The thermal stability of helium burning on accreting neutron stars

Transcription

1 doi: /mnras/stu1927 The thermal staility of helium urning on accreting neutron stars Michael Zamfir, Andrew Cumming and Caroline Niquette Department of Physics, McGill University, 3600 rue University, Montréal, Quéec H3A 2T8, Canada Accepted 2014 Septemer 13. Received 2014 Septemer 3; in original form 2014 July 28 1 INTRODUCTION Thermonuclear urning of hydrogen H) and helium He) on the surface of an accreting neutron star is expected to undergo a transition from eing thermally unstale to thermally stale at a critical accretion rate Ṁ crit gs 1 close to the Eddington accretion rate; Hansen& vanhorn1975; Fujimoto, Hanawa & Miyaji 1981). The transition occurs ecause the temperature-dependence of the He urning reactions ecomes less steep at higher urning temperatures, so that at a high enough accretion rate the reactions are no longer temperature dependent enough to overcome the stailizing radiative cooling of the layer. Oservationally, unstale nuclear urning is seen as type I X-ray ursts, right flashes in X-rays with a typical duration of sec that recur on time-scales of hours to days Lewin, van Paradijs & Taam 1993). Consistent with the idea that the urning stailizes, the rate of type I X-ray ursts drops dramatically in several sources aove a persistent luminosity L X erg s 1 see also Clark et al. 1977; van Paradijs, Penninx & Lewin 1988; Cornelisse et al. 2003), and the urst energetics clearly point to most of the accreted fuel urning in a stale manner van Paradijs et al. 1988; Galloway et al. 2008). Other oserved phenomena also point to stale urning mzamfir@physics.mcgill.ca ABSTRACT Thermonuclear urning on the surface of accreting neutron stars is oserved to stailize at accretion rates almost an order of magnitude lower than theoretical models predict. One way to resolve this discrepancy is y including a ase heating flux that can stailize the layer. We focus our attention on pure helium accretion, for which we calculate the effect of a ase heating flux on the critical accretion rate at which thermonuclear urning stailizes. We use the MESA stellar evolution code to calculate ṁ crit as a function of the ase flux, and derive analytic fitting formulae for ṁ crit and the urning temperature at that critical accretion rate, ased on a one-zone model. We also investigate whether the critical accretion rate can e determined y examining steady-state models only, without time-dependent simulations. We examine the argument that the staility oundary coincides with the turning point dy urn /dṁ = 0inthe steady-state models, and find that it does not hold outside of the one zone, zero ase flux case. A linear staility analysis of a large suite of steady-state models is also carried out, which yields critical accretion rates a factor of 3 larger than the MESA result, ut with a similar dependence on ase flux. Lastly, we discuss the implications of our results for the ultracompact X-ray inary 4U Key words: instailities stars: neutron X-rays: inaries X-rays: ursts X-rays: individual: 4U at high accretion rates. Stale H/He urning is required in models for superursts to produce the caron fuel that is elieved to drive those events Schatz et al. 2003; Woosley et al. 2004; Stevens et al. 2014), and is manifested in the energetics of type I X-ray ursts oserved from superurst sources in t Zand et al. 2003). The mhz quasiperiodic oscillations QPOs) oserved in some sources Revnivtsev et al. 2001; Altamirano et al. 2008; Linares et al. 2012) have een identified with an oscillatory mode of nuclear urning that emerges when the urning is marginally stale, i.e. transitioning etween stale and unstale Paczynski 1983a; Narayan & Heyl 2003; Heger, Cumming & Woosley 2007a; Keek, Cyurt & Heger 2014). Despite this qualitative agreement, a long-standing puzzle has een that the oserved accretion rate at which the onset of stale urning occurs is Ṁ gs 1, an order of magnitude lower than theory predicts given standard assumptions for the thermal state of the crust Bildsten 2000; Brown 2000; Keek et al. 2014). Several mechanisms have een suggested to account for this discrepancy, including a change in urning mode to slowly propagating fires around the neutron star surface Bildsten 1995), partial covering of the accreted fuel Bildsten 1998), mixing of fuel driven y rotational instailities Fujimoto et al. 1987; Piro & Bildsten 2007; Keek, Langer & in t Zand 2009), and strong heating of the layer associated with spin-down and spreading of the fuel following disc accretion Inogamov & Sunyaev 1999, 2010), or other sources of heating Bildsten 1995; Narayan & Heyl 2002, 2003; Keek et al. 2009). C 2014 The Authors Pulished y Oxford University Press on ehalf of the Royal Astronomical Society

2 The proposal that the unstale urning is quenched y heating is intriguing ecause evidence has accumulated that the outer crust and ocean of accreting neutron stars are strongly heated y an unknown shallow heat source. One piece of evidence is from superursts, whose oserved ignition properties require temperatures of K e achieved at column depths of gcm 2 in the neutron star ocean, requiring an additional source of heat e added to models Brown 2004; Cumming et al. 2006). This prolem has een exasperated recently with oservations of superursts in transient systems Keek et al. 2008; Altamirano et al. 2012). The second piece of evidence is from modelling of the thermal relaxation of transiently accreting neutron stars in quiescence. Brown & Cumming 2009) found that the temperatures oserved in KS and MXB approximately one month into quiescence required an inward heat flux into the neutron star crust and a corresponding strong shallow heat source. Degenaar, Wijnands & Miller 2013) reached a similar conclusion ased on rapid cooling of XTE J after a short 10 week outurst. Schatz et al. 2014) showed that a strong neutrino cooling source may operate in the outer crust, emphasizing the need for additional heating at shallow depths. Finally, modelling of X-ray urst recurrence times in a numer of sources has suggested that outward fluxes of 0.3 MeV per nucleon 1 or more heat the accumulating H/He layer Cumming 2003; Galloway & Cumming 2006). Determining the dependence of Ṁ crit on the ase flux is critical to assess whether shallow heating could also e the reason for stailization of type I X-ray