DEPENDENCE OF X-RAY BURST MODELS ON NUCLEAR REACTION RATES

The 1D multi-zone X-ray burst model used in this study is based on the hydrodynamics code Kepler (Weaver et al. 1978), and is discussed in detail in Woosley et al. (2004). This model couples the energy generation of a complete, adaptive nuclear reaction network of over 1300 isotopes to a one-dimensional hydrodynamic simulation of the accretion on an adaptive Lagrangian grid, nuclear burning, radiative and convective energy transport, and mixing of the isotopic composition both during and between bursts. A "smooth" accretion scheme as well as an updated implementation of electron conduction are employed (Keek & Heger 2011). The radial dimension of the accreted layer is resolved into typically around 200 zones. The model follows a sequence of bursts, taking into account steady-state nuclear burning in between bursts and compositional inertia effects, where ashes from preceding bursts partially mix with freshly accreted material and affect the nuclear processes in subsequent bursts.

The specific accretion model used for these studies is model ZM from Woosley et al. (2004), with material of solar metallicity (hydrogen mass fraction X = 0.7048, helium mass fraction Y = 0.2752, and ¹⁴N mass fraction 0.0200) accreted at a rate of 1.75 × 10⁻⁹ M_⊙ yr⁻¹, roughly 10% of the Eddington mass accretion rate. This model represents a typical burster powered by mixed hydrogen and helium burning, producing longer bursts with timescales well beyond 10 s, up to minutes (Galloway et al. 2004). A version of this model at slightly lower accretion rate produced the best agreement with observations to date (Heger et al. 2007).

The neutron star is taken to be 10 km in radius and to have a mass of 1.4 M_⊙. Since this study involves no comparison to observational data, the general relativistic correction from the frame of the neutron star to an observer at infinity will be neglected (Woosley et al. 2004). All times are therefore given in the reference frame of the neutron star surface.

The multi-zone X-ray burst model was first run for a long sequence of bursts (∼50) using the unmodified baseline reaction rate database, ReaclibV1.0. The long sequence of bursts serves as a baseline and ensures enough burst statistics to characterize steady-state burst properties, including burst-to-burst variations that affect the identification of sensitivities to changes in reaction rate. The models with varied reaction rates were run for shorter times for a set number of simulation time steps. This limits the computational time needed for each run. Because the size of the time step in Kepler is dynamically adjusted to optimally resolve the time-dependent behavior, the total simulation time varies among the multi-zone simulations. The multi-zone model calculations with individual reaction rate variations generally produced about 14 bursts, but in all cases, a continuous sequence of at least 12 bursts was simulated. This is sufficient to ensure convergence into a steady-state bursting behavior.

The burst light curves are shown in Figures 1 and 4. The first burst is special, because it occurs atop an inert neutron star substrate (assumed to be iron for the purpose of heat conduction) though in this model it turns out to be quite similar to the following bursts. Already the second burst is very similar to the remaining bursts and there is no evidence for systematic variation beyond the second burst, indicating that reasonable steady-state equilibrium is achieved beginning with the third burst. At this point bursts occur rather regularly, with a recurrence time of t_rec = 175 ± 3 minutes, peak luminosity of L_peak = (1.7 ± 0.1) × 10³⁸ erg s⁻¹, and a total energy released during the burst of E_tot = (6.5 ± 0.1) × 10³⁹ erg (all numbers given in the reference frame of the local neutron star surface). The quoted uncertainties are standard deviations and stem from the remaining intrinsic burst-to-burst variations. These variations do not decrease with the number of bursts and are a feature of the steady-state behavior of the model. In a steady state, bursts have a rise time of 5.3 ± 0.1 s from 10% to 90% of L_peak, and a duration of τ = E_tot/L_peak = 39 ± 3 s. The burst duration from 10% (rise) to 10% (decline) luminosity is 73 ± 5 s.

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In Figure 4 the burst luminosity profiles are shown, shifted in time to align the peak luminosities of all bursts. Burst-to-burst variations affect not only recurrence time and peak luminosity, but also the shape of the light curve.

This study requires direct comparison of two light curve sequences calculated with different nuclear physics. The burst-to-burst variations need to be taken into account for this comparison. In order to define a light curve for comparison, we take the ensemble of bursts starting with the third burst, shift each burst to align the peak luminosity in time, and resample all bursts on a common time grid using linear interpolation. The surface resolution of our model is finer than what was used by Heger et al. (2007), with the mass of the outer zone being up to 100 times lower. At times when the burst is brightest, an individual zone near the surface may get a "kick," and its behavior then differs briefly from the rest of the outer zones. To prevent this from influencing the simulated light curve, we average the luminosity of the outermost nine zones and smooth over a scale of one second to remove any numerical noise. At each point in time