Low-mass White Dwarfs with Hydrogen Envelopes as a Missing Link in the Tidal Disruption Menu

Since WDs have an inverse mass–radius relationship, the lowest mass WDs will be able to probe the highest mass BHs. Let us estimate the lowest mass WD available through single-star evolution. Setting the main sequence lifetime equal to the age of the universe ($\approx 13.8\,\mathrm{Gyr};$ Hurley et al. 2000) using an analytic formula for the MS lifetime from Hinshaw et al. (2013) gives ${M}_{{\rm{i}}}\approx 0.9\,{M}_{\odot }$. Using this mass in an empirical initial–final mass relation for WDs from Catalán et al. (2008) for ${M}_{{\rm{i}}}\lt 2.7\,{M}_{\odot }$,

$$\begin{eqnarray}&&{M}_{{\rm{f}}}=(0.096\pm 0.005){M}_{{\rm{i}}}+(0.429\pm 0.015),\end{eqnarray} \tag{ 3 }$$

we find that the minimum WD mass possible through single-star evolution is ${M}_{\mathrm{WD}}\approx 0.5\,{M}_{\odot }$.

WDs that are less massive than roughly half of a solar mass will have formed through binary interactions, barring cases of extreme metallicity (Kilic et al. 2007). Low-mass WDs can be formed either through stable Roche-lobe overflow mass transfer or common-envelope evolution (e.g., Driebe et al. 1998; Sarna et al. 2000; Nelson et al. 2004; Althaus et al. 2013; Nandez et al. 2015). A helium-core WD forms if one component of the binary loses its hydrogen envelope before helium burning. This object has a degenerate helium core and is formed with an extended hydrogen envelope supported by a thin hydrogen burning layer.

The maximum mass of an He WD is approximately $0.46\,{M}_{\odot }$, and only He WDs are formed below this mass (Sweigart et al. 1990; Nelemans et al. 2001). The final mass of the He WD depends on the mass of the progenitor and the binary orbital properties (e.g., Nelemans et al. 2001). The progenitor star needs a zero-age main sequence mass below $2.3\,{M}_{\odot }$, as more massive stars do not form helium cores. The strict minimum timescale for formation of an He WD is therefore the MS lifetime of a $2.3\,{M}_{\odot }$ star, ${t}_{\mathrm{MS}}\approx 1.16\,\mathrm{Gyr}$ (Hurley et al. 2000).

Istrate et al. (2014, 2016) performed calculations of He WD formation via stable mass transfer; we quote some results below. After detachment from Roche-lobe overflow, the progenitor star enters a "bloated" proto-WD phase where much of the hydrogen in the envelope is burned in stable hydrogen shell burning. The mass of hydrogen left after Roche-lobe detachment is on the order of ${10}^{-2}\,{M}_{\odot }$, yet this can fuel a proto-WD phase lasting up to 2.5 Gyr for the lowest mass ($M\lesssim 0.20\,{M}_{\odot }$) WDs.

Istrate et al. (2014) derived a timescale for hydrogen burning,

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{\mathrm{proto}}\simeq 400\ \mathrm{Myr}{\left(\displaystyle \frac{0.20{M}_{\odot }}{{M}_{\mathrm{WD}}}\right)}^{7},\end{eqnarray} \tag{ 4 }$$

which describes the star's contraction from Roche-lobe detachment to its maximum effective temperature on the cooling track. Istrate et al. (2016) also defined a cooling timescale, ${t}_{\mathrm{cool},{{\rm{L}}}_{-2}}$, which is the time from detachment to reaching $\mathrm{log}(L/{L}_{\odot })=-2$ on the cooling track. This timescale is set primarily by the mass of the hydrogen envelope left at the end of the proto-WD phase. Generally, a shorter orbital period at the onset of mass transfer leads to a lower proto-WD mass and a higher final envelope mass.