ursts at oserved accretion rates Ṁ gs 1. Most calculations of the critical accretion rate Ṁ crit in the literature are for a fixed ase flux, typically Q 0.1 MeV per nucleon taken from models of the gloal thermal state of the neutron star; e.g. Brown 2000) for which Ṁ crit gs 1 e.g. Heger et al. 2007a; Keek et al. 2014). Bildsten 1995) calculated the effect of a flux from deep caron urning on the staility of the He shell using a one-zone approach and Fushiki & Lam 1987) also included the ase temperature as a parameter in their onezone study. Using a linear staility analysis of models with fixed temperatures set elow the accreted layer, Narayan & Heyl 2002, 2003) found transitions to stale urning for a solar mixture of H and He at accretion rates of around Ṁ Edd. Keek et al. 2009) calculated the staility oundary for pure He accretion using detailed multizone models for several different ase fluxes, showing that an increased heating rate decreases Ṁ crit. They found that a ase luminosity of L crust erg s 1 approximately 1 MeV per nucleon at 0.1 Eddington) lowered the critical accretion rate to gs 1. Analogous simulations varying ase flux for H/He accretion have not een carried out. When the accreted material contains a significant amount of H, the urning proceeds via the rapid proton capture process rp-process) involving hundreds of nuclei Wallace & Woosley 1981) and so calculations are much more numerically intensive and so far have een carried out only for specific choices of ase flux Schatz et al. 1998; Woosley et al. 2004; Keek et al. 2014). In this paper, we take some further steps towards calculating and understanding the variation of Ṁ crit with ase flux. For simplicity, we consider only pure He accretion, ut with the goal of developing techniques that can e readily applied to the mixed H/He accretion case later. We first use the stellar evolution code Modules for Experiments in Stellar Astrophysics MESA; Paxton et al. 2011, 2013) 1 Throughout the paper, we will measure the heat flux in units of the equivalent energy per accreted nucleon Q in MeV per nucleon, so that the flux is F = Q ṁ,whereṁ is the local accretion rate ṁ = Ṁ/4πR 2. The thermal staility of helium urning 3279 to confirm the results of Keek et al. 2009) for pure He accretion. We then extend the one-zone model of Bildsten 1998) to include a ase flux, which we use to understand the shape of the relation etween Ṁ crit and Q, and to derive useful fitting formulae. In the second part of the paper, we investigate two different methods that have een proposed to determine the staility of the nuclear urning ased purely on a steady-state model at a given accretion rate, rather than running time-dependent simulations. This is potentially very powerful ecause steady-state models can e calculated quickly even when rp-process urning is included e.g. see the large grid of steady-state models recently calculated y Stevens et al. 2014). An outline of the paper is as follows. The time-dependent simulations of He accretion and one-zone analysis are presented in Section 2. In Section 3, we discuss the relation etween the urning depth in steady-state models and the thermal staility of the model. In Section 4, we develop a linear staility analysis of steady-state models and compare to the time-dependent results from MESA. We conclude in Section 5, where we also discuss the application of our results to the ultracompact X-ray inary 4U THE EFFECT OF BASE HEATING ON THE STABILITY BOUNDARY We start in this section y calculating the critical accretion rate Ṁ crit for pure He accretion as a function of the ase flux Q. The results of our time-dependent simulations are presented in Section 2.1, and a one-zone model is developed in Section 2.2 to help to understand the results. 2.1 Time-dependent calculations with MESA One of the exciting developments in stellar astrophysics in recent years has een the release of the open source stellar evolution code MESA Paxton et al. 2011, 2013). MESA solves the equations of stellar evolution in a fully coupled way, and includes the relevant microphysics for the outer layers of a neutron star relevant for type I X-ray ursts. Indeed, a sequence of He flashes on an accreting neutron star was modelled in Paxton et al. 2011), and accretion on to a neutron star is a standard test case in the MESA distriution. We apply MESA here to determine the staility oundary for pure He accretion. We view this as a straightforward first step to developing MESA as a general tool to study X-ray ursts on accreting neutron stars. Here, we will present only our results on the staility oundary, leaving a detailed analysis of urst sequences and the evolution of the urning layers during a urst for a future paper. We used the MESA release 6596 for our simulations. To enale a meaningful comparison with one-zone models and linear staility analysis Section 4), we used a simplified nuclear network that takes into account only the triple-α reaction 3α 12 C, so that only two species, He and caron, were present. For determining the staility oundary, this is in fact a good approximation: we also tried using the approx 21 network that includes a sequence of He urning reactions to heavier elements, and found that the critical accretion rate changed y 10 per cent with the change of network. The reason for this is that the urning temperature at the staility oundary, T K, is small enough that the urning does not proceed significantly past caron e.g. Brown & Bildsten 1998). From this point onwards, we will use local values for the accretion rate. We adopt a standard value for the local Eddington accretion rate, ṁ Edd = gcm 2 s 1 the equivalent gloal accretion rate is Ṁ Edd = gs 1 = M yr 1 ). This corresponds to the Eddington rate for solar composition; we use it

