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

Sierra Casten Department of Physics and Astronomy, Michigan State University,
567 Wilson Road, East Lansing, MI 48824, USA  Tod E. Strohmayer Astrophysics
Science Division and Joint Space-Science Institute, NASA’s Goddard Space
Flight Center, Greenbelt, MD 20771, USA  Peter Bult Department of Astronomy,
University of Maryland, College Park, MD 20742, USA  Astrophysics Science
Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA

###### Abstract

We present a study of weak, thermonuclear X-ray bursts from the accreting
millisecond X-ray pulsar SAX J1808.4–3658. We focus on a burst observed with
the Neutron Star Interior Composition Explorer on 2019 August 9, and describe
a similar burst observed with the Rossi X-ray Timing Explorer in 2005 June.
These bursts occurred soon after outburst onset, $2.9$ and $1.1$ days, after
the first indications of fresh accretion. We measure peak burst bolometric
fluxes of $6.98\pm 0.50\times 10^{-9}$ and $1.54\pm 0.10\times 10^{-8}$ erg
cm-2 s-1, respectively, which are factors of $\approx 30$ and $15$ less than
the peak flux of the brightest, helium-powered bursts observed from this
source. From spectral modeling we estimate accretion rates and accreted
columns at the time of each burst. For the 2019 burst we estimate an accretion
rate of $\dot{M}\approx 1.4\textrm{--}1.6\times 10^{-10}$ $M_{\odot}$ yr-1,
and a column in the range $3.9\textrm{--}5.1\times 10^{7}$ g cm-2. For the
2005 event the accretion rate was similar, but the accreted column was half of
that estimated for the 2019 burst. The low accretion rates, modest columns,
and evidence for a cool neutron star in quiescence, suggest these bursts are
triggered by thermally unstable CNO cycle hydrogen-burning. The post-burst
flux level in the 2019 event appears offset from the pre-burst level by an
amount consistent with quasi-stable hydrogen-burning due to the temperature-
insensitive, hot-CNO cycle, further suggesting hydrogen-burning as the primary
fuel source. This provides strong observational evidence for hydrogen-
triggered bursts. We discuss our results in the context of previous
theoretical modeling.

stars: neutron — X-rays: binaries — X-rays: bursts — X-rays: individual (SAX
J1808.4$-$3658) — radiation: dynamics

††software: NICERDAS (v8), XSPEC (Arnaud 1996)

##  1 Introduction

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-$\alpha$
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
$(p,\gamma)$ reactions on carbon and nitrogen (Fowler & Hoyle, 1965). Also
operating in the cycle are several rate-limiting $\beta^{+}$ decays dependent
on slower weak reaction rates. The cycle is closed by the $(p,\alpha)$
reaction that converts ${}^{15}{\rm N}$ back to ${}^{12}{\rm C}$. This physics
leads to two regimes in which the CNO cycle may operate in this context. Above
temperatures of about $8\times 10^{7}$ K, the rate of the so-called hot-CNO
cycle is limited by the $\beta^{+}$ decays that take ${}^{15}{\rm O}$ and
${}^{14}{\rm O}$ to ${}^{15}{\rm N}$ and ${}^{14}{\rm N}$, respectively. In
this regime the energy generation rate becomes temperature insensitive, that
is, thermally stable. However, for temperatures below $\approx 8\times 10^{7}$
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-$\alpha$ 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 (Wijnands & van der Klis, 1998;
Chakrabarty & Morgan, 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 $3.5\pm 0.1$ 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 ten 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., 2022).

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 §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 §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 NuSTAR observations that began on 2019 August
10 and in which several brighter bursts were detected. We conclude in §4 with
a summary, a brief discussion of relevant uncertainties and other possible
interpretations, and the outlook for future efforts.