There is a growing body of observations of these low-mass objects: the targeted survey for extremely low-mass (ELM; $M\lt 0.3\,{M}_{\odot }$) WDs has found 76 binaries to date, with a median primary mass of $\approx 0.18\,{M}_{\odot }$ (Brown et al. 2016a, 2016b). Many of these WDs appear to be bloated, and this bloated state can persist for a long time: Macias et al. (2015) found that roughly half of these systems will still be burning hydrogen when they merge. One object in this sample is the binary system NLTT 11748 (Kaplan et al. 2014), which contains a helium-core hydrogen-envelope WD of mass $0.17\,{M}_{\odot }$ and radius $0.043\,{R}_{\odot }$, whereas a standard WD mass–radius relation for this mass would give a radius of $0.02\,{R}_{\odot }$. This object's bloated size allows it to be disrupted by a BH of up to $3.8\times {10}^{6}\,{M}_{\odot }$. This WD has a cooling age of 1.6–1.7 Gyr; younger He WDs can have much more extended envelopes, allowing them to be disrupted by even ${10}^{7}\,{M}_{\odot }$ or ${10}^{8}\,{M}_{\odot }\,\mathrm{BHs}$. As an example of this more extreme bloating, observations and astroseismological studies of the eclipsing binary J0247−25 find an He WD with mass $0.186\pm 0.002\,{M}_{\odot }$ and radius $0.368\pm 0.005\,{R}_{\odot }$ (Maxted et al. 2013). Note that this He WD has a larger radius than a MS star of its mass. In a study of the Galactic WD binary population, Maoz et al. (2012) found that roughly half of WDs in binaries are He WDs, and that the probability density distribution for He WDs is relatively flat below $0.4\,{M}_{\odot }$.

It is difficult to estimate the typical age of a He WD upon disruption by a MBH, as these objects are formed from a range of progenitor masses and undergo a binary interaction of uncertain timescale. We do know that nuclear star clusters exhibit a wide range of stellar ages. For example, observations of the nearby S0 galaxy NGC 404 show that half of the mass of the nuclear star cluster is from stars with ages of $\approx 1\,\mathrm{Gyr}$, while the bulge is dominated by much older stars (Seth et al. 2010). In our own Galactic center, roughly 80% of the stars formed over 5 Gyr ago and the remaining 20% formed in the last 0.1 Gyr (Pfuhl et al. 2011). In addition, TDEs have so far been found preferentially in post-starburst galaxies, with significant 1 Gyr old or younger stellar populations (Arcavi et al. 2014; French et al. 2016). Another consideration is that in a study of a population of He WDs in the globular cluster NGC 6397, Hansen et al. (2003) found that the progenitor binaries of the He WDs very likely underwent an exchange interaction within the last Gyr. Finally, we note that the two-body relaxation time is $\approx 0.1\,\mathrm{Gyr}$ for a ${10}^{5}\,{M}_{\odot }\,\mathrm{BH}$ and $\approx 1.8\,\mathrm{Gyr}$ for a ${10}^{6}\,{M}_{\odot }\,\mathrm{BH}$. Motivated by the above considerations, in our disruption simulations we take the radius of the He WD at 1 Gyr after formation (i.e., since Roche-lobe detachment).

For the tidal disruption calculations in this work, we construct a $0.17\,{M}_{\odot }$ He WD consisting of a $0.16\,{M}_{\odot }$ degenerate helium core and a $0.01\,{M}_{\odot }$ hydrogen envelope using the MESA stellar evolution code (Paxton et al. 2011, 2013, 2015). This envelope mass is consistent with theoretical predictions of hydrogen retention (Althaus et al. 2001; Serenelli et al. 2001; Panei et al. 2007). The left panel in Figure 2 shows the relative abundance of helium and hydrogen as a function of radius for this object. The hydrogen envelope extends to roughly 10 times the radius of the core, and is supported by a thin hydrogen burning shell. This snapshot is at 1 Gyr after formation.

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We also calculate the radius as a function of time since formation (through a binary interaction) for several He WDs in MESA. This is shown for our $0.17\,{M}_{\odot }$ object in the right panel of Figure 2. We show the radius of a core-only WD of the same mass for comparison in dashed blue (here we show a fixed radius that does not evolve with time). In a similar calculation for a $0.15\,{M}_{\odot }$ WD, we find that a very extended envelope persists for $\gt 10\,\mathrm{Gyr}$.

The particular tidal disruption rates of different types of objects depend on the detailed dynamics and evolution of the dense stellar system surrounding the central BH. Given these uncertainties, here we make a simple estimate of the relative rate of He WD disruptions. We find that several factors could increase the rate from that suggested by these objects' low population fraction. We can decompose the observed rate into (1) the fractional disruption rate and (2) the rate of luminous flares.

3.2.1. Disruption

3.2.2. Flaring

Using the MESA stellar evolution code, we construct a $0.17\,{M}_{\odot }$ white dwarf with a $0.16\,{M}_{\odot }$ degenerate helium core and a $0.01\,{M}_{\odot }$ hydrogen envelope. As noted in the previous section, there is a growing population of observed objects in this mass range. In these low-mass objects, the extended envelope lasts for a long time, as ${\rm{\Delta }}{t}_{\mathrm{proto}}\propto {M}_{\mathrm{WD}}^{-7}$ (Equation (4)). We might therefore be more likely to see flares from the stripped envelopes of objects close to this mass, as they exist in a bloated state for longer than their higher-mass cousins.