3 3280 M. Zamfir, A. Cumming and C. Niquette here as a standard value even though our simulations are for pure He accretion. We assume a 1.4 M, 10 km neutron star, which has a surface gravity of g = cm s 2. This is the Newtonian value for the surface gravity, and does not include the general relativistic correction; however the dependence of the critical accretion rate on gravity is weak see Section 4.2). To find the critical accretion rate, we followed these steps. For each choice of Q and ṁ, we first accrete a column gcm 2 of caron, allowing the model to thermally adjust to the ase luminosity. We then accrete an additional column of gcm 2 of pure He. Since the urning depth is 10 8 gcm 2, this means that we accrete a column of roughly one hundred urning depths which allows the initial transient ehaviour to die away at the eginning of the run. We then assess whether the urning has stailized y looking at the range of luminosities in the last 10 per cent of the light curve. If the luminosity variation is smaller than a factor of L/L = 2 then we classify the urning as stale. We have checked that our derived staility oundary does not significantly change if we use another value for L/L. To find convergence in the staility oundary, we used a value of 0.1 for the parameter mesh_delta_coeff in the MESA code. We found that using a value 10 times larger for this parameter yielded critical accretion rates that differed y 10 per cent. For each Q, we start at a large accretion rate and run successive models with accretion rate reduced in steps of log 10 ṁ/ṁ Edd = 0.025, until the urning ecomes unstale, which means that we have located the staility oundary. Fig. 1 shows that, for a ase flux Q = 0.1 MeV per nucleon, the initially stale ehaviour transforms to a sequence of ursts elow 2 ṁ Edd, with the urst recurrence time and amplitude growing as the accretion rate is lowered further. The staility oundary as a function of ase flux is shown in Fig. 2. We see a smooth decrease in ṁ crit with Q, reaching 0.1 ṁ Edd at Q 0.7 MeV. The results of Keek et al. 2009) areshownas a comparison note that Keek et al present these results as luminosity against ṁ, see their fig. 11, here, we have divided the luminosity y ṁ to convert the luminosity to MeV per nucleon units). The agreement is good, with typical deviations of tens of per cent, although the point at Q 0.8 MeV from Keek et al. 2009) is a factor of 2 higher than the MESA result. 2.2 One-zone model To understand the shape of the ṁ crit Q ) relation, it is helpful to consider a one-zone model with a ase flux included. In a one-zone treatment of the urning layer, we follow the layer temperature and column depth according to c P dt dt dy dt = ɛ 3α ɛ cool + Q ṁ y, 1) = ṁ ɛ 3α E 3α y 2) Paczynski 1983a; Bildsten 1998; Heger et al. 2007a), where ɛ 3α is the heating rate and E 3α is the energy per unit mass released from 3α reactions, and the one-zone cooling rate is ɛ cool act 4 /3κy 2. Heating from eneath the layer is represented y the third term on the right-hand side of equation 1) Heger et al. 2007a). To derive the staility oundary, we consider steady-state solutions of equations 1) and 2) and pertur them, taking the perturations to e at constant pressure and column depth since column depth y = P/g in a thin layer). We follow Bildsten 1998) and assume an ideal gas equation of state, so that δρ/ρ = δt/t at Figure 1. Light-curve profiles generated y MESA showing stale urning and various ursting ehaviours at different accretion rates. All models are computed with a ase flux of Q = 0.1 MeV per nucleon. In going from ṁ = 1.9toṁ = 2.0ṁ Edd, the amplitude aruptly changes from L/L 10 to roughly zero, illustrating the fact that modest variations to the instaility criterion we used L/L > 2) do not strongly affect the location of the staility oundary. Figure 2. The staility oundary we find using MESA lack crosses) agrees well with the oundary found y Keek et al. 2009, red circles). The analytically derived one-zone estimate to the staility oundary equation 9) is shown as a green dotted curve. The one-zone curve shares a similar shape to, ut overestimates the values of the MESA data. By making small adjustments to equation 9) see text), we otain an analytic fit to the MESA results equation 10), shown as the lue dotted curve.

4 The thermal staility of helium urning 3281 constant pressure. This gives dδt c P = δɛ 3α δɛ cool dt = δt T [ɛ 3α ν η) ɛ cool 4 κ T )], 3) where we have expressed the heating rate as ɛ 3α ρ η T ν,andκ T = ln κ/ ln T P. For triple-α urning, ν 44/T 8 ) 3andη = 2, where T 8 = T/10 8 K Hansen & Kawaler 1994). When the ase heat flux is much smaller than the energy generated inside the layer, the steady-state oeys ɛ 3α ɛ cool, and the condition for instaility dδt/dt > 0) is ν η 4 + κ T = κ T > 0 4) T 8 e.g. Bildsten 1995, 1998; Yoon, Langer & van der Sluys 2004). When the ase flux is significant, ɛ 3α is no longer equal to ɛ cool, and in fact is smaller since the ase flux Q now contriutes to the heating of the layer. The instaility condition is ν η ɛ cool 4 κ T ) > 0. 5) ɛ 3α In steady state, equation 1) gives ɛ cool = ɛ 3α + ṁq /y, and the urning depth is given y equation 2) as y/ṁ = E 3α /ɛ 3α.Therefore, ɛ cool ɛ 3α = 1 + Q Q = 1 + E 3α 0.61 MeV. 6) We see that when Q is significant, the cooling term in the instaility criterion is enhanced. This implies that to trigger a urst in the presence of a ase flux, the temperature in the layer must e lower than without the ase flux, so that ν is larger and ale to overcome the cooling term. The effect of Q is therefore to lower ṁ crit compared to its Q = 0 value, as seen in the MESA results in Fig. 2. Setting the equality in equation 5) gives an expression for the critical temperature elow which He urning ecomes unstale in the one-zone model, T crit, 8 = 4.9 [ κ T )] Q 1. 7) 0.61 MeV Setting Q and κ T to zero, we recover the critical temperature for stale He urning found y Bildsten 1998), T crit, 8 = 4.9. Equation 7) confirms that the inclusion of a ase heating flux reduces the critical temperature required for the onset of unstale urning. As Keek et al. 2009) noticed, T 8 = 4.9 is well aove the urning temperature at marginal staility in multizone time-dependent calculations see Fig. 3). However since the shapes of the one-zone and MESA curves agree well, we can adjust the prefactor in equation 7) from 4.9 to 3.6 to otain a simple analytical expression that descries the urning temperature at the critical accretion rate in MESA shown as a lue dotted curve in Fig. 3). We can now find ṁ crit at a given Q y calculating the accretion rate at which the urning temperature is equal to the value in equation 7). To do so, we can use the following expression Bildsten 1998, equation 19) which gives the temperature at the He urning depth in steady state, ṁ T urn = K ṁ Edd ) 1/5 1 + ) Q 3/20, 8) 0.61 MeV where we assume pure He composition and other appropriate parameters μ = 4/3, E 18 = 0.58, g 14 = 1.9) and have written the flux heating the