Tidal Disruptions of Main-sequence Stars of Varying Mass and Age: Inferences from the Composition of the Fallback Material

If a star with mass M_⋆ and radius R_⋆ is on a parabolic orbit around an SMBH of mass M_bh with pericenter distance, r_p, less than the tidal radius, ${r}_{{\rm{t}}}={R}_{\star }{({M}_{\mathrm{bh}}/{M}_{\star })}^{1/3}={R}_{\star }{q}^{-1/3}$, the star will be tidally disrupted. Here q ≡ M_⋆/M_bh is the mass ratio.

When a star is disrupted, the debris moves on approximately ballistic trajectories, with a spread in specific orbital energy that is roughly frozen at r_t. This spread arises because at the time of disruption, the leading portions of the star are deeper in the potential of the SMBH than the trailing portions, which are farther away. The spread in specific energy of the debris, E_t, can be approximated by taking the Taylor expansion of the SMBH's potential at the star's location:

$$\begin{eqnarray}&&{E}_{{\rm{t}}}={{GM}}_{\mathrm{bh}}{R}_{\star }/{r}_{{\rm{t}}}^{2}={q}^{-1/3}{E}_{\star },\end{eqnarray} \tag{ 1 }$$

where E_⋆ = GM_⋆/R_⋆ is the specific self-binding energy of the star. Because most stars that are tidally disrupted in galactic nuclei approach the SMBH on nearly zero energy orbits, E_t determines the fallback timescale for the most tightly bound debris

$$\begin{eqnarray}{t}_{{\rm{t}}} & = & \displaystyle \frac{\pi }{{M}_{\star }}{\left(\displaystyle \frac{{M}_{\mathrm{bh}}{R}_{\star }^{3}}{2G}\right)}^{1/2}\\ & = & 0.1\,\mathrm{year}{\left(\displaystyle \frac{{M}_{\mathrm{bh}}}{{10}^{6}{M}_{\odot }}\right)}^{1/2}{\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)}^{-1}{\left(\displaystyle \frac{{R}_{\star }}{{R}_{\odot }}\right)}^{3/2}.\end{eqnarray} \tag{ 2 }$$

In order to form an accretion flow, the bound stellar debris must lose a significant amount of energy by viscous dissipation (Guillochon & Ramirez-Ruiz 2015; Shiokawa et al. 2015; Bonnerot et al. 2016; Hayasaki et al. 2016). If the viscosity is large enough to allow accretion onto the SMBH on a timescale shorter than t_t, the luminosity of the flare is expected to follow the rate of mass fallback $\dot{M}=({dM}/{dE})({dE}/{dt})\propto {t}^{-5/3}$, where dM/dE = M_⋆/(2E_t) for a star on an initially parabolic orbit and q ≪ 1 (Rees 1988; Phinney 1989). The t^−5/3 dependence of TDE light curves relies on the assumption that the specific energy distribution of stellar debris dE/dM is roughly flat with orbital specific energy, which is only valid at late times (Guillochon & Ramirez-Ruiz 2013). At early times, the assumption of constant dM/dE is incorrect and depends sensitively on the structure of the disrupted star (Lodato et al. 2009; Ramirez-Ruiz & Rosswog 2009) and the strength of the tidal interaction (Laguna et al. 1993; Guillochon et al. 2009; Guillochon & Ramirez-Ruiz 2013).

Lodato et al. (2009) and Kesden (2012) moved beyond this simple description by constructing models that explicitly calculate the energy distribution of the disrupted stellar debris to ${ \mathcal O }({q}^{1/3})$ for stars described by a self-gravitating, spherically symmetric, polytropic fluid. By solving the Lane–Emden equation they determined the density profile of the star, which in turn allowed them to calculate dM/dE. In this paper, we build on their work and show how their formalism can be easily extended to estimate the rate at which the debris falls back to pericenter and is subsequently accreted for tidally disrupted stars with realistic profiles.

