# On the Rigidly Precessing, Eccentric Gas disk Orbiting the white dwarf SDSS
J1228+1040

Ates Goksu David A. Dunlap Department of Astronomy & Astrophysics, University
of Toronto  Taylor Kutra David A. Dunlap Department of Astronomy &
Astrophysics, University of Toronto  Lowell Observatory  Yanqin Wu David A.
Dunlap Department of Astronomy & Astrophysics, University of Toronto

###### Abstract

Metal pollution onto white dwarfs is a wide-spread phenomenon that remains
puzzling. Some of these white dwarfs also harbour gaseous debris disks.
Emission lines from these disks open a unique window to the physical
properties of the polluting material, lending insights to their origin. We
model the emission line kinematics for the gas disk around SDSS J1228+1040, a
system that has been monitored for over two decades. We show that the disk
mass is strongly peaked at $1R_{\odot}$ (modulo the unknown inclination), and
the disk eccentricity decreases from a value of $0.44$ at the inner edge, to
nearly zero at the outer edge. This eccentricity profile is exactly what one
expects if the disk is in a global eccentric mode, precessing rigidly under
general relativity and gas pressure. The precession period is about two
decades. We infer that the mass of the gas disk is roughly equivalent to that
of a 50-km rocky body, while the mass of the accompanying dust disk is likely
insignificant. The disk eccentricity confirms an origin in tidal disruption,
while the short disk diffusion time suggests that the disruption event
happened a few centuries ago. Moreover, we argue that the initial orbit for
the disrupted body, and that of its putative planetary perturber, fall within
an AU around the white dwarf. The total mass of the source population is
likely orders of magnitude more massive than our own Asteroid belt, and does
not seem to exist around main-sequence stars.

Circumstellar Debris Disks — Near white dwarf environment — SDSS J1228+1040 —
Ca II triplet emission profile

##  1 Introduction

About a third of all white dwarfs show signs of on-going or recent accretion
of heavy metals (e.g., Zuckerman et al., 2003). And starting from the first
example of white dwarf G29-38 (Koester, 1987; Graham et al., 1990; Jura,
2003), many are now known to also exhibit infrared excesses, signs of
circumstellar dust disks (e.g. Kilic et al., 2005, 2006; Farihi et al., 2009;
Debes et al., 2011, 2012). These disks are likely metallic in composition and
are responsible for the pollution. It is now commonly believed that large
asteroids (and/or comets) around these stars are, for some reason, excited to
high eccentricities and are tidally disrupted when they approach the white
dwarfs within the Roche radius. The resulting debris forms the dust disk.
However, many key elements in this story, including the source and the orbital
excitation for these bodies, remain mysterious (for reviews, see Farihi, 2016;
Veras, 2016, 2021).

Interestingly, a few percent of these dusty white dwarfs are also known to
possess gaseous debris disks (Manser et al., 2020). First discovered from SDSS
data by Gänsicke et al. (2006) around the white dwarf SDSS J122859.93+104032.9
(short-named as J1228 below), about two dozens such disks are now known
(Gänsicke et al., 2006, 2007, 2008; Farihi et al., 2012; Gentile Fusillo et
al., 2021a). These disks manifest as double-peaked emission lines in the
spectra, most conspicuously in Ca II infrared triplets. They are found
exclusively around white dwarfs hotter than $\sim 13,000\,\rm K$, likely
because only such stars are luminous enough to sublimate dust at a distance of
$\sim 1R_{\odot}$. The rarity of gaseous disk is likely explained by the
rarity of these hot stars (constituting only a few percent of all white
dwarfs).

Unlike dust disks which reveal little about their kinematics, compositions, or
density distributions (e.g., see a discussion in Nixon et al., 2020), gaseous
disks open a lucky window. The characteristic double-peaked emission lines
from these disks contain information about their radial extent, eccentricity,
surface density and temperature profiles. Interestingly, some of them  exhibit
asymmetric lines that are most easily interpreted as the signature of an
eccentric disk. Moreover, these disks also appear to be time variable (e.g.,
Wilson et al., 2014, 2015; Dennihy et al., 2018; Manser et al., 2021). For
instance, J1228, the best monitored system (Manser et al., 2016, 2019), shows
gradual variations of its line profile over a timescale of decades. This is
much longer than the local Keplerian timescale, which is of order hours.

This kinematic information offers the hope of understanding the origin of
these debris disks. Here, we undertake a study of the J1228 gaseous disk, with
an aim to answer the following specific questions.

First, how does the disk manage to retain its eccentric shape? This disk is
known to extend radially (with a width $\Delta r\sim r$). If so, general
relativistic effects precess the gas differentially, and would have led to its
total circularization within a few decades. In fact, such a consideration
motivates Hartmann et al. (2011); Metzger et al. (2012); Cauley et al. (2018);
Fortin-Archambault et al. (2020) to propose that the observed line profiles
are not due to an eccentric disk, but are instead due to non-axisymmetric
brightness patterns (a vortex, or a spiral wave) on a circular disk. An
alternative possibility is laid out by Miranda & Rafikov (2018) wherein they
argue that gas pressure in the disk can resist GR and maintain the disk in a
state of rigid precession. We resolve this issue by first fitting a physically
motivated disk model (§2) against detailed observations (§3), and then show
that the disk is indeed in a state of pressure-maintained rigid precession
(§4), as is proposed by Miranda & Rafikov (2018).

Second, the kinematic information we extract from the emission lines allows us
to set unique constraints on the origin of the gas disk, and more generally on
the origin of white dwarf pollution (§5).

##  2 Physical Model

We will construct a Keplerian disk model to reproduce the observed double-
peaked line profiles in the Ca II infrared triplet. We assume that the gaseous
disk region is free of solid grains. As we argue below (§5.3), one should not
expect dust to co-exist with an eccentric gas disk.

In such an exercise, as one only measures the line-of-sight velocity, and is
ignorant of the orbital period, one can only determine the length combination,
$a/\sin^{2}i$, where $i$ is the orbital inclination relative to the line of
sight (with $i=0$ being face-on). This differs from the usual radial velocity
literature where one also knows the orbital period and can determine $a\sin
i$. We suppress the factor $1/\sin^{2}i$ in this section, but re-introduce it
in later discussions.

Figure 1: An illustration of our disk model. We represent the disk using a
discreet set of elliptical, aligned, Keplerian rings, with a surface density
profile (left panel) that is composed of two broken power-laws, and an
eccentricity profile (middle panel) that is linear in radius. The right panel
is a birds-eye view of the disk with the colour indicating normalized
emission. This figure is produced using our best fit parameters. The phase
angle $\phi$ takes the value of $0$ if the disk long-axis lies on the plane of
the sky.

Furthermore, in order to translate line flux into gas density, we need to
consider the physics of emission. The Ca II triplet are most likely
recombination lines, namely, spontaneous emission from Ca II ions at excited
levels, arriving after photo-ionization and recombination. The line fluxes
should, therefore, scale linearly with the rate of recombination, which, at
equilibrium, equals the rate of photo-ionization. So the emissivity should
scale as $\propto n_{\gamma}\,n_{\rm CaII}$, where $n_{\gamma}$ is the number
density of ionizing photons from the white dwarf, and $n_{\rm CaII}$ that of
Ca II. The above scaling remains valid even when the disk is very optically
thick to the recombination lines (as is the case for our disk).

This consideration allows us to determine the local gas density, under some
simplifying assumptions. In particular, we will assume that $n_{\gamma}$
depends on the radial distance from the white dwarf as $n_{\gamma}\propto
r^{-q}$, with $q$ being a free parameter. We expect $q=2$ when the disk is
optically thin to the ionizing photons, while $q>2$ when the radial optical
depth to these photons is larger than unity. We will also assume that the disk
has a negligible vertical extent ($H$) and is not being viewed nearly edge-on
(i.e., $H\ll r\cos i$) These allow us to determine the local column density
from the local height-integrated emission, without worrying about the disk
vertical structure. Lastly, we assume that Ca II perfectly traces the local
gas density.

We model the gas disk around J1228 as an assembly of $20$ tightly packed, co-
focal elliptical Keplerian rings. Their semi-major axes are evenly distributed
between $a_{\rm in}$ and $a_{\rm out}$. The gas surface density profile is
assumed to be a broken power-law with a transition radius $a_{\rm break}$:

${\Sigma}\propto\begin{cases}a^{p_{1}}\,,\,\hskip 36.135pta_{\rm in}\leq
a<a_{\rm break}\,;\\\ a^{p_{2}}\,,\hskip 36.135pta_{\rm break}\leq a\leq
a_{\rm out}\,.\end{cases}$ (1)

Such a broken profile is motivated by the observed line profiles, which also
suggest that $p_{1}>0$ and $p_{2}<0$.

