On the Rigidly Precessing, Eccentric Gas Disk Orbiting the White Dwarf SDSS J1228+1040

Manser et al. (2016) have gathered spectra of J1228 from 2003 March to 2015 May in a total of 18 epochs using the Very Large Telescope, the SDSS telescope, and the William Herschel Telescope. 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 (km s⁻¹) using the atomic rest wavelengths and a systemic velocity of +22 km s⁻¹. This value is within the range reported by Manser et al. (2016): +19 ± 4 km s⁻¹. ⁴ It is chosen so that the emission lines at the June 2007 epoch, which have very similar amplitudes in the blue- and redshifted peaks, are also symmetric in the velocity space. We coadd 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: 2007 June, 2011 June, and 2015 May. We assign a phase of ϕ = π/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 ∼20 yr (see below). Our initial attempt did not produce a satisfactory fit to the June 2007 data, so we proceeded to include two more epochs, 2007 April and 2007 July, with their corresponding ϕ values, into the procedure. This serves to strengthen the model constraint around ϕ = π/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 ∈ [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)

$$\begin{eqnarray}&&\left|e(a)+\displaystyle \frac{d\,e(a)}{d\,\mathrm{ln}(a)}\right|\lt 1.\end{eqnarray} \tag{ 4 }$$

We run emcee with 100 walkers and iterate each for 4000 steps. This ensures that the autocorrelation 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 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 Figure 3 illustrates them for a continuum of phases.

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While the overall comparison is satisfactory, we note that the data show a conspicuous shortage of emission in the blue wing at ϕ = π/2 (bottom panel in Figure 2, yellow box in Figure 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 (2007 April and July). The latter rules out the possibility that the choice of our ϕ = π/2 epoch is the cause of such an asymmetry. We have no explanation for this deficit.

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_in ∼ 0.6 R_⊙ and a_out ∼ 1.2 R_⊙. Our solutions are broadly consistent with their values, with a_in = 0.57 R_⊙ and a_out = 1.7 R_⊙. Bear in mind that the physical lengths are smaller than these by ${\sin }^{2}i$. Our value for the inner eccentricity (e_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 ± 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 Figure 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 Section 4.

With our inferred surface density profile (power-law indexes p₁ ∼ 1.8, p₂ ∼ −1.9), the mass of the disk is strongly concentrated around a_break ∼ 1 R_⊙. We also find that q ∼ 2.4 is 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 (p₁, p₂), since all of them describe radial dependencies (q on the radius, while p on the semimajor axis), Figure 4 convincingly shows that the degeneracy is broken, likely due to the fact that the rings are substantially eccentric.

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An eccentric ring that is in proximity to the white dwarf experiences GR precession. Let the complex eccentricity be

$$\begin{eqnarray}&&E=e\,\exp (i\varpi ),\end{eqnarray} \tag{ 5 }$$

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

$$\begin{eqnarray}&&{\dot{\varpi }}_{\mathrm{GR}}=\displaystyle \frac{3{{GM}}_{* }{\rm{\Omega }}}{{c}^{2}a(1-{e}^{2})}\approx \displaystyle \frac{2\pi }{84\,\mathrm{yr}}\,\displaystyle \frac{1}{(1-{e}^{2})}{\left(\displaystyle \frac{a}{1{R}_{\odot }}\right)}^{-5/2},\end{eqnarray} \tag{ 6 }$$

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

Since our gas disk extends radially(with a width Δr ∼ r), the above equation suggests that the eccentric disk should have been markedly twisted after only 10 yr, the precession period for the innermost 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 in the same eccentric shape for over 20 yr.

For this reason, previous studies have argued that J1228 and other similar white dwarfs do not harbor eccentric disks but instead host circular disks with nonaxisymmetric 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 is not known to be self-sustaining.

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,

$$\begin{eqnarray}&&{\dot{\varpi }}_{p}\sim {\left(\displaystyle \frac{{c}_{s}}{{v}_{{kep}}}\right)}^{2}{\rm{\Omega }}\sim \displaystyle \frac{2\pi }{84\mathrm{yr}}{\left(\displaystyle \frac{a}{1{R}_{\odot }}\right)}^{-3/2}{\left(\displaystyle \frac{{c}_{s}/{v}_{\mathrm{kep}}}{0.0028}\right)}^{2},\end{eqnarray} \tag{ 7 }$$

for a ring with width Δr ∼ r (Goodchild & Ogilvie 2006; Teyssandier & Ogilvie 2016). To be competitive against GR (Equation (6)), we only need c_s /v_kep ≥ 0.0024, or a gas temperature ⁵

$$\begin{eqnarray}&&T\geqslant 1150\,{\rm{K}}\times \left(\displaystyle \frac{\mu }{9}\right),\end{eqnarray} \tag{ 8 }$$

where we have evaluated at a = 1 R_⊙ 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 photoionization and ranges from 5000 to 9000 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 misdemeanor.