We approximate the core and envelope as nested polytropes (e.g., Rappaport et al. 1983; MacLeod et al. 2012; Liu et al. 2013), using polytropic indices ${n}_{\mathrm{core}}=1.5$ and ${n}_{\mathrm{env}}\,=3.8$. Figure 3 shows the density versus radius profile of this object from MESA as well as from the nested polytrope that we matched. We use this nested polytrope as an input to our hydrodynamical simulations as it provides a reasonable description of the object's structure, and makes possible comparisons with non-hydrogen-envelope WD disruption calculations using polytropic equations of state (MacLeod et al. 2014).

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A single polytrope is unstable to small variations in pressure p₀ and volume V₀ if ${(\partial p/\partial V)}_{0}$ is positive, and this occurs for polytropic indices of $n\gt 3$. However, it is difficult to derive simple stability criteria for our nested polytrope structure, as it is not differentiable across the core-envelope discontinuity. We instead ensure that two heuristic tests of stability are satisfied: (1) the entropy increases with radius, or $\partial S/\partial r\gt 0$, and (2) the star does not contract or relax significantly when placed on our hydrodynamical grid structure for 20 dynamical timescales of the full star. The dynamical timescale for the full star is ${t}_{\mathrm{dyn}}^{\mathrm{full}}\simeq \sqrt{{R}^{3}/{GM}}=535\,{\rm{s}}$. In this work we will often refer to the dynamical timescale of the He core of this WD for comparison; this is ${t}_{\mathrm{dyn}}^{\mathrm{core}}=22.5\,{\rm{s}}$.

Our simulations of tidal disruption are performed with the basic framework and code described in detail in Guillochon et al. (2009, 2011), MacLeod et al. (2012), Liu et al. (2013), and Guillochon & Ramirez-Ruiz (2013). We use FLASH (Fryxell et al. 2000), a 3D adaptive mesh grid-based hydrodynamics code including self-gravity. Hydrodynamics equations are solved using the using the piecewise parabolic method (Colella & Woodward 1984). We refine the grid mesh on the value of the density, and derefine by one level every decade in density below $\rho ={10}^{-4}\ {\rm{g}}\,{\mathrm{cm}}^{-3}$. All of the simulations presented here are resolved by at least ${R}_{\star }/{\rm{\Delta }}{r}_{\min }\gt 130$, where ${\rm{\Delta }}{r}_{\min }$ is the size of the smallest cells. We note that adaptive mesh refinement is well suited for disruption calculations of an object with this core and envelope structure, as the envelope occupies a large volume yet has a very low mass fraction.

We perform our calculations in the rest-frame of the star to avoid introducing artificial diffusivity by moving the star rapidly across the grid structure. We solve the self-gravity of the star using a multipole expansion about the center of mass of the star with ${l}_{\max }=10$. We then evolve the orbit based on the center of mass of the star and the position of a point-mass black hole (see the Appendix of Guillochon et al. 2011 for details). We use Newtonian gravity for the black hole, which is a reasonable approximation as our star's closest approach in any of our simulations is $\gt 10{r}_{{\rm{g}}}$, in the weak field regime. Cheng & Bogdanović (2014) showed that general relativistic effects in tidal disruption simulations should be small is this regime. Note that, because we use Newtonian gravity, by construction, the encounters we simulate are outside of our rapid circulation condition defined in Section 2. The effect of relativistic encounters is discussed in Section 7.2.

We run our simulations using the $0.17\,{M}_{\odot }$ He WD described above and a ${10}^{5}\,{M}_{\odot }\,\mathrm{BH}$. We input the MESA profile, matched as a nested polytrope (Figure 3), into FLASH. We use two different fluids in the simulation: one for the helium core and one for the hydrogen envelope. Both have the same equation of state, with a ${\gamma }_{\mathrm{fluid}}=5/3$. This setup has an envelope composition of 100% hydrogen. More accurately, the envelope has a residual helium abundance that will migrate toward the core over time depending on the relative strength of mixing and gravitational settling. We relax the object onto the grid for $5\ {t}_{\mathrm{dyn}}$ before sending the BH toward it. We use an eccentricity $e\approx 1$, as most disrupted stars originate from orbits scattered from the sphere of influence (Magorrian & Tremaine 1999; Wang & Merritt 2004). As discussed in Guillochon & Ramirez-Ruiz (2013), for a given stellar structure, we can understand the vast majority of disruptions by surveying in impact parameter $\beta ={r}_{{\rm{t}}}/{r}_{{\rm{p}}}$, as all other parameters obey simple scaling relations when relativistic effects are unimportant. Similar to the dynamical timescale, we can define β with respect to the tidal radius of the full star or the degenerate core. We survey in ${\beta }_{\mathrm{full}}$ from 1 to 10 in 12 runs. This corresponds to ${\beta }_{\mathrm{core}}$ of $\approx 0.1$ to 1.2. We run our simulations for $21\ {t}_{\mathrm{dyn}}^{\mathrm{full}}=500\ {t}_{\mathrm{dyn}}^{\mathrm{core}}$, well into the self-similar decay portion of the mass fallback rate.