##  2 NICER Observations of J1808

In late July 2019, 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 August 1, 2019 (Bult et al., 2020). NICER is an X-ray
observatory that operates on the International Space Station (ISS). It
observes across the 0.2–12 keV X-ray band and provides low-background, high-
throughput ($\approx 1900$ 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), 205026010$mm$, and 25840101$nn$,
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 $<54^{\prime\prime}$, the Earth elevation angle was $>15^{\circ}$,
the elevation angle with respect to the bright Earth limb was $>30^{\circ}$,
and the instrument was not in the South Atlantic Anomaly (SAA). 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 August 6, 2019 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.

Figure 1:  Light curve of the 2019 July–August outburst of J1808 in the 0.4–7
keV band observed with NICER. Count rates were computed in 2 s intervals. Note
the logarithmic scale. The first, weaker burst is evident near day 3. The
bright burst near day 14 was reported on by Bult et al. (2019a).

###  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.

Table 1: Spectral Model Parameters for SAX J1808: Persistent Emission Parameter | 2584010102 (Pre-burst) | 2050260110 | 2050260109
---|---|---|---
$n_{H}$ ($10^{22}$ cm-2) | $0.131\pm 0.016$ | $0.160\pm 0.025$ | $0.177\pm 0.028$
$kT_{in}$ (diskbb, keV) | $0.849\pm 0.026$ | $0.679\pm 0.024$ | $0.528\pm 0.017$
Norm (diskbb) | $27.33\pm 5.36$ | $25.19\pm 6.46$ | $10.82\pm 1.79$
$kT$ (bbodyrad, keV) | $2.03\pm 0.15$ | $1.67\pm 0.15$ |  $\cdots$
Norm (bbodyrad, keV) | $0.98\pm 0.47$ | $0.604\pm 0.336$ |  $\cdots$
Index (power) | $1.96\pm 0.37$ | $2.50\pm 1.05$ | $2.01\pm 0.23$
Norm (power) | $0.0266\pm 0.0137$ | $4.64\times 10^{-3}$ | $4.23\pm 1.6\times 10^{-3}$
E (gauss, keV) | $0.990\pm 0.009$ | $0.935\pm 0.014$ |  $\cdots$
$\sigma_{E}$ (gauss, keV) | $0.015$ | $0.015$ |  $\cdots$
Norm (gauss) | $8.44\pm 2.03\times 10^{-4}$ | $3.51\pm 1.00\times 10^{-4}$ |  $\cdots$
$f_{0.1-20}$ (erg cm-2 s-1) | $7.06\pm 0.23\times 10^{-10}$ | $2.08\pm 0.40\times 10^{-10}$ | $5.35\pm 0.81\times 10^{-11}$
$f_{bol}$ (erg cm-2 s-1) | $7.9\times 10^{-10}$ | $2.5\times 10^{-10}$ | $6.4\times 10^{-11}$
$\chi^{2}$ (dof) | 117.9 (112) |  $98.9$ (97) | $94.5(106)$
Rate ($s^{-1}$, 0.5-10 keV) | $181.9\pm 0.4$ | $58.70\pm 0.25$ | $13.46\pm 0.07$
Epoch (d) | 2.89 | 1.45 | 0.59
Exposure (s) | 740 | 927 | 2807

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), that 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 $\chi^{2}=117.9$
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 $7.06\pm 0.23\times 10^{-10}$ erg cm-2 s-1. If we extend
the bandpass to estimate a bolometric flux we find a value (0.1 $-$ 100 keV)
of $7.9\times 10^{-10}$ erg cm-2 s-1. The average count rate in the fitted
energy band (0.5 $-$ 10 keV) is $181.9\pm 0.5$ s-1, so we estimate a flux per
NICER count rate (0.5 $-$ 10 keV) of $4.34\times 10^{-12}$ erg cm-2 s-1 $({\rm
counts}\;{\rm s}^{-1})^{-1}$ for this interval.