layer in terms of Q. The scalings in this expression indicate that a reduction in the critical temperature required for Figure 3. The temperature at the triple-α urning depth along the MESA staility oundary is shown in lack crosses. Sharing a similar shape ut with higher values, the one-zone analytical estimate to the critical temperature equation 7) is shown as a dotted green line. By changing the prefactor in equation 7) from 4.9 to 3.5, we find a good fit to the MESA data, allowing us to estalish a simple analytical expression for the urning temperature at the critical accretion rate. instaility implies a reduction in the accretion rate, for a constant ase flux. Equating the temperatures in equations 7) and 8), we find the critical accretion rate ṁ crit = 16 ṁ Edd 1 + Q 0.61 MeV ) 3/4 1 + ) Q 5, 1.37 MeV 9) where we again set κ T = 0. We have checked equation 9) y running time-dependent onezone models, solving equations 1) and 2) in time. We include electron scattering, free free, and conductive opacities following Schatz et al. 1999) and Stevens et al. 2014), the 3α urning rate from Fushiki & Lam 1987a), and we used fitting formulae for the contriutions of degenerate and relativistic electrons to the equation of state from Paczynski 1983). We use a similar method to the MESA runs descried in Section 2.1 to determine from the light curve whether the urning is stale or unstale. We find that the analytic expression in equation 9) underestimates the time-dependent one-zone ṁ crit y per cent across the range of Q.These differences are mostly due to the assumptions of constant opacity κ = cm 2 g 1 and ideal gas equation of state that go into equation 8). We have confirmed this y running time-dependent models that adopt the same assumptions. At low accretion rates and fluxes Q 1 MeV, another source of error is that the approximation exp 44/T 8 ) T 8 /3.38) 13 used y Bildsten 1998) to expand the triple-α urning rate as a power law egins to reak down. We find that adjusting the prefactor in equation 9) to 23) ṁ Edd reproduces the full time-dependent one-zone model to within per cent, and this is plotted as a green dotted curve in Fig Analytic expression for ṁ crit Q ) Comparing the one-zone result with the MESA calculation in Fig. 2 shows that the overall shape of the curve is reproduced well y the one-zone model, ut the magnitude of ṁ crit is overestimated

5 3282 M. Zamfir, A. Cumming and C. Niquette y a factor of approximately 5. This factor is similar to the difference etween the ṁ crit = 16 ṁ Edd found y Bildsten 1995) and the ṁ crit 3 ṁ Edd found y Keek et al. 2009). The inaccuracy of the one-zone model comes from the instaility criterion equation 5) which overestimates the critical temperature for stale urning equation 7). Equation 8) for the urning temperature of the layer, which comes from an analytic integration of the temperature profile in the layer, is quite accurate. For example, at the low flux staility oundary ṁ = 3.5 ṁ Edd, equation 8) predicts T 8 = 3.6 which agrees well with the urning temperature see Fig. 3). To otain an analytic fit to the MESA results, we rescaled equation 9) y adjusting the prefactor and making a small adjustment to the numerical constant inside the final term to improve the fit at intermediate values of Q.Thefinalresult,showninFig.2as a lue dotted curve, is Q ) 3/4 ṁ crit = 4.6 ṁ Edd MeV 1 + Q ) 5, 10) 0.9MeV which reproduces the MESA results to within 18 per cent for Q 1MeV. 3 THE RELATION BETWEEN THE STEADY-STATE BURNING DEPTH AND STABILITY In this section, we investigate the relation etween the urning depth in steady-state models and thermal staility. Paczynski 1983a) pointed out that in one-zone models, the urning depth y urn decreases with ṁ for unstale models dy urn /dṁ <0), ut increases with ṁ in stale models dy urn /dṁ >0). The staility oundary is therefore at the turning point dy urn /dṁ = 0. Narayan & Heyl 2003) argued that the same criterion should apply to multizone models also. If this result is generally true, it would e a very powerful way to determine the staility oundary without doing any time-dependent calculations, and large grids of steady-state models already exist as functions of ṁ, Q and accreted composition He fraction; Stevens et al. 2014). We first discuss the one-zone case in Section 3.1, extending the arguments of Paczynski 1983a) to the case with Q > 0. We then consider multizone models in Section 3.2. We show that in oth cases dy urn /dṁ is non-zero at marginal staility, and so dy urn /dṁ = 0 can e used to locate the marginally stale point only for one-zone models with Q = The turning point and staility of one-zone models First consider the case studied y Paczynski 1983a), a sequence of one-zone models with increasing ṁ, andq = 0. From equations 1) and 2), these models must oey ɛ 3α = ɛ cool, 11) ṁ = ɛ 3α y, 12) E 3α in steady state. At the accretion rate where dy urn /dṁ = 0, two neighouring steady-state models which differ in accretion rate y an amount ṁ have the same urning depth, so y urn = 0. Equation 12) then gives ṁ = y/e 3α ) ɛ 3α. Since the column depth remains unchanged, the difference in accretion rates etween the two models is accommodated y a change in the urning rate driven y a temperature difference at constant pressure column depth), ɛ 3α = ν η)ɛ 3α T/T. The temperature difference etween the two models also implies a difference in cooling rates ɛ cool = 4 κ T )ɛ cool T/T, and so setting ɛ 3α = ɛ cool as must e the case for two steady-state models, we arrive at ν η) = 4 κ T ), 13) exactly the criterion for marginal staility see equation 4). Therefore, we have shown that the steady-state model with dy urn /dṁ = 0 is marginally stale. When a ase flux is included, equation 11) ecomes ɛ 3α + Q ṁ = ɛ cool. 14) y Two neighouring models at dy urn /dṁ = 0 are still related y ṁ = y/e 3α ) ɛ 3α ecause equation 12) has not changed, ut from equation 14), they must now satisfy ɛ 3α + Q y ṁ = ɛ 3α 1 + Q ) = ɛ cool 15) E 3α or ɛ 3α 1 + Q ) ν η) ɛ cool 4 κ T ) = 0. 16) E 3α But the steady-state model oeys ɛ 3α 1 + Q /E 3α ) = ɛ cool equation 6), giving again ν η) = 4 κ T ) 17) at the accretion rate where dy urn /dṁ = 0. But for Q > 0this is no longer the condition for marginal staility see equation 5). Therefore, the turning point for y urn no longer specifies the staility oundary when Q > 0. It is curious that at the turning point for any value of Q,the criterion for marginal staility at Q = 0 equation 17) is satisfied. This means that models where dy urn /dṁ = 0 for any value of Q will have the same urning temperature T 8 = 4.9 at which equation 17) is satisfied. 