
measured with respect to the line joining both stars and increases counter-
clockwise in the eclipse maps of Fig. 4. (gas stream) and another for the
remaining azimuths ($\theta=90^{\circ}-360^{\circ}$), and by computing median
brightness temperatures in anulli of width $0.05\,R_{L_{1}}$ for each slice.

Azimutally averaged radial brightness temperature distributions of the
combined (disc+stream+WD) and of the WD-subtracted surface brightness
distributions are shown in Fig. 6. The WD at disc center has $T_{b}=(19\pm
1)\times 10^{3}\,K$, consistent at the 2-$\sigma$ limit with the assumed WD
temperature and indicating that $T_{b}\simeq T_{\mathrm{eff}}$ in this case.
It is surrounded by a faint, cool accretion disc ($T_{b}\simeq 6000$-$3500\,K$
in the range $0.1$-$0.5\,R_{L_{1}}$) with enhanced emission along a hotter gas
stream ($T_{b}\simeq 7000$-$4500\,K$).

The observed disc radial temperature distribution is much flatter than the
$T\propto R^{-3/4}$ law and is reminiscent to those of quiescent dwarf novae
(e.g., OY Car, Wood et al., 1989). From the brightness temperatures in the
outer disc regions ($R=0.5$-$0.6\,R_{L1}$) we estimate a disc mass input rate
of $\dot{M}=(9\pm 1)\times 10^{-12}\,M_{\odot}\,yr^{-1}$, more than an order
of magnitude lower than the $\dot{M}$ expected from binary evolution with
conservative mass transfer ($\simeq 1.8\times 10^{-10}\,M_{\odot}\,yr^{-1}$,
Z14). We put forward a few possible explanations for this discrepancy. It may
be that the outer disc is optically thin and the gas effective temperatures
(and corresponding $\dot{M}$ values) are significantly larger than the
inferred $T_{b}$ values; multicolour (or spectral) eclipse mapping would be
helpful to clarify this issue. It may also be that the observed period
increase is not a consequence of evolution with conservative mass transfer,
but part of a cyclical period change which could hide a long term period
increase at a much slower pace (a $\dot{P}\sim 1.5\times 10^{-14}\,s/s$ would
be expected from the above mass transfer rate estimate). A third explanation
would be that the present mass transfer rate of J0926 is significantly lower
than its secular average. This raises the possibility of the existence of
large amplitude, long-term modulations in mass transfer rates of AM CVn stars
similar to those which presumably occur in cataclysmic variables (the
hibernation scenario of novae, see Kovetz et al., 1988). However, while in CVs
these modulations seem a consequence of their recurrent nova eruptions, in AM
CVn stars a different mechanism would be required since their hydrogen-
deficient discs cannot drive hydrogen-burning thermonuclear runaways at the WD
surface.

Figure 6: Azimuthally averaged radial brightness temperature distributions for
the gas stream ($\theta=0^{\circ}-90^{\circ}$, blue), and for the full
(‘disc+WD’, green) and WD-subtracted (‘disc’, red) eclipse maps
($\theta=90^{\circ}-360^{\circ}$) for an assumed distance of 465 pc. The
dotted curves show the 1$\sigma$ limits on the corresponding distributions.
Effective temperature distributions of opaque, steady-state disc models for
mass accretion rates of $5\times 10^{-12},10^{-11}$, and $1.8\times
10^{-10}\,M_{\odot}\,yr^{-1}$ are shown as dotted lines.

##  4 Conclusions

We observed J9026 at five occasions on 2012 February-March, during a 4.6 yr
long period without recorded outbursts. The light curves of all nights
consistently show the same morphology, brightness level, eclipse shape and
depth, with no evidence of superhumps or of an orbital hump produced by
anisotropic emission from a bright spot at disc rim. We combined our median
eclipse timing with those in the literature to revise the binary ephemeris and
to confirm that the orbital period is increasing at a rate $\dot{P}=(3.2\pm
0.4)\times 10^{-13}\,s/s$. Eclipse mapping of the average light curve shows a
hot white dwarf at disc centre surrounded by a faint, cool accretion disc with
a residual bright spot (at $R_{d}\simeq 0.65\,R_{L1}$) and enhanced emission
along the ballistic stream trajectory well beyond the disc rim, suggesting the
occurence of gas stream overflow or penetration at that epoch. For an assumed
distance of 465 pc, the estimated disc mass input rate of $\dot{M}=(9\pm
1)\times 10^{-12}\,M_{\odot}\,yr^{-1}$ is more than an order of magnitude
lower than that expected from binary evolution with conservative mass
transfer.

## Acknowledgements

We are grateful to Michael Bode for granting LRT time for these observations,
and to Iain Steele and Robert Smith for much appreciated help with the data
reduction process. W. S. acknowledges financial support from INCT-A/Brazil and
CNPq/Brazil. This paper is based on observations made with the Liverpool
Telescope operated on the island of La Palma by Liverpool John Moores
University in the Spanish Observatorio del Roque de los Muchachos of the
Instituto de Astrofisica de Canarias with financial support from the UK
Science and Technology Facilities Council. This research has made use of the
SIMBAD database, operated at CDS, Strasbourg, France.

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