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 apocenters. The radial pressure gradient there tends to precess the inner streamline backward, while the outer streamlines precess forward. This equilibrates their differential GR rates (Equation (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, 3D disk. For the case of a power-law disk, where the surface density scales as Σ ∝ r^p , and where the temperature also obeys a power law,

$$\begin{eqnarray}&&T(r)={T}_{\mathrm{in}}{\left(\displaystyle \frac{r}{{r}_{\mathrm{in}}}\right)}^{-\gamma },\end{eqnarray} \tag{ 9 }$$

their result is simplified by Miranda & Rafikov (2018) into a 1D ordinary differential equation,

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{\partial }^{2}E}{\partial {r}^{2}}+\displaystyle \frac{(3-p)}{r}\displaystyle \frac{\partial E}{\partial r}+\left[\displaystyle \frac{6-\gamma (\gamma +2)-p(\gamma +1)}{{r}^{2}}\right.\\ \,\left.+\displaystyle \frac{6{r}^{2}{{\rm{\Omega }}}^{4}}{{c}^{2}{c}_{s}^{2}}-\displaystyle \frac{2{\rm{\Omega }}{\omega }_{{prec}}}{{c}_{s}^{2}}\right]E\,=\,0,\end{array}\end{eqnarray} \tag{ 10 }$$

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 μ = 9 (see Section 5).

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

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{\partial }^{2}E}{\partial {\tilde{r}}^{2}}+\displaystyle \frac{(3-p)}{\tilde{r}}\displaystyle \frac{\partial E}{\partial \tilde{r}}+\left[\displaystyle \frac{6-\gamma (\gamma +2)-p(\gamma +1)}{{\tilde{r}}^{2}}\right.\\ \,\left.+\displaystyle \frac{6{\tilde{r}}^{2}{\tilde{{\rm{\Omega }}}}^{4}}{{c}^{2}{c}_{s}^{2}{\sin }^{4}i}-\displaystyle \frac{2\tilde{{\rm{\Omega }}}{\omega }_{{prec}}\sin i}{{c}_{s}^{2}}\right]E\,=\,0,\end{array}\end{eqnarray} \tag{ 11 }$$

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

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

$$\begin{eqnarray}&&{\left.\displaystyle \frac{\partial E}{\partial r}\right|}_{{{\rm{a}}}_{\mathrm{in}}}\,=\,0;E{| }_{{{\rm{a}}}_{\mathrm{out}}}\,=\,0.\end{eqnarray} \tag{ 12 }$$

These differ from that in Miranda & Rafikov (2018), where they also took ∂E/∂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 Equation (11) from the two boundaries toward ${\tilde{a}}_{\mathrm{break}}$ and insist that E and ∂E/∂r remain continuous across ${\tilde{a}}_{\mathrm{break}}$. 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 Figure 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_in and γ, we experiment within some sensible ranges. These calculations are performed for i = 90°, but the conclusion remains largely the same, given our uncertainties in T_in and γ, for inclinations as low as ∼60^o .

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We conclude with two findings. First, within the relevant range of T_in, and sensible temperature profiles (γ), we find that the theoretical modes have periods comparable to the observed value (20.5 yr). Second, even though we have only modeled the disk using a linear profile (the simplest choice), this profile agrees very well with the shape of the linear eigenfunctions. These two quantitative agreements strongly support the hypothesis that J1228 hosts a rigidly precessing gas disk under the combined effects of GR and gas pressure.

We estimate a mass for the gas disk, M_gas, by assuming that the viscous spreading of the gas disk supplies the observed accretion onto J1228.