The geometrical setup envisioned here is shown in Figure 1. To calculate $\dot{M}$, we begin by using the standard assumption that the star freezes in at the moment of disruption at r_t. The specific binding energy of a fluid element in this case depends on its position, and dM/dE can be expressed in terms of the star's initial density profile ρ_⋆. The mass of a slice of stellar debris dM, defined here as having the same orbital energy, is found by integrating

$$\begin{eqnarray}&&\displaystyle \frac{{dM}}{{dx}}={\int }_{0}^{{H}_{x}}{\rho }_{\star }(h)2\pi h\ {dh},\end{eqnarray} \tag{ 3 }$$

where x is measured from the center of the star, H_x is the radius of the slice at a given x, and h is the rescaled height coordinate. If the orbital period t of a given slice is given in terms of its orbital binding energy dE/dx, then the rate dM/dt at which mass falls back to pericenter can be calculated by numerically integrating Equation (3). Using this framework, we calculate the accretion rate history for a large number of realistic stars, whose density profiles we generate using the Modules for Experiments in Stellar Astrophysics (MESA) code. The reader is referred to Section 2.2 for a description of our MESA setup.

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The use of this analytic method allows for an extensive study of $\dot{M}$ arising from the disruption of different stars. While this formalism leads to a large reduction in computational expense, it is nonetheless restricted as it relies on the assumption of a spherically symmetric star at the time of disruption. Contrary to what can be predicted by the simple analytical models used in this paper, the rate at which material falls back depends strongly on the strength of the encounter, which can be measured by the penetration factor β ≡ r_t/r_p.

This is because varying β changes the amount of mass lost by the star, which affects the rate at which the liberated stellar debris returns to pericenter (e.g., Law-Smith et al. 2017a). In Figure 2, we compare fallback curves calculated using the analytical model (thick dark blue line) to those calculated using simulations (thin colored lines). For the purpose of comparison, both models use a 1M_⊙ star with adiabatic index γ = 5/3 and a 10⁶ M_⊙ SMBH. We find that the broad features of $\dot{M}$ are reasonably well captured by the simple model (the same holds true for stars constructed with γ = 4/3), as was also argued by Lodato et al. (2009) and Kesden (2012). This fact is extremely powerful in that it permits a reasonable characterization of TDE signatures without the need to run many computationally expensive simulations on the large set of stars we study here.

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Furthermore, for a fixed β, the time evolution of the forces applied is identical, regardless of the ratio of the star's mass to the mass of the SMBH. This is because the ratio of the time the star takes to cross pericenter to the star's own dynamical time depends only on β. Therefore, as long as q ≪ 1, the tidal disruption problem is self-similar, and our results can be scaled to predict how the time (Equation (2)) of peak accretion rate, t_peak, and its corresponding magnitude ${\dot{M}}_{\mathrm{peak}}$ change with M_bh, M_⋆, and R_⋆:

$$\begin{eqnarray}&&{\dot{M}}_{\mathrm{peak}}\propto {M}_{\mathrm{bh}}^{-1/2}{M}_{\star }^{2}{R}_{\star }^{-3/2},\end{eqnarray} \tag{ 4 }$$

and

$$\begin{eqnarray}&&{t}_{\mathrm{peak}}\propto {M}_{\mathrm{bh}}^{1/2}{M}_{\star }^{-1}{R}_{\star }^{3/2}.\end{eqnarray} \tag{ 5 }$$

This fact is extremely powerful in that it permits us to completely characterize the properties of a disruption of a given star with one calculation. An exception to these simple scalings is if the star penetrates deeply enough such that r_p is comparable to the Schwarzschild radius r_g. In this case, general relativistic effects can alter the outcome, especially if the black hole is spinning (Laguna et al. 1993; Kesden 2012).

We remind the reader that the exact value of the time of peak accretion rate t_peak and its corresponding magnitude ${\dot{M}}_{\mathrm{peak}}$ are not precisely determined. Most of these differences arise from how the problem was originally formulated, in which the star's self-gravity is ignored, and only the spread in binding energy across the star at pericenter is assumed to be important for determining $\dot{M}$. Our primary goal in this paper is to develop a robust formalism for calculating the rate of fallback and its associated chemical composition as well as conducting a preliminary survey of the key stellar evolution parameters associated with this problem. The formalism presented in this section is well suited to this goal.

