Hydrogen-triggered X-Ray Bursts from SAX J1808.4−3658? The Onset of Nuclear Burning

Thermonuclear (Type I) X-ray bursts occur when an accreted layer of matter on a neutron star undergoes a thermonuclear runaway (Hansen & van Horn 1975; Strohmayer & Bildsten 2006; Galloway & Keek 2021). The thin shell thermal instability that triggers these bursts occurs when the energy generation rate due to nuclear burning exceeds the local cooling rate in the accreted shell. Depending on the accreted fuel composition there can be a large number of nuclear reactions that contribute to the energy production that powers bursts; however, two reactions are thought to be the primary triggers for the thermal instability. The first is the highly temperature-sensitive triple-α reaction, which burns three helium nuclei to carbon. The other relevant process is the CNO cycle that burns hydrogen to helium via a series of (p, γ) reactions on carbon and nitrogen (Fowler & Hoyle 1965). Also operating in the cycle are several rate-limiting β⁺ decays dependent on slower weak reaction rates. The cycle is closed by the (p, α) reaction that converts ¹⁵N back to ¹²C. This physics leads to two regimes in which the CNO cycle may operate in this context. Above temperatures of about 8 × 10⁷ K, the rate of the so-called hot-CNO cycle is limited by the β⁺ decays that take ¹⁵O and ¹⁴O to ¹⁵N and ¹⁴N, respectively. In this regime the energy generation rate becomes temperature insensitive, that is, thermally stable. However, for temperatures below ≈8 × 10⁷ K the CNO energy generation rate remains temperature sensitive, so that, in principle, a hydrogen-burning thermal instability can operate if the accreting shell is cool enough.

The implications of these physical processes for the production of X-ray bursts were explored in several early, "classic" papers. Fujimoto et al. (1981) were among the first to delineate the possible bursting regimes based on the mass accretion rate, but also see Joss (1978) and Taam & Picklum (1979). They identified three bursting regimes with decreasing mass accretion rate. At higher accretion rates a helium shell grows via accretion and thermally stable CNO cycle burning that converts some of the accreted hydrogen to helium. The accretion rate is high enough that the base of the shell reaches ignition conditions before all the hydrogen can be burned to helium. The triple-α instability is initiated in a shell with a significant hydrogen abundance, leading to so-called mixed H/He bursts. The presence of hydrogen allows for the more complex set of nuclear reactions known as the rapid-proton capture process to occur, which can enhance and delay the energy release, leading to longer duration bursts (Schatz et al. 2001). The well-studied "clocked" bursts from GS 1826−238 are prototypical of this type (Ubertini et al. 1999; Galloway et al. 2004; Heger et al. 2007; Zamfir et al. 2012). For lower accretion rates there will be a rate such that, prior to ignition, the CNO burning has just had sufficient time to burn all the hydrogen in the accreting shell. At this critical accretion rate the helium burning is initiated in a pure helium shell. These "pure helium" bursts are characterized by a rapid intense energy release, are typically of shorter duration than the mixed H/He bursts, and often reach the Eddington limit, as exhibited by photospheric radius expansion (PRE). The bright PRE bursts observed from the accreting millisecond X-ray pulsar (AMXP) SAX J1808.4−3658 are examples of such bursts (Galloway & Cumming 2006; in't Zand et al. 2013; Bult et al. 2019a). At low accretion rates, temperatures in the accumulating shell may be low enough that steady CNO hydrogen burning essentially switches off but can proceed in the unstable, temperature-sensitive regime. This can, in principle, lead to unstable ignition of the hydrogen in the accumulating shell. Two possible paths have been discussed in this case. First, ignition of hydrogen raises the temperature in the shell, and if the column depth is large enough the heated shell may cross the helium instability curve, producing a prompt, mixed H/He burst. Second, if the hydrogen ignition depth is too shallow to cross the helium instability curve, then fuel will continue to accumulate until helium ignition can occur. At these low accretion rates it is also likely that gravitational sedimentation of heavier elements relative to hydrogen and helium will play a role in setting the conditions for unstable burning (Peng et al. 2007).

However, observational evidence for this hydrogen ignition regime is limited, as there have been to our knowledge few published reports of X-ray bursts that can be clearly attributed to unstable hydrogen shell flashes. In one such case, Boirin et al (2007) reported on the first observations of triple, short recurrence time (SRT) bursts from the high inclination, eclipsing source EXO 0748-676. They suggested that the initial bursts of singles, pairs or triples (they call these the long waiting time, LWT, bursts), could be attributed to either helium-triggered, mixed H/He bursts at moderate accretion rates (10% of Eddington), or perhaps hydrogen-triggered bursts at lower accretion rates (1% of Eddington). Because the LWT bursts appeared somewhat under-luminous compared with mixed H/He bursts in 1-d Kepler models and the well-known example of such bursts from GS 1826-238 they suggested that this might be explained by the latter, hydrogen-triggered mechanism. They further suggested that the SRT events, with waiting times close to 10–12 minutes, were likely caused by the re-ignition of unburned fuel, but they did not have a detailed explanation of how this occurs. More recently, Keek & Heger (2017) have outlined a theoretical mechanism to account for SRT bursts. Using detailed, 1-d Kepler hydrodynamic simulations they showed that such events can be produced by opacity-driven convective mixing that transports fresh fuel to the ignition depth, and they also argued that this mechanism can produce simulated burst events that are "strikingly similar" (in their words) to the SRT bursts seen from EXO 0748-676. If this mechanism is indeed at work, then it would further argue for the higher accretion rate (10% of Eddington), helium-triggered scenario in EXO 0748-676, as warmer envelopes, naturally produced by higher accretion rates, were required to produce the SRT events in their burst simulations. Moreover, they also showed that the fraction of fuel burned in the LWT events dropped as the envelope became hotter, and this relatively low fuel burning fraction could also naturally explain the apparently under luminous LWT bursts noted by Boirin et al (2007). Thus, while Boirin et al (2007) suggest that a hydrogen-triggered mechanism is possible for the LWT bursts from EXO 0748-676, we would characterize the current, overall evidence in support of that conclusion as tentative, particularly given the remaining uncertainties in the distance and anisotropy factors for this source. Indeed, in support of this we note that in their recent review of the field, Galloway & Keek (2021) also comment that, "No observations matching case I or case II bursting have been identified." Here, cases I and II refer to the two hydrogen ignition paths at low accretion rates that we sketched above.