Figure 2:  Light curve of the first, weak burst from J1808 in the 0.4 - 7 keV
band observed with NICER. Main panel: The count rates were computed in 1 s
intervals, and the vertical dashed and dotted lines denote the intervals used
to extract spectra for the pre- and post-burst spectral modeling,
respectively. Inset panel: The same data are used, but the time bins are 16 s,
and the logarithmic scale highlights the offset in count rate between the pre-
and post-burst emission. The dashed red line is a constant value fit to the
pre-burst level, and is meant as a guide to the eye.  Figure 3:  Energy
spectrum of the pre-burst interval, modeled with a phenomenological model
similar to that employed by Patruno et al. (2009), which includes diskbb,
bbodyrad and powerlaw components, in addition to the line at 1 keV. See the
text in §2.1 for further details.

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 $2.85\times 10^{-12}$ and $3.61\times 10^{-12}$ erg cm-2 s-1 $({\rm
counts}\;{\rm s}^{-1})^{-1}$, 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 $\chi^{2}$
of 31.3, and the ratio of the line normalization to its 1$\sigma$ uncertainty
is $\approx 4.2$. The line is also evident in OBSID 2050260110, though at
lower significance, with the ratio of the line normalization to its 1$\sigma$
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 $\approx
0.09$ keV (3$\sigma)$. 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 since 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 $E_{p}=7.92\times 10^{-5}$ 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,

$L_{p}=4\pi d^{2}\xi_{p}f_{p}=\frac{z\dot{M}c^{2}}{(1+z)^{3}}\;,$ (1)

where ${(1+z)=(1-2{GM}/{c}^{2}R)}^{-1/2}$, $\dot{M}$, $\xi_{p}$, $f_{p}$, $M$
and $R$ are the surface red-shift, 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, $\xi_{p}$, can be thought of as the solid angle into which
the radiation is emitted, normalized by $4\pi$, thus, isotropic emission is
characterized by $\xi_{p}=1$. We write the accreted mass column in the local
neutron star frame as,

$y_{a}=\frac{M_{a}}{4\pi R^{2}}=\frac{\int\dot{M}(t^{\prime})dt^{\prime}}{4\pi
R^{2}}\;,$ (2)

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

$y_{a}=\frac{d^{2}\xi_{p}(1+z)^{2}}{zc^{2}R^{2}}\int
f_{p}(t)(1+z)dt^{\prime}=\frac{d^{2}\xi_{p}(1+z)^{2}}{zc^{2}R^{2}}E_{p}\;,$
(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 $E_{p}=7.92\times
10^{-5}$ erg cm-2, we can write the accreted column as,

$y_{a}=1.03\times
10^{7}\;\left(\frac{\xi_{p}(1+z)^{2}}{zR_{10}^{2}}\right)\;\;\;{\rm g}\;{\rm
cm}^{-2}\;,$ (4)

where $R_{10}$ is the neutron star radius in units of 10 km. For
$M=1.4M_{\odot}$, $R=11$ km, and adopting $\xi_{p}=1$ we find
$y_{a}=5.12\times 10^{7}$ g cm-2. With $M=2.0M_{\odot}$ and $R=11$ km we find
that $y_{a}$ decreases slightly to $3.91\times 10^{7}$ 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
$7.9\times 10^{-10}$ erg cm-2 s-1, and a distance of $3.5$ kpc, we find,

$\dot{M}=2.03\times
10^{-11}\;\left(\frac{\xi_{p}(1+z)^{3}}{z}\right)\;M_{\odot}\;{\rm
yr}^{-1}\;.$ (5)

Using the same parameter assumptions as above, we find estimates of
$\dot{M}=1.56\times 10^{-10}$ and $1.38\times 10^{-10}$ $M_{\odot}$ 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 $\chi^{2}$ are shown in Figure 4. We found a peak
bolometric burst flux of $f_{b}=6.98\pm 0.50\times 10^{-9}$ erg s-1 cm-2.
Using trapezoidal numerical integration of the flux, we calculated a
bolometric fluence of $7.05\pm 1.16\times 10^{-8}$ erg cm-2. The burst
luminosity is defined as $L_{b}=4\pi d^{2}\xi_{b}f_{b}$, where $\xi_{b}$
characterizes the anisotropy of the burst emission. Adopting $\xi_{b}=1$, and
with $d=3.5$ kpc, we can then estimate that the total energy released during
the burst was $1.03\times 10^{38}$ ergs.