We use the open source MESA code (Paxton et al. 2011) to calculate the structure and composition of the stars that will be disrupted. We generated 192 solar metallicity stellar profiles ranging in mass from 0.8 to 3.0 M_⊙ and evolutionary state from the zero-age main sequence (ZAMS) to the near terminal-age main sequence (TAMS). Profiles are spaced in intervals of 0.05 in central hydrogen fraction.

The MESA setup used here is described below.⁴ We begin with a pre-MS model, use the mesa_49 nuclear network with the jina rates preference, the Asplund et al. (2009) abundances (X = 0.7154, Y = 0.2703, and Z = 0.0142), and mixinglengthalpha = 2.0. The final profile, which we call TAMS, is at a central hydrogen fraction of 10⁻³. Time steps are limited to a maximum change in central hydrogen fraction of 1%.

We consider the mass range of 0.8–3.0 M_⊙ as stars with masses below 0.8 M_⊙ will not evolve appreciably over the age of the universe, and stars with masses above 3 M_⊙, with MS lifetimes <300 Myr, are unlikely to be disrupted (the relaxation time for most galactic nuclei is ≫300 Myr).

We do not consider evolved stars for two reasons. First, the contribution of evolved stars to the current and near-future tidal disruption population is expected to be modest (MacLeod et al. 2012). Second, studies of the tidal disruption of evolved stars such as MacLeod et al. (2012) have shown that even for large β, giant stars are effective at retaining envelope mass and effectively retaining their cores (where the differences in composition arise from MS and post-MS evolution). In this paper, we are interested in the evolved material in the innermost layers of stars that can be reasonably revealed during a TDE and thus we do not focus on significantly evolved stars.

Here we briefly discuss the stellar evolution features that are central to our study; these arise from changes in mass and evolutionary state along the MS. The two main burning processes in MS stars, the p–p chain and the CNO cycle, are highly sensitive to interior temperatures (Kippenhahn et al. 2012) and contribute differently to stars of varying mass. The p–p chain, which increases the abundance of ${}^{4}\mathrm{He}$ in stars, roughly dominates for masses ≲1.5 M_⊙. For masses ≳1.5 M_⊙ the CNO cycle dominates. During the CNO cycle, fusing hydrogen to helium results in an increase (decrease) of ¹⁴N (¹⁶O) abundance, with ¹²C acting as a catalyst for the entire cycle. As argued by Kochanek (2016), strong compositional variations are expected in the fallback material of MS stars. In this paper, we trace the abundance variations of the following elements: ¹H, ⁴He, ¹⁶O, ¹²C, ²⁰Ne, and ¹⁴N. These elements make up at least 99.6% of each star's total mass. The ³⁴S contribution and abundance ratio is very similar to that of ²⁰Ne and is thus not explicitly shown in this paper. In what follows, we present abundances relative to solar.

As an example, in Figure 3, we show the compositional variations along the MS for a 1 M_⊙ star with solar abundance at ZAMS. The differently styled lines correspond to different stellar ages as defined by f_H, the fraction of central hydrogen burned. A star will have f_H = 0 at ZAMS and f_H = 0.99 near the end of its MS lifetime. At ZAMS, the star has solar composition (dotted lines) and is roughly homogeneous. After 4.8 Gyr (dashed lines), when more than half of the central hydrogen has been processed (f_H = 0.60), the following abundance variations are seen: a significant increase of ¹⁴N, a modest increase (decrease) of ⁴He (¹H), a significant decrease of ¹²C, and a roughly unchanged abundance of ²⁰Ne and ¹⁶O. At TAMS (solid lines), where most of the central hydrogen has been processed (f_H = 0.99), a depletion in ¹⁶O abundance is also observed. At this late stage, there is also a secondary increase in ¹⁴N in the core of the star.

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In summary, we see that ¹H, ⁴He, and ¹⁶O abundances evolve gradually, slowly extending to larger parts of the star and encompassing larger radii, while ¹²C and ¹⁴N abundances evolve rapidly across the burning region. All stars follow a similar trend. The most massive star in this study (3 M_⊙) has, at TAMS, large compositional changes across roughly half of its mass (or about 20% of its radius). As discussed by Kochanek (2016), in the fallback material from a TDE, we expect ¹²C and ¹⁴N abundance anomalies to be more noticeable and appear at earlier times than the other elemental anomalies.