As the simplest approximation, we assume that the ring eccentricities vary
linearly with the semi-major axis as

$e(a)=e_{\rm in}+{{de}\over{da}}\times(a-a_{\rm in})\,.$ (2)

We do not specify the sign of the eccentricity gradient ($de/da$). The rings
are assumed to remain apse-aligned at all times. This requires the disk to
precess rigidly, a working assumption we justify in §4.

The line emissivity $\epsilon$, from a ring segment of length $d\ell$, is
therefore

$\epsilon={\rm
const}\times\,r^{-q}\,\times\Sigma\,\times\left(\frac{\frac{1}{v}{d\ell}}{\oint{\frac{1}{v}d\ell}}\right)\,,$
(3)

with $v$ being the Keplerian velocity of the segment. The first factor
($r^{-q}$) describes the radial dependence for the ionizing flux, and the last
factor describes the fractional mass within the line segment. This scales
inversely with the local Keplerian velocity as mass is conserved along a
Keplerian streamline. This behaviour is behind the so-called ‘apocentre glow’
in debris disks(Pan et al., 2016; MacGregor et al., 2017). The overall
normalization constant is discussed in §5, where we show that our model disk
can also account for the total observed flux, in addition to the line profile.

To assign a Doppler velocity to the above line segment, we introduce a phase
angle $\phi$, where $\phi=0$ corresponds to the case where the orbital long
axis lie on the plane of the sky. The resultant line profile is symmetric at
this phase, while the line is at its most asymmetric when $\phi=\pi/2$.

All our model parameters are illustrated in Fig. 1. In the following, we
proceed to search for the best model parameters for the observed Ca-II line
data from Manser et al. (2016).

##  3 MCMC And Results

###  3.1 Markov Chain Monte-Carlo

Manser et al. (2016) have gathered spectra of J1228 from March 2003 to May
2015 in a total of 18 epochs, using the VLT, the SDSS telescope and the WHT.
C. Manser has kindly provided us with the data. Some subsequent observations
are presented in Manser et al. (2019) but are not used for fitting.

To prepare the emission profiles for analysis, we convert the Ca-II triplet
data from wavelength (Å) to velocity (${\rm km}\,{\rm s}^{-1}$), using the
atomic rest-wavelengths, and a systemic velocity of $+22\,\rm km/\,\rm s$.
This value is within the range reported by Manser et al. (2016): $+19\,\pm
4\,{\rm km}\,{\rm s}^{-1}$.111One can also determine the systemic velocity
from the white dwarf spectra, after accounting for a gravitational redshift of
$35\,\rm km/\,\rm s$ for lines emitted from the surface of the white dwarf. It
is chosen so that the emission lines at the June 2007 epoch, which have very
similar amplitudes in the blue- and red-shifted peaks, are also symmetric in
the velocity space. We co-add the three lines to produce a joint line profile.

The precession of the disk means we have the good fortune to observe it from
different vantage points, each giving some unique constraints on the disk
model. We decided, initially, to focus on data from three (equally spaced)
epochs: June 2007, June 2011, and May 2015. We assign a phase of $\phi=\pi/2$
to the first epoch (most symmetric), and phases of $0.31$ and $-0.92$ to the
remaining two. This choice is motivated by our inferred precession period of
$\sim 20\,{\rm yrs}$ (see below). Our initial attempt did not produce a
satisfactory fit to the June 2007 data, so we proceed to include two more
epochs, April 2007 and July 2007, with their corresponding $\phi$ values, into
the procedure. This serves to strengthen the model constraint around
$\phi=\pi/2$.

We now determine the best fit model parameters, using the emcee (Foreman-
Mackey et al., 2013) implementation of the Markov Chain Monte-Carlo (mcmc)
method.

This procedure requires appropriate priors. For each of our 8 parameters, we
choose a flat prior over a wide range (Table 1). Our prior on the eccentricity
profile warrants some comments. First, we posit that $e\in[0,1)$ everywhere.
Second, we insist that streamlines cannot cross within the disk. As streamline
crossing likely occurs at highly supersonic speed, the resultant shock will
remove much of the orbital energy from the gas and cause rapid in-spiral,
effectively truncating the disk (i.e. Artymowicz & Lubow, 1994). This leads us
to impose the condition (Goldreich & Tremaine, 1979)

$\bigg{|}e(a)+\frac{\mathrm{d}\,e(a)}{\mathrm{d}\,\mathrm{ln}(a)}\bigg{|}<1\,.$
(4)

We run emcee with 100 walkers and iterate each for 4000 steps. This ensures
that the auto-correlation time is a sufficiently small fraction of the total
run. We then trim the first 2000 steps to minimize the effects of the initial
conditions. The full results are presented in Figure 4. There is a good
convergence for all parameters, though some show a slight (but unimportant)
bimodal distributions in their posteriors. The maximum-likelihood parameters,
and their corresponding uncertainties, are presented in Table 1. Furthermore,
the resultant line profiles for the three chosen epochs are presented in
Figure 2; while Fig. 3 illustrates them for a continuum of phases.

While the overall comparison is satisfactory, we note that the data show a
conspicuous shortage of emission in the blue wing at $\phi=\pi/2$ (bottom
panel in Fig. 2, yellow box in Fig. 3). This is the phase where we expect
symmetric emission, and indeed the blue and the red peaks do look symmetric.
The deficit is only in the wing, and it persists in observations from nearby
epochs (April and July 2007). The latter rules out the possibility that the
choice of our $\phi=\pi/2$ epoch is the cause of such an asymmetry. We have no
explanation for this deficit.

Figure 2: Comparing the emission profiles between model and data, for three
phases. At $\phi=\pi/2$, we expect symmetry but the data shows a deficit of
emission in the blue wing. Figure 3:  Observed and modelled emission profiles
as functions of the precession phase (vertical axis). The data plot is
generated using epochs from April 2007 to May 2015 (Manser et al., 2016) and
summing over the Ca II infrared triplet. While the overall resemblance is
good, we highlight one notable exception. Near phase $\phi=\pi/2$, the
observed profile is less symmetric than the model and shows a sharper cutoff
at the blue peak (yellow boxes).

###  3.2 Disk Properties

Figure 4: Corner plot for the 1- and 2-D marginalized posterior distributions
for our model parameters, with the first 2000 steps discarded as burn-in. We
see good overall convergence for our parameters.

We now review the properties of our best-fit model. Gänsicke et al. (2006) and
Hartmann et al. (2016) have previously determined the inner and outer radii
for the gas disk: $a_{\rm in}\sim 0.6$ $R_{\odot}$ and $a_{\rm out}\sim
1.2\,R_{\odot}$. Our solutions are broadly consistent with their values, with
$a_{\rm in}=0.57R_{\odot}$ and $a_{\rm out}=1.7R_{\odot}$. Bear in mind that
the physical lengths are smaller than these by $\sin^{2}i$. Our value for the
inner eccentricity ($e_{\rm in}=0.44$) is also consistent with that inferred
by Manser et al. (2016, 2019) This value is higher than the $e=0.021$ value in
the original discovery paper (Gänsicke et al., 2006), because they happened to
catch the lines when they were more symmetric.

Our most interesting result from the MCMC fit is the eccentricity gradient. We
find a significant and negative eccentricity gradient: $de/da=-0.42\pm 0.059$.
Compared to the inner disk, the outer disk is substantially more circular, and
is in fact consistent with being circular. This result can be intuitively
understood by looking at the top panel of Fig. 2: the sharp spike in the blue
wing comes about because the rings are compressed together at apoapse (both in
physical space and in velocity space). To do so, the inner part of the disk
has to be more eccentric. A negative eccentricity gradient is a characteristic
feature of a rigidly precessing gas disk (Miranda & Rafikov, 2018).  We return
to this gradient in §4.

Table 1: Most likely model parameters and their $1-\sigma$ uncertainties. Parameter | Prior | Solution
---|---|---
|  | Mean ($\mu$) | Uncertainty ($\sigma$)
$a_{\rm in}\,[R_{\odot}]$ | $\in[0.3,4]$ | 0.57 | $5.5\%$
$a_{\rm break}\,[R_{\odot}]$ | $\in[a_{\rm in},a_{\rm out}]$ | 1.0 | $2.8\%$
$a_{\rm out}\,[R_{\odot}]$ | $\in[a_{\rm in},7]$ | 1.7 | $4.1\%$
$e_{\rm in}$ | $\in[0,1)$ | 0.44 | $3.9\%$
$\nabla e$ | Eqn. 4 | -0.42 | $14\%$
$q$ | $>0$ | 2.4 | $6.8\%$
$p_{1}$ | $\in(0,5)$ | 1.8 | $17\%$
$p_{2}$ | $\in(-5,0)$ | -1.9 | $10\%$

With our inferred surface density profile (power-law indexes $p_{1}\sim
1.8,\,p_{2}\sim-1.9$), the mass of the disk is strongly concentrated around
$a_{\rm break}\sim 1R_{\odot}$. We also find that $q\sim 2.4$, slightly
steeper than our expectation for an optically thin disk ($q=2$) and represents
a disk that is radially optically thick to the ionizing photons. It is worth
commenting that, while one naively expects a degeneracy between $q$ and
$(p_{1},p_{2})$, since all of them describe radial dependencies ($q$ on
radius, while $p$ on semi-major axis), Fig. 4 convincingly shows that the
degeneracy is broken, likely due to the fact that the rings are substantially
eccentric.