Figure 4 shows the time evolution of the star for a ${\beta }_{\mathrm{core}}=0.7$ encounter in 2D slices in density through the 3D simulation box, zoomed in on the star. Time is labeled in terms of the dynamical time of the core. In this moderately plunging encounter, the star is distorted through pericenter, evolving into a surviving remnant and two tidal tails—one bound and one unbound from the BH.

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As we increase the impact parameter, the star is perturbed closer to its center. For mildly plunging encounters, only the hydrogen envelope is stripped, while the core survives intact. For more deeply plunging encounters, both the core and envelope are disrupted and fed to the BH. We can see this qualitatively in Figure 5, where we show slices through the simulation box zoomed in on the star for ${\beta }_{\mathrm{core}}=0.5$, 0.7, and 0.9 encounters. We plot the ratio of the core material to envelope material density. The different spatial distributions of core and envelope material will result in different fallback times to the BH, which will result in observed light curves dominated by material of different compositions at different times. Because of their different structures—the envelope has a steeper density gradient than the core—these two fluids react to losing mass in characteristically different ways, as we will see below.

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Figure 6 shows the mass lost from the star as a function of impact parameter, calculated at the last timestep of our simulations. We run our simulations long enough so that the mass lost calculated from this final timestep is asymptotically close to the final mass lost. Note that half of the lost mass will return to the black hole and half is ejected as an unbound debris stream. The object is smoothly disrupted with the impact parameter, albeit with two components from the envelope and the core. This is different from giant star disruptions (MacLeod et al. 2012), where the core is never disturbed, and likely arises because the density contrast between the core and envelope is in general larger for giants than it is for He WDs.

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A fitting formula from Guillochon & Ramirez-Ruiz (2013, 2015b) for a ${\rm{\Gamma }}=5/3$ polytrope fits the mass lost from the core well. This is expected, as once the core has been penetrated, the envelope has negligible dynamical effect, and the disruption will proceed as if for a typical WD. Full disruption occurs at ${\beta }_{\mathrm{core}}\approx 0.9$.

This n = 1.5 polytrope has a lower critical β (for full disruption) compared to higher index polytropes, as the mass is distributed more evenly. In addition to this, a n = 1.5 polytrope has an inverse mass–radius relation, and so expands when mass is removed, making the object more vulnerable to disruption. We model the envelope, on the other hand, as an n = 3.8 polytrope, which reacts to mass removal by contracting—"protecting" itself. Because the envelope has a steeper density gradient, its critical β is higher than for a ${\rm{\Gamma }}=5/3$ polytrope. We can see this in the shallower slope of ${\rm{\Delta }}M/M$ versus β for envelope material relative to core material.

We calculate the spread in binding energy of the star's material to the BH, dM/dE versus E, over time. We compute the specific binding energy of the material in each cell of the simulation, which depends on its distance and velocity relative to the center of mass of the star and to the black hole. Details of the calculation are presented in Guillochon & Ramirez-Ruiz (2013). Only material that is bound to the BH and not bound to the star will contribute to the mass fallback onto the BH. We compute the specific binding energy of the material in each cell of the simulation, which depends on its distance and velocity relative to the center of mass of the star and to the black hole.

Figure 7 shows the spread in dM/dE over time for ${\beta }_{\mathrm{core}}=0.5$, 0.7, and 0.9 encounters, with the contribution from material unbound to the star in solid black and contributions from the core and envelope of the remnant in red and blue. We see that impact parameter drastically changes the spread in binding energy through and following disruption, both for the bound and unbound material. Grazing encounters leave the core relatively unperturbed and are able to retain more envelope material, while deeper encounters leave a compact remnant that has been all but stripped of its envelope.