As a star evolves along the MS, its average density, ${\bar{\rho }}_{\star }$, decreases and its core density, ρ_core, increases. This is illustrated in Figure 4, where we show the evolution of the density profile for a 1M_⊙ star with initial solar abundance from ZAMS to TAMS. Since the star's radius increases with age, while its mass remains nearly constant, ${\bar{\rho }}_{\star }$ decreases with age. The effects of ${\bar{\rho }}_{\star }$ on the star's vulnerability to tidal deformations can be readily seen by rewriting ${r}_{{\rm{t}}}$ as ${r}_{{\rm{t}}}\cong {M}_{\mathrm{bh}}^{1/3}{\bar{\rho }}_{\star }^{-1/3}$. This scaling implies that as the star evolves, it becomes progressively more vulnerable to tidal deformations and mass loss. However, this scaling is unable to accurately capture the exact impact parameter required to fully disrupt a star. This is because as the star evolves a denser core, a surviving core is likely to persist for a disruption at r_t (β = 1), which is the penetration factor assumed for the analytical calculations. Nonetheless, we expect the time and magnitude of the peak accretion rate to be reasonably well captured by the simple formalism described here.

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Here we analyze how the tidal radius, r_t,⋆, evolves with stellar mass and age along the MS for the stars in our study. The left panel of Figure 5 shows r_t,⋆ normalized to the tidal radius of the same star at ZAMS, r_t,ZAMS. We plot this ratio as a function of f_H, the fraction of central hydrogen burned, and stellar mass M_⋆. As expected, we find that the tidal radius increases with age and evolves more dramatically with f_H throughout the lifetime of more massive stars. For example, the tidal radius of a 3 M_⊙ star increases by roughly a factor of two over its MS lifetime. As stars move along the MS, they become progressively more vulnerable to tidal dissipation and mass stripping.

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Next, we discuss how the vulnerability of regions with processed element abundances compares to that of the entire star. The right panel of Figure 5 shows the ratio of ${r}_{{\rm{t}},\star }$ to r_t,burn, where r_t,burn is defined as the tidal radius of material within the regions of a star that exhibit active nuclear burning. This region of active nuclear burning is defined to be where the specific power from nuclear reactions is greater than 1 erg g⁻¹ s⁻¹. This is a consistent way of defining the burning region throughout all of the stellar profiles calculated here. As expected, this region is located at small radii where the density is much higher than ${\bar{\rho }}_{\star }^{-1/3}$ and thus deeper penetrations are required in order to observe the evolved element abundances in the fallback material. Also, as this region is located within the innermost layers of the star, the processed elements will be revealed in the fallback material only at later times.

Figure 6 shows the mass fallback rate arising from the full disruption of a 1M_⊙ star at two different evolutionary states: at ZAMS (dotted lines) and after 4.8 Gyr (dashed lines), when more than half of the central hydrogen has been processed (f_H = 0.60). These curves are normalized to the peak fallback rate and peak time of the corresponding ZAMS star: ${\dot{M}}_{\mathrm{peak},\mathrm{ZAMS}}$ and t_peak,ZAMS, respectively. The compositions of the stars before disruption are shown in Figure 3 as dotted (ZAMS) and dashed (f_H = 0.60) lines. The disruption of the TAMS 1M_⊙ star, whose composition is shown by the solid lines in Figure 3, is expected to be similar in shape to the disruption of the f_H = 0.60 star, with an enhancement in ¹⁴N and depletion in ¹²C.

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The smooth behavior of the fallback rates for all the plotted elements during the disruption of the ZAMS star (dotted lines in Figure 3) is the result of the nearly homogeneous elemental composition within the star. The fallback rates for the f_H = 0.60 star (dashed lines in Figure 3), on the other hand, contain information about the varying nature of its elemental composition. In the fallback rates, we can see an obvious increase in ¹⁴N, a decrease in ¹²C, and a slight increase in ⁴He, which is consistent with the compositional structure of the star before disruption. These results are in agreement with Kochanek (2016). We note that the fallback curves for the f_H = 0.60 star have no abundance variations at t ≲ t_peak. These compositional anomalies might provide insight into the nature of the progenitor star near or after the most luminous time of the tidal disruption flare.