In this paper we present a study of an apparently rarer class of weak X-ray bursts observed from SAX J1808.4−3658 (hereafter, J1808) that we argue show the hallmarks of being associated with the hydrogen ignition regime. This object was the first AMXP discovered (Chakrabarty & Morgan 1998; Wijnands & van der Klis 1998), and hosts a neutron star in a 2.1 hr orbit with a low-mass brown dwarf (Bildsten & Chakrabarty 2001). Its distance has been estimated at 3.5 ± 0.1 kpc (Galloway & Cumming 2006), and it is likely that the donor provides a hydrogen-rich mix of matter to the neutron star during outbursts (Galloway & Cumming 2006; Goodwin et al. 2019). To date, J1808 has been observed extensively during 10 outbursts. While it is not our intention here to provide a broad observational overview of the source—for the purposes of this paper we focus on issues relevant to its thermonuclear bursting behavior—readers can find elsewhere some recent studies on coherent pulse timing (Sanna et al. 2017; Bult et al. 2020; Illiano et al. 2022), X-ray spectral properties (Di Salvo et al. 2019), and aperiodic timing behavior (Bult & van der Klis 2015; Sharma et al. 2023).

Observations of J1808 have revealed two types of thermonuclear bursts that show dramatically different peak fluxes and fluences. The bright PRE bursts (mentioned above) show significantly higher total energy release and peak X-ray flux. The less frequently observed weak bursts produce much less energy and show peak fluxes about a factor of 25 less than the bright events, as such they are not Eddington-limited. When these weak bursts have been observed, they appear to be confined to earlier portions of the outbursts and occurred before the bright bursts were seen. This suggests there may be a window of occurrence for these bursts associated with the initial onset of accretion after a period of quiescence. This is particularly intriguing in the context of J1808 because it is known that the neutron star cools dramatically in quiescence (Heinke et al. 2009), and the unstable hydrogen-burning regime requires cooler temperatures in the accumulating layer. There has been more observational and theoretical research exploring the nature of the bright bursts than the weak class.

Here we present a detailed study of one of these weak bursts that was observed with the Neutron Star Interior Composition Explorer (NICER) during the recent, 2019 August outburst from J1808. We also provide a briefer description of a similar burst observed with the Rossi X-ray Timing Explorer (RXTE) in 2005 June. The paper is organized as follows. In Section 2 we introduce the NICER data and present light curves focusing on the initial part of the 2019 outburst, showing a single weak burst. We also present a spectral study of the persistent and burst emission (for the weak burst) in order to understand its energetics and to constrain the mass accretion rate and the likely accreted mass column at the time of its ignition. We present a discussion in Section 3 of a likely physical scenario that results in the weak burst, arguing that the initial accretion onto a cool neutron star at the onset of the outburst naturally places the accumulating layer in the thermally unstable regime for CNO hydrogen ignition. Here, we also describe the 2005 June RXTE event, and we also report a brief summary of Nuclear Spectroscopic Telescope Array (NuSTAR) observations that began on 2019 August 10 and in which several brighter bursts were detected. We conclude in Section 4 with a summary, a brief discussion of relevant uncertainties and other possible interpretations, and the outlook for future efforts.

In late 2019 July, it was reported that the optical flux from J1808 had increased, perhaps presaging a new X-ray outburst (Russell et al. 2019; Goodwin et al. 2020). This initiated an extensive monitoring campaign with NICER, which began on 2019 August 1 (Bult et al. 2020). NICER is an X-ray observatory that operates on the International Space Station. It observes across the 0.2–12 keV X-ray band and provides low-background, high-throughput (≈1900 cm² at 1.5 keV), and high-time-resolution capabilities (Gendreau et al. 2012). The data obtained prior to the onset of the outburst, and up to and including the first observed X-ray burst are organized under observation IDs (OBSIDs) 205026010mm and 25840101nn, where mm and nn run from 03–10 and 01–02, respectively. We used the standard screening criteria and NICERDAS version 8 to produce cleaned event lists. This means we retained only those epochs during which the pointing offset was <54'', the Earth elevation angle was >15°, the elevation angle with respect to the bright Earth limb was >30°, and the instrument was not in the South Atlantic Anomaly. We used HEASOFT Version 6.29c to produce the light curves and spectra for the analyses reported here. The initial observations of the campaign did not reveal evidence of J1808 in X-ray outburst. The first indication that an accretion-driven flux was present occurred on 2019 August 6 at approximately 21:59 TT (Bult et al. 2019b). Figure 1 shows the light curve (0.4–7 keV) of the outburst over approximately 20 days from the observed onset of significant X-ray activity. Time zero in the plot refers to the time of outburst onset, 58701.91597 MJD (TT). The two detected X-ray bursts are evident as "spikes" in the count rate near days 3 and 14, respectively. The much brighter second burst (near day 14) was reported on by Bult et al. (2019a). Here we focus on a study of the much weaker first burst, which occurred at 58704.80764 MJD (TT) and is present in OBSID 2584010102.