Figure 4:  Evolution of the weak X-ray burst derived from spectral modeling in
the $0.5-10$ keV band. We show from the top down: the bolometric flux,
blackbody temperature, blackbody radius (at 3.5 kpc), and reduced $\chi^{2}$,
respectively. The error bars indicate 1-$\sigma$ confidence intervals.

##  3 Physical Scenario and Interpretation

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 ($\gtrsim 2M_{\odot}$) 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
$7.2-7.7\times 10^{6}$ K, for neutron star masses between 1.4 and 2.0
$M_{\odot}$. Due to the high thermal conductivity in the core and crust (Brown
& Cumming, 2009), it is thus very likely that when accretion begins in J1808
after a period of quiescence, the accumulating layer starts out at
temperatures $\lesssim 1\times 10^{7}$ 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, $\dot{M}_{st}(B)$) as
$2.7\times 10^{-10}$ $M_{\odot}$ yr-1, for a neutron star mass and radius of
$1.41M_{\odot}$ 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
that 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 $\approx
10\%$, which would bring the value to $\approx 3\times 10^{-10}$ $M_{\odot}$
yr-1. Expressed as a fraction of the Eddington accretion rate,
$\dot{M}_{Edd}$, and adopting the value of $\dot{M}_{Edd}=1.8\times 10^{-8}$
$M_{\odot}$ yr-1 (Cumming, 2004), this is then equivalent to
$\dot{M}=0.0167\;\dot{M}_{Edd}$. Note also that Cumming (2004) quotes a value
of $\dot{M}\gtrsim 0.01\;\dot{M}_{Edd}$ 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 $\lesssim 0.003\;\dot{M}_{Edd}$.

Above, we estimated an accretion rate at the time of the weak NICER burst in
the range from $\approx 1.38-1.56\times 10^{-10}$ $M_{\odot}$ yr-1
($\dot{M}\approx 0.0077-0.0087\;\dot{M}_{Edd}$). 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 §2 above we estimated the accreted column to be in the range
$\approx 3.91-5.12\times 10^{7}$ g cm-2, and we measured an energy fluence in
the burst of $\approx 1\times 10^{38}$ 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 Fig. 1), Galloway & Keek (2021,
see their Fig. 2), and Peng et al. (2007, see their Fig. 4). Based on these
curves, we can estimate that a column of this size would be ignited at a
temperature in the range from $\approx 4-5\times 10^{7}$ 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, $\Delta X$, of the hydrogen needs to burn in order
to raise the temperature. For a temperature change of $10^{8}$ K, they
estimate $\Delta X\approx 0.002$ (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 $10^{8}$ K would render the shell unstable to
helium ignition, promptly producing a mixed H/He burst. We note that the work
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
$\dot{m}=0.002\;\dot{m}_{Edd}$ Cooper & Narayan (2007, see their Fig. 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 $\sim 2\times 10^{8}$ K are likely. Peng et al. (2007, see
their Fig. 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 $E_{h}=6.4\times 10^{18}$ erg g-1 (Clayton,
1983), we would require $m_{h}=1.6\times 10^{19}\;(1+z)$ 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, $y_{h}=m_{h}/4\pi R^{2}$, and assuming $R=11$ km, we find
$y_{h}=1.05\times 10^{6}\;(1+z)$ 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,

$f_{h}=\frac{y_{h}}{y_{a}X}=0.105\;\frac{(1+z)}{Y_{a}X}\;,$ (6)

where $Y_{a}$ is the estimated accreted column in units of $10^{7}$ g cm-2.
Taking $Y_{a}$ in the range from $3.9-5.1$, 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
$0.04-0.06$. 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 $\Delta T=10^{8}$ K; we don’t 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, since 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 $Q_{nuc}=(1.3+5.8X)\times
10^{18}$ erg g-1 (Galloway & Cumming, 2006), and again adopting $X=0.7$ we
would expect $\approx 3.2-4.2\times 10^{39}$ ergs 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 Fig. 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
$\approx 6.9\times 10^{-9}$ erg cm-2 s-1. If we scale this by the peak flux
($2.3\times 10^{-7}$ erg cm-2 s-1) of the Eddington-limited burst observed
later in the outburst (Bult et al., 2019a), we find a ratio of
$\log(0.03)=-1.52$. Looking at Fig. 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?