##  4 Rigidly Precessing Disk

Here, we first argue that the disk cannot be differentially precessing, as
general relativity would have it. We then show, using the tool of linear
eigen-mode calculations, that it has the appropriate gas pressure to resist
differential GR precession. In fact, both the observed precession period and
the eccentricity profile agree with theoretical expectations. We thus firmly
establish a long-suspected behaviour, that the J1228 disk is rigidly
precessing.

We first establish a new estimate for the observed precession period.
Previously, Manser et al. (2016) reported a period of 24-30 yrs, based on data
up to May 2015. More recent monitoring extends the data to May 2018 (Manser et
al., 2019) . From these, one infers that the triplet evolves through a
symmetric profile around Oct. 2017. The last time they did so was around July
2007. This leads us to refine the precession period to a value of 20.5 yrs.

We now also insert the factor of $1/\sin^{2}i$ where necessary. Previously,
Gänsicke et al. (2006) have assigned an inclination of $i\sim 70^{\circ}$ to
the disk, based on the crude arguments that the disk is far from being face-on
(double-peaked emission), and is also not edge-on (no self-absorption). This
value remains uncertain, so we keep it as a variable.

###  4.1 GR Makes It Differential

An eccentric ring that is in proximity of the white dwarf experiences general
relativistic (GR) precession. Let the complex eccentricity be

$E=e\,\exp(i{\varpi})\,,$ (5)

where $\varpi$ is the longitude of pericentre measured relative to a fixed
direction in space. GR acts to advance $\varpi$ at a rate

$\displaystyle{\dot{\varpi}}_{\rm GR}$ $\displaystyle=$
$\displaystyle{\frac{3GM_{*}\Omega}{c^{2}a(1-e^{2})}}\approx\frac{2\pi}{84\,{\rm
yrs}}\,\frac{1}{(1-e^{2})}\,\left(\frac{a}{1R_{\odot}}\right)^{-5/2}\,,$ (6)

where $\Omega=\left(GM_{*}/a^{3}\right)^{1/2}$ is the Keplerian frequency.

Since our gas disk extends radially (with a width $\Delta r\sim r$), the above
equation suggests that the eccentric disk should have been markedly twisted
after only $10$ yrs, the precession period for the inner most orbit. If so,
streamlines from different orbits could have crossed, and the resulting
dissipation should have circularized and shrunk the disk. In contrast, the
sharp spikes seen in the line profiles suggest a significant eccentricity. In
fact, the disk has been observed to remain the same eccentric shape for over
20 yrs.

For this reason, previous studies have argued that J1228 and other similar
white dwarfs do not harbour eccentric disks, but instead host circular disks
with non-axisymmetric brightness patterns (e.g., spiral wave, vortex, Hartmann
et al., 2011; Metzger et al., 2012).There are also suggestions of eccentric
disks but with misaligned apses (Cauley et al., 2018; Fortin-Archambault et
al., 2020). These proposals, while being able to provide reasonable fits to
the data, are not physically motivated. An asymmetric pattern on a circular
disk can be rapidly sheared out on the Keplerian timescale (even faster than
GR); and an eccentric disk with misaligned apses are not known to be self-
sustaining.

Figure 5: Comparing our disk model to the linear eigen-mode calculations,
assuming an inclination of $90\deg$. Solid curves in the left panel show the
calculated precession period as a function of temperature at the inner edge,
for a few choices of $\gamma$ (eq. 9). The grey band corresponds to the
precession period determined from data, and the blue dotted line that of GR
period at the disk inner edge. The right panel shows the calculated
eccentricity eigen-functions in solid lines (they are too close to tell apart)
corresponding to the three starred positions on the left panel. Our best-fit
eccentricity profile is plotted as the thick blue line. The theoretical modes
are normalized to have the same eccentricity at the inner edge as the observed
one.

###  4.2 Pressure Keeps It Rigid

In our work, we opt to model the disk as a series of apse-aligned rings that
rigidly precess. We now confirm that this is physically motivated.

First, the radial pressure gradient also causes precession, at a rate that is,
to order of magnitude,

${\dot{\varpi}}_{p}\sim\left({{c_{s}}\over{v_{kep}}}\right)^{2}\Omega\sim\frac{2\pi}{84{\rm
yrs}}\,\left(\frac{a}{1R_{\odot}}\right)^{-3/2}\,\left(\frac{c_{s}/v_{\rm
kep}}{0.0028}\right)^{2}\,,$ (7)

for a ring with width $\Delta r\sim r$ (Goodchild & Ogilvie, 2006; Teyssandier
& Ogilvie, 2016). To be competitive against GR (eq. 6), we only need
$c_{s}/v_{\rm kep}\geq 0.0024$, or a gas temperature222Cooler disks may still
maintain rigid precession, but will require a very steep eccentricity
gradient.

$T\geq 1150{\,\rm K}\times\left(\frac{\mu}{9}\right)\,,$ (8)

where we have evaluated at $a=1R_{\odot}$ and have scaled the mean-molecular
weight against that for singly-ionized metallic gas (see below). The
temperature of the gas disk is likely controlled by photo-ionization and
ranges from $5000$ to $9000\,\rm K$ (Melis et al., 2010).We have also
confirmed this independently using the spectral synthesis code CLOUDY (Ferland
et al., 2017). So the whole disk can easily communicate via pressure, and can
smooth out any precessional mis-demeanour.

The negative eccentricity gradient we report here supports the hypothesis that
the disk is rigidly precessing. Such a configuration means that the rings are
more compressed together at their apocentres. The radial pressure gradient
there tends to precess the inner streamline backwards, while the outer
streamlines precess forwards. This equilibrates their differential GR rates
(eq. 6).

We now make the above arguments more quantitative. We follow Miranda & Rafikov
(2018) to compute the eccentricity eigenmode, the global coherent response of
the disk to an eccentricity perturbation. Teyssandier & Ogilvie (2016) have
studied the linear response of a locally isothermal, 3-D disk. For the case of
a power-law disk, where the surface density scales as $\Sigma\propto r^{p}$,
and where the temperature also obeys a power-law,

$T(r)=T_{\rm in}\left(\frac{r}{r_{\rm in}}\right)^{-\gamma}\,,$ (9)

their result is simplified by Miranda & Rafikov (2018) into a one-dimensional
ODE,

$\displaystyle\frac{\partial^{2}E}{\partial
r^{2}}+\frac{(3-p)}{r}\frac{\partial E}{\partial
r}+\left[\frac{6-\gamma(\gamma+2)-p(\gamma+1)}{r^{2}}\right.$
$\displaystyle\left.+\frac{6r^{2}\Omega^{4}}{c^{2}c_{s}^{2}}-\frac{2\Omega\omega_{prec}}{c_{s}^{2}}\right]E=0\,,$
(10)

where $E=E(r)$ is the (apse-aligned) eccentricity eigenfunction, and
$w_{prec}$ the frequency of global precession. The isothermal sound speed is
$c_{s}=\sqrt{kT/\mu m_{H}}$ and we adopt $\mu=9$ (see §5).

For our problem, since we only measure the length combination
${\tilde{r}}=r/\sin^{2}i$, we transform the above equation to

$\displaystyle\frac{\partial^{2}E}{\partial{\tilde{r}}^{2}}+\frac{(3-p)}{\tilde{r}}\frac{\partial
E}{\partial{\tilde{r}}}+\left[\frac{6-\gamma(\gamma+2)-p(\gamma+1)}{{\tilde{r}}^{2}}\right.$
$\displaystyle\left.+\frac{6{\tilde{r}}^{2}{\tilde{\Omega}}^{4}}{c^{2}c_{s}^{2}\sin^{4}i}-\frac{2{\tilde{\Omega}}\omega_{prec}\sin
i}{c_{s}^{2}}\right]E=0\,,$ (11)

where ${\tilde{\Omega}}=\sqrt{(}GM_{*}/{\tilde{r}})$ and
$c_{s}^{2}=c_{s0}^{2}({\tilde{r}}/{\tilde{r}}_{0})^{-\gamma}$.