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2.1. Persistent Spectrum, Fluence, and Accreted Mass

To explore the weak burst energetics and ignition conditions we aim to constrain the total accreted mass from the beginning of the outburst up to the onset of the first burst. To do this we model the spectrum of the persistent emission to determine its flux and then integrate from the outburst onset to just prior to the burst. This integral provides an estimate of the energy fluence produced via accretion, which can then be converted to an accreted mass using standard assumptions for the accretion luminosity produced by the release of gravitational potential energy of the accreted matter.

In practice we find that the shape of the persistent spectrum gradually changes during this portion of the outburst, with the spectrum showing a modest hardening over time. We therefore measure the flux at a few intervals along the outburst rise, and use these measurements to estimate the flux per unit NICER count rate. We then use simple linear interpolation and the trapezoidal rule to integrate the flux from outburst onset to the first burst to estimate the energy fluence.

The light curve in Figure 2 shows a close-up of the epoch around the first burst. We extracted a spectrum prior to the burst, the "pre-burst" interval (marked by the vertical dashed lines in Figure 2) and modeled its spectrum using XSPEC version 12.12.1 (Arnaud 1996). We produced response files with NICERDAS version 8, and we used the 3C50 background model, nibackgen3c50 (Remillard et al. 2022), to produce a background spectrum appropriate for spectral modeling within XSPEC. We employed a phenomenological model similar to that discussed by Patruno et al. (2009), which includes thermal disk, power-law, and blackbody continuum components. In addition, and similarly to Bult et al. (2019a), we find evidence for narrow-line emission near 1 keV, and we include a Gaussian component to model this. In XSPEC notation the model has the form, phabs ∗ (diskbb + powerlaw + bbodyrad + Gaussian), where phabs represents the line-of-sight photoelectric absorption model parameterized by the column density of neutral hydrogen, n_H. This absorption model uses cross sections from Verner et al. (1996) and the chemical abundances from Anders & Grevesse (1989). We fit this model across the 0.5–10 keV bandpass and find that it provides an excellent fit, with a minimum χ² = 117.9 for 112 degrees of freedom. The best-fitting model parameters are given in Table 1, and Figure 3 shows the unfolded photon spectrum (top), the observed count-rate spectrum and best-fitting model (middle), and the fit residuals in units of standard deviations (bottom). This model gives an unabsorbed flux (0.1–20 keV) of 7.06 ± 0.23 ×10⁻¹⁰ erg cm⁻² s⁻¹. If we extend the bandpass to estimate a bolometric flux we find a value (0.1–100 keV) of 7.9 × 10⁻¹⁰ erg cm⁻² s⁻¹. The average count rate in the fitted energy band (0.5–10 keV) is 181.9 ± 0.5 s⁻¹, so we estimate a flux per NICER count rate (0.5–10 keV) of 4.34 ×10⁻¹² erg cm⁻² s⁻¹ ${(\mathrm{counts}\,{{\rm{s}}}^{-1})}^{-1}$ for this interval.

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Table 1. Spectral Model Parameters for SAX J1808: Persistent Emission

Parameter	2584010102 (Pre-burst)	2050260110	2050260109
nH (1022 cm−2)	0.131 ± 0.016	0.160 ± 0.025	0.177 ± 0.028
kTin (diskbb, keV)	0.849 ± 0.026	0.679 ± 0.024	0.528 ± 0.017
Norm (diskbb)	27.33 ± 5.36	25.19 ± 6.46	10.82 ± 1.79
kT (bbodyrad, keV)	2.03 ± 0.15	1.67 ± 0.15	⋯
Norm (bbodyrad, keV)	0.98 ± 0.47	0.604 ± 0.336	⋯
Index (power)	1.96 ± 0.37	2.50 ± 1.05	2.01 ± 0.23
Norm (power)	0.0266 ± 0.0137	4.64 × 10−3	4.23 ± 1.6 × 10−3
E (Gauss, keV)	0.990 ± 0.009	0.935 ± 0.014	⋯
σE (Gauss, keV)	0.015	0.015	⋯
Norm (Gauss)	8.44 ± 2.03 × 10−4	3.51 ± 1.00 × 10−4	⋯
f0.1–20 (erg cm−2 s−1)	7.06 ± 0.23 × 10−10	2.08 ± 0.40 × 10−10	5.35 ± 0.81 × 10−11
fbol (erg cm−2 s−1)	7.9 × 10−10	2.5 × 10−10	6.4 × 10−11
χ2 (dof)	117.9 (112)	98.9 (97)	94.5(106)
Rate (s−1, 0.5–10 keV)	181.9 ± 0.4	58.70 ± 0.25	13.46 ± 0.07
Epoch (days)	2.89	1.45	0.59
Exposure (s)	740	927	2807

Note. Parameter uncertainties are estimated as 1σ values. For OBSID 2050260109 the bbodyrad and Gauss line components were not required in the fit. The "⋯" symbols in this case indicate that these parameters were not included in the fit. For additional context, the "Epoch" specifies the center time of the interval in which the spectra were extracted, and the value refers to the time axis of Figure 1.