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Given dM/dE and a pericenter distance, we can calculate the mass fallback rate onto the BH by Kepler's third law,

$$\begin{eqnarray}&&\displaystyle \frac{{dM}}{{dt}}=\displaystyle \frac{{dM}}{{dE}}\displaystyle \frac{{dE}}{{dt}}=\left(\displaystyle \frac{{dM}}{{dE}}\right)\displaystyle \frac{1}{3}{(2\pi {{GM}}_{\mathrm{bh}})}^{2/3}{t}^{-5/3}.\end{eqnarray} \tag{ 5 }$$

The left panel of Figure 8 shows the spread in specific binding energy dM/dE versus E at the last timesteps of our simulations for all impact parameters. We verify that the binding energy has effectively "frozen in," or converged to its final distribution, by this timestep. The right panel shows dM/dE mapped onto dM/dt across time for the same impact parameters, with the Eddington limit for this BH shown in dashed black. We take ${\dot{M}}_{\mathrm{Edd}}\,=0.02\ (\eta /0.1)\ ({M}_{\mathrm{bh}}/{10}^{6}\,{M}_{\odot })\ {M}_{\odot }\,{\mathrm{yr}}^{-1}$ with $\eta =0.1$. Feeding rates peak at ${t}_{\mathrm{peak}}\sim 5\times {10}^{4}\ \mathrm{to}\ {10}^{6}\ {\rm{s}}\approx 0.6\mbox{--}11\ \mathrm{days}$ depending on β. Weakly plunging encounters peak later, while deeply plunging encounter peak earlier. Note that ${t}_{\mathrm{peak}}$ evolves strongly with β (it spans more than an order of magnitude), in contrast to single polytrope solutions where the evolution in ${t}_{\mathrm{peak}}$ is much more gradual (e.g., Guillochon & Ramirez-Ruiz 2013). This means that the He WD disruptions—and disruptions of other objects with this core and extended envelope structure—probe a much wider range of potential transient characteristics for a given BH mass. We see that even for very weakly plunging encounters, for which only a fraction of the envelope is stripped (see mass lost in Figure 6), the mass fallback rate is super-Eddington. Encounters only stripping the envelope appear to have a shallower slope in early-time mass fallback and smoother evolution near peak than encounters penetrating the core. This is due to their different polytropic structures.

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Guillochon & Ramirez-Ruiz (2013) presented a fitting formula for the peak fallback rate of material onto the BH, where ${\dot{M}}_{\mathrm{peak}}=f({M}_{\mathrm{bh}},\beta ,\gamma )$. Figure 9 shows ${\dot{M}}_{\mathrm{peak}}$ values from our simulations of the disruption of an $0.17\,{M}_{\odot }$ He WD compared with those from this fitting formula for a ${\rm{\Gamma }}=5/3$ non-hydrogen-envelope WD with a mass of $0.155\,{M}_{\odot }$, the mass of the core of the He WD. We expect this functional form to match for disruptions that penetrate the core. In low β encounters, the hydrogen envelope provides mass return rates that are unavailable to WDs without envelopes.

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We track the core and envelope material separately in our simulations, which allows us to track the composition of the debris falling onto the BH. In Figure 10 we show $\dot{M}$ as a function of time for ${\beta }_{\mathrm{core}}=0.5$, 0.6, and 0.8, with absolute and fractional contributions from the helium core in red and the hydrogen envelope in blue. The mass fallback rate from weakly plunging encounters can be super-Eddington and hydrogen-dominated. In more deeply plunging encounters, the early rise of the mass fallback rate is fed almost entirely by the hydrogen envelope, while the peak and late time evolution are fed by the helium core; the nature of this transition depends on β. Note that the disruption turns the star inside out: the material that is removed first accretes first, and is then buried underneath the material that is removed last and accretes last. The diffuse envelope material feeds a qualitatively slower rise in the mass fallback curve compared to the core material.

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The first evidence that a range of stellar or spectral properties might be represented in TDEs was the discovery of a helium-rich TDE, PS1-10jh (Gezari et al. 2012). Gezari et al. explained its hydrogen-free spectrum as the result of the tidal disruption of the helium-rich core of a star, similar in structure to an He WD progenitor. Arcavi et al. (2014) noted that TDEs observed thus far show a continuum of helium-rich to hydrogen-rich spectral features; there is an ongoing debate over the origin of the strong helium emission. Kochanek (2016a) found that stellar evolution can play a role in producing this spectral diversity. Roth et al. (2016) modeled the emission from TDEs through an extended, optically thick envelope formed from stellar debris. They found that due to optical depth effects, hydrogen Balmer line emission is often strongly suppressed relative to helium line emission. For MS stars, for example, it is possible for the hydrogen emission lines to be absent. Having said this, the specific composition of the material is expected to have consequences on the detailed line ratios. Line diagnostics from disruptions of He WDs could transition from hydrogen to helium smoothly with β. An encounter stripping only the envelope could provide a rare, (nearly) pure hydrogen-powered mass fallback. If an optically thick reprocessing envelope exists, however, observational evidence of this type of encounter could be variable.