In Figure 7, we show the fractional contribution to the total fallback rate arising from each element during the disruption of a 1 M_⊙ star at three different evolutionary stages. From left to right, these panels correspond to the ZAMS (dotted), f_H = 0.60 (dashed), and TAMS (solid) composition profiles in Figure 3, respectively. In each panel, we calculate the ratio of the fallback rate for each element, ${\dot{M}}_{{\rm{X}}}$, to the total mass fallback rate, ${\dot{M}}_{\mathrm{full}}$.

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For the disruption of a 1 M_⊙ star, it might be challenging to distinguish its evolutionary stage using spectral information if it is only obtained at t ≲ t_peak (though the exact values of ${\dot{M}}_{\mathrm{peak}}$ and t_peak are expected to be distinct; Figure 12). This is, however, not the case after t_peak.

Figure 8 shows the abundance of the fallback material relative to solar following the disruption of a 1 M_⊙ at two different evolutionary stages: f_H = 0.60 (left panel) and TAMS (right panel). Elemental abundances relative to solar are calculated here using

$$\begin{eqnarray}&&\displaystyle \frac{X}{{X}_{\odot }}=\displaystyle \frac{{\dot{M}}_{{\rm{X}}}/{\dot{M}}_{{\rm{H}}}}{{M}_{{\rm{X}}}/{M}_{{\rm{H}},\odot }},\end{eqnarray} \tag{ 6 }$$

where ${\dot{M}}_{{\rm{X}}}$ is the fallback rate for a selected element, ${\dot{M}}_{{\rm{H}}}$ is the fallback rate of ¹H, and ${M}_{{\rm{X}}}/{M}_{{\rm{H}},\odot }$ is the abundance mass ratio relative to solar of element X. The disruptions of a f_H = 0.60 and a TAMS star each show a significant increase in ¹⁴N and ⁴He after t_peak. As expected, these features are more prominent for the TAMS star. Near t = 10t_peak, Figure 8 shows steeper abundance gradients in the right panel compared to the left in all elements except ¹²C. We note that these values are relative to ¹H. This is important in the case of ¹⁶O and ²⁰Ne, where we see an increase in their abundance. This is because while ¹H is depleted at every evolutionary stage, ¹⁶O and ²⁰Ne abundances remain relatively constant for a star of this mass, which results in higher solar ratios. However, this behavior is also altered by the mass of the star as we discuss in the following section.

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For reasons discussed previously, it seems likely that the evolutionary state of a star might be revealed by charting the compositional evolution of the fallback material, which might be inferred from particular features in the spectra of the resulting luminous flare. The association of a significant fraction of TDEs with post-starburst galaxies (Arcavi et al. 2014; French et al. 2016, 2017; Law-Smith et al. 2017b) has suggested the likely presence of evolved stars in the nuclei of TDE hosts, or at least a subset thereof. Much of our effort in this section will thus be dedicated to determining the state of the fallback material after the tidal disruption of stars of a wide range of ages and masses.

In Figure 9, we show the relative abundances of the fallback material for three representative MS star disruptions. The first row of panels shows the abundance of the fallback material for a 0.8 M_⊙ star tidally disrupted at three different evolutionary stages: f_H = 0.3, f_H = 0.6, and f_H = 0.99. The abundances shown are similar to those shown in Figure 8 for a 1 M_⊙ star. At these low masses, we expect the abundance anomalies to be present in the fallback material at a few times t_peak.