Previous theoretical studies concluded that the ignition of the hydrogen flash
will raise the temperature in the layer to at least the stable burning regime,
and likely higher. Thus, hot-CNO cycle burning would be expected to continue
for some period of time after the unstable ignition. Can we see evidence for
such stable burning in the NICER data? Interestingly, there is a clear
“offset” between the pre- and post-burst flux levels. This offset can be seen
in Figure 2. Note that the inset panel uses a larger time bin size and log
scale to emphasize the persistent count rate levels, to more clearly highlight
the offset. We also plot the average count rate value for the pre-burst level
(red dashed line) as a guide to the eye. To explore this question further we
used the same spectral model to characterize the post-burst data as we used
for the pre-burst and other persistent emission intervals. The time interval
used for the post-burst spectral extraction is marked by the vertical dotted
lines in Figure 2 (main panel). We first tried to fit the post-burst spectrum
using the same spectral shape as obtained from the pre-burst interval,
allowing for the constant, $f_{a}$ parameter to make up the flux difference.
This did not provide an acceptable fit, and suggests the presence of an
additional spectral component in the post-burst interval. To explore this
further we subtracted the pre-burst spectrum from the post-burst and found
that the remaining excess could be well fit by a soft thermal spectrum,
characterized as a blackbody with $kT=0.51\pm 0.02$ keV, normalization of
$82.5\pm 12.0$, and bolometric flux of $6.1\pm 0.2\times 10^{-11}$ erg cm-2
s-1. This is equivalent to a luminosity of $\approx 8.9\times 10^{34}$ erg s-1
(at 3.5 kpc).

If the hydrogen burns stably at the same rate as it is accreted, then we would
estimate a hydrogen-burning luminosity of $L_{h}=X\dot{m}E_{h}$, where $X$ is
the mass fraction of hydrogen in the accreted fuel, $\dot{m}$ is the mass
accretion rate at the burst onset, and $E_{h}$ is the energy production per
gram due to hydrogen-burning. With $\dot{m}=1.4\times 10^{-10}$ $M_{\odot}$
yr-1, $X=0.7$, and $E_{h}=6.4\times 10^{18}$ erg g-1, we would predict a
stable hydrogen-burning luminosity of $\approx 4\times 10^{34}$ erg s-1, which
is a good fraction of the measured offset. Perhaps a better estimate can be
obtained by evaluating the energy production rate associated with the
saturated, hot-CNO burning rate as, $L_{CNO}=4\pi R^{2}y_{a}\epsilon_{CNO}$,
where $\epsilon_{CNO}$, $y_{a}$, and $R$, are the energy production rate due
to hot-CNO burning, the accreted column depth, and the neutron star radius,
respectively. With $\epsilon_{CNO}=5.8\times 10^{13}(Z_{CNO}/0.01)$ erg g-1
s-1 (Cumming & Bildsten, 2000), $y_{a}=4.5\times 10^{7}$ g cm-2, and $R=11$
km, we find $L_{CNO}\approx 4\times 10^{34}(Z_{CNO}/0.01)$ erg s-1. Here,
$Z_{CNO}$ is the abundance of the CNO catalyzing elements. Employing the solar
value $Z_{CNO}=0.016$, we find $L_{CNO}=6.4\times 10^{34}$ erg s-1, however,
as noted above, at these low accretion rates sedimentation is very likely to
be effective in enhancing the abundance of CNO elements near the base of the
accreted fuel layer. For example, Peng et al. (2007, see their Figs. 2 & 3)
report enhancements in CNO element abundances by factors of 2 to 5, depending
on the accretion rate.