3.2 The turning point and staility of multizone models To locate the turning point dy urn /dṁ = 0 in the multizone case, we constructed a set of steady-state models of the He urning layer as a function of Q and ṁ. We solve for the temperature T, Hemass fraction Y,andfluxFas a function of column depth y y integrating see Brown & Bildsten 1998) dt dy = 3κF 18) 4acT 3 df dy = ɛ 3α + ṁc PT ad ) 19) y and dy dy = 12ɛ 3α, 20) ṁq 3α where Q 3α = MeV is the energy release from one triple-α reaction. We assume that He urns to caron only, with the triple-α generation rate ɛ 3α, opacity κ and equation of state calculated in the same way as in Section 2.2. The oundary conditions are Y = 1at the top of the layer, and F = Q ṁ at the ase. The flux at the top has contriutions from Q, the nuclear urning, and the compressional

6 The thermal staility of helium urning 3283 Figure 4. Contour lines for constant values of urning column depth y urn ; lue lines), that is, the depth at which the triple-α urning rate peaks, and temperature at the urning depth T urn ; red lines). The numered laels on the contours show the ase-10 logarithm of the respective quantities. Note that the log 10 T urn = 8.7 contour line appears to very nearly pass through the stationary points of the y urn contour curves, that is, where dy urn /dṁ = 0. This is addressed in Section 3.1. heating, descried y the term involving ad on the right-hand side of equation 19). Since the compressional heating depends on the temperature profile, it is necessary to iterate the solution until the assumed compressional heating is self-consistent. For our steady-state models, we set the lower oundary at y = gcm 2. This is deep enough that He urning is complete at the ase. As Fig. 4 shows, the He urning depth is typically 10 8 gcm 2, ut can reach gcm 2 at ṁ 0.01 ṁ Edd and Q 0.1 MeV per nucleon. Our inner oundary also lies aove the depth where caron is likely to urn Brown & Bildsten 1998). Furthermore, the temperature profile is expected to turn over at some point, with heat eing transported into the crust and core. We stop our integrations at a depth shallower than oth the temperature turn over point, and the caron ignition depth. The value of Q should e interpreted as the outward flux evaluated at the lower oundary depth, y = gcm 2. The compressional heating gives some sensitivity to the choice of the location of the lower oundary. Beneath the He urning depth, the layer is close to isothermal, is much smaller than ad,and c P T ad is roughly constant allowing an estimate of the contriution to the flux from compressional heating, dq comp dlog 10 y = MeV nucleon T 9 c p erg g 1 K 1 ) ad 0.4 ), 21) where we take a typical value of c P from our numerical models. Every additional decade in column depth included elow the He urning depth contriutes an extra MeV per nucleon. This means that models with small Q 0.1 MeV per nucleon actually have a flux heating the He urning layer that is sustantially larger than Q. In other words, compressional heating in the ocean sets an effective lower limit on the ase heating of the He urning layer. The contriutions to the total compressional heat flux are roughly evenly divided etween depths elow and aove the He urning depth. From equation 21), we estimate the contriution from elow to e 0.04 MeV nucleon 1 for a typical urning depth and our choice of lower oundary, which gives a total Q comp = 0.08 MeV nucleon 1. Fig. 4 shows contours of the urning depth and temperature. We define the urning depth y urn as the location where the 3α urning rate is maximal. The urning temperature T urn is defined as the temperature at the depth y urn. The staility oundary as calculated in Section 2.1 is shown. Clearly, dy urn /dṁ <0 along the staility oundary. Interestingly, the locus of points where dy urn /dṁ = 0 where the lue contours turn over) follows closely the temperature contour where log 10 T = 8.7 or T 8 = 5, as the arguments from the one-zone model indicated. We conclude that the correspondence etween dy urn /dṁ = 0 and marginal staility does not carry over into multizone models for any value of Q. 4 LINEAR STABILITY ANALYSIS In this section, we carry out a linear staility analysis of the steadystate models descried in Section 3.2. A similar technique was used y Narayan & Heyl 2003), although applied to artificially truncated steady-state models in an attempt to calculate ignition conditions in the unstale regime. Here, we are interested in locating the staility oundary and so pertur full steady-state models that urn to completion. This technique should reproduce the staility oundary, since we will identify those values of ṁ and Q where the steady-state model is unstale. We first derive the perturation equations and oundary conditions in Section 4.1, and present the results in Section Perturation equations For the perturation analysis, we use pressure or equivalently column depth as the independent coordinate pressure and column depth are related y P = gy in a thin layer, where g is the constant gravity). At each pressure P, wesett T + δt and F F + δf, where the perturations have a time-dependence e γ t. With the choice of pressure coordinates, we are adopting Lagrangian perturations. In the appendix, we derive the perturation equations from an Eulerian approach, in which vertical displacements are followed explicitly. We assume that on the time-scale of the thermal perturation, the composition does not change δy = 0, since only a small amount of He need urn for a large change in temperature see equations [1] and [2]). Putting the time-dependent term c P T/ t ack into equation 19) and perturing, we find d dy δf = c P γ ɛ ) 3αɛ T δt, 22) T where ɛ T = ln ɛ 3α / ln T P and we have neglected the compressional heating term. The radiative diffusion equation 18) gives [ ] ) d dt δf δt = dy dy F + κt 3 δt, 23) T where κ T = ln κ/ ln T P. For a given steady-state model Ty), Yy), Fy)) otained y integrating equations 18) 20), the perturation equations 22) and 23) form an eigenvalue prolem for γ, i.e. the perturation equations and their oundary conditions will e satisfied only for particular choices of the growth or decay) rate γ.