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The second row of panels in Figure 9 shows the relative abundances of the fallback material for a disrupted 2 M_⊙ star. The abundance patterns are broadly similar to those seen for the 0.8 M_⊙ and 1 M_⊙ stellar disruptions. However, there are three main differences. First, in contrast to the observed increase of ¹⁶O seen in the 0.8 M_⊙ and 1.0 M_⊙ disruptions, a significant decrease in ¹⁶O abundance is observed. This is an indication of the increased CNO activity in the 2 M_⊙ star. Second, two distinct bumps are seen in the evolution of the ¹⁴N abundance, contrary to its steady increase in the smaller mass disruptions. The first increase in the ¹⁴N abundance (and the corresponding ¹²C depletion) is due to the local maximum of CNO burning that is located at roughly 20% of the star's radius. There is also significant CNO and p–p chain activity in the star's core, which is revealed at later times in the fallback material, and leads to the relatively delayed increase in ⁴He and ²⁰Ne, the corresponding decrease of ¹⁶O, and a secondary increase in ¹⁴N. Third, abundance variations are observed significantly closer to t_peak in the 2 M_⊙ disruptions than in the 0.8 M_⊙ disruptions. This is a result of the more extended burning region within the star, whose material is revealed at earlier times following the disruption.

In the bottom row of panels in Figure 9, we show the composition of the fallback material following the disruption of a 3 M_⊙ star. The abundance variations in these fallback curves closely resemble those for the 2 M_⊙ star, but with larger variations appearing at earlier times. The abundance variations presented in Figure 9 for the few representative stars accurately describe the overall trends in our sample. These trends are illustrated in Figure 10, in which various elemental abundances are shown at the time that the mass fallback rate has reached one-tenth of its peak value, ${t}_{0.1}\gt {t}_{\mathrm{peak}}$.

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The fallback abundances at t_0.1 are plotted in Figure 10 as a function of the star's fractional main-sequence lifetime, t/t_MS, and stellar mass. In Figure 11, we show the same abundance values as in Figure 10 but presented with the evolutionary age of the star in years. Some key points should be emphasized. We find carbon decrements to be indicative of stellar mass, while helium enhancements are indicative of age. ${(X/{X}_{\odot })}_{{}^{14}{\rm{N}}}\gtrsim 5.0$ occurs only for masses greater than 1.5 M_⊙ and develops early in the star's evolution. This is due to the enhanced CNO activity inside the more massive stars in our sample. We also find oxygen abundances to be primarily dependent on stellar mass.

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The processes discussed here suggest that TDEs may have a more complex spectrum and time-structure than simple models suggest. The effects are especially interesting when the accretion rate is high, as this gives rise to high luminosities, and thus can more readily offer clues to the nature of the disrupted star. The specific values of ${\dot{M}}_{\mathrm{peak}}$ and t_peak can further aid in distinguishing the properties of the progenitor star before disruption. This is illustrated in Figure 12 where we show abundances of carbon, helium, nitrogen, and oxygen (relative to solar) in the fallback debris as a function of ${\dot{M}}_{\mathrm{peak}}$ and t_peak. Each panel in Figure 12 corresponds to a different element, the different lines correspond to different stars in our study (0.8 M_⊙, 1.0 M_⊙, 1.2 M_⊙, 1.4 M_⊙, 2.0 M_⊙, and 3.0 M_⊙), the points are different stages in the stars' evolution on the MS (roughly equally spaced in time), and the color of the points is the abundance of the fallback debris at the time that $\dot{M}$ falls to one-tenth of its peak value, t_0.1. We used the fitting formulas presented in Guillochon & Ramirez-Ruiz (2013), which give ${\dot{M}}_{\mathrm{peak}}$ and t_peak given β, γ, M_⋆, and R_⋆. We used γ = 4/3 and its corresponding penetration factor for full disruption (β = 1.85) given by Guillochon & Ramirez-Ruiz (2013). The values of M_⋆ and R_⋆ were taken from the MESA profiles and we have assumed ${M}_{\mathrm{bh}}={10}^{6}{M}_{\odot }$ (the reader is referred to Equations (4) and (5) for the scalings of ${\dot{M}}_{\mathrm{peak}}$ and t_peak with M_bh, respectively). The abundance values are the same as in Figure 10.