One seems to have some flexibilities in choosing the boundary condition (see,
e.g. Miranda & Rafikov, 2018). We adopt the following set

$\left.\frac{\partial E}{\partial r}\right|_{\rm a_{\rm in}}=0\,;\hskip
36.135ptE|_{\rm a_{\rm out}}=0\,.$ (12)

These differ from that in Miranda & Rafikov (2018), where they also took
$\partial E/\partial r=0$ at the outer boundary. Our adopted one is more
descriptive of our best-fit solution. In any case, this does not much affect
the precession rate.

For our broken power-law disk, we integrate Eq. 11 from the two boundaries
towards ${\tilde{a}}_{\rm break}$ and insist that $E$ and $\partial E/\partial
r$ remain continuous across ${\tilde{a}}_{\rm break}$. We then look for
eigenmodes of the lowest radial order. These have the smoothest eccentricity
profiles, and hence the lowest dissipation rates. They are most likely to
persist.

We present in Fig. 5 the results of these calculations, adopting the best-fit
model parameters from Table 1.As we do not have information on the values of
$T_{\rm in}$ and $\gamma$, we experiment within some sensible ranges. These
calculations are performed for $i=90^{o}$, but the conclusion remains largely
the same, given our uncertainties in $T_{\rm in}$ and $\gamma$, for
inclinations as low as $\sim 60^{o}$.

We conclude with two findings. First, within the relevant range of $T_{\rm
in}$, and sensible temperature profiles ($\gamma$), we find that the
theoretical modes have periods comparable to the observed value ($20.5$ yrs).
Second, even though we have only modelled the disk using a linear profile (the
simplest choice), this profile agrees very well with the shape of the linear
eigen-functions. These two quantitative agreements strongly supports the
hypothesis that J1228 hosts a rigidly precessing gas disk, under the combined
effects of GR and gas pressure.

##  5 Insights on Origin

We now discuss what our results imply for the origin of the gas disk around
J1228, as well as for the pollution of white dwarfs in general. We base our
discussions on the current favourite model for white dwarf pollution, tidal
disruption of highly eccentric planetesimals (see reviews by Farihi, 2016;
Brouwers et al., 2022).

###  5.1 Disk Mass and the Progenitor Mass

We estimate a mass for the gas disk, $M_{\rm gas}$, by assuming that viscous
spreading of the gas disk supplies the observed accretion onto J1228.

The gas scale height is

$\frac{H}{r}=\frac{c_{s}}{v_{\rm kep}}\sim
0.006\left({\mu\over{9}}\right)^{-1/2}\left(\frac{T}{5000{\rm
K}}\right)^{1/2}\,\left(\frac{r}{1R_{\odot}}\right)^{1/2}\,.$ (13)

If the disk is accreting under a constant viscosity parameter $\alpha$
(Shakura & Sunyaev, 1973), the viscous diffusion time is

$\displaystyle{\tau_{\rm diff}}$ $\displaystyle\approx$
$\displaystyle{\alpha^{-1}\Omega^{-1}\left(\frac{H}{r}\right)^{-2}}$ (14)
$\displaystyle\sim$ $\displaystyle 200{\rm
yrs}\,\left(\frac{r}{1R_{\odot}}\right)^{3/2}\,\left(\frac{\alpha}{10^{-2}}\right)^{-1}\,\left(\frac{H/r}{0.006}\right)^{-2}\,\,.$

We adopt the accretion rate as determined by Dwomoh & Bauer (2023)333This
updated rate accounts for diffusion from thermohaline mixing and is much
greater than the estimate of in $\dot{M}=5.6\times 10^{8}\,\rm g/\,\rm s$ in
Gänsicke et al. (2012), . and estimate the current disk mass by assuming
${\dot{M}}\sim M_{\rm gas}/\tau_{\rm diff}$, to obtain

$\displaystyle M_{\rm gas}$ $\displaystyle\approx$ $\displaystyle 2\times
10^{21}{\rm g}\left(\frac{\dot{M}}{2\times 10^{11}{\rm g}/{\rm
s}}\right)\left(\frac{\alpha}{10^{-2}}\right)^{-1}$ (15)
$\displaystyle\times\left(\frac{T}{5000{\rm
K}}\right)^{-1}\left(\frac{r}{1R_{\odot}}\right)^{1/2}\,.$

We argue that this estimate likely also reflects the total mass of the
original disrupted planetesimal. The current disk is significantly eccentric –
it avoids rapid streamline crossing and circularization by organizing itself
into a coherent eccentric mode. But over the viscous timescale, the
eccentricity should be gradually damped as the disk spreads radially. So the
current disk has likely weathered no more than a few viscous times. Or, its
current mass is close to its original mass. If true, this means the original
disrupted plantesimals has a radius $R_{p}\sim 50\,\rm km$ (assuming a bulk
density of $\rho_{p}\sim 5\,\rm g/\,\rm cm^{3}$).

To substantiate the above estimate for the disk mass, we also consider whether
it is consistent with the observed Ca II emissions. Let us adopt the same
chemical composition as that for CI chrondrite from Palme et al. (2014), and
assume that all metals are singly ionized, the mean nuclei weight of the gas
is $15$ and the mean molecular weight is $9$ (in unit of hydrogen mass). So
the above mass estimate corresponds to a surface density $\Sigma\sim M_{\rm
gas}/r^{2}\sim 0.13\,\rm g/\,\rm cm^{2}$, and a midplane density of $\rho\sim
3\times 10^{-10}\,\rm g/\,\rm cm^{3}$. In the meantime, Ca nuclei are only
$1/280$ of the total nucleus number. We therefore arrive at a radial column
density for Ca of $\sim 3\times 10^{21}\,\rm cm^{-2}$, and a vertical one at
$\sim 2\times 10^{19}\,\rm cm^{-2}$. We find:

  * •

if all Ca is in Ca II,444Ca I, with an ionization potential of $6.11$eV, is
easily ionized; while Ca II, with a potential of $11.87$eV, is much harder to
ionize., with a photo-ionization cross-section of $5\times 10^{-19}\,\rm
cm^{2}$, the optical depth for a Ca II ionizing photon is $\tau\sim 1600$. So
the gas midplane is very opaque to the Ca II ionizing photons. This may
explain why we obtain a steeper fall-off of the ionizing flux
($n_{\gamma}\propto r^{-2.4}$), than is expected for an optically thin disk
($n_{\gamma}\propto r^{-2}$).

  * •

the vertical optical depth for the Ca II infrared triplet is $\sim 10^{6}$
(CLOUDY result). This explains why the line ratios within the CaII triplet do
not reflect their individual strengths, but approach those of a blackbody
(Melis et al., 2010).

  * •

the above mass estimate allows us to explain the total flux observed in Ca II
triplet. For J1228, about $5.6\%$ of its energy are in photons that can ionize
Ca II (ionization energy $11.87$ eV). Let the disk be optically thick to these
photons from the midplane up to $n$ scale heights. The total energy
intercepted by Ca II is then $2n\times(H/r)\times 5.6\%\times L_{\rm wd}$. As
Ca II is ionized and recombined, a fraction of the ionization energy is
emitted in Ca II triplets (photon energy $\sim 1.5$ eV). So we expect a total
line flux of $\sim 2n(H/r)\times 5.6\%\times(1.5/11.87)\times L_{\rm wd}$. The
observed total line flux is $\sim 3\times 10^{-4}L_{\rm wd}$ (Manser et al.,
2016), or we require $n\sim 3.5$. In comparison, using the above midplane
radial optical depth ($\tau\sim 1600$), and assuming that the disk is in
vertical hydro-static equilibrium, we find that the disk can capture these
photons up to a height multiple of $n\sim 3.8$. In other words, the observed
CaII line fluxes can be explained by our inferred disk density.

###  5.2 The Three Radii

We aim to draw clues on the progenitor by considering the three radii we
inferred for the disk, $a_{\rm in}$, $a_{\rm break}$ and $a_{\rm out}$.

With our values of $p_{1}$ and $p_{2}$, most of the disk mass lies closely
around $a=a_{\rm break}\simeq 1\times\sin^{2}i\,R_{\odot}$. This suggests that
$a_{\rm break}$ may be a special radius for the progenitor body. Gas deposited
here may then viscously spread both outwards and inwards, forming the extended
disk. In particular, Metzger et al. (2012); Rafikov (2016) showed that the
surface density of an isothermal accretion disk should scale as $r^{-2}$,
similar to our value of $p_{2}=-1.9$.

We then consider two physical radii, one for dust sublimation and the other
for tidal disruption.