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We extracted spectra from two other OBSIDs along the outburst rise, 2050260109 and 2050260110, and analyzed these spectra in the same manner as for the pre-burst interval just described. Results of these spectral fits are also reported in Table 1. For these intervals we estimate flux per NICER count-rate values of 2.85 × 10⁻¹² and 3.61 × 10⁻¹² erg cm⁻² s⁻¹ ${(\mathrm{counts}\ {{\rm{s}}}^{-1})}^{-1}$, respectively.

For completeness we make a few additional comments regarding the 1 keV line component included in the spectral model. For the pre-burst interval (OBSID 2584010102), removing the Gaussian line results in an increase in χ² of 31.3, and the ratio of the line normalization to its 1σ uncertainty is ≈4.2. The line is also evident in OBSID 2050260110, though at lower significance, with the ratio of the line normalization to its 1σ uncertainty now at 3.5. For OBSID 2050260109, the spectrum extracted closest to the outburst onset and at the lowest observed flux, we no longer find evidence for the line. When detected the line is narrow in the sense that it is unresolved and we can only place an upper limit on its width of ≈0.09 keV (3σ). Finally, in this work our primary focus is to model the X-ray spectrum to infer the broadband flux. Excluding the 1 keV line from the spectral fits only changes the inferred flux at the few percent level, so including it, or not, does not significantly alter our inferences regarding the source flux. We elected to include it as doing so provides a better overall statistical description of the data.

To estimate the outburst fluence we use simple linear interpolation between data gaps, and we also apply linear interpolation of the flux per unit count rates, based on the spectral results discussed above. We employ the trapezoidal rule to integrate the total counts. We find a persistent emission energy fluence of E_p = 7.92 × 10⁻⁵ erg cm⁻², representing an estimate of the total energy associated with accretion from the outburst onset up to the initiation of the first observed burst.

Assuming the observed, accretion-driven luminosity for spherical accretion,

$$\begin{eqnarray}&&{L}_{p}=4\pi {d}^{2}{\xi }_{p}{f}_{p}=\displaystyle \frac{z\dot{M}{c}^{2}}{{\left(1+z\right)}^{3}},\end{eqnarray} \tag{ 1 }$$

where ${(1+z)=(1-2{GM}/{c}^{2}R)}^{-1/2}$, $\dot{M}$, ξ_p , f_p , M, and R are the surface redshift, mass accretion rate (as measured at the neutron star surface), persistent emission anisotropy factor, observed bolometric flux, neutron star mass, and radius, respectively. We can use this equation to estimate the total accreted mass required to produce the observed energy fluence (Johnston 2020). We emphasize that L_p and f_p are the luminosity and flux as measured by an observer far from the neutron star. The anisotropy factor, ξ_p , can be thought of as the solid angle into which the radiation is emitted, normalized by 4π; thus, isotropic emission is characterized by ξ_p = 1. We write the accreted mass column in the local neutron star frame as,

$$\begin{eqnarray}&&{y}_{a}=\displaystyle \frac{{M}_{a}}{4\pi {R}^{2}}=\displaystyle \frac{\int \dot{M}(t^{\prime} ){dt}^{\prime} }{4\pi {R}^{2}},\end{eqnarray} \tag{ 2 }$$

where we use $t^{\prime}$ to emphasize that the $\dot{M}$ integral is over time as measured at the neutron star surface. With the use of Equation (1) this becomes,

$$\begin{eqnarray}&&{y}_{a}=\displaystyle \frac{{d}^{2}{\xi }_{p}{\left(1+z\right)}^{2}}{{{zc}}^{2}{R}^{2}}\int {f}_{p}(t)(1+z){dt}^{\prime} =\displaystyle \frac{{d}^{2}{\xi }_{p}{\left(1+z\right)}^{2}}{{{zc}}^{2}{R}^{2}}{E}_{p},\end{eqnarray} \tag{ 3 }$$

where t is the time measured in the observer's frame, and we note that ${dt}=(1+z){dt}^{\prime}$, and thus $\int {f}_{p}(t)(1+z){dt}^{\prime} =\int {f}_{p}(t){dt}={E}_{p}$ is just the observed energy fluence.

If we assume d = 3.5 kpc (Galloway & Cumming 2006) and use E_p = 7.92 × 10⁻⁵ erg cm⁻², we can write the accreted column as,

$$\begin{eqnarray}&&{y}_{a}=1.03\times {10}^{7}\ \left(\displaystyle \frac{{\xi }_{p}{\left(1+z\right)}^{2}}{{{zR}}_{10}^{2}}\right)\ \ \ {\rm{g}}\ {\mathrm{cm}}^{-2},\end{eqnarray} \tag{ 4 }$$

where R₁₀ is the neutron star radius in units of 10 km. For M = 1.4M_⊙, R = 11 km, and adopting ξ_p = 1 we find y_a = 5.12 × 10⁷ g cm⁻². With M = 2.0M_⊙ and R = 11 km we find that y_a decreases slightly to 3.91 × 10⁷ g cm⁻².

We can also use Equation (1) to estimate the mass accretion rate, $\dot{M}$, at the time of the weak burst onset. Using the estimated pre-burst flux of 7.9 × 10⁻¹⁰ erg cm⁻² s⁻¹, and a distance of 3.5 kpc, we find,

$$\begin{eqnarray}&&\dot{M}=2.03\times {10}^{-11}\ \left(\displaystyle \frac{{\xi }_{p}{\left(1+z\right)}^{3}}{z}\right)\ {M}_{\odot }\ {\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 5 }$$

Using the same parameter assumptions as above, we find estimates of $\dot{M}=1.56\times {10}^{-10}$ and 1.38 × 10⁻¹⁰ M_⊙ yr⁻¹, respectively.