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The variation in elemental abundances is accompanied by a wide range in ${\dot{M}}_{\mathrm{peak}}$ and a moderate range in t_peak; a combination of these different pieces of information can help characterize the progenitor stars of TDEs. For example, the disruption of a 3M_⊙ star has similar t_peak values to that of a 2 M_⊙ star. While their ¹²C and ¹⁶O abundances are very similar, the 3M_⊙ star's disruption results in a higher abundance in ¹⁴N and ⁴He at every stage in its evolution, along with a higher ${\dot{M}}_{\mathrm{peak}}$. In the lower mass stars (0.8–1.4 M_⊙) there are many degeneracies in ${\dot{M}}_{\mathrm{peak}}$ and t_peak values. Here, the ¹⁴N, ¹⁶O, and ⁴He abundances are similar (over the age of the universe) but the ¹²C abundances vary at the early stages in these stars' MS evolution. Compositional information, combined with reprocessing and radiative transfer calculations (e.g., Roth et al. 2016), can thus be used to discern the stellar mass and age of the disrupted star.

Motivated by the work of Kochanek (2016), we have modeled the tidal disruption of MS stars of varying mass and age. We adopted the analytic formalism originally presented in Lodato et al. (2009) to study, for the first time, the time evolution of the composition of the fallback debris onto the SMBH. We compared the analytic method to hydrodynamic simulations in Figure 2 and found, similarly to Lodato et al. (2009) and Kesden (2012), that the broad features of the fallback curves are reasonably well captured by it.⁵ We quantify the variations in composition arising from the disruption of 12 different stars with masses of 0.8–3.0 M_⊙ at 16 different evolutionary stages along the MS. The main results of our study are the following.

• 1  
  We predict an increase in nitrogen and depletion in carbon abundance in the fallback debris with MS evolution for all stars in our sample (in agreement with Kochanek 2016). We find a decrease in oxygen with MS evolution for M_⋆ ≳ 1.5M_⊙, and an increase for M_⋆ < 1.5 M_⊙.
• 2  
  For all of the TDEs modeled in this study, we find that the time during the fallback rate curve when anomalous abundance features are present, ${t}_{\mathrm{burn}}$, is after the time of peak fallback rate t_peak.
• 3  
  Abundance variations are more significant and t_burn/t_peak is smaller for stars of larger mass.
• 4  
  Some key variations in the compositional evolution are highlighted, along with the types of observations that would help to discriminate between different stellar disruptions. In particular, we find carbon and oxygen abundances to be strongly dependent on stellar mass for M_⋆ ≲ 2 M_⊙, while helium abundances are found to be correlated with stellar age for all masses. ${(X/{X}_{\odot })}_{{}^{14}{\rm{N}}}\gtrsim 5.0$ occurs only for masses greater than 1.5 M_⊙ and is observed early in the star's evolution.
• 5  
  Studying the compositional variation in the fallback debris provides a clear method for inferring the properties of the progenitor star before disruption.

It is evident from the results described above that the evolution of the interior structure of stars during their MS lifetimes is very rich. Even in the simplest case of a Sun-like star, complex behavior with multiple abundance transitions in the fallback material may be observed. The resulting TDE spectra are expected to depend fairly strongly on the abundance properties of the fallback material (Roth et al. 2016). This implies that if one can be very specific about the times at which we expect to see such transitions in the observed emission, one can better constrain the properties of the disrupted star.

Motivated by this, in Figure 13, we plot the fallback time t_burn, relative to t_peak, at which we expect to see anomalous abundance variations. Here t_burn is defined as the time at which the abundances of ¹²C and ¹⁴N in the fallback material, as presented in Figures 8 and 9, both deviate from unity. t_burn/t_peak is shown in Figure 13 as a function of stellar mass and age (characterized by f_H). At fixed f_H, we see that non-solar abundances in the fallback debris begin to appear systemically closer to t_peak as stellar mass increases. For the 3.0 M_⊙ star, t_burn ≈ 1.2t_peak for f_H ≲ 0.3. For constant M_⋆, t_burn/t_peak increases mildly with f_H for stars with M_⋆ > 1.6 M_⊙. For stars with ${M}_{\star }\lt 1.6{M}_{\odot }$, this ratio remains fairly constant throughout the star's evolution. In summary, t_burn depends strongly on M_⋆ but has a relatively weak dependence on stellar age. It is important to note that independently of the mass and age of the disrupted star, no anomalous abundances are expected to be observed before t_peak.

