Super-Eddington Winds from Type I X-Ray Bursts

In Figure 1 we show the evolution of the thermal profile of model y3n21 during the burst rise.² As the base temperature T_b rises due to He burning, a convective zone forms and begins to extend outward to lower pressure (smaller y) on a timescale of ∼1 ms (for y1n21 it is about 50 times longer). Initially, T_b rises so quickly that there is not enough time for the radiative layer above the convective zone to thermally adjust. As a result, the thermal profile in the radiative region is unchanged from the pre-ignition profile (see also WBS, Paxton et al. 2011). This can be seen in Figure 1 at times t < −28.9 ms, where t = 0 corresponds to when the wind turns on.

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Over most of the convective zone, the convection is highly subsonic and efficient and the temperature profile very nearly follows an adiabat $T\propto {y}^{n}$, with $n\simeq 2/5$ (i.e., close to the adiabatic index of an ideal gas). In the overlying radiative region, the temperature profile is shallower and since the opacity varies only slightly with column depth, $T\propto {y}^{1/4}$.

Eventually, the top of the convective zone reaches low enough y that the local thermal timescale of the overlying radiative layer equals the heating timescale at the base. The radiative flux can then diffuse outward through the radiative region without being overtaken by the growing convective region. This flux begins to heat the radiative region and the convective zone gradually retreats downward to larger y (Figure 1 at t > −28.9 ms). As the radiative region heats up, the radiative luminosity ${L}_{\mathrm{rad}}$ in the shallower layers begins to approach the local Eddington limit

$$\begin{eqnarray}&&{L}_{\mathrm{Edd}}(T)=\displaystyle \frac{4\pi {cGM}}{\kappa (T)}.\end{eqnarray} \tag{ 1 }$$

The opacity $\kappa (T)$ is dominated by electron scattering and is temperature-dependent due to Klein–Nishina (i.e., special relativistic) corrections. It varies approximately as (Paczynski 1983; MESA uses a more exact form)

$$\begin{eqnarray}&&\kappa (T)={\kappa }_{0}{\left[1+{\left(\displaystyle \frac{T}{0.45\mathrm{GK}}\right)}^{0.86}\right]}^{-1},\end{eqnarray} \tag{ 2 }$$

where κ₀ ≃ 0.2(1+X_H) cm² g⁻¹ and X_H is the hydrogen mass fraction. Since κ is larger for smaller T, smaller y have smaller L_Edd. As a result, a given luminosity L can be sub-Eddington in the deep hotter layers, but becomes super-Eddington as the radiation diffuses upward into the shallow cooler layers (Ebisuzaki et al. 1983). Indeed, as we will show in the hydrodynamic simulations (see Section 3.1 and Figure 5), the base of the wind is initially at small y, moves to larger y as the deeper layers heat up, and finally moves back outward to smaller y as the layers begin to cool.

We run the hydrostatic simulations until the moment the luminosity first exceeds the local Eddington limit (defined as t = 0). As can be seen in Figure 1, at t = 0 the convective zone has retreated and the atmosphere is almost fully radiative.

As the convective zone extends outward, it efficiently mixes the ashes of burning up to lower column depths. The minimum column depth ${y}_{c,\min }$ reached by the convective zone, and hence reached by the ashes of burning, is shown by the solid squares in Figure 2 for five of the burst models. For model y3n21, which has the deepest ignition and thus the largest energy release, ${y}_{c,\min }\,\approx \,{10}^{3}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$, while for the other models, ${y}_{c,\min }\,\approx \,{10}^{4}\mbox{--}{10}^{5}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$ (consistent with WBS). We also see that a more complete reaction network (n21 compared to n9) results in slightly smaller ${y}_{c,\min }$ due to the increased energy release; comparing the y2 and y3 models, we find that this difference becomes more significant at larger ignition depths.

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In Section 3 we describe the time-dependent wind and show that the column depth of the wind's base ${y}_{\mathrm{wb}}\gg {y}_{c,\min }$. As a result, ashes are ejected by the wind and exposed. In Figure 3 we show the composition profiles of models y1n21, y2n21, and y3n21 at the end of the hydrostatic phase, just before the wind is launched. The dashed vertical lines indicate y_wb. At a given y, the composition is determined by the burning stage at the moment ${y}_{c}(t)=y$, where y_c(t) is the location of the top of the retreating convective zone. For y1n21 (${y}_{{\rm{b}}}=5\times {10}^{8}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$), we see that the wind will be dominated by light elements, primarily ⁴He with a small amount of ¹²C. However, for y2n21 (${y}_{{\rm{b}}}=1.5\times {10}^{9}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$), the wind will be dominated by heavy elements such as ⁴⁸Cr and ⁵²Fe, while for models ignited at even deeper depths (y3n21; ${y}_{{\rm{b}}}=5\times {10}^{9}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$) the wind will be primarily ⁵⁶Ni.³

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In Figures 4 and 5 we show the temperature and luminosity as a function of both density and column depth at four different times during the hydrodynamic wind phase of model y2n21. The location where ${L}_{\mathrm{rad}}$ first exceeds L_Edd corresponds to the wind base. Note that L never exceeds L_Edd by more than a few percent, since any excess luminosity is used to expel matter to infinity (Ebisuzaki et al. 1983; Kato 1983; Paczynski & Proszynski 1986). At early times ($t=0.07\,{\rm{s}}$), the column depth of the wind base ${y}_{\mathrm{wb}}\approx {10}^{4}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$. As the wind evolves during the next ≈10 s, the location where ${L}_{\mathrm{rad}}\gt {L}_{\mathrm{Edd}}$ moves to larger y and ρ and thus higher T, eventually reaching as far down as ${y}_{\mathrm{wb}}\simeq {10}^{6}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$. By $t=29\,{\rm{s}}$, the NS surface layers have cooled, y_wb has moved back to shallower depths, and the wind dies down. As we explain in Section 3.2, the profiles at depths greater than that of the wind base approximately follow power-law relations $T\propto {\rho }^{1/3}\propto {y}^{1/4}$. The $T\propto {\rho }^{1/3}$ relation also holds in regions sufficiently above the wind base.

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In Figure 6 we show the wind structure of model y2n21 in more detail. We plot profiles as a function of r rather than y or ρ in order to more clearly reveal the structure of the tenuous outer regions of the wind out to $r\sim {10}^{3}\,\mathrm{km}$. The open circles indicate the location of the photosphere r_ph. We see that there is a large radius expansion, with the photosphere reaching a maximum of ${r}_{\mathrm{ph}}\simeq 200\,\mathrm{km}$ (see also Figure 12). The pluses indicate the location of the isothermal sonic point, defined as the radius where the velocity satisfies ${v}^{2}={kT}/\mu {m}_{p}$, where μ is the mean molecular weight and m_p is the proton mass (see, e.g., Quinn & Paczynski 1985; Joss & Melia 1987); the equilibrium sonic point where ${v}^{2}={dP}/d\rho$ occurs at $\tau \lt 1$ and is thus beyond our simulated region.

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As the wind gains strength during the first 10 seconds, the mass-loss rate ${\dot{M}}_{{\rm{w}}}$, temperature, density, and optical depth all increase throughout the wind. The velocity, which never exceeds $\sim 0.01c$, decreases during this time, since ${\dot{M}}_{{\rm{w}}}\simeq 4\pi {r}^{2}\rho v$ only changes by order unity whereas ρ increases significantly. At $t\simeq 10\,{\rm{s}}$ the wind settles into a steady state and for the next ≈15 s the profiles change very little. For model y2n21, ${\dot{M}}_{{\rm{w}}}\simeq 2\times {10}^{18}\,{\rm{g}}\,{{\rm{s}}}^{-1}$ at its maximum. By $t=29\,{\rm{s}}$, the energy and mass supply have dwindled and ${\dot{M}}_{{\rm{w}}}$ decreases. As a result, the temperatures and densities drop and the photosphere begins to fall back to the NS surface.

We find that aside from differences in duration, the wind profiles of our other burst models are all similar to that of model y2n21 despite the significant range of ignition depth. This is because the wind structure is largely determined by ${\dot{M}}_{{\rm{w}}}$ (Kato 1983; Quinn & Paczynski 1985; Paczynski & Proszynski 1986), and the different models all have very similar ${\dot{M}}_{{\rm{w}}}(t)$ up until the wind terminates. We illustrate this in the bottom panel of Figure 7 for four of the models. In Section 3.3, we explain why ${\dot{M}}_{{\rm{w}}}$ is a weak function of y_b.

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In the top panel of Figure 8 we show the ratio of the total mass ejected by the wind ${M}_{\mathrm{ej}}$ at the end of the PRE to the accreted mass ${M}_{\mathrm{accr}}\simeq 4\pi {R}^{2}{y}_{b}$. We find

$$\begin{eqnarray}&&\eta \equiv \displaystyle \frac{{M}_{\mathrm{ej}}}{{M}_{\mathrm{accr}}}\simeq 2.5\times {10}^{-3}\end{eqnarray} \tag{ 3 }$$

almost independent of y_b. Burst energetics set an upper bound of η ≲ 8 × 10⁻³, which is given by the ratio of the nuclear energy release per nucleon $\simeq 1.6\,\mathrm{MeV}\,{\mathrm{nucleon}}^{-1}$ (i.e., the difference in binding energy between ⁴He and ⁵⁶Ni) to the gravitational binding energy per nucleon ${GM}/R\simeq 200\,\mathrm{MeV}\,{\mathrm{nucleon}}^{-1}$. The value $\eta \simeq 2.5\times {10}^{-3}$ implies that ≈30% of the nuclear energy goes to unbinding matter from the NS, independent of ignition depth.

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Since the flow is subsonic at radii smaller than the equilibrium sonic point (which is located at an optical depth $\tau \lt 1$), the structure throughout the modeled region is nearly in hydrostatic equilibrium at each instant. Therefore, the evolution approximately follows a sequence of steady-state solutions (i.e., quasi-static profiles) determined by the instantaneous ${\dot{M}}_{{\rm{w}}}(t)$. Indeed, our profiles are qualitatively similar to those of steady-state wind models in which ${\dot{M}}_{{\rm{w}}}$ is treated as a free parameter (Ebisuzaki et al. 1983; Kato 1983; Paczynski & Proszynski 1986; Joss & Melia 1987). In steady-state, the time-dependent terms vanish and the mass, momentum, and energy equations are

$$\begin{eqnarray}&&{\dot{M}}_{{\rm{w}}}=4\pi {r}^{2}\rho v=\mathrm{constant},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&v\displaystyle \frac{{dv}}{{dr}}=-\displaystyle \frac{1}{\rho }\displaystyle \frac{{dP}}{{dr}}-g,\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\dot{E}}_{{\rm{w}}}={\dot{M}}_{{\rm{w}}}\left(\displaystyle \frac{{v}^{2}}{2}-\displaystyle \frac{{GM}}{r}+h\right)+{L}_{\mathrm{rad}}=\mathrm{constant},\end{eqnarray} \tag{ 6 }$$

where ${\dot{E}}_{{\rm{w}}}$ is the energy-loss rate of the wind, $h=(U+P)/\rho$ is the enthalpy, and U is the energy density. Over a large region between the wind base and the equilibrium sonic point, we find that ${dP}/{dr}\simeq -\rho g$, radiation pressure dominates so that P ∝ T⁴ and $h\simeq 4P/\rho$, and ${L}_{\mathrm{rad}}(r\gg R)\simeq {L}_{\mathrm{Edd},0}\simeq {\dot{E}}_{{\rm{w}}}$, where ${L}_{\mathrm{Edd},0}=4\pi {cGM}/{\kappa }_{0}$. Together these imply that over this region $P\propto {\rho }^{4/3}$ and the fluid behaves as if it has an adiabatic index γ = 4/3, as also noted by Kato (1986).⁴ As a result, $\rho \propto {r}^{-3}$, T ∝ r⁻¹, and $v\propto r$, as shown in Figures 4 and 6 (see also Figure 5 in Paczynski & Proszynski 1986).

Given that $\rho \propto {r}^{-3}$, the optical depth $\tau \simeq {\tau }^{* }/2$, where ${\tau }^{* }=\kappa \rho r$ is the effective optical depth used in the steady-state wind calculations of Quinn & Paczynski (1985) and Paczynski & Proszynski (1986). In Figure 7 we show that at the photosphere $\tau \simeq 3.5$, nearly independent of time and ignition model, which is comparable to the values found by Quinn & Paczynski (1985) and Paczynski & Proszynski (1986).

Since ${L}_{\mathrm{rad}}$ is much larger than the kinetic power, X-ray burst winds are in the opposite regime from massive star winds, which Quataert et al. (2016) studied. Their analytic steady-state model is therefore not directly applicable here. More recently, Owocki et al. (2017) derived semi-analytic steady-state wind solutions that bridge the two regimes. Although we have not attempted to implement their solutions, they should be applicable to the steady-state regime of PRE bursts.

The duration of the wind increases with ignition depth y_b, but its structure is nearly independent of y_b. This is because ${\dot{M}}_{{\rm{w}}}(t)$, which effectively sets the wind structure (see, e.g., Kato 1983; Paczynski & Proszynski 1986), depends only weakly on y_b. We can understand this weak dependence by appealing to energy conservation, assuming a steady-state wind. Just above the base of the wind (${r}_{\mathrm{wb}}\simeq R$), the enthalpy and kinetic energy are small compared to the binding energy, and by Equation (6)

$$\begin{eqnarray}&&{\dot{E}}_{{\rm{w}}}\simeq -\displaystyle \frac{{GM}{\dot{M}}_{{\rm{w}}}}{R}+{L}_{\mathrm{rad}}({r}_{\mathrm{wb}}),\end{eqnarray} \tag{ 7 }$$

where ${L}_{\mathrm{rad}}({r}_{\mathrm{wb}})\simeq 4\pi {cGM}/{\kappa }_{\mathrm{wb}}$, i.e., the Eddington luminosity at the wind base, with ${\kappa }_{\mathrm{wb}}=\kappa [T({r}_{\mathrm{wb}})]$. At $r\gg R$, the flow of mechanical energy is small compared to ${L}_{\mathrm{rad}}$, and

$$\begin{eqnarray}&&{\dot{E}}_{{\rm{w}}}\simeq {L}_{\mathrm{rad}}(r\gg R)\simeq \displaystyle \frac{4\pi {cGM}}{{\kappa }_{0}},\end{eqnarray} \tag{ 8 }$$

where the last equality follows because the luminosity at large r only slightly exceeds the local Eddington limit. During the steady state, ${\dot{E}}_{{\rm{w}}}$ is constant throughout the wind, and we can equate Equations (7) and (8) to find

$$\begin{eqnarray}&&{\dot{M}}_{{\rm{w}}}\simeq \displaystyle \frac{4\pi {cR}}{{\kappa }_{0}}\left[\displaystyle \frac{{\kappa }_{0}}{{\kappa }_{\mathrm{wb}}}-1\right]\simeq \displaystyle \frac{4\pi {cR}}{{\kappa }_{0}}{\left(\displaystyle \frac{{T}_{\mathrm{wb}}}{0.45\mathrm{GK}}\right)}^{0.86}\end{eqnarray} \tag{ 9 }$$

(Paczynski & Proszynski 1986 derive a similar expression). We now show that T_wb (and hence ${\dot{M}}_{{\rm{w}}}$) is a weak function of y_b by first estimating the peak temperature at the ignition base ${T}_{b}({y}_{b})$ and then relating T_b to T_wb.

The base temperature T_b rises until it becomes radiation-pressure-dominated, which lifts the degeneracy and stifles the burning (Fujimoto et al. 1981; Bildsten 1998). At its maximum,

$$\begin{eqnarray}&&{T}_{b}\simeq f{\left(\displaystyle \frac{3{{gy}}_{b}}{a}\right)}^{1/4}\simeq 2.3\,{y}_{b,9}^{1/4}\,\mathrm{GK},\end{eqnarray} \tag{ 10 }$$

where ${y}_{b,9}={y}_{b}/{10}^{9}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$ and at f = 1 radiation pressure completely dominates. In the numerical expression here and below we set f = 0.8 based on our numerical calculations (see Figure 4 and also iZW10). During the wind phase, the bound layers are fully radiative and satisfy $T\propto {y}^{1/4}$. When the wind is at its peak strength, ${y}_{\mathrm{wb}}\simeq \eta {y}_{b}$ and ${T}_{\mathrm{wb}}\simeq {\eta }^{1/4}{T}_{b}$, where η is given by Equation (3). We thus find

$$\begin{eqnarray}&&{y}_{\mathrm{wb}}\simeq 2.5\times {10}^{6}\,{y}_{b,9}\,{\rm{g}}\,{\mathrm{cm}}^{-2},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{T}_{\mathrm{wb}}\simeq 0.5\,{y}_{b,9}^{1/4}\,\mathrm{GK},\end{eqnarray} \tag{ 12 }$$

where here and below we set $\eta =2.5\times {10}^{-3}$ (see top panel of Figure 8). In practice, this leads to a slight overestimate of y_wb, since η is determined by the total ejected mass at the end of the burst and therefore η ≳ y_wb/y_b. Plugging Equation (12) into Equation (9), we find that during the approximate steady-state phase

$$\begin{eqnarray}&&{\dot{M}}_{{\rm{w}}}\simeq 2.1\times {10}^{18}{y}_{b,9}^{0.22}\,{\rm{g}}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 13 }$$

As we show in Figure 8, this estimate agrees reasonably well with the wind simulation results (the simulations show a somewhat smaller ${\dot{M}}_{{\rm{w}}}$ and an even weaker y_b dependence). We thus see that ${\dot{M}}_{{\rm{w}}}$ and therefore the wind structure is nearly independent of y_b.

Given ${\dot{M}}_{{\rm{w}}}$, we can estimate the wind duration

$$\begin{eqnarray}&&{t}_{{\rm{w}}}=\displaystyle \frac{{M}_{\mathrm{ej}}}{\dot{M}}=15\,{y}_{b,9}^{0.79}\,{\rm{s}}.\end{eqnarray} \tag{ 14 }$$

This compares well with the numerically calculated value of the total wind duration, shown in the bottom panel of Figure 8. The latter is slightly larger because Equation (13) overestimates ${\dot{M}}_{{\rm{w}}}$, especially near the beginning and end of the wind.

In Figure 9 we show the wind composition as a function of radius (top axis) and column depth (bottom axis) for model y2n21 at $t=10\,{\rm{s}};$ by this time, the wind has settled into a steady state. We find that the wind at that time is dominated by heavy-element ashes, particularly ⁴⁸Cr, ⁴⁴Ti, and ⁵²Fe, whose mass fractions are about 0.5, 0.2 and 0.1, respectively. Comparing with the pre-wind profile, we see that this is the material that initially resided at a column depth $y\simeq {10}^{6}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$ (see dashed vertical line in middle panel of Figure 3). This is because ${\dot{M}}_{{\rm{w}}}\approx {10}^{18}\,{\rm{g}}\,{{\rm{s}}}^{-1}$ and thus after $t=10\,{\rm{s}}$, the wind has ablated the surface layers down to a depth ${\dot{M}}_{{\rm{w}}}t/4\pi {R}^{2}\approx {10}^{6}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$.

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An interesting feature of the pre-wind profile is that for deep ignitions (${y}_{b,9}=1\mbox{--}5$) the column depth at which the composition transitions from mostly light to mostly heavy elements is almost a constant value of ${y}_{\mathrm{ash}}\approx {10}^{5}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$. In Figure 3 we indicate y_ash with a vertical dotted line, where we formally define y_ash as the location where the total mass fraction of elements with mass number A > 40 equals 0.5 (for $y\lt {y}_{\mathrm{ash}}$, the elements consist predominantly of ⁴He and ¹²C). We can then define the timescale to expose the heavy ashes

$$\begin{eqnarray}{t}_{\mathrm{ash}} & = & \displaystyle \frac{4\pi {R}^{2}{y}_{\mathrm{ash}}}{{\dot{M}}_{{\rm{w}}}}\\ & \simeq & (1.3\,{\rm{s}})\left(\displaystyle \frac{{y}_{\mathrm{ash}}}{{10}^{5}\,{\rm{g}}\,{\mathrm{cm}}^{-2}}\right)\left(\displaystyle \frac{{10}^{18}\,{\rm{g}}\,{{\rm{s}}}^{-1}}{{\dot{M}}_{{\rm{w}}}}\right).\end{eqnarray} \tag{ 15 }$$

In Figure 10 we plot t_ash as a function of y_b. Starting from ${y}_{b,9}=0.5$, we find that t_ash first decreases sharply as y_b increases, but then for ${y}_{b,9}\gtrsim 3$ it plateaus at ${t}_{\mathrm{ash}}\simeq 1\,{\rm{s}}$. This is because y_ash plateaus at ${y}_{\mathrm{ash}}\approx {10}^{5}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$, while ${\dot{M}}_{{\rm{w}}}$ depends very weakly on y_b (see Section 3.3). In Section 4.2, we describe how this might explain a feature of superexpansion bursts.

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In Figure 11 we show a related result: the ion mean molecular weight ${\mu }_{\mathrm{ion}}={({\sum }_{i}{X}_{i}/{A}_{i})}^{-1}$ at the photosphere as a function of time, where X_i and A_i are the mass fraction and mass number of element i. For reference, a mixture consisting of 50% ⁴He and 50% ⁵⁶Ni has ${\mu }_{\mathrm{ion}}=7.5$. For the y2n21 and y3n21 models, it takes a few seconds for ${\mu }_{\mathrm{ion}}$ to increase above 7.5, agreeing well with the t_ash estimate. Note that for $t\gt 10\,{\rm{s}}$, the y3n21 model has a smaller ${\mu }_{\mathrm{ion}}$ than the y2n21 model. This is because a larger amount of ⁴He is left unburned in y3n21 than in y2n21.⁵ To illustrate this effect, the dashed lines in Figure 11 show the ion mean molecular weight excluding ⁴He. The value for the y3n21 model approaches 56 since it is dominated by ⁵⁶Ni, while the y2n21 model approaches 48 since it is dominated by ⁴⁸Cr.

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In Figure 12 we show the evolution of the bolometric luminosity, photospheric radius r_ph, and photospheric temperature T_ph, for different ignition models (these are related by ${L}_{\mathrm{rad}}=4\pi {r}_{\mathrm{ph}}^{2}\sigma {T}_{\mathrm{ph}}^{4}$). Depending on the ignition depth, the PRE phase lasts from $\simeq 5\,{\rm{s}}$ to $\simeq 100\,{\rm{s}}$ (see Equation (14)), during which the bolometric luminosity is nearly constant at ${L}_{\mathrm{rad}}\simeq {L}_{\mathrm{Edd}}$. Initially, all the models follow similar tracks: from t = 0 s to t ≈ 3 s, the photosphere expands from $10\,\mathrm{km}$ to ≳100 km and the temperature ${T}_{\mathrm{ph}}\propto {r}_{\mathrm{ph}}^{-1/2}$ drops from about 2 to 0.5 keV. Due to the larger nuclear energy release, the deeper ignition models expand outward for longer and reach slightly larger radii (150–200 km). Furthermore, unlike model y1n21, which shows a fairly abrupt contraction after reaching maximum expansion, models y2n21 and y3n21 have a long, approximately steady phase during which r_ph remains near its maximum for ≃20 s and ≃90 s, respectively. Comparing models y2n9 and y2n21, we see that the latter expands faster and reaches a larger maximum r_ph because it has a more complete reaction network and thus a larger energy release.

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In Figure 13 we show the approximate count rate that RXTE-PCA and NICER would detect for a source located at a distance of 10 kpc. We assume a blackbody spectrum and use our calculation of ${L}_{\mathrm{rad}}[{r}_{\mathrm{ph}}(t)]$ and ${T}_{\mathrm{ph}}(t)$ to estimate the count rate integrated over the effective area of the detector. For RXTE-PCA, we take the effective area from Jahoda et al. (2006) and assume that two out of the five proportional counting units are working, as was typical during its operations (iZW10). For NICER we adopt the effective area given on the mission website.⁶ The effective collecting area of RXTE-PCA decreases significantly below 2 keV and therefore its count rate drops during a PRE, as the spectrum shifts to lower T_ph. By contrast, the effective collecting area of NICER remains large down to 0.3 keV and its count rate actually increases during a PRE. Moreover, since T_ph ≲ 1 keV throughout the PRE, the NICER count rate is always significantly larger that RXTE's (note the different scales in Figure 13). Both of these effects have been reported in the NICER observation of 4U 1820-30 (Keek et al. 2018).

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In Section 3.4 we showed that the wind ejects heavy elements synthesized during the burst. These ejected ashes include ⁴⁸Cr, ⁴⁴Ti, and ⁵⁶Ni at mass fractions X ≳ 0.1. WBS showed that the expanded photosphere is sufficiently cool that heavy elements such as these will bind with electrons and imprint significant photoionization edges on the burst spectra. However, they assumed that the wind base is located at ${y}_{\mathrm{wb}}=0.01{y}_{b}$, whereas our calculations explicitly determine ${y}_{\mathrm{wb}}(t)$ and show that ${y}_{\mathrm{wb}}\approx \mathrm{few}\times {10}^{-3}{y}_{b}$ at its maximum (see Figure 8). WBS estimated that during the PRE the edges should have equivalent widths EW ∼ 0.1 keV. Using their approach for estimating the edge strengths and the abundances from our wind calculation, we also find EW ∼ 0.1 keV during the PRE for bursts with y_b ≳ 5 × 10⁸ g cm⁻².

Our ${r}_{\mathrm{ph}}(t)$ results are broadly consistent with observations of PRE bursts, although there are some notable differences. According to iZW10, the vast majority (≳99%) of PRE bursts have photospheres that do not expand beyond 10³ km (the exceptions are superexpansion bursts, which we discuss below). This is consistent with our result that ${r}_{\mathrm{ph}}\sim 100\,\mathrm{km}$ nearly independent of ignition depth. There are weak PREs with maximum r_ph that are only a factor of a few larger than R (Galloway et al. 2008). These PREs may be weaker because they are igniting in mixed H/He layers, and thus by assuming a pure He layer our simulations do not capture this population.

One of the most well sampled measurements of ${r}_{\mathrm{ph}}(t)$ is from the recent burst detected with NICER from 4U 1820-30 (Keek et al. 2018). This source is an ultracompact X-ray binary (UCXB) that is thought to be accreting He-rich material (Cumming 2003). As Figure 13 illustrates, NICER is ideally suited to follow the entire PRE phase of bursts due its sub-keV sensitivity. Keek et al. (2018) found that the entire PRE phase lasts $\simeq 3\,{\rm{s}}$ and reaches a maximum expansion radius ${r}_{\mathrm{ph}}=190\pm 10\,\mathrm{km}$. These are consistent with the duration and expansion radius of our y1n21 model (${y}_{b}=5\times {10}^{8}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$).

On the other hand, the temporal variation of the count rate and r_ph of the 4U 1820-30 burst look somewhat different from our y1n21 model (compare Figure 3 in Keek et al. 2018 and our Figures 12 and 13). The observed expansion timescale at the start of the PRE is only $\simeq 0.1\,{\rm{s}}$ compared to our $\simeq 1\,{\rm{s}}$. Also, after reaching maximum expansion, the observed photosphere falls back to the NS surface somewhat more slowly than ours (over $\simeq 3\,{\rm{s}}$ compared to $\simeq 1\,{\rm{s}}$). A possible explanation for why our expansion is too slow and contraction is too fast is that we ignore general relativistic effects. Paczynski & Proszynski (1986) showed that at small ${\dot{M}}_{{\rm{w}}}$, relativistic models predict a larger r_ph than Newtonian models (see their Figures 10 and 11). For example, at ${\dot{M}}_{{\rm{w}}}\simeq 4\times {10}^{17}\,{\rm{g}}\,{{\rm{s}}}^{-1}$ we find ${r}_{\mathrm{ph}}\simeq 35\ \mathrm{km}$ (see Figure 6 at t = 0.07 s) whereas Paczynski & Proszynski (1986) find ${r}_{\mathrm{ph}}\simeq 100\ \mathrm{km}$. (The difference is much smaller at ${\dot{M}}_{{\rm{w}}}\gtrsim {10}^{18}\,{\rm{g}}\,{{\rm{s}}}^{-1}$ and thus our maximum r_ph should be reasonably accurate). As a result, our simulations probably underestimate r_ph at the small ${\dot{M}}_{{\rm{w}}}$ that applies near the start and end of the PRE, which would mean we underestimate (overestimate) the rate of expansion (contraction).

Superexpansion bursts (${r}_{\mathrm{ph}}\gt {10}^{3}\,\mathrm{km}$) provide another interesting point of comparison. According to iZW10, there have been 32 superexpansion bursts detected from 8 sources (of these, 22 were from 4U 1722-30). Of the superexpansion bursts that have been identified with an object (7 out of 8), all are from candidate UCXBs. The neutron star in an UCXB accretes hydrogen-deficient fuel and the bursts tend to be longer (several tens of minutes rather than seconds, i.e., intermediate duration bursts; in't Zand et al. 2005; Cumming et al. 2006). In two superexpansion bursts observed with RXTE, iZW10 detected strong absorption edges. The edge energies and depths are consistent with large abundances of iron-peak elements and support our finding that the wind can eject significant amounts of heavy-element ashes.

Superexpansion bursts always show two distinct phases: a superexpansion phase during which r_ph ≳ 10³ km, followed by a moderate expansion phase during which ${r}_{\mathrm{ph}}\sim 30\mbox{--}50\,\mathrm{km}$ and ${L}_{\mathrm{rad}}\simeq {L}_{\mathrm{Edd}}$ (iZW10). Interestingly, the duration of the superexpansion phase is always a few seconds, independent of the ignition depth y_b. By contrast, the duration of the moderate expansion phase ranges from short (≈10–100 s) for ${y}_{b}\,\sim {10}^{9}\,{\rm{g}}\,{\mathrm{cm}}^{-2}$ to intermediate (≳10³ s) for ${y}_{b}\sim {10}^{10}\,{\rm{g}}\,{\mathrm{cm}}^{-2}.$ iZW10 speculated that the superexpansion phase always lasts a few seconds because it corresponds to a transient stage in the wind's development. In this stage, they argue, a shell of initially opaque material is ejected to large radii by the sudden onset of super-Eddington flux deep below the photosphere. Within a few seconds the expanding shell reaches such large radii ($\gt {10}^{3}\,\mathrm{km}$) that it becomes optically thin and the observer suddenly sees the underlying photosphere of the already formed steady-state wind, which is located at ${r}_{\mathrm{ph}}\sim 30\mbox{--}50\,\mathrm{km}$. According to this picture, this marks the onset of the moderate expansion phase, whose duration equals that of the steady-state wind and thus correlates with y_b (see Equation (14)).

We do not, however, see evidence of a shell ejection in our simulations. This might be because our treatment of radiative transfer in the low optical depth regions (τ ≲ 3) is too simplistic, because we ignore relativistic effects, or because of unaccounted for dynamics during the transition from the hydrostatic rise to the hydrodynamic wind (e.g., in't Zand et al. 2014 reported two bursts with exceptionally short precursors, which they argue may indicate a detonation initiated by the rapid onset of the ¹²C(p, γ)¹³N(α, p)¹⁶O reaction sequence). However, there is a feature in our simulations that suggests an alternative explanation for why the superexpansion phase always lasts a few seconds, regardless of y_b. As we described in Section 3.4, the timescale t_ash to expose heavy elements in the wind is also a few seconds, and plateaus at ${t}_{\mathrm{ash}}\simeq 1\,{\rm{s}}$ for ${y}_{b}\gtrsim 3\times {10}^{9}\ {\rm{g}}\,{\mathrm{cm}}^{-2}$. It suggests that the transition from superexpansion to moderate expansion after ≈1 s might be due to the wind's composition changing from light to heavy elements (and not due to shell ejection). As discussed in iZW10, the heavy elements will only be partially ionized and the radiative force acting on the bound electrons above the photosphere will be ≳100 times larger than the force acting on the free electrons. Line-driving in the outer parts of the wind may therefore boost the outflow velocity. By mass conservation (Equation (4)), this would decrease the overlying density and bring the photosphere inward to smaller radii. This could explain why the photosphere transitions from very large r_ph during the superexpansion (when the ejecta are mostly light elements) to smaller values of r_ph ∼ 30–50 km during the moderate expansion, and why the the timescale for such transitions is always a few seconds. Evaluating this in detail likely requires accounting for line-driving in the wind.