Using the most updated stellar parameters from Koester et al. (2014),
$M_{*}=0.705M_{\odot}$, $R_{*}=0.0117R_{\odot}$, $T_{\rm eff}=20,713{\rm K}$,
a blackbody at distance $r$ is heated to a temperature

$T_{\rm bb}=\left(\frac{1}{4}\right)^{1/4}\left(\frac{R_{\rm
wd}}{r}\right)^{1/2}T_{\rm wd}\sim 1300{\rm
K}\,\left(\frac{r}{1.5R_{\odot}}\right)^{-1/2}\,,$ (16)

where $1300{\rm K}$ is roughly the sublimation temperature of silicate grains
under our disk midplane density, $\rho\sim 3\times 10^{-10}\,\rm g/\,\rm
cm^{3}$ (Pollack et al., 1994).555The fit provided by Isella & Natta (2005) is
$T_{\rm evap}\sim 1600\,{\rm K}\left(\frac{\rho}{10^{-5}{\rm g}{\rm
cm}^{-3}}\right)^{0.0195}$. This will place the sublimation radius at around
$1.5R_{\odot}$, much beyond our inferred break radius ($a_{\rm
break}=1\sin^{2}i\,R_{\odot}$), but close to our inferred outer edge for the
gas disk, $a_{\rm out}=1.7\sin^{2}i\,R_{\odot}$. It is therefore likely that
the gas disk is truncated near the sublimation boundary. There may well be a
dust component lying beyond it (see discussions below).

On the other hand, the tidal disruption radius typically lies closer than the
sublimation radius. For a body with negligible internal strength666A finite
internal strength will allow the body to survive closer to the white dwarf
(see, e.g. Zhang et al., 2021). and on a parabolic orbit ($e\approx 1$), this
is located at a peri-centre distance of (Sridhar & Tremaine, 1992; Watanabe &
Miyama, 1992)

$\displaystyle\left.r_{\rm roche}\right|_{e\approx 1}$ $\displaystyle\approx$
$\displaystyle 1.7R_{\rm wd}\times\left({{\rho_{\rm wd}}\over{\rho_{\rm
p}}}\right)^{1/3}\,$ (17) $\displaystyle\approx
1.0R_{\odot}\left(\frac{\rho_{\rm p}}{5\,\,\rm g/\,\rm
cm^{3}}\right)^{-1/3}\,,$

where $\rho_{p}$ is the body’s bulk density. If the orbit is less eccentric,
this radius moves outwards, reaching a maximum of $1.5R_{\odot}$ at $e=0$
(Chandrasekhar, 1961).

Given our observed break radius ($a_{\rm break}=1\sin^{2}iR_{\odot}$), it
seems reasonable to suggest that the progenitor enters the zone of tidal
disruption, is threaded apart, and its debris then sublimates to form the
observed disk (also see McDonald & Veras, 2021). There is much energy
dissipation during this process, so the resultant disk is expected to be more
compact and less eccentric. This so-called ’circularization’ process has been
discussed by Veras et al. (2015a); O’Connor & Lai (2020); Malamud et al.
(2021a) and Trevascus et al. (2021). Of particular note is the Trevascus et
al. (2021) work, where they investigated the formation of a gaseous disk under
an eccentric, out-gassing parent body. Unfortunately, current simulations can
only integrate for hundreds of orbits, or in physical time, $\sim$ days. This
is too short to predict the final disk shape (eccentricity and mass) which
evolves on precession timescales (decades).

Lastly, the observed line profiles clearly indicate that the gas disk has an
abrupt inner cutoff at $a_{\rm in}\sim 0.57\sin^{2}i\,R_{\odot}$. What is the
origin of this cutoff? Metzger et al. (2012) suggested that stellar magnetic
field may be able to truncate the disk, much like that around in T Tauri
stars. However, we suspect a different explanation may be at work here. In the
inner part of the disk, rigid precession under the stronger GR precession
demands a steeper eccentricity gradient. If the disk extends closer to the
white dwarf than is observed, the implied high eccentricity will be
challenging for its survival – nonlinear effects may disrupt the pattern of
rigid precession and cause streamline crossing. We hypothesize that disk
eccentricity, rather than stellar magnetic field, truncates our disk at the
observed $a_{\rm in}$. More detailed study is required.

###  5.3 A Co-spatial Dust Disk?

The spectral energy distribution of J1228 shows the presence of a dusty
component, with a total luminosity of $L_{\rm dust}\sim 6\times 10^{-3}L_{\rm
wd}$, and blackbody temperatures that range from $450\,\rm K$ to $1700\,\rm K$
(Brinkworth et al., 2009). If the dust lies in a geometrically flat disk and
sees the central star unobstructed, it should be illuminated to a temperature
(Chiang & Goldreich, 1997; Jura, 2003)

$\displaystyle T_{\rm dust}$ $\displaystyle=$
$\displaystyle\left({2\over{3\pi}}\right)^{1/4}\left(\frac{R_{\rm
wd}}{r}\right)^{3/4}T_{\rm wd}$ (18) $\displaystyle\sim 350\,\rm
K\left(\frac{r}{1.5R_{\odot}}\right)^{-3/4}\,.$

So the above dust temperatures translate to a range of $0.2R_{\odot}$ to
$1.2R_{\odot}$.

But such a model would place the dust component in the same annulus as the gas
disk. The eccentricity of the gas disk makes this problematic. The gas and
dust components, if orbiting at different eccentricities and precessing
independently (i.e., dust experiences GR but not gas pressure), will encounter
each other at enormous speeds, of order a few hundred km/s. This would lead to
circularization of the gas disk, if the dust mass is high enough. In fact,
this un-welcomed prospect led Metzger et al. (2012) to suggest that the gas
disk cannot be eccentric, a proposition now amply refuted by our analysis.

One way to resolve this is if the dust component do not lie in a flat, opaque
disk, a proposal that is also supported by dust observations of GD 56 (Jura et
al., 2007), WD J0846+5703, (Gentile Fusillo et al., 2021b) and G29-38
(Ballering et al., 2022). If the dust grains are not in a flat disk and do not
block each other’s view to the star, their temperaturets will then be
described by eq. 16, and they can remain hot out to larger distances. The
observed blackbody may then arise from a region beyond $0.9R_{\odot}$, largely
avoiding the most eccentric part of the gas disk. These grains can lie even
further away, if they are smaller than the wavelengths of their own thermal
radiation and are thus super-heated.

Such a situation (free-floating grains) can arise if the grains are short-
lived and have not yet undergone collisional flattening (into a thin disk).
Because of the proximity of the gas disk to the sublimation radius, these
grains may be in condensation/sublimation equilibrium with the gas disk, and
are transiently formed and destroyed. In this case, one can obtain a lower-
limit to the dust mass by assuming that the observed dust luminosity is
produced by grains of size $s$, bulk density $\rho_{\rm bulk}$ and temperature
$T$,

$\displaystyle M_{\rm dust}$ $\displaystyle\geq$ $\displaystyle\frac{L_{\rm
dust}\,s\,\rho_{\rm bulk}}{3\sigma T^{4}}\sim 2\times 10^{17}\,\rm
g\,\,\left(\frac{L_{\rm dust}}{6\times 10^{-3}L_{\rm wd}}\right)\times$ (19)
$\displaystyle\left({T\over{1600\,\rm K}}\right)^{-4}\left(\frac{s}{1\mu
m}\right)\left(\frac{\rho_{\rm bulk}}{5\,\rm g/\,\rm cm^{3}}\right)\,.$

In other words, only a minute amount of dust is needed to reproduce the
observed SED. Most of the progenitor mass may instead lie in the gas disk.

In this scenario, the dust component (and its Poynting-Robertson drag) will be
irrelevant for the evolution and accretion of the gas disk, differing from the
proposal by Rafikov (2011); Metzger et al. (2012).

###  5.4 The Progenitor

Here, we remark on how the J1228 disk informs on the origin of white dwarf
pollution.

The observed gas disk is markedly eccentric. Barring the possibility that the
eccentricity is excited after formation (see, e.g., proposal from Miranda &
Rafikov, 2018), this points to a very eccentric orbit for the progenitor. This
is expected in the hypothesis of tidal disruption.

We can infer the original peri-centre approach, by assuming that the orbital
angular momentum ($\sqrt{a(1-e^{2})}$) is largely conserved when the tidal
debris is circularized into the observed disk. This yields a pericenter
distance, $r_{p}=a_{\rm orig}(1-e_{\rm orig})\sim a_{\rm break}*(1-e_{\rm
break}^{2})/(1+e_{\rm orig})\sim 0.47\sin^{2}i\,R_{\odot}$, for $e_{\rm
orig}\approx 1$ and $e_{\rm break}\sim 0.26$ at $a_{\rm break}$. This is a
couple times smaller than the Roche radius (eq. 17).