Based on these estimates it appears plausible that most or all of the observed
flux offset can be accounted for by quasi-steady, hot-CNO burning of hydrogen.
We note that the thermal nature of the spectral excess, and its $\approx 0.5$
keV temperature, similar to that at late times during the weak burst, is also
consistent with this interpretation. This conclusion is also consistent with
the hydrogen flash temperature and flux evolution calculations of Cooper &
Narayan (2007). As an example, the hydrogen flashes shown in their Fig. 4
indicate that at ignition the flux rises abruptly, but then shows a “plateau-
like” phase which decays over a timescale of several hours. The average flux
levels near the beginning of these events are approximately consistent with
the stable hydrogen-burning luminosity we estimated above. Once ignited, these
flashes are burning hydrogen to helium in the fuel layer at essentially the
saturated, hot-CNO cycle rate. We suggest that the two-zone model of Cooper &
Narayan (2007) (with H and He zones) probably does not adequately track and
resolve the fast, initial hydrogen-burning when the thermal instability is
initiated, but better predicts the longer timescale, thermally stable burning.
The hydrogen-only ignition modeled by Peng et al. (2007, see their Fig. 7)
also appears at least approximately similar to what is observed for the weak
NICER burst. Indeed, the ratio of the peak burst bolometric flux to the
persistent, pre-burst flux is $7\times 10^{-9}/7.9\times 10^{-10}\approx 8.8$,
which is similar to the peak value of $F_{cool}/F_{acc}$ for the initial,
burst-like flux increase shown in their Figure 7 (middle panel), and the
overall burst duration appears consistent with the observed burst as well.

More detailed radially resolved, and perhaps multidimensional calculations
will likely be needed to more accurately track the rapid hydrogen ignition
phase which we suggest may account for the weak NICER burst. To briefly
summarize, the weak NICER burst and post-burst flux offset appear to be
consistent with the onset of a hydrogen-triggered shell flash in the cool,
temperature-sensitive regime of the CNO cycle. The ignition column was likely
shallow enough that the subsequent temperature increase was not sufficient to
also promptly ignite a helium-burning instability.

###  3.2 Missed bursts?

While NICER was able to begin observations quite close to the onset of
accretion in the 2019 August outburst, the overall on-source coverage from
onset to the time of the first observed burst was still rather modest, with a
duty-cycle of about 4%. Thus, if other bursts occurred it is conceivable that
the NICER observations simply missed them. However, based on our estimate of
the size of the accreted column, as well as current theoretical estimates of
the hydrogen ignition curve, we argue that likely only a few such bursts might
have been missed. Firstly, while we don’t know the precise temperature
trajectory of the initial accumulating layer, it cannot plausibly be $\lesssim
2\times 10^{7}$ K because at such low temperatures only columns much larger
($\gtrsim 2-3\times 10^{8}$ gm cm-2) than our estimate of the accreted column
at the time of the weak burst ($3.9-5.1\times 10^{7}$ g cm-2) would be needed
to ignite unstable burning, and such an ignition would also very likely lead
to a bright, mixed H/He burst, which was not observed, though could have
perhaps been missed. Secondly, as the temperature of the fuel layer increases
the size of the unstable column decreases, however, above temperatures of
about $8\times 10^{7}$ K the hydrogen-burning will stabilize, precluding
bursts. This sets a minimum combustible column for hydrogen ignition which is,
using the ignition curve in Cumming (2004) as a guide, $\approx 1\times
10^{7}$ g cm-2. Based on our estimated accreted column this would set a limit
of not more than about five such bursts potentially being produced, as that
would just about exhaust the total column accreted at the time of the weak
burst. Another benchmark can be set by the accretion rate. We estimated a
value of $\dot{m}=1.4-1.6\times 10^{-10}$ $M_{\odot}$ yr-1 at the time of the
weak burst. If we take half of this value as more representative of the mean
rate during the 72 hrs prior to the weak burst, we can estimate the time
required to accrete the minimum unstable column of $1\times 10^{7}$ g cm-2.
For $\dot{m}=7\times 10^{-11}$ $M_{\odot}$ yr-1, and assuming a radius $R=11$
km, we find it would take 9.5 hr to accumulate such a column. Since the weak
burst was observed after about 2.9 days, this also suggests an upper limit of
$\sim 7$ to the total number of such weak bursts. We suggest that the actual
temperature trajectory is probably somewhere between the two extremes
described above, perhaps consistent with an unstable column on the order of
$\sim 2-3\times 10^{7}$ gm cm-2. If this is correct it would suggest that the
NICER observations may have missed one or two such weak 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 2 June at approximately 00:42:30 TT, and is evident near
0.25 days 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 $1.54\pm 0.11\times
10^{-8}$ 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 $1.6\times 10^{-8}$ erg cm-2 s-1, and a fluence of
$1.67\times 10^{-7}$ erg cm-2 (Galloway et al., 2020).