Here we compare ${t}_{\mathrm{peak}}$ values from simulations to those of two particularly rapidly rising TDE candidates, Dougie (Vinkó et al. 2015) and PTF10iya (Cenko et al. 2012), accounting for luminosity and BH mass constraints. Vinkó et al. (2015) estimated Dougie's peak bolometric luminosity as ${L}_{\mathrm{peak}}\,\approx 5(\pm 1)\times {10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and its rise time as ${t}_{\mathrm{rise}}\sim 10\,\mathrm{days}$. They estimated a central BH mass of a few 10⁶ to ${10}^{7}\,{M}_{\odot }$ for Dougie's host galaxy. Cenko et al. (2012) estimated 10iya's peak bolometric luminosity as ${L}_{\mathrm{peak}}\approx (1-5)\,\times {10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ and placed a limit on its rise time of ${t}_{\mathrm{rise}}\lt 5\,\mathrm{days}$. They constrained the central BH mass via the observed bulge luminosity versus BH mass relation as $\mathrm{log}{M}_{\mathrm{BH}}/{M}_{\odot }\lesssim 7.5$.

In order to constrain the kinds of disruptions that can produce such rapid flares, we construct a histogram of ${t}_{\mathrm{peak}}$ for the $0.17\,{M}_{\odot }$ He WD disruptions presented in this work as well as for regular WDs, MS stars, BDs, and planets. We model regular WDs, MS stars, BDs, and planets with $M\lt 0.3\,{M}_{\odot }$ as ${\rm{\Gamma }}=5/3$ polytropes. We model MS stars with $M\gt 0.3\,{M}_{\odot }$ as ${\rm{\Gamma }}=4/3$ polytropes. The mass at which we transition from 5/3 to 4/3 does not affect our conclusions significantly, as their ${t}_{\mathrm{peak}}$ values overlap. Giant star disruptions have longer timescales than we are interested in here.

We draw from flat distributions in ${M}_{\mathrm{obj}}$, with white dwarf masses of $0.2\,{M}_{\odot }\lt {M}_{\mathrm{WD}}\lt 1\,{M}_{\odot }$, MS star masses of $0.085\,{M}_{\odot }\lt {M}_{\mathrm{MS}}\lt 3\,{M}_{\odot }$, BD masses of $13\,{M}_{\mathrm{Jup}}\lt {M}_{\mathrm{BD}}\,\lt 0.085\,{M}_{\odot }$, and planet masses of $1\,{M}_{\mathrm{Jup}}\lt {M}_{\mathrm{pl}}\lt 13\,{M}_{\mathrm{Jup}}$. We use only the one $0.17\,{M}_{\odot }$ He WD mass. We draw from a flat distribution in BH mass with ${10}^{6}\lt {M}_{\mathrm{bh}}/{M}_{\odot }\lt {10}^{7}$, roughly the BH mass constraints for Dougie and 10iya. We draw from a flat distribution in β, discarding encounters where ${r}_{{\rm{p}}}\lt {r}_{\mathrm{ibco}}$. We estimate the peak luminosity from each encounter as ${L}_{\mathrm{peak}}=\min (0.1{\dot{M}}_{\mathrm{peak}}{c}^{2},{L}_{\mathrm{Edd}})$ for the given BH, as these events were observed in the optical/UV and we expect accretion luminosity to be Eddington limited. We discard encounters with ${L}_{\mathrm{peak}}\lt 3\times {10}^{44}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, which is comfortably below the errors in Dougie's peak luminosity.

In Figure 13, we show the outcome of the above exercise. We find that the only objects that satisfy the luminosity requirement are MS stars, BDs, and our prototypical He WD. MS stars and BDs, however, cannot reproduce the rapid timescales of Dougie and 10iya from $\dot{M}$ alone. Thermal TDEs such as PS1-10jh show a good correspondence between the observed luminosity and the fallback rate (Guillochon et al. 2014). This simplicity makes the disruption of He WDs an appealing explanation for rapidly rising nuclear transients.