The gas scale height is

$$\begin{eqnarray}&&\displaystyle \frac{H}{r}=\displaystyle \frac{{c}_{s}}{{v}_{\mathrm{kep}}}\sim 0.006{\left(\displaystyle \frac{\mu }{9}\right)}^{-1/2}{\left(\displaystyle \frac{T}{5000\,{\rm{K}}}\right)}^{1/2}{\left(\displaystyle \frac{r}{1{R}_{\odot }}\right)}^{1/2}.\end{eqnarray} \tag{ 13 }$$

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

$$\begin{eqnarray}\begin{array}{rcl}{\tau }_{\mathrm{diff}} & \approx & {\alpha }^{-1}{{\rm{\Omega }}}^{-1}{\left(\displaystyle \frac{H}{r}\right)}^{-2}\\ & & \sim 200\,\mathrm{yr}{\left(\displaystyle \frac{r}{1{R}_{\odot }}\right)}^{3/2}{\left(\displaystyle \frac{\alpha }{{10}^{-2}}\right)}^{-1}{\left(\displaystyle \frac{H/r}{0.006}\right)}^{-2}.\end{array}\end{eqnarray} \tag{ 14 }$$

We adopt the accretion rate as determined by Dwomoh & Bauer (2023) ⁶ and estimate the current disk mass by assuming $\dot{M}\sim {M}_{\mathrm{gas}}/{\tau }_{\mathrm{diff}}$, to obtain

$$\begin{eqnarray}\begin{array}{rcl}{M}_{\mathrm{gas}} & \approx & 2\times {10}^{21}{\rm{g}}\left(\displaystyle \frac{\dot{M}}{2\times {10}^{11}{\rm{g}}/{\rm{s}}}\right){\left(\displaystyle \frac{\alpha }{{10}^{-2}}\right)}^{-1}\\ & & \times {\left(\displaystyle \frac{T}{5000{\rm{K}}}\right)}^{-1}{\left(\displaystyle \frac{r}{1{R}_{\odot }}\right)}^{1/2}.\end{array}\end{eqnarray} \tag{ 15 }$$

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 planetesimals have a radius R_p ∼ 50 km (assuming a bulk density of ρ_p ∼ 5 g 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 chondrite 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 units of hydrogen mass). So the above mass estimate corresponds to a surface density Σ ∼ M_gas/r² ∼ 0.13 g cm⁻², and a midplane density of ρ ∼ 3 × 10⁻¹⁰ g 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 ∼3 × 10²¹ cm⁻², and a vertical one at ∼2 × 10¹⁹ cm⁻². We find:

• 1.  
  if all Ca is in Ca ii, ⁷ , with a photoionization cross section of 5 × 10⁻¹⁹ cm², the optical depth for a Ca ii ionizing photon is τ ∼ 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_γ ∝ r^−2.4) than is expected for an optically thin disk (n_γ ∝ r⁻²).
• 2.  
  the vertical optical depth for the Ca ii infrared triplet is ∼10⁶ (CLOUDY result). This explains why the line ratios within the Ca ii triplet do not reflect their individual strengths but approach those of a blackbody (Melis et al. 2010).
• 3.  
  the above mass estimate allows us to explain the total flux observed in Ca ii triplet. For J1228, about 5.6% of its energy is 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 × (H/r) × 5.6% × L_wd. As Ca ii is ionized and recombined, a fraction of the ionization energy is emitted in Ca ii triplets (photon energy ∼1.5 eV). So we expect a total line flux of ∼2n(H/r) × 5.6% × (1.5/11.87) × L_wd. The observed total line flux is ∼3 × 10⁻⁴ L_wd (Manser et al. 2016), or we require n ∼ 3.5. In comparison, using the above midplane radial optical depth (τ ∼ 1600) and assuming that the disk is in vertical hydrostatic equilibrium, we find that the disk can capture these photons up to a height multiple of n ∼ 3.8. In other words, the observed Ca ii line fluxes can be explained by our inferred disk density.

We aim to draw clues on the progenitor by considering the three radii we inferred for the disk, a_in, a_break, and a_out. With our values of p₁ and p₂, most of the disk mass lies closely around $a={a}_{\mathrm{break}}\simeq 1\times {\sin }^{2}{{iR}}_{\odot }$. This suggests that a_break may be a special radius for the progenitor body. Gas deposited here may then viscously spread both outward and inward, forming the extended disk. In particular, Metzger et al. (2012) and Rafikov (2016) showed that the surface density of an isothermal accretion disk should scale as r⁻², similar to our value of p₂ = −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_⊙, R_* = 0.0117 R_⊙, T_eff = 20, 713K, a blackbody at distance r is heated to a temperature