Figure 5:  Light curve from RXTE data (PCU 2, 3–30 keV) of the 2005 outburst
from J1808. Note the logarithmic scale. A weak X-ray burst is seen early in
this outburst. Much brighter and energetic bursts are seen near days 4 and 8.
Note that the burst near day 8 was truncated by the RXTE exposure, and almost
certainly the brightest part of this event was missed.

The first evidence of active accretion for this outburst was provided by
RXTE/PCA Galactic bulge scan observations on 31 May at 23:00:00 UTC, and
indicated a persistent 2 - 10 keV X-ray flux level of $\approx 3$ 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 $8.6\times 10^{-10}$ 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 $3.8\times 10^{-5}$ 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 $2.0-2.6\times
10^{7}$ 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 $40\arcsec$ 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 hours after the weak NICER
burst, while the second occurred another 11 hours later. This light curve is
shown in Figure 6. We emphasize that though some of the NICER exposure was
simultaneous with NuSTAR, this did not include these two bursts, and they were
only observed with NuSTAR.

Figure 6: Light curves from NICER (black, left axis) and NuSTAR (red, right
axis) around the time of the weak X-ray burst at t=0. Both light curves are
calculated using an 8-s time resolution.

We first investigate the persistent emission by extracting a spectrum from a
$100$ second 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 $1.47\pm 0.05\times 10^{-9}$ 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 $1.3\times 10^{-4}$ erg cm-2 and
$5.8\times 10^{-5}$ erg cm-2 for the two bursts, respectively. Converting
these measurements to column depths, we use equation 4 to find $8.4\times
10^{7}$ and $3.7\times 10^{7}$ g cm-2, respectively, where we again assumed a
1.4 $M_{\odot}$ 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 seconds 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 $5\times 10^{-7}$ erg cm-2 and $3\times 10^{-7}$ 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 ten 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 $2\sim 4\times 10^{-6}$ 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) | $7\times 10^{-9}$ | $7\times 10^{-8}$ | $4\times 10^{-8}$ | $3\times 10^{-7}$
Burst fluence (erg cm-2) | $7\times 10^{-8}$ | $5\times 10^{-7}$ | $3\times 10^{-7}$ | $2\times 10^{-6}$
Accretion fluence (erg cm-2) | $8\times 10^{-5}$ | $1.3\times 10^{-4}$ | $5.8\times 10^{-5}$ |  $\cdots$
Peak kT (keV) | 1.0 | 2.3 | 1.7 | 2.5

##  4 Summary, Caveats & Outlook

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, $\lesssim 5\times 10^{7}$ 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 $\approx 5\times 10^{7}$ K this
will occur at a column depth of about $3\times 10^{7}$ 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 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, multi-dimensional 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 $\approx 3.5$ hrs
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 $\approx 3\times 10^{7}$ g cm-2, such an initial hydrogen
flash is unlikely to produce a 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