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In order to explain Dougie as a MS star disruption, models require a strong wind component with a functional form that may not directly reflect $\dot{M}$ (Vinkó et al. 2015). A wind that carries a significant amount of kinetic and thermal energy may be produced if the accretion rate onto the BH exceeds its Eddington limit (Strubbe & Quataert 2009; Lodato & Rossi 2011; Jiang et al. 2016; Metzger & Stone 2016). While this scenario could explain Dougie and other rapidly rising TDEs such as PTF10iya, their timescales can be naturally explained by the $\dot{M}$ from He WD disruptions.

We note that Vinkó et al. (2015) found that Dougie appears offset $\approx 3.9\,\mathrm{kpc}$ from the photometric center of its host galaxy. This initially seems to disfavor a TDE interpretation. However, the photometric center of a galaxy is not necessarily its dynamical center. Vinkó et al. also noted that lower-mass off-center BHs are rare yet not unprecedented (e.g., Barth et al. 2008; Reines & Deller 2012), making the TDE hypothesis tenable.

Our study focuses on a single example of the disruption of a prototypical $0.17\,{M}_{\odot }$ He WD. However, as we saw in Section 3.1, these objects can have a wide range of masses and radii, and the radius evolution even for a single mass is appreciable (see Figure 2). The inclusion of hydrogen-bearing He WDs with a larger range of core masses and envelope masses could potentially explain events with shorter or longer timescales than the prototypical encounters presented here.

In this work, we model the interaction between only a single He WD and a BH. We expect, however, that many He WDs will be in binary systems as they approach the BH, composed of either two He WDs or one He WD and one CO/ONe WD. This suggests that some disruptions of He WDs involve two stars instead of one (Antonini et al. 2011). The interaction of the binary with the BH can shift the distribution in binding energy of the debris, and cause the time of peak accretion to occur either earlier or later depending on the sign of the energy shift (a similar effect is seen in disruptions of stars on elliptical orbits; Hayasaki et al. 2012; Dai et al. 2013). In extreme cases, this interaction can bind all of the material to the BH (as opposed to just half), allowing the BH to accrete the whole star; alternatively, all of the material can become unbound, preventing any accretion onto the BH. If the binary separation is of order the tidal radius, double tidal disruptions are possible (Mandel & Levin 2015). However, our single-star calculations are still applicable for double disruptions, as the hydrodynamics of the disruption are independent for each of the components of the binary. In cases where the outgoing debris streams from the two disrupted stars do not interact with one another, the fallback resulting from a binary disruption can be mimicked by applying simple shifts to the binding energy distribution of the debris of the single-star case.

We do not consider general relativistic effects in our disruption calculations—the gravitational potential of our point mass is purely Newtonian. Cheng & Bogdanović (2014) investigated relativistic effects on the fallback rate of debris. For highly relativistic encounters, they found a more gradual rise and delayed peak of the fallback compared to the Newtonian result. For a $1\,{M}_{\odot },1\,{R}_{\odot }$ MS star encounter with a ${10}^{7}\,{M}_{\odot }\,\mathrm{BH}$, where ${r}_{{\rm{p}}}/{r}_{{\rm{g}}}\approx 10$, they found a difference in ${\dot{M}}_{\mathrm{peak}}$ of $\approx 18 \%$ and a difference in ${t}_{\mathrm{peak}}$ of $\approx 10 \%$ between Newtonian and relativistic simulations. For a $0.6\,{M}_{\odot }$ WD encounter with a ${10}^{5}\,{M}_{\odot }\,\mathrm{BH}$, where ${r}_{{\rm{p}}}/{r}_{{\rm{g}}}\approx 4.6$, the difference in ${\dot{M}}_{\mathrm{peak}}$ is $\approx 69 \%$ and the difference in ${t}_{\mathrm{peak}}$ is $\approx 49 \%$.

For the He WD encounters presented in this work, the critical β of full disruption has ${r}_{{\rm{p}}}/{r}_{{\rm{g}}}\approx 12$, and the transition between an envelope-stripping encounter and one penetrating the core occurs at ${r}_{{\rm{p}}}/{r}_{{\rm{g}}}\approx 19$. Thus, relativistic corrections to our results should be small. In scaling to higher BH masses, however, our errors will increase. However, this will not weaken (and will in fact strengthen) our conclusions regarding the ability of He WDs to achieve the peak timescales of rapidly rising TDE candidates such as Dougie and PTF10iya though $\dot{M}$ alone, as the relativistic effect is to lengthen the peak fallback timescale.