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Information regarding the nature of the disrupted star should be imprinted on the properties of the TDE light curve (e.g., t_peak and ${\dot{M}}_{\mathrm{peak}}$) and spectrum (particularly at t ≳ t_burn). Current observations of TDEs show clear differences in their rise and decay properties as well as in their spectral evolution. Peculiar emission features have been observed in their spectra, which include an array of helium, hydrogen, and nitrogen broad line emission features. The origin of these features as well as their associated line ratios have caused significant debate. The extreme helium to hydrogen line ratio observed in the transient event PS1-10jh was initially proposed to be the result of the tidal disruption of a helium-rich star (Gezari et al. 2012). However, such line ratios have also been shown to arise from the reprocessing of radiation through the fallback debris of a disrupted Sun-like star (Roth et al. 2016). As for the additional presence of rare nitrogen features, Kochanek (2016) first proposed that the disruption of MS stars with evolved stellar compositions could lead to enhanced nitrogen (as well as anomalous helium and carbon abundances).

In Figure 14, we show compositional features in the spectra of 10 observed TDEs. We place each spectrum in the light curve of each event, relative to its peak luminosity and peak time. Symbols indicate features present in the spectra. Bolometric light-curve fits for each event are from Mockler et al. (2018). Data is taken from Gezari et al. (2012, 2015), Chornock et al. (2014), Arcavi et al. (2014), Brown et al. (2016, 2017, 2018), Blagorodnova et al. (2017), Hung et al. (2017), Holoien et al. (2014, 2016a, 2016b), Cenko et al. (2016), and Wyrzykowski et al. (2017). Note that this figure shows TDEs with well-sampled light curves and existing spectroscopic observations. Several TDE spectra show compositional features at or near the peak in their light curve. Our calculations (in particular, see Section 3.3 and Figure 13) predict no compositional abundance changes (relative to solar) in the fallback material at or near peak due to the star. This implies that the strong suppression of hydrogen Balmer line emission relative to helium line emission should occur even at solar composition, as argued by Roth et al. (2016), due to optical depth effects alone. For observations at t > t_burn, we expect the reprocessing material to be enhanced in helium, yet the optical depth effects are expected to be less important (Guillochon et al. 2014). As such, radiation transfer calculations are needed before firm conclusions can be derived from observations of evolving line ratios in a given TDE.

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Nitrogen emission lines, on the other hand, are only currently detected at t ≳ 1.2t_peak. If their presence is primarily attributed to a drastic increase in nitrogen abundance, then based on the results shown in Figures 12 and 13, one would conclude that M_⋆ ≳ 1.8 M_⊙ for the star whose disruption triggered the ASASSN-14li flaring event and M_⋆ ≳ 3.0 M_⊙ for the star whose disruption triggered iPTF16fnl. However, early UV spectra that show a lack of nitrogen emission lines at early times are needed to support constraints such as these. In addition, the line ratio variability of, for example, CIII and NIII, can be used to infer the abundance evolution (as these lines have similar ionization potentials). The timescale for chemical enrichment (i.e., t_burn) can thus provide a direct observational test of which stars are being disrupted by the central SMBH.

Much progress has been made in understanding how the feeding rate onto an SMBH proceeds after the disruption of a particular star, and in deriving the generic properties of the flares that follow from this. There still remain a number of mysteries, especially concerning the identity of the star, the nature of the energy dissipation mechanism, and the timescales involved. The modeling of the flare itself (i.e., the dissipation mechanism and the radiation processes) is a formidable challenge to theorists and to numerical techniques. It is also a challenge for observers, in their quest to detect fine details in distant, fading sources. The class of models we have presented here predict that the spectral properties of the fading signals will turn out to be even more telling and fascinating that initially anticipated.

Future work will include a more detailed exploration of the parameters governing the abundance of the fallback material, including hydrodynamical calculations (e.g., Law-Smith et al. 2017a) as well as radiative transfer calculations (e.g., Roth et al. 2016) evolved over time for different properties of the reprocessing material. Studies of this sort, in comparison with improved spectral observations of TDEs, will undoubtedly help clarify the physics governing these transient sources.