Who can place the progenitor on such an odd orbit, with an implausibly small
pericenter approach? The most likely scenario left is planetary perturbations
(see review by Veras, 2021). One often invoked mechanism is the so-called
Kozai-Lidov oscillations (Kozai, 1962; Lidov, 1962), wherein secular torque
from a strongly misaligned planetary perturber kneads the progenitor orbit
into an almost needle-like radial orbit.777Compared to the Kozai-Lidov
process, other known dynamical channels do not favour such extreme
eccentricities and have much smaller probabilities of success. However, such a
process could be easily suppressed by GR precession (e.g. Eggleton & Kiseleva-
Eggleton, 2001). To avoid the suppression, the secular perturbations may need
to act in cohort with mean-motion interactions and/or close encounters (the
so-called ’planet scattering’, see, e.g., Nagasawa et al., 2008). This then
requires that the orbital separation of the progenitor to be within a factor
of few from that of the planet.

Where could the putative planet lie? We invoke the following estimate to argue
that it must not be much further than an AU. Such a new constraint, un-seen
before in literature, is enabled by the very short diffusion time of the gas
disk. Things have to happen in a hurry.

After the progenitor is shredded apart, its debris fly on different orbits,
but ones that have long axis well aligned with the original one. They then
start on the process of circularization. In principle, differential GR
precession can scramble the orbital orientations and produce crossing orbits,
which then circularize as mutual collisions dissipate the kinetic energy.
However, if the progenitor’s semi-major axis is $a$, and its pericentre
distance $r_{p}\sim 0.47R_{\odot}$, the GR precession time is of order $\sim
2.5\times 10^{5}(a/{\rm AU})^{3/2}$ yrs (eq. 6). This is so long, it seems
hard to circularize the debris disk before the gaseous component diffuses away
(eq. 14, $\sim 200$ yrs). In other words, GR alone will not circularize the
debris disk fast enough to allow much gas accumulation.

On the other hand, if a planetary perturber lies within an AU, it is able to
quickly scramble the debris orbits. We come to this conclusion as follows. To
reduce the orbital energy by order unity (to achieve an order unity shrinkage
of the semi-major axis), the debris orbits need to be misaligned by of order
unity. This can be induced either by secular perturbations from the planet, or
by successive scatterings. Both take about the same time for a particle that
crosses the planet’s orbit. So we will simplify the argument by setting the
scrambling time to be the secular time,

$T_{\rm secular}\approx\left({{M_{*}}\over{M_{\rm planet}}}\right)P_{\rm
planet}\,.$ (20)

This is consistent with results from Li et al. (2021), where a Neptune-massed
planet at 10AU takes $\sim 10^{6}$ yrs to scramble the orbits.

To replenish the gas disk within its short diffusion time ($200$yrs), we
therefore require the scrambling time to be shorter than the diffusion time,
or

$a_{\rm planet}\leq 0.3{\rm au}\,\left({{M_{\rm planet}}\over{M_{\rm
Jup}}}\right)^{2/3}\,,$ (21)

where $M_{\rm Jup}$ is the mass of Jupiter.

Such a close-in planet is at tension with planet survival around an evolved
star. Previous calculations (Villaver & Livio, 2009; Kunitomo et al., 2011;
Mustill & Villaver, 2012; Adams & Bloch, 2013; Nordhaus & Spiegel, 2013;
Villaver et al., 2014; Madappatt et al., 2016; Ronco et al., 2020) have shown
that as a star evolves through the giant phases and then loses its envelope,
the closest distance one expects a planet (or planetesimals) to survive should
lie outside a few AU – reaching $\sim 7$au for an initial stellar mass of
$3M_{\odot}$ (Fig. 10 of Mustill & Villaver, 2012).

It is unclear yet how to resolve this tension. If the perturber lies much
further away, one has to come up with an alternative way to rapidly
circularize the debris disk. Past proposals have included radiative drag
(Veras et al., 2015b) or collisional drag with a pre-existing disk (Malamud et
al., 2021b). However, these are not effective unless the debris have been
ground down to microscopic sizes, and even then it is hard to satisfy the
$200$ yr constraint.

We now turn to the source population that gives rise to the progenitor.

If the gas disk disappears in a few diffusion time ($\sim 200$yrs), that means
the disruption event must have occurred only recently, within the past few
centuries. We then estimate how common such an event occurs during the
lifetime of J1228. White dwarfs that are of a similar temperature as J1228 (so
that they can sublimate grains and maintain a gaseous accretion disk) have a
space density of $\sim 4\times 10^{-5}/{\rm pc}^{3}/{\rm mag}$ (Leggett et
al., 1998). So within the distance of J1228 ($\sim 127$pc), there are about
$300$ such white dwarfs. Among these, about two dozen gas disks have been
reported (Gänsicke et al., 2006, 2007, 2008; Farihi et al., 2012; Gentile
Fusillo et al., 2021a). So the occurrence rate of gas disks among hot white
dwarfs is $\sim 10\%$, not dissimilar to the occurrence rate of dust disks
among all white dwarfs.888We note that Manser et al. (2020) estimated an
occurrence rate of $0.067\pm^{0.042}_{0.025}\%$ for gaseous disks among alll
white dwarfs. This is comparable to our estimate here, since hot white dwarfs
like J1228 are of order $1\%$ of the local white dwarf population(Leggett et
al., 1998). So the total mass of (disrupted) progenitors (all assumed to be
$R_{p}=50\,\rm km$) over the cooling age of J1228 ($\sim 10^{7}$ yrs) 999Here,
we ignore events that may have occurred during the main sequence phase. is,

$\displaystyle M_{\rm disrupted}$ $\displaystyle\approx{{4\pi}\over
3}\rho_{p}R_{p}^{3}\,\times{{{10\%}\times{10^{7}{\rm yrs}}}\over{t_{\rm
diff}}}$ (22) $\displaystyle\sim 10^{25}\,\rm
g\,\times\left({{R_{p}}\over{50\,\rm
km}}\right)^{3}\left({{\rho_{p}}\over{5\,\rm g/\,\rm cm^{3}}}\right).$

This mass is similar to the total mass of our own asteroid belt ($\sim 2\times
10^{24}\,\rm g$). However, the disrupted bodies are those that are able to
reach the Roche radius, and so likely only constitute a small fraction of the
total source disk. Moreover, evidence on older white dwarfs suggests that
metal pollution continues well past the cooling age of J1228. Both these
arguments imply that, for us to witness a rapidly draining gas disk around
J1228, there must exist planetesimal disks around these white dwarfs that are
much more massive than what one expects from their main-sequence counter
parts. This, a massive belt at au distance, is again at tension with our
current knowledge. While a planetesimal belt at the Kuiper belt distance can
remain massive for longer, due to a slower collision timescale, a belt at the
Asteroid belt distance should grind down quickly during the main-sequence
lifetime of a star, even if it started massive (see, e.g. Wyatt et al., 2007).

Lastly, we comment on the report of a planetesimal embedded in the gas disk.
Manser et al. (2019) discovered periodic variations in the line profiles of
J1228, with a period of $123.4$min (corresponding to an orbital semi-major
axis of $0.73R_{\odot}$). They interpreted these disturbances as due to an
out-gassing planetesimal on an eccentric orbit that overlaps with the gas
disk. If such an object does exist and is massive (radius $R\geq 50\,\rm km$),
our above arguments on the disk origin can fail, because the disk can now be
continuously fed and can survive much longer than the diffusion time.101010Our
main results on disk profiles and precession properties remain valid. However,
there are multiple reasons to suspect the planetesimal interpretation. First,
as remarked in Manser et al. (2019), the survival of such a body well inside
the nominal Roche radius is problematic: the body has to have an unphysically
high bulk density ($\rho_{p}\geq 40$g/cm3 for circular orbit; higher value
needed for eccentric orbit; see eq. 17); or the body has to have a high
internal strength $\sim 10^{8}(R_{p}/50\,\rm km)^{2}\,\,\rm dyne/\,\rm cm^{2}$
(see eq. 4.78 of Murray & Dermott, 1999), comparable to that of a monolithic
rock. Moreover, if the planetesimal precesses at a different rate than the gas
disk (which is physically likely), there will be continuous streamline
crossings between the existing gas disk and the newly sublimated gas,
resulting in a disk that is nearly circular and much more compact than the
parent body. Lastly, it is hard to imagine how a massive planetesimal can
evolve from a nearly parabolic orbit to its current close orbit, short of any
sensible dissipation mechanism.

##  6 Conclusion

We undertake a detailed modelling of the gaseous disk around the white dwarf
SDSS J1228+1040. We find that the disk has a surface density profile that
peaks around $1\,\sin^{2}i\,R_{\odot}$, and an eccentricity profile that
decreases outward. The latter, we show, uncannily reproduces the theoretical
profile of a disk that precesses rigidly under the combined forces of general-
relativity and gas pressure. In other words, the observed disk is in an
eccentric eigen-state. This explains why the disk can be eccentric yet long-
lived. As we expect the eccentricity to be dissipated in the viscous timescale
($\sim 200$ yrs), the current disk should have formed fairly recently.