7 3284 M. Zamfir, A. Cumming and C. Niquette At the top of the layer, the oundary condition comes from perturing a radiative zero solution F = act 4 /3κy) for the outer layers, giving δf F = 4 κ T ) δt T, 24) where the choice of δt/t at the top is aritrary and sets the overall normalization. At the ase of the layer, the usual approach would e to set δt = 0, which is appropriate when the thermal time-scale at the ase is much longer than the growth rate of the mode, γ 1. At marginal staility, however, the growth time-scale ecomes very long and exceeds the thermal time-scale at the ase, in which case it is not clear how to set the lower oundary condition. In fact, we find that the results do not depend sensitively on the choice of lower oundary condition. For the results shown, we fix the flux at the ase δf = 0. We find that changing the oundary condition from δf = 0toδT = 0 at the ase lowers ṁ crit y <10 per cent for Q < 0.5 MeV per nucleon. The differences in ṁ crit ecome larger, roughly a factor of 2, for Q > 1 MeV per nucleon. 4.2 Results At any given accretion rate and ase flux, there are many stale γ <0) eigenmode solutions and at most one unstale mode. The unstale mode, if present, transitions to staility at a specific accretion rate this defines the staility oundary. As an example, Fig. 5 shows the values of γ as a function of ṁ for the first six eigenmodes, for a ase flux Q = 0.2 MeV per nucleon. The lowest order mode has γ > 0 unstale) for ṁ 4 ṁ Edd and γ < 0 stale) for ṁ 4 ṁ Edd. Fig. 6 shows the eigenmodes at ṁ = 3 ṁ Edd, just elow the staility oundary in the unstale region, again for Q = 0.2 MeV per nucleon. The unstale mode has a single peak in δt at the triple-α urning depth, since this is the location at which the thermal runaway occurs during the onset of a urst. The stale cooling) modes show oscillations, with an increasing numer of nodes associated with decreasing larger negative) values of γ. The cooling eigenmodes with the most negative γ decay most quickly, due to the e γ t time dependence of the perturations. Figure 5. The change in eigenvalue γ with accretion rate for the first few eigenmodes using a Q ase = 0.2 MeV. The first eigenmode lack line) transitions from unstale γ >0) to stale as the accretion rate increases past 4ṁ Edd. The other eigenmodes are stale across all accretion rates. A horizontal dotted lack line is used to highlight the location of the transition, γ = 0. Figure 6. Example of the first few pertured temperature top panel) and flux ottom panel) eigenmodes for ṁ = 3ṁ Edd and Q ase = 0.2 MeVper nucleon. The eigenmodes displayed have a correspondence with those shown in Fig. 5 at ṁ = 3ṁ Edd, sharing the same line styles and colours. The urning depth, that is, the location at which the triple-α urning rate is a maximum in the steady-state models, is represented y the vertical dotted line at roughly z = 5 m. A horizontal dashed lack line is used to highlight the location of δt/t = 0 in the top panel, and δf/f = 0 in the ottom panel. The normalization is chosen so that δt/t = 0.1 at the top of the layer. In Narayan & Heyl 2003), a slightly different instaility criterion was used, namely that the unstale mode growth time-scale 1/γ e shorter than three times the accretion time-scale y ign /ṁ. The values of ṁ crit we found with this criterion were not sustantially different from those found using γ>0. For example, in the case of Q = 0.2 MeV per nucleon, the Narayan & Heyl 2003) instaility criterion here, roughly γ > 10 3 s 1 ) leads to a 10 per cent decrease in the value of ṁ crit.asisshowninfig.5, this modest decrease can e understood from the sudden steep rise in γ with decreasing ṁ for the dominant mode solid lack curve), elow ṁ/ṁ Edd 4. Rather than searching for γ = 0 y varying ṁ, we locate ṁ crit at each Q y setting γ = 0 and then treating ṁ as the eigenvalue. The resulting staility oundary is shown in the top panel of Fig. 7, as the solid green curve. We have also included staility curves for the same calculation ut with different lower oundaries, y ase = and gcm 2. This illustrates the effect of compressional heating: a deeper layer has additional compressional heating, increasing the flux heating the He layer and stailizing the urning, moving ṁ crit to lower values. As can e seen, the effect is not large, with an 20 per cent change in ṁ crit over the factor of 100 change in y ase. To correct for the effect of compressional heating on the staility oundary, in the lower panel of Fig. 7 we show ṁ crit against the sum of Q and Q comp, which gives the total flux heating the He urning layer, plus a contriution to Q comp coming from depths shallower than He urning. In other words, Q + Q comp represents the total non-nuclear heating flux which emerges at the top of the accreted layer. The linear staility curves now lie on top of one another for all choices of y ase. This plot also emphasizes that compressional heating in the ocean sets a minimum value for the effective ase flux of 0.1 MeV nucleon 1, similar to the estimate of the total compressional heating that we found in Section 3.2. Recall that in arriving at equation 22), we did not include perturations of the compressional heating terms from equation 19). We checked the effect of including these terms on the staility oundary,