We use a nested polytrope matched to a MESA profile of the He WD as the initial condition in our disruption calculations. We also track only two fluids—one for the core and one for the envelope—in the simulation, and make the simple choice to model the core as fully helium and the envelope as fully hydrogen. A more realistic treatment might use the MESA profile directly in the disruption calculations, and track the composition of the object more fully. For the particular object used in the simulations in this work, however, the gains in accuracy (aside from composition information) in using the MESA profile directly may be minimal, as the nested polytrope profile is very close to the true profile.

We have modeled the tidal disruption of a new class of object: the low-mass He WD with an extended hydrogen envelope. These objects are a missing link both hydrodynamically and in terms of BH masses probed through prompt tidal disruption flares. In summary, we find that:

• 1.  
  Because of their lower density cores and extended envelopes, these objects extend the potential BH masses probed by single-star evolution WDs. In general, their peak fallback timescales will be longer that those of typical WDs and shorter than those of MS stars.
• 2.  
  Grazing encounters that strip only the envelope will be hydrogen dominated, and—for a very small amount of mass removed—can provide high and often super-Eddington fallback.
• 3.  
  Encounters penetrating the core generally have a fallback rate that is hydrogen-dominated in its rise and helium-dominated in its peak and decline, with relative composition versus time a function of impact parameter.
• 4.  
  The typical peak accretion rate of He WD disruptions is a few times larger than that of a typical MS disruption. This likely makes these disruptions observable to larger distances, which would make them a larger fraction of the observed total than suggested by their relative population.

These objects are perhaps the last missing piece of a theoretical tidal disruption menu that includes WDs, MS stars, planets, and evolved stars. Constructing this menu is key to better understanding tidal disruptions. The reader is referred to Figures 1, 11, and 12 for a summary of the phase space of the menu.

This work may have particular bearing on two puzzling observational aspects of TDEs that have emerged in the past few years. The first is their rates. There is a great deal of uncertainty in the properties of the nuclear star clusters from which stars are fed into disruptive orbits. Most calculations make standard assumptions of a spherical single-mass nuclear star cluster that feeds stars to the BH by a two-body relaxation-driven random walk in angular momentum space. These calculations predict disruption rates of $\gtrsim {10}^{-4}\,{\mathrm{yr}}^{-1}$ per galaxy (Magorrian & Tremaine 1999; Wang & Merritt 2004; Stone & Metzger 2016), and are in general in tension with the lower observationally derived rates of roughly ${10}^{-5}\,{\mathrm{yr}}^{-1}$ (e.g., van Velzen & Farrar 2014). However, there can be several complicating effects—such as secular relaxation, or the presence of a triaxial potential, rings or disks of stars, and/or a second massive body—and there is a lack of understanding of their relative importance in local galaxies. In addition, we need to better understand the mass spectrum of disrupted stars, in particular given mass segregation (e.g., MacLeod et al. 2016b).

The second puzzling observation is that a significant fraction of tidal disruptions may arise from unique stellar populations. We are learning that tidal disruption flares may occur preferentially in post-starburst galaxies (Arcavi et al. 2014; French et al. 2016), and that these types of galaxies are overrepresented as TDE hosts. This remains a mystery. Post-starburst galaxies are elliptical-type galaxies that have experienced a star formation burst that has stopped within the past ∼1 Gyr, leaving these galaxies with both old and very young stars.

If only certain types of stars (which are a small fraction of the population) produce prompt flares for BH masses of $\sim {10}^{6}\,{M}_{\odot }$ due to circularization effects, this could alleviate some of the tension in the observed flaring versus disruption rate. As we have argued in Section 3.2, the rate of luminous flare production can be distinct from the disruption rate itself. We have shown in Sections 2 and 6 that different stellar types probe distinct islands of BH mass when we consider prompt flares. This is strong evidence for a connection between stellar population details and the disruption flare rates. The post-starburst galaxy preference may be due to the production of particular stellar species in the nuclei of these galaxies, rather than in the dynamics of their nuclei. We caution, however, that the stellar population of a galaxy as a whole does not necessarily reflect its nuclear population.

In this work, we have argued that to effectively use TDEs to constrain the mass function of BHs, we need to acknowledge that not all disruptions produce luminous flares. Moving forward likely involves understanding the intersection of nuclear region stellar dynamics, stellar populations, and stellar evolution, along with the hydrodynamics of the disruptions themselves. Targeting the observational characteristics of certain TDEs might offer a way to identify the presence of BHs at the low end of the supermassive BH mass range.