Based on the high accretion rate onto the white dwarf (current estimate
$\dot{M}\sim 2\times 10^{11}\,\rm g/\,\rm s$), we infer a mass of $\sim
10^{21}\,\rm g$ for the gaseous disk. Such a mass estimate is also consistent
with the emission measures in the Ca II triplets. The young age of the current
disk then implies that it still contains most of the source mass, pegging the
progenitor at a size of $R\sim 50\,\rm km$.

Given the eccentricity of the gas disk, there is unlikely to be a massive
dusty debris disk that is co-spatial. Rather, we suggest that the observed
dust emission may arise from small amounts of grains that are in
condensation/sublimation equilibrium with the gas disk.

The eccentricity of the remnant disk supports the hypothesis of tidal
disruption. The progenitor body, after being excited to a nearly parabolic
orbit and reaching a peri-centre distance of $0.47\sin^{2}iR_{\odot}$, was
tidally disrupted by the white dwarf. The debris then circularizes into the
current moderately eccentric and compact disk.

The short diffusion time of the gas disk allows us to set unique and stringent
constraints on the origin of the system. By insisting that the tidal debris
circularizes into the current size in a time shorter than the gas disk
diffusion time, we find that both the planetesimal and its planetary perturber
likely orbit around the white dwarf within an AU. This is in tension with our
current knowledge of planet survival around white dwarf progenitors.

Moreover, to account for the observed rate of gaseous disks among similarly
hot white dwarfs, we estimate that the source disk needs to contain at least
the mass of our asteroid belt, and likely many orders of magnitude more in
reality. Such a massive disk does not seem to exist around the main-sequence
counter-parts of these white dwarfs.

Our study in this work is preliminary in nature. We have not investigated in
detail the temperature structure and the emission mechanism of the metallic
disk. Emission line diagnostics may be used to constrain the disk inclination,
which may lead to further insights. We also fall short of analyzing the
‘circularization’ process after tidal disruption. This latter seems a
promising route to infer the nature of the progenitor and the architecture of
planetary system around J1228.

Such careful studies are clearly warranted. J1228 is likely not unusual.
First, many white dwarfs with gaseous disks show variable emissions,
indicating eccentric, precessing disks (Gänsicke et al., 2008; Melis et al.,
2010; Wilson et al., 2014; Cauley et al., 2018; Manser et al., 2021). Similar
dynamics as we reveal here for J1228 may be in play in all of these disks.
Second, while J1228 is hot and can sublimate rocks at around $1R_{\odot}$,
cooler white dwarfs will only harbour fully dusty disks. Such a disk reveals
no information on its kinematics and surface density. J1228 offers us a lucky
window into these otherwise obscure disks and may well be the Rosetta stone to
decipher the mystery of white dwarf pollution.

We acknowledge NSERC for funding. We also thank C. Manser for providing the
line data, Renu Malhotra and Elliot Lynch for discussions. Lastly, we thank
the anonymous referee for a careful critique of the paper.