8 The thermal staility of helium urning 3285 Figure 7. a) The staility oundary for He urning as found using linear staility analysis 4) is shown as a green solid curve. In addition, the staility curves for the same calculation are shown, using different lower oundary depths, namely y ase = gcm 2 red dashed), y ase = gcm 2 lue dotted).the MESA staility oundary is represented y lack crosses. ) The same staility oundaries shown in a), this time plotted against Q + Q comp, the sum of Q and the total contriution to compressional heating across the layer. The linear staility analysis curves with different ase depths now overlap each other. and found only a small 6 7 per cent increase in the value of the critical accretion rate. In addition, we also evaluated the effect of changing the surface gravity. A surface gravity of g 14 = 1.0 cm s 2 yielded an 30 per cent decrease in the value of the critical accretion rate, while g 14 = 3.0 cm s 2 yielded an 25 per cent increase. scaling found y Bildsten 1998). Fig. 7 shows that the ṁ crit calculated with linear staility analysis is a factor of 3 greater than the ṁ crit determined from the timedependent MESA simulations. The reason for this discrepancy is not clear. We have compared our steady-state models with MESA for values of ṁ and Q at which MESA achieves a steady solution, while our linear staility analysis predicts instaility, and find excellent agreement see Fig. 8). There are small differences in the opacity profile and κ T lower panel of Fig. 8), ut these differences make only a These results agree very well with the ṁ crit g 1/2 14 Figure 8. A comparison of steady urning model calculations using MESA lue) and the model presented in Section 3.2, with ṁ = 4ṁ Edd and Q = 0.1 MeV per nucleon. At this accretion rate and ase flux, the MESA simulation indicates that this model is stale, while our linear staility analysis Section 4.1) indicates that the model is unstale. Profiles for the temperature, flux, opacity, and opacity and urning rate derivatives κ T, ɛ T oth taken at constant pressure) are shown. small change in the growth rate. For example, we calculated the linear growth rate for a model with ṁ = 4ṁ Edd and Q = 0.1 MeV per nucleon using the κ T profile from MESA, and compared it to the growth rate found using the κ T profile from our steady-state models, ut found only a small difference, γ = s 1 compared to γ = s 1. Therefore, the difference in κ T profile is not the reason that the model is stale in MESA ut unstale according to the linear staility analysis. Lastly, while the profiles for ɛ T diverge dramatically at depths y 10 5 gcm 2, this parameter is irrelevant at these depths since the He urning rate is negligile. Another possile reason for the difference could e that we have not allowed changes in composition in our linear staility analysis, setting δy = 0.

9 3286 M. Zamfir, A. Cumming and C. Niquette However, we do not expect these extra terms to significantly change the results, since the thermal time-scale is shorter than the time-scale to change composition y a factor Q 3α /c P T 10 for T SUMMARY AND DISCUSSION The main result of the paper is a new calculation of the critical accretion rate ṁ crit at which He urning stailizes on accreting neutron stars. We used the MESA stellar evolution code to calculate ṁ crit as a function of the ase flux heating the He layer, written in terms of the energy per nucleon Q F = ṁq ). Equation 10) gives an analytic expression for ṁ crit Q ) in units of the local Eddington rate ṁ Edd = gcm 2 s 1, which should e useful for applications. In agreement with Keek et al. 2009), we find that the critical accretion rate at low fluxes, ṁ crit 4 ṁ Edd is sustantially smaller than the rate ṁ crit 20 ṁ Edd predicted y one-zone models Bildsten 1995, 1998). The difference arises ecause the one-zone instaility criterion equation 5) overestimates the urning temperature at marginal staility, which is close to Kin multizone models ut predicted to e K in a one-zone model. We also investigated whether the critical accretion rate can e determined y examining steady-state models only, without running a time-dependent simulation. Paczynski 1983a) showed that a one-zone model with Q = 0 has a turning point in the urning depth dy/dṁ = 0 at marginal staility. We find that this result does not hold in one-zone models when a ase flux is included, and does not hold for any value of Q in multizone models. This is contrary to the findings of Narayan & Heyl 2003), who studied multizone models with fixed temperatures as a lower oundary. We then carried out a linear staility analysis of steady-state urning models to determine the staility oundary. Linear staility analysis has een applied to nuclear urning on neutron stars efore e.g. Narayan & Heyl 2003), ut not compared directly to time-dependent simulations. Although the shape of the ṁ crit Q ) curve is reproduced quite well Fig. 7), the linear staility analysis overestimates ṁ crit y a factor of aout 3. We were not ale to identify the reason for the discrepancy; for now we must take the results of linear staility analysis as approximate. Narayan & Heyl 2003) assumed a solar composition, and so cannot e compared with our results. Heger et al. 2007) discussed a further prediction of theoretical models, that close to marginal staility, the eigenvalue of thermal perturations ecomes complex Paczynski 1983a), leading to an oscillatory mode of urning which has een identified with mhz frequency QPOs oserved from three X-ray inaries Revnivtsev et al. 2001; Altamirano et al. 2008; Linares et al. 2012). By considering only thermal perturations in this paper, we have confined our attention to the real part of the eigenvalue, neglecting the compositional perturations that are important in marginally stale urning. This is a straightforward extension of the method presented here, and remains to e addressed in a future paper. It would e interesting to apply the linear staility analysis to steady-state models of solar composition, which include H urning y the rp-process. A large grid of models were recently pulished as a function of Q and He fraction Y Stevens et al. 2014). Heger et al. 2007a) found the staility oundary ṁ crit = ṁ Edd for Q = 0.15 MeV per nucleon in simulations with the KEPLER code. Keek et al. 2014) extend these calculations to investigate the sensitivity of ṁ crit to nuclear reaction uncertainties. For their standard set of rates, they have ṁ crit 1.1 ṁ Edd. Bildsten 1998) estimated ṁ crit = 0.74 ṁ Edd for solar composition using the one-zone ignition criterion. In that case, the one-zone estimate appears to give a much more accurate estimate than for pure