In estimating accretion rates and accreted columns we allowed for variation in
the neutron star mass, however, there are other uncertainties which complicate
such estimates. These include the source distance, anisotropy factors,
bolometric corrections, and the line-of-sight absorption. We note that the
more recent work of Goodwin et al. (2019) reports a slightly closer distance
of $3.3^{+0.3}_{-0.2}$ kpc for J1808. While their quoted uncertainty range
includes the 3.5 kpc value we have adopted, a decrease from 3.5 to 3.3 kpc
would reduce our estimates by a factor of $0.9$. These authors also provide
estimates of the anisotropy factors for both persistent and burst emission,
finding $\xi_{p}=0.87^{+0.12}_{-0.10}$ and $\xi_{b}=0.74^{+0.10}_{-0.10}$.
Applying these values would also reduce the estimated accretion rate and
column, by a factor of 0.87, and decrease our estimate of the burst peak
luminosity and fluence, by a factor of 0.74. Adopting the best values reported
by Goodwin et al. (2019) for both $d$ and $\xi_{p}$ would reduce the estimated
accretion rate and accreted column by a factor of 0.77.

We have argued that the weak bursts result from, principally, hydrogen-
burning, but are there other possibilities involving the unstable burning of
helium? One scenario that can produce weak or underluminous bursts is the
phenomenon of short recurrence time (SRT) bursts (Keek et al., 2010). The idea
behind SRT events is that they burn fuel remaining from a preceding burst. In
the present case, a preceding, larger burst would have had to occur (and been
missed) for this idea to be workable. In principle, this could account for the
observed weak bursts, but there are some difficulties with this
interpretation. First, the estimated accreted columns are uncomfortably low.
This scenario would require that a relatively bright, mixed H/He burst would
have occurred prior to the observed weak events, and been missed in each case.
As discussed above, this would require relatively large ignition columns,
likely $\gtrsim 2\times 10^{8}$ g cm-2, which is much larger than the
estimated columns present just prior to each weak event. Our accretion column
estimates would have to be underestimated by factors of 4 - 5 for this to be
more plausible. Second, J1808 is not currently known to produce SRT events.
There has been reasonably good coverage of past J1808 outbursts, and no SRT
events have been definitively observed. For example, the compilation of SRT
burst observations by Keek et al. (2010) does not include J1808, and we also
note that the 401 Hz spin frequency for J1808 is less than the faster,
$\gtrsim 500$ Hz, spins associated with some of the known SRT sources. Third,
the flux offset between the pre- and post-burst emission seems to make more
sense in the context of stable hydrogen-burning than what might be expected
from an SRT event, for which one would not typically expect to find such a
flux offset. We note also that the theoretical mechanism of opacity-driven
convective mixing explored by Keek & Heger (2017) to account for SRT bursts
occurs for ignition in relatively hot envelopes, which seems less applicable
to the low accretion rate regime near burst onset that we have described
above. It is difficult to completely rule out the SRT scenario, but we think
the considerations above argue against it.

We have argued that the early accretion outburst evolution onto a “cool”
neutron star in J1808 provides a unique environment to explore the physics of
nuclear burning on neutron stars, and most interestingly, the ignition of
unstable hydrogen-burning in the temperature-sensitive regime of the CNO
cycle. We suggest that the weak bursts seen by NICER and RXTE in the 2019 and
2005 outbursts, respectively, may result from this process. More complete,
continuous observational coverage of the first 4-5 days of subsequent
outbursts from J1808 could definitively test this hypothesis. Such data would
also provide for detailed physical comparisons with new theoretical efforts to
track the outcome of time-varying accretion onto neutron stars and the
subsequent nuclear burning of the accreted matter. This could provide
interesting constraints on such things as the accretion rate, the thermal
profile of the accreting matter and the nuclear energy generation and
subsequent heat transport in the accreted layers.

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).
NICER, ADS, HEASARC

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