$$\begin{eqnarray}&&{T}_{\mathrm{bb}}\,=\,{\left(\displaystyle \frac{1}{4}\right)}^{1/4}{\left(\displaystyle \frac{{R}_{\mathrm{wd}}}{r}\right)}^{1/2}{T}_{\mathrm{wd}}\sim 1300\,{\rm{K}}{\left(\displaystyle \frac{r}{1.5{R}_{\odot }}\right)}^{-1/2},\end{eqnarray} \tag{ 16 }$$

where 1300 K is roughly the sublimation temperature of silicate grains under our disk midplane density, ρ ∼ 3 × 10⁻¹⁰ g cm⁻³ (Pollack et al. 1994). ⁸ This will place the sublimation radius at around 1.5 R_⊙, much beyond our inferred break radius (${a}_{\mathrm{break}}\,=\,1{\sin }^{2}i\,{R}_{\odot }$), but close to our inferred outer edge for the gas disk, ${a}_{\mathrm{out}}\,=\,1.7{\sin }^{2}i\,{R}_{\odot }$. 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 strength ⁹ and on a parabolic orbit (e ≈ 1), this is located at a pericenter distance of (Sridhar & Tremaine 1992; Watanabe & Miyama 1992)

$$\begin{eqnarray}&&{\left.{r}_{\mathrm{roche}}\right|}_{e\approx 1}\approx 1.7{R}_{\mathrm{wd}}\times {\left(\displaystyle \frac{{\rho }_{\mathrm{wd}}}{{\rho }_{{\rm{p}}}}\right)}^{1/3}\,\approx 1.0{R}_{\odot }{\left(\displaystyle \frac{{\rho }_{{\rm{p}}}}{5\,\,{\rm{g}}\,\,{\mathrm{cm}}^{-3}}\right)}^{-1/3},\end{eqnarray} \tag{ 17 }$$

where ρ_p is the body's bulk density. If the orbit is less eccentric, this radius moves outward, reaching a maximum of 1.5 R_⊙ at e = 0 (Chandrasekhar 1961).

Given our observed break radius (${a}_{\mathrm{break}}\,=\,1{\sin }^{2}{{iR}}_{\odot }$), 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. (2015), O'Connor & Lai (2020), Malamud et al. (2021) 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, outgassing parent body. Unfortunately, current simulations can only integrate for hundreds of orbits, or in physical time, ∼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 ${a}_{\mathrm{in}}\sim 0.57{\sin }^{2}i\,{R}_{\odot }$. What is the origin of this cutoff? Metzger et al. (2012) suggested that a 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 streamlined crossing. We hypothesize that disk eccentricity, rather than stellar magnetic field, truncates our disk at the observed a_in. A more detailed study is required.

The spectral energy distribution (SED) of J1228 shows the presence of a dusty component, with a total luminosity of L_dust ∼ 6 × 10⁻³ L_wd, and blackbody temperatures that range from 450 to 1700 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 & Goldreich1997; Jura 2003)

$$\begin{eqnarray}&&{T}_{\mathrm{dust}}\,=\,{\left(\displaystyle \frac{2}{3\pi }\right)}^{1/4}{\left(\displaystyle \frac{{R}_{\mathrm{wd}}}{r}\right)}^{3/4}{T}_{\mathrm{wd}}\sim 350\,{\rm{K}}{\left(\displaystyle \frac{{\rm{r}}}{1.5{{\rm{R}}}_{\odot }}\right)}^{-3/4}.\end{eqnarray} \tag{ 18 }$$

So the above dust temperatures translate to a range of 0.2 R_⊙ to 1.2 R_⊙.

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 the circularization of the gas disk if the dust mass is high enough. In fact, this unwelcomed 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 does 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. 2021) 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 of the star, their temperatures will then be described by Equation (16), and they can remain hot out to larger distances. The observed blackbody may then arise from a region beyond 0.9 R_⊙, 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 superheated.

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 ρ_bulk and temperature T,

$$\begin{eqnarray}\begin{array}{rcl}{M}_{\mathrm{dust}} & \geqslant & \displaystyle \frac{{L}_{\mathrm{dust}}\,s\,{\rho }_{\mathrm{bulk}}}{3\sigma {T}^{4}}\sim 2\times {10}^{17}\,{\rm{g}}\left(\displaystyle \frac{{{\rm{L}}}_{\mathrm{dust}}}{6\times {10}^{-3}{{\rm{L}}}_{\mathrm{wd}}}\right)\\ & & \times {\left(\displaystyle \frac{T}{1600\,{\rm{K}}}\right)}^{-4}\left(\displaystyle \frac{s}{1\mu m}\right)\left(\displaystyle \frac{{\rho }_{\mathrm{bulk}}}{5\,{\rm{g}}\,\,{\mathrm{cm}}^{-3}}\right).\end{array}\end{eqnarray} \tag{ 19 }$$

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 to the evolution and accretion of the gas disk, differing from the proposal by Rafikov (2011) and Metzger et al. (2012).