## References

  * Adams & Bloch (2013) Adams, F. C., & Bloch, A. M. 2013, ApJ, 777, L30, doi: 10.1088/2041-8205/777/2/L30
  * Artymowicz & Lubow (1994) Artymowicz, P., & Lubow, S. H. 1994, ApJ, 421, 651, doi: 10.1086/173679
  * Ballering et al. (2022) Ballering, N. P., Levens, C. I., Su, K. Y. L., & Cleeves, L. I. 2022, ApJ, 939, 108, doi: 10.3847/1538-4357/ac9a4a
  * Brinkworth et al. (2009) Brinkworth, C. S., Gänsicke, B. T., Marsh, T. R., Hoard, D. W., & Tappert, C. 2009, ApJ, 696, 1402, doi: 10.1088/0004-637X/696/2/1402
  * Brouwers et al. (2022) Brouwers, M. G., Bonsor, A., & Malamud, U. 2022, MNRAS, 509, 2404, doi: 10.1093/mnras/stab3009
  * Cauley et al. (2018) Cauley, P. W., Farihi, J., Redfield, S., et al. 2018, ApJ, 852, L22, doi: 10.3847/2041-8213/aaa3d9
  * Chandrasekhar (1961) Chandrasekhar, S. 1961, Hydrodynamic and hydromagnetic stability
  * Chiang & Goldreich (1997) Chiang, E. I., & Goldreich, P. 1997, ApJ, 490, 368, doi: 10.1086/304869
  * Debes et al. (2012) Debes, J. H., Hoard, D. W., Farihi, J., et al. 2012, ApJ, 759, 37, doi: 10.1088/0004-637X/759/1/37
  * Debes et al. (2011) Debes, J. H., Hoard, D. W., Wachter, S., Leisawitz, D. T., & Cohen, M. 2011, ApJS, 197, 38, doi: 10.1088/0067-0049/197/2/38
  * Dennihy et al. (2018) Dennihy, E., Clemens, J. C., Dunlap, B. H., et al. 2018, ApJ, 854, 40, doi: 10.3847/1538-4357/aaa89b
  * Dwomoh & Bauer (2023) Dwomoh, A., & Bauer, E. B. 2023, arXiv e-prints, arXiv:2306.03864
  * Eggleton & Kiseleva-Eggleton (2001) Eggleton, P. P., & Kiseleva-Eggleton, L. 2001, ApJ, 562, 1012, doi: 10.1086/323843
  * Farihi (2016) Farihi, J. 2016, New A Rev., 71, 9, doi: 10.1016/j.newar.2016.03.001
  * Farihi et al. (2012) Farihi, J., Gänsicke, B. T., Steele, P. R., et al. 2012, MNRAS, 421, 1635, doi: 10.1111/j.1365-2966.2012.20421.x
  * Farihi et al. (2009) Farihi, J., Jura, M., & Zuckerman, B. 2009, ApJ, 694, 805, doi: 10.1088/0004-637X/694/2/805
  * Ferland et al. (2017) Ferland, G. J., Chatzikos, M., Guzmán, F., et al. 2017, Rev. Mexicana Astron. Astrofis., 53, 385, doi: 10.48550/arXiv.1705.10877
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067
  * Fortin-Archambault et al. (2020) Fortin-Archambault, M., Dufour, P., & Xu, S. 2020, ApJ, 888, 47, doi: 10.3847/1538-4357/ab585a
  * Gänsicke et al. (2012) Gänsicke, B. T., Koester, D., Farihi, J., et al. 2012, MNRAS, 424, 333, doi: 10.1111/j.1365-2966.2012.21201.x
  * Gänsicke et al. (2008) Gänsicke, B. T., Koester, D., Marsh, T. R., Rebassa-Mansergas, A., & Southworth, J. 2008, MNRAS, 391, L103, doi: 10.1111/j.1745-3933.2008.00565.x
  * Gänsicke et al. (2007) Gänsicke, B. T., Marsh, T. R., & Southworth, J. 2007, MNRAS, 380, L35, doi: 10.1111/j.1745-3933.2007.00343.x
  * Gänsicke et al. (2006) Gänsicke, B. T., Marsh, T. R., Southworth, J., & Rebassa-Mansergas, A. 2006, Science, 314, 1908, doi: 10.1126/science.1135033
  * Gentile Fusillo et al. (2021a) Gentile Fusillo, N. P., Manser, C. J., Gänsicke, B. T., et al. 2021a, MNRAS, 504, 2707, doi: 10.1093/mnras/stab992
  * Gentile Fusillo et al. (2021b) —. 2021b, MNRAS, 504, 2707, doi: 10.1093/mnras/stab992
  * Goldreich & Tremaine (1979) Goldreich, P., & Tremaine, S. 1979, AJ, 84, 1638, doi: 10.1086/112587
  * Goodchild & Ogilvie (2006) Goodchild, S., & Ogilvie, G. 2006, MNRAS, 368, 1123, doi: 10.1111/j.1365-2966.2006.10197.x
  * Graham et al. (1990) Graham, J. R., Matthews, K., Neugebauer, G., & Soifer, B. T. 1990, ApJ, 357, 216, doi: 10.1086/168907
  * Hartmann et al. (2011) Hartmann, S., Nagel, T., Rauch, T., & Werner, K. 2011, A&A, 530, A7, doi: 10.1051/0004-6361/201116625
  * Hartmann et al. (2016) —. 2016, A&A, 593, A67, doi: 10.1051/0004-6361/201628403
  * Isella & Natta (2005) Isella, A., & Natta, A. 2005, A&A, 438, 899, doi: 10.1051/0004-6361:20052773
  * Jura (2003) Jura, M. 2003, ApJ, 584, L91, doi: 10.1086/374036
  * Jura et al. (2007) Jura, M., Farihi, J., & Zuckerman, B. 2007, ApJ, 663, 1285, doi: 10.1086/518767
  * Kilic et al. (2005) Kilic, M., von Hippel, T., Leggett, S. K., & Winget, D. E. 2005, ApJ, 632, L115, doi: 10.1086/497825
  * Kilic et al. (2006) —. 2006, ApJ, 646, 474, doi: 10.1086/504682
  * Koester (1987) Koester, D. 1987, ApJ, 322, 852, doi: 10.1086/165779
  * Koester et al. (2014) Koester, D., Gänsicke, B. T., & Farihi, J. 2014, A&A, 566, A34, doi: 10.1051/0004-6361/201423691
  * Kozai (1962) Kozai, Y. 1962, AJ, 67, 591, doi: 10.1086/108790
  * Kunitomo et al. (2011) Kunitomo, M., Ikoma, M., Sato, B., Katsuta, Y., & Ida, S. 2011, ApJ, 737, 66, doi: 10.1088/0004-637X/737/2/66
  * Leggett et al. (1998) Leggett, S. K., Ruiz, M. T., & Bergeron, P. 1998, ApJ, 497, 294, doi: 10.1086/305463
  * Li et al. (2021) Li, D., Mustill, A. J., & Davies, M. B. 2021, MNRAS, 508, 5671, doi: 10.1093/mnras/stab2949
  * Lidov (1962) Lidov, M. L. 1962, Planet. Space Sci., 9, 719, doi: 10.1016/0032-0633(62)90129-0
  * MacGregor et al. (2017) MacGregor, M. A., Matrà, L., Kalas, P., et al. 2017, ApJ, 842, 8, doi: 10.3847/1538-4357/aa71ae
  * Madappatt et al. (2016) Madappatt, N., De Marco, O., & Villaver, E. 2016, MNRAS, 463, 1040, doi: 10.1093/mnras/stw2025
  * Malamud et al. (2021a) Malamud, U., Grishin, E., & Brouwers, M. 2021a, MNRAS, 501, 3806, doi: 10.1093/mnras/staa3940
  * Malamud et al. (2021b) —. 2021b, MNRAS, 501, 3806, doi: 10.1093/mnras/staa3940
  * Manser et al. (2020) Manser, C. J., Gänsicke, B. T., Gentile Fusillo, N. P., et al. 2020, MNRAS, 493, 2127, doi: 10.1093/mnras/staa359
  * Manser et al. (2016) Manser, C. J., Gänsicke, B. T., Marsh, T. R., et al. 2016, MNRAS, 455, 4467, doi: 10.1093/mnras/stv2603
  * Manser et al. (2019) Manser, C. J., Gänsicke, B. T., Eggl, S., et al. 2019, Science, 364, 66, doi: 10.1126/science.aat5330
  * Manser et al. (2021) Manser, C. J., Dennihy, E., Gänsicke, B. T., et al. 2021, MNRAS, 508, 5657, doi: 10.1093/mnras/stab2948
  * McDonald & Veras (2021) McDonald, C. H., & Veras, D. 2021, MNRAS, 506, 4031, doi: 10.1093/mnras/stab1906
  * Melis et al. (2010) Melis, C., Jura, M., Albert, L., Klein, B., & Zuckerman, B. 2010, ApJ, 722, 1078, doi: 10.1088/0004-637X/722/2/1078
  * Metzger et al. (2012) Metzger, B. D., Rafikov, R. R., & Bochkarev, K. V. 2012, MNRAS, 423, 505, doi: 10.1111/j.1365-2966.2012.20895.x
  * Miranda & Rafikov (2018) Miranda, R., & Rafikov, R. R. 2018, ApJ, 857, 135, doi: 10.3847/1538-4357/aab9a2
  * Murray & Dermott (1999) Murray, C. D., & Dermott, S. F. 1999, Solar System Dynamics, doi: 10.1017/CBO9781139174817
  * Mustill & Villaver (2012) Mustill, A. J., & Villaver, E. 2012, ApJ, 761, 121, doi: 10.1088/0004-637X/761/2/121
  * Nagasawa et al. (2008) Nagasawa, M., Ida, S., & Bessho, T. 2008, ApJ, 678, 498, doi: 10.1086/529369
  * Nixon et al. (2020) Nixon, C. J., Pringle, J. E., Coughlin, E. R., Swan, A., & Farihi, J. 2020, arXiv e-prints, arXiv:2006.07639, doi: 10.48550/arXiv.2006.07639
  * Nordhaus & Spiegel (2013) Nordhaus, J., & Spiegel, D. S. 2013, MNRAS, 432, 500, doi: 10.1093/mnras/stt569
  * O’Connor & Lai (2020) O’Connor, C. E., & Lai, D. 2020, MNRAS, 498, 4005, doi: 10.1093/mnras/staa2645
  * Palme et al. (2014) Palme, H., Lodders, K., & Jones, A. 2014, in Planets, Asteriods, Comets and The Solar System, ed. A. M. Davis, Vol. 2, 15–36
  * Pan et al. (2016) Pan, M., Nesvold, E. R., & Kuchner, M. J. 2016, ApJ, 832, 81, doi: 10.3847/0004-637X/832/1/81
  * Pollack et al. (1994) Pollack, J. B., Hollenbach, D., Beckwith, S., et al. 1994, ApJ, 421, 615, doi: 10.1086/173677
  * Rafikov (2011) Rafikov, R. R. 2011, MNRAS, 416, L55, doi: 10.1111/j.1745-3933.2011.01096.x
  * Rafikov (2016) —. 2016, ApJ, 830, 7, doi: 10.3847/0004-637X/830/1/7
  * Ronco et al. (2020) Ronco, M. P., Schreiber, M. R., Giuppone, C. A., et al. 2020, ApJ, 898, L23, doi: 10.3847/2041-8213/aba35f
  * Shakura & Sunyaev (1973) Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337
  * Sridhar & Tremaine (1992) Sridhar, S., & Tremaine, S. 1992, Icarus, 95, 86, doi: 10.1016/0019-1035(92)90193-B
  * Teyssandier & Ogilvie (2016) Teyssandier, J., & Ogilvie, G. I. 2016, MNRAS, 458, 3221, doi: 10.1093/mnras/stw521
  * Trevascus et al. (2021) Trevascus, D., Price, D. J., Nealon, R., et al. 2021, MNRAS, 505, L21, doi: 10.1093/mnrasl/slab043
  * Veras (2016) Veras, D. 2016, Royal Society Open Science, 3, 150571, doi: 10.1098/rsos.150571
  * Veras (2021) —. 2021, in Oxford Research Encyclopedia of Planetary Science, 1, doi: 10.1093/acrefore/9780190647926.013.238
  * Veras et al. (2015a) Veras, D., Leinhardt, Z. M., Eggl, S., & Gänsicke, B. T. 2015a, MNRAS, 451, 3453, doi: 10.1093/mnras/stv1195
  * Veras et al. (2015b) —. 2015b, MNRAS, 451, 3453, doi: 10.1093/mnras/stv1195
  * Villaver & Livio (2009) Villaver, E., & Livio, M. 2009, ApJ, 705, L81, doi: 10.1088/0004-637X/705/1/L81
  * Villaver et al. (2014) Villaver, E., Livio, M., Mustill, A. J., & Siess, L. 2014, ApJ, 794, 3, doi: 10.1088/0004-637X/794/1/3
  * Watanabe & Miyama (1992) Watanabe, S.-I., & Miyama, S. M. 1992, ApJ, 391, 318, doi: 10.1086/171348
  * Wilson et al. (2015) Wilson, D. J., Gaensicke, B. T., Koester, D., et al. 2015, in Astronomical Society of the Pacific Conference Series, Vol. 493, 19th European Workshop on White Dwarfs, ed. P. Dufour, P. Bergeron, & G. Fontaine, 279
  * Wilson et al. (2014) Wilson, D. J., Gänsicke, B. T., Koester, D., et al. 2014, MNRAS, 445, 1878, doi: 10.1093/mnras/stu1876
  * Wyatt et al. (2007) Wyatt, M. C., Smith, R., Greaves, J. S., et al. 2007, ApJ, 658, 569, doi: 10.1086/510999
  * Zhang et al. (2021) Zhang, Y., Liu, S.-F., & Lin, D. N. C. 2021, ApJ, 915, 91, doi: 10.3847/1538-4357/ac00ae
  * Zuckerman et al. (2003) Zuckerman, B., Koester, D., Reid, I. N., & Hünsch, M. 2003, ApJ, 596, 477, doi: 10.1086/377492