He. The transition to stale urning is elieved to explain the oserved quenching of type I X-ray ursts following a superurst Cumming & Bildsten 2001; Kuulkers et al. 2002; Cumming & Maceth 2004; Keek, Heger & in t Zand 2012). Cumming & Maceth 2004) assumed that the critical flux that would quench urning is Q 0.7 MeV per nucleon, independent of accretion rate. In fact, as we showed in this paper, we expect the Q required to stailize urning to depend strongly on ṁ. Superurst sources are not pure He accretors in general, ut we can compare our results with Keek et al. 2012), who ran time-dependent simulations of superursts and studied quenching for the pure He case. They found that urning ecame unstale as the luminosity dropped through L erg s 1 for accretion at 0.3 ṁ Edd. Sutracting the nuclear urning flux, this is in good agreement with Fig. 7 which predicts a critical flux of Q 0.5 MeV per nucleon for this accretion rate. The fact that this is close to the value assumed y Cumming & Maceth 2004) suggests that their results may not e strongly affected y their assumption that Q is independent of ṁ. Our results can e immediately applied to 4U , an ultracompact inary that most likely accretes pure He. It displays regular type I X-ray ursts in its low state, which disappear when the accretion rate increases and the source enters the soft state Clark et al. 1977; Cornelisse et al. 2003). Cumming 2003) found that at the local rate of ṁ X = gcm 2 s 1 = 0.14 ṁ Edd as inferred from the X-ray luminosity of the source when ursts are seen), a flux from elow of Q = 0.4 MeV per nucleon was necessary to explain the short 3 h urst recurrence times. For this value of Q,we find that urning will stailize aove ṁ = 0.35 ṁ Edd using equation 10). This can e accommodated in the range of accretion rates oserved in the six month cycle of 4U , which is aout a factor of 3. Therefore, it may e possile to make a consistent model of the urst recurrence time and the quenching of ursts at higher accretion rates y including a ase flux of the appropriate size that is always present. An alternative is that the flux switches on at a critical rate, quenching the urning, ut this would have difficulty explaining the short recurrence times when ursts are seen. Time-dependent simulations, e.g. with the MESA code, are required to test whether a self-consistent model of the ursting ehaviour of can e made. One issue for explaining the transition to stale urning is the time-scale on which ursts appear or disappear as the accretion rate changes. in t Zand et al. 2012) noted that the urst ehaviour in 4U changes within a day or two of entering or leaving the low state. They suggest that this implies that the shallow heat source must lie at a depth where the thermal time is 1 d, corresponding to a density of ρ 10 9 gcm 2, so that it can adjust to the changing accretion rate. Otherwise, for example, when the accretion rate dropped into the low state, the luminosity from the crust would remain as it was in the high state, not having time to thermally adjust, and X-ray ursts would remain quenched. Instead, we want the luminosity to adjust to a new value of Q ṁ so that ursting activity can resume. The fact that the staility oundaries for pure He and solar composition are closer than previously thought ased on one-zone models, in which they are more than an order of magnitude different) may help to explain why urning stailizes in 4U at a

10 similar accretion rate to other low mass X-ray inary neutron stars that accrete H rich material. ACKNOWLEDGEMENTS We thank L. Keek and G. Ushomirsky for useful discussions. We are grateful for support from the National Sciences and Engineering Research Council NSERC) of Canada. AC is an Associate Memer of the CIFAR Cosmology and Gravity program. MZ and AC are memers of an International Team in Space Science on thermonuclear ursts sponsored y the International Space Science Institute in Bern, Switzerland. REFERENCES Altamirano D., van der Klis M., Wijnands R., Cumming A., 2008, ApJ, 673, L35 Altamirano D. et al., 2012, MNRAS, 426, 927 Bildsten L., 1995, ApJ, 438, 852 Bildsten L., 1998, in Buccheri R., van Paradijs J., Alpar A., eds, NATO ASIC Proc. 515: The Many Faces of Neutron Stars. Kluwer, Dordrecht, p. 419 Bildsten L., 2000, in Holt S. S., Zhang W. W., eds, AIP Conf. Proc. Vol. 522, Cosmic Explosions: Tenth Astrophysics Conference. Am. Inst. Phys., New York, p. 359 Brown E. 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E., 1981, ApJS, 45, 389 Woosley S. E. et al., 2004, ApJS, 151, 75 Yoon S.-C., Langer N., van der Sluys M., 2004, A&A, 425, 207 APPENDIX A: EULERIAN PERTURBATIONS In Section 4.1, we derived the perturation equations using pressure coordinates, a Lagrangian approach. Here, we instead use an Eulerian approach, where perturations are taken at fixed spatial position, and show that the perturation equations reduce to those derived in Section 4.1 when written in terms of Lagrangian quantities. We follow the convention of Cox 1980) y denoting Eulerian perturations using the prime symol. For example, T represents the Eulerian temperature perturation. The Lagrangian temperature perturation is then δt = T + ξ z T/ z, whereξ z is the vertical displacement. The displacement oeys the continuity equation d dy ξ z = δρ ρ = χ T δt 2 ρχ ρ T, A1) wherewehavesetδp = 0. Perturing equation 18) using Eulerian perturations gives 1 T = 1 [ T F ρ z ρ z F 3 T ] T + κ κ + ρ. A2) ρ Now to rewrite this in terms of Langrangian perturations. The gradient of the Lagrangian temperature perturation is δt = T z z + ) T ξ z = T z z z T δρ z ρ + ξ 2 T z z, 2 A3) where we used the continuity equation dξ z /dz = δρ/ρ to sustitute for ξ z. Comining this with equation A2) gives δt = T [ δf z z F 3 δt T + δκ ] κ [ )] T Fκρ ξ z ln. A4) z z T 3 T z The last term in equation A3) vanishes since the expression inside the logarithm is a constant, giving δt = dt [ δf y dy F + κt 3 T ) δt We have recovered equation 23) from Section 4.1. ]. A5)