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., the 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 pericenter 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_orig(1 − e_orig) $\sim {a}_{\mathrm{break}}* (1-{e}_{\mathrm{break}}^{2})/(1+{e}_{\mathrm{orig}})$ $\sim 0.47{\sin }^{2}i\,{R}_{\odot }$, for e_orig ≈ 1 and e_break ∼ 0.26 at a_break. This is a couple of times smaller than the Roche radius (Equation (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. ¹⁰ However, such a process could be easily suppressed by GR precession (e.g., Eggleton & Kiseleva-Eggleton 2001). To avoid suppression, the secular perturbations may need to act in a 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 be within a factor of a 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 astronomical unit. Such a new constraint, unseen 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 flies on different orbits, but ones that have a long axis well aligned with the original one. They then start 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 semimajor axis is a, and its pericentre distance r_p ∼ 0.47 R_⊙, the GR precession time is of order ∼2.5 × 10⁵(a/au)^3/2 yr (Equation (6)). This is so long that it seems hard to circularize the debris disk before the gaseous component diffuses away (Equation (14), ∼200 yr). 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 astronomical unit, 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 semimajor 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,

$$\begin{eqnarray}&&{T}_{\mathrm{secular}}\approx \left(\displaystyle \frac{{M}_{* }}{{M}_{\mathrm{planet}}}\right){P}_{\mathrm{planet}}.\end{eqnarray} \tag{ 20 }$$

This is consistent with results from Li et al. (2021), where a Neptune-massed planet at 10 au takes ∼10⁶ yr to scramble the orbits.

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

$$\begin{eqnarray}&&{a}_{\mathrm{planet}}\leqslant 0.3\mathrm{au}{\left(\displaystyle \frac{{M}_{\mathrm{planet}}}{{M}_{\mathrm{Jup}}}\right)}^{2/3},\end{eqnarray} \tag{ 21 }$$

where M_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 astronomical unit—reaching ∼7 au for an initial stellar mass of 3M_⊙ (Figure 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. 2015) or collisional drag with a pre-existing disk (Malamud et al. 2021). However, these are not effective unless the debris has 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 times (∼200 yr), 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 ∼4 × 10⁻⁵ pc⁻³ mag⁻¹ (Leggett et al. 1998). So within the distance of J1228 (∼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. 2021). So the occurrence rate of gas disks among hot white dwarfs is ∼10%, not dissimilar to the occurrence rate of dust disks among all white dwarfs. ¹¹ So the total mass of (disrupted) progenitors (all assumed to be R_p = 50 km) over the cooling age of J1228 (∼10⁷ yr) ¹² is,

$$\begin{eqnarray}&&\begin{array}{l}{M}_{\mathrm{disrupted}}\approx \displaystyle \frac{4\pi }{3}{\rho }_{p}{R}_{p}^{3}\,\times \displaystyle \frac{10 \% \times {10}^{7}\,\mathrm{yr}}{{t}_{\mathrm{diff}}}\sim {10}^{25}\,{\rm{g}}\\ \,\,\,\times {\left(\displaystyle \frac{{R}_{p}}{50\,\mathrm{km}}\right)}^{3}\left(\displaystyle \frac{{\rho }_{p}}{5\,{\rm{g}}\,\,{\mathrm{cm}}^{-3}}\right).\end{array}\end{eqnarray} \tag{ 22 }$$

This mass is similar to the total mass of our own asteroid belt (∼2 × 10²⁴ 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 counterparts. This, a massive belt at au distance, is again in 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 semimajor axis of 0.73 R_⊙). They interpreted these disturbances as due to an outgassing planetesimal on an eccentric orbit that overlaps with the gas disk. If such an object does exist and is massive (radius R ≥ 50 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. ¹³ 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 (ρ_p ≥ 40 g cm⁻³ for circular orbit; higher value needed for eccentric orbit; see Equation (17)); or the body has to have a high internal strength $\sim {10}^{8}{({R}_{p}/50\,\mathrm{km})}^{2}\,\mathrm{dyne}\,\,{\mathrm{cm}}^{-2}$ (see Equation 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.