2.2. Burst Spectral Evolution: Peak Flux and Fluence

We first segmented the burst light curve into intervals of approximately 500 counts using a 1/8 s time bin. We modeled the segmented spectra in the 0.5–10 keV band by adding a blackbody component to the pre-burst persistent emission model. The parameters of the persistent emission model were frozen to their best-fit values, given in Table 1, only allowing the added blackbody component to vary, so that our model is

phabs(constant(diskbb + bbodyrad + powerlaw + Gaussian) + bbodyrad).

We first tried multiplying the persistent emission model by a constant (Worpel et al. 2013), but found it was not statistically necessary, as it was possible to get a good fit with it left frozen at 1.0. The resulting evolution of the bolometric flux, the free parameters of the blackbody temperature and blackbody radius (at 3.5 kpc), along with the resulting χ² are shown in Figure 4. We found a peak bolometric burst flux of f_b =6.98 ± 0.50 × 10⁻⁹ erg s⁻¹ cm⁻². Using trapezoidal numerical integration of the flux, we calculated a bolometric fluence of 7.05 ± 1.16 × 10⁻⁸ erg cm⁻². The burst luminosity is defined as L_b = 4π d² ξ_b f_b , where ξ_b characterizes the anisotropy of the burst emission. Adopting ξ_b = 1 and d = 3.5 kpc, we can then estimate that the total energy released during the burst was 1.03 × 10³⁸ erg.

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Transient systems like J1808 provide an interesting laboratory to explore the different predicted regimes of nuclear burning on neutron stars. Deep X-ray spectroscopic studies of the object in quiescence suggest rapid cooling of the neutron star core, perhaps by a form of enhanced neutrino emission such as Direct Urca (Heinke et al. 2009), which also provides some tentative evidence for a more massive neutron star (≳2M_⊙) in the system. Using the surface effective temperature constraints in quiescence from Heinke et al. (2009) and the theoretical results of Potekhin et al. (1997), Mahmoodifar & Strohmayer (2013) estimated a core temperature for J1808 in the range from 7.2 to 7.7 × 10⁶ K, for neutron star masses between 1.4 and 2.0 M_⊙. Due to the high thermal conductivity in the core and crust (Brown & Cumming 2009), it is very likely that when accretion begins in J1808 after a period of quiescence, the accumulating layer starts out at temperatures ≲1 × 10⁷ K, that is, well below the temperature at which CNO cycle hydrogen burning becomes thermally stable (Fujimoto et al. 1981; Cumming 2004; Galloway & Keek 2021). In this temperature-sensitive regime hydrogen burning proceeds at very low levels, and the thermal profile of the accumulating layer will be set principally by compressional heating. This is a much less efficient heat source than the energy released from hot-CNO cycle burning in the stable burning regime.

Fujimoto et al. (1981) estimate the accretion rate required to maintain a stable hydrogen-burning shell (see their Table 1; ${\dot{M}}_{\mathrm{st}}(B)$) as 2.7 × 10⁻¹⁰ M_⊙ yr⁻¹, for a neutron star mass and radius of 1.41M_⊙ and 6.57 km. We note that the somewhat older neutron star models employed by Fujimoto et al. (1981) have rather small radii compared to those suggested by more recent modeling (Miller et al. 2019; Riley et al. 2019). For a more typical radius of, say, 11 km (which we employed above), we would expect that the estimated rate would increase modestly by ≈10%, which would bring the value to ≈3 × 10⁻¹⁰ M_⊙ yr⁻¹. Expressed as a fraction of the Eddington accretion rate, ${\dot{M}}_{\mathrm{Edd}}$, and adopting the value of ${\dot{M}}_{\mathrm{Edd}}=1.8\times {10}^{-8}$ M_⊙ yr⁻¹ (Cumming 2004), this is then equivalent to $\dot{M}=0.0167\ {\dot{M}}_{\mathrm{Edd}}$. Note also that Cumming (2004) quotes a value of $\dot{M}\gtrsim 0.01\ {\dot{M}}_{\mathrm{Edd}}$ for stable, hot-CNO cycle hydrogen burning. In addition, Cooper & Narayan (2007) used a two-zone model to carry out a linear stability analysis to specifically explore the conditions under which hydrogen-triggered bursts can occur at low accretion rates, and found that they occur for rates $\lesssim 0.003\ {\dot{M}}_{\mathrm{Edd}}$.

Above, we estimated an accretion rate at the time of the weak NICER burst in the range from ≈1.38–1.56 × 10⁻¹⁰ M_⊙ yr⁻¹ ($\dot{M}\approx 0.0077\mbox{--}0.0087\ {\dot{M}}_{\mathrm{Edd}}$). This is less than the required rates estimated by Cumming (2004) and Fujimoto et al. (1981) for stable CNO burning, but slightly higher than the accretion rate obtained by Cooper & Narayan (2007). These considerations provide strong evidence that, in the initial outburst stage, accretion onto J1808 proceeds in an $\dot{M}$ range consistent with what Fujimoto et al. (1981) refer to as case 3 shell flashes. In this regime the accumulating layer remains cool enough that CNO hydrogen burning proceeds in the temperature-sensitive regime, that is, very little hydrogen is burned until the layer reaches the conditions for unstable ignition. Indeed, following Fujimoto et al. (1981, see their Equation (11)), we would estimate that only about 1%–2% of the hydrogen would be burned prior to ignition. Further insights are provided by our estimates of the total column of matter accreted at the time of the burst and its total energy fluence. In Section 2 above we estimated the accreted column to be in the range ≈3.91–5.12 × 10⁷ g cm⁻², and we measured an energy fluence in the burst of ≈1 × 10³⁸ erg (both at 3.5 kpc). For the following discussion we refer the reader to the illustrative hydrogen ignition curves presented by Cumming (2004, see their Figure 1), Galloway & Keek (2021, see their Figure 2), and Peng et al. (2007, see their Figure 4). Based on these curves, we can estimate that a column of this size would be ignited at a temperature in the range from ≈4–5 × 10⁷ K. What happens upon ignition of the hydrogen? The unstable burning will quickly heat the layer, raising the temperature to at least that at which the CNO energy generation rate saturates, but likely somewhat higher. Fujimoto et al. (1981) estimate that only a small fraction, ΔX, of the hydrogen needs to burn in order to raise the temperature. For a temperature change of 10⁸ K, they estimate ΔX ≈ 0.002 (see their Equation (12)).

After ignition of the hydrogen, two subsequent paths have been described in the literature. First, if the ignition column is small enough, then an increase in its temperature may not cause it to cross the helium ignition curve, and additional accretion and/or an increase in the helium fraction is required before it will ignite. Alternatively, for deeper ignition columns, a temperature increase of a few 10⁸ K would render the shell unstable to helium ignition, promptly producing a mixed H/He burst. We note that the works of Cooper & Narayan (2007) and Peng et al. (2007) also predict these two paths, and their calculations provide estimates of the hydrogen ignition columns that are broadly consistent with our estimate of the column accreted at the time of the weak burst. For example, at an accretion rate of $\dot{m}=0.002\ {\dot{m}}_{\mathrm{Edd}}$ Cooper & Narayan (2007, see their Figure 4, right column) find behavior consistent with the first scenario, a sequence of weak hydrogen flashes occurs until the helium column grows sufficiently to reach ignition conditions. These calculations also provide an estimate of the temperature increase produced by the unstable hydrogen ignition and suggest that changes of ∼2 × 10⁸ K are likely. Peng et al. (2007, see their Figure 7) also find a regime where hydrogen ignition does not lead to prompt ignition of a He burst. They also explore the effect of sedimentation on hydrogen-triggered bursts, which enhances the amount of CNO nuclei at the ignition depth and causes a sharper temperature rise. Sedimentation is likely to play an important role in setting the ignition conditions for the weak burst given the low estimated accretion rate.

Measurement of the burst fluence enables us to estimate the fraction, f_h , of accreted hydrogen needed to burn in order to produce that much energy. For an energy release (per gram) of E_h = 6.4 × 10¹⁸ erg g⁻¹ (Clayton 1983), we would require m_h = 1.6 × 10¹⁹ (1 + z) g of hydrogen to burn, where the factor of (1 + z) is included because we are interested in the energy released at the neutron star surface. Expressed as a column on the neutron star, y_h = m_h /4π R², and assuming R = 11 km, we find y_h = 1.05 × 10⁶ (1 + z) g cm⁻². The amount of hydrogen present in the accreted column is y_a X, where X is the mass fraction of hydrogen in the accreted material. We thus have,

$$\begin{eqnarray}&&{f}_{h}=\displaystyle \frac{{y}_{h}}{{y}_{a}X}=0.105\ \displaystyle \frac{(1+z)}{{Y}_{a}X},\end{eqnarray} \tag{ 6 }$$

where Y_a is the estimated accreted column in units of 10⁷ g cm⁻². Taking Y_a in the range from 3.9 to 5.1, a fractional hydrogen abundance in the accreted fuel of X = 0.7, and using the same M and R assumptions employed above to evaluate (1 + z), we find f_h in the range from 0.04 to 0.06. Note that this value should be considered a lower limit, as it assumes that the estimated total accreted column produced only a single such burst, and the mass fraction would likely be reduced further if sedimentation is present (Peng et al. 2007; see below for additional discussion regarding potentially missed bursts). This value is larger than the estimate given by Fujimoto et al. (1981) for the fraction of hydrogen needed to burn to raise the temperature of the fuel by ΔT = 10⁸ K; we do not know the actual temperature increase however, and the estimate of Fujimoto et al. (1981) should be thought of as a lower limit to our estimate from the measured burst energy fluence, as burning will continue at the stable CNO burning rate.

Alternatively, we can ask the question, how much energy would we expect in the burst if the entire accreted column were to burn to heavy elements? The energy release per gram, Q_nuc, would depend on the details of the nuclear burning pathways; however, employing the value of Q_nuc = (1.3 + 5.8X) × 10¹⁸ erg g⁻¹ (Galloway & Cumming 2006) and again adopting X = 0.7 we would expect ≈3.2–4.2 × 10³⁹ erg liberated at the neutron star surface by burning all the fuel. This estimate also assumes that the total accreted column produces a single burst. This is a factor of 30–40 larger than the observed fluence in the weak X-ray burst, and also argues that the weak burst is likely not a mixed H/He burst. Rather, our analysis suggests that it likely represents the unstable ignition of a modest fraction of the hydrogen in the accreted layer, which constitutes strong observational evidence for such a weak "hydrogen-only" flash.

Interestingly, in their two-zone model Cooper & Narayan (2007) compute the peak energy fluxes produced during such weak hydrogen flashes. The range of fluxes that their model can produce is summarized in their Figure 7 (bottom panel), where the peak flux is given as a fraction of the Eddington flux. Working backward, we measured a peak flux during the weak X-ray burst of ≈6.9 × 10⁻⁹ erg cm⁻² s⁻¹. If we scale this by the peak flux (2.3 × 10⁻⁷ erg cm⁻² s⁻¹) of the Eddington-limited burst observed later in the outburst (Bult et al. 2019a), we find a ratio of $\mathrm{log}(0.03)=-1.52$. Looking at Figure 7 in Cooper & Narayan (2007; bottom panel), we can find hydrogen flashes (the dotted–dashed lines in the figure) in a narrow range of accretion rate that reach this flux level.

3.1. Stable Burning after the Burst?

3.2. Missed Bursts?

3.3. Other Examples: RXTE Observations of the 2005 Outburst

We searched the literature and previous observations of J1808 to try and identify similar examples of weak bursts. We found a quite similar event early in the 2005 June outburst that was observed with RXTE. We show in Figure 5 the light curve of this outburst as obtained from RXTE pointed observations. This burst occurred on June 2 at approximately 00:42:30 TT, and is evident near 0.25 day in the figure. We carried out a time-resolved spectral analysis of this event, and found qualitatively similar properties for this burst as for the weak NICER burst. It reaches a peak bolometric flux of 1.54 ± 0.11 × 10⁻⁸ erg cm⁻² s⁻¹, about a factor of 2 greater than the NICER burst. It also had a peak blackbody temperature of 1.25 keV, which is about 25% larger than that of the NICER burst. We note that this burst appears in the Multi-Instrument Burst Archive (MINBAR) catalog, with a reported peak bolometric flux of 1.6 × 10⁻⁸ erg cm⁻² s⁻¹, and a fluence of 1.67 × 10⁻⁷ erg cm⁻² (Galloway et al. 2020).

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The first evidence of active accretion for this outburst was provided by RXTE/PCA Galactic bulge scan observations on May 31 at 23:00:00 UTC, and indicated a persistent 2–10 keV X-ray flux level of ≈3 mCrab (Markwardt et al. 2005). This flux value is similar to that measured with NICER for OBSID 2050260109 during the 2019 outburst (see Table 1). The X-ray burst was observed approximately 25.7 hr later, and MINBAR reports a persistent flux at the time of the burst of 8.6 × 10⁻¹⁰ erg cm⁻² s⁻¹, just a bit larger than the value estimated prior to the 2019 NICER burst (again, see Table 1). We can use the pre-burst flux value reported by MINBAR and the earliest RXTE observations of the 2005 outburst reported by Markwardt et al. (2005) and Wijnands et al. (2005) to estimate the persistent, accretion-driven fluence prior to the weak 2005 burst. Evaluating a simple trapezoidal sum gives a value of 3.8 × 10⁻⁵ erg cm⁻² that is approximately half of the estimated fluence prior to the 2019 NICER event. This then suggests a total accreted column just prior to the 2005 RXTE event of about half that estimated for the 2019 NICER burst. Simply scaling our value estimated for the 2019 NICER burst suggests a range of 2.0–2.6 × 10⁷ g cm⁻² for the total accreted column prior to the 2005 RXTE event.

3.4. Subsequent Bursts Detected with NuSTAR

Additional observations of J1808 were collected with NuSTAR between 2019 August 10 and 11 (MJD 58705.5–58706.5). While these data do not cover the time of the weak X-ray burst observed with NICER, NuSTAR did catch two subsequent bursts, providing some additional, interesting context to this early phase of the outburst. We processed the NuSTAR data (ObsID 90501335002) using nustardas version 2.1.2. Source data were extracted in the 3–79 keV energy range from a 40'' circular region centered on the source coordinates. The background was extracted using the same approach, but with the extraction region positioned in the background field. The NuSTAR light curve reveals two X-ray bursts, the first of which occurred 24.8 hr after the weak NICER burst, while the second occurred another 11 hr later. This light curve is shown in Figure 6. We emphasize that although some of the NICER exposure was simultaneous with NuSTAR, this did not include these two bursts, and they were only observed with NuSTAR.

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We first investigate the persistent emission by extracting a spectrum from a 100 s window just prior to the first NuSTAR burst. As can be seen in Figure 6, this epoch was simultaneously observed with NICER, so we also extracted the contemporaneous NICER spectrum to obtain broadband energy coverage. We model this spectrum using the same persistent emission used previously (see Table 1), allowing for a constant cross calibration factor between NICER and FPMA/B of NuSTAR. In keeping with the analysis of the NICER burst, we extrapolated the best spectral model over 0.1–100 keV to find a bolometric flux estimate of 1.47 ± 0.05 × 10⁻⁹ erg s⁻¹ cm⁻².

From the recurrence times between the observed bursts, we obtain an estimate of the fluence due to the accretion of 1.3 × 10⁻⁴ erg cm⁻² and 5.8 × 10⁻⁵ erg cm⁻² for the two bursts, respectively. Converting these measurements to column depths, we use Equation (4) to find 8.4 × 10⁷ and 3.7 × 10⁷ g cm⁻², respectively, where we again assumed a 1.4 M_⊙ neutron star mass and an 11 km stellar radius. These column depths are of the same order as the one we calculated for the initial NICER burst. Indeed, given the observed 11 hr recurrence time between the two NuSTAR bursts, and the relatively constant persistent flux (and hence accretion rate), it is conceivable that a similar burst was missed in the gap between the weak NICER burst and the first NuSTAR burst. If so, then the accretion fluence for the two NuSTAR bursts would be essentially consistent with each other.

To explore the burst spectra, we proceeded by dividing the bursts into multiples of 1/8 s such that each bin contains at least 500 counts. We extract a spectrum for each bin and model it using an absorbed blackbody in addition to the fixed persistent emission. The inferred burst properties obtained from these fits are listed in Table 2. The two NuSTAR bursts had fluences of 5 × 10⁻⁷ erg cm⁻² and 3 × 10⁻⁷ erg cm⁻², respectively. This means that they are about a factor of 4–7 times more energetic than the weak X-ray burst observed with NICER. The first NuSTAR burst reached a peak flux 10 times greater than that of the weak NICER burst, and it was also significantly "hotter," reaching a peak blackbody temperature of 2.3 keV. At the same time, these bursts remain much fainter than the Eddington-limited bursts observed at later times in the outbursts of J1808, which typically have fluences of 2 ∼ 4 × 10⁻⁶ erg cm⁻² (Galloway et al. 2008; in't Zand et al. 2013).

Table 2. Burst Parameters

Parameter	NICER 1	NuSTAR 1	NuSTAR 2	NICER 2
Onset (MJD, TT)	58704.8068	58705.8459	58706.3058	58716.0861
Peak flux (erg s−1 cm−2)	7 × 10−9	7 × 10−8	4 × 10−8	3 × 10−7
Burst fluence (erg cm−2)	7 × 10−8	5 × 10−7	3 × 10−7	2 × 10−6
Accretion fluence (erg cm−2)	8 × 10−5	1.3 × 10−4	5.8 × 10−5	⋯
Peak kT (keV)	1.0	2.3	1.7	2.5

Note. The properties of the second NICER burst are taken from Bult et al. (2019a) as an example of a bright Eddington-limited X-ray burst from J1808.

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Based on the considerations above we suggest a scenario similar to that discussed in the work of Cooper & Narayan (2007) and Peng et al. (2007) as a working hypothesis to account for the weak bursts observed by NICER and RXTE during the early days of the 2019 and 2005 outbursts of J1808. As accretion begins, the neutron star is cool enough and the accretion rate is low enough that CNO hydrogen burning in the accumulating layer occurs in the temperature-sensitive regime. At these lower temperatures, ≲5 × 10⁷ K, very little hydrogen is burned. Significant burning of hydrogen will only begin when the accumulated column reaches the conditions for the thermal instability to set in. For a temperature of ≈5 × 10⁷ K this will occur at a column depth of about 3 × 10⁷ g cm⁻². This value is not too dissimilar from the column estimated just prior to the 2005 event. When the initial accumulating layer reaches the ignition depth, the hydrogen instability occurs, triggering a hydrogen flash. We suggest that the initial rapid increase in the nuclear energy generation rate ultimately results in the "heat pulse" that is observed as the weak X-ray burst; however, we think that more sophisticated, multidimensional theoretical calculations of the time-dependent nuclear energy generation coupled with the subsequent heat and radiation transport, will be needed to test the details of this hypothesis. After the initial hydrogen ignition, the burning layer will reach a high enough temperature that subsequent hydrogen burning can proceed at the thermally stable level appropriate to the hot-CNO cycle. Above, we have argued that the observed offset between the pre- and post-burst flux levels of the 2019 event is consistent with this "quasi-steady" burning phase. This source of heat will keep the layer warm enough for burning to continue for a time, likely measured in hours if conditions are not too dissimilar from those modeled by Cooper & Narayan (2007). During this time the quasi-stable burning will increase the helium fraction of the layer. Given the gaps in NICER coverage after the weak burst, we cannot say how long this "quasi-steady" burning may have persisted, but we note that observations ≈3.5 hr after the burst show a count rate and flux approximately consistent with the pre-burst level. For the conditions described above, that is, a hydrogen ignition column of ≈3 × 10⁷ g cm⁻², such an initial hydrogen flash is unlikely to produce prompt helium ignition, simply because at that column depth the helium will not be thermally unstable (Cumming 2004).

As accretion continues, the hydrogen layer or layers that initially flashed will be pushed deeper, to higher column depths. The freshly accreted material above it will also reach the hydrogen ignition depth, and if so, produce another hydrogen flash, assuming its temperature is low enough. In this way, a sequence of hydrogen flashes could be produced. Eventually, the helium-enriched layers will likely reach column depths where the helium will ignite, producing more energetic, mixed H/He bursts. We suggest that the observed NuSTAR bursts are the result of this process. The steadily increasing accretion rate will also be an important variable, as this will tend to increase the temperature of the accreting layers. More complete theoretical modeling of this process will have to include the time-varying accretion rate (Johnston et al. 2018).

If the above scenario is approximately correct we can try to speculate further regarding a few other details of the observations. The 2005 event observed with RXTE was the earlier event in terms of the time since outburst onset, occurring approximately 1 day after onset. Other things being equal one would expect the accreting layer to be cooler than at later times, such as the 2.9 days post-outburst from the 2019 event. A cooler shell will have a larger unstable column, so that this could perhaps explain the fact that the RXTE event is the more energetic of the two weak bursts. This also provides some tentative evidence that the 2019 NICER event may have been preceded by at least one additional weak burst that was missed.

4.1. Remaining Uncertainties and Alternative Interpretations

This work was supported by NASA through the NICER mission and the Astrophysics Explorers Program. This research also made use of data and/or software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA/GSFC and the High Energy Astrophysics Division of the Smithsonian Astrophysical Observatory. We thank Ed Brown, Andrew Cumming, and Duncan Galloway for comments that helped to improve this manuscript. P.B. acknowledges support from NASA through the Astrophysics Data Analysis Program (80NSSC20K0288) and the CRESST II cooperative agreement (80GSFC21M0002).

Facilities: NICER - , ADS - , HEASARC. -

Software: NICERDAS (v8), XSPEC (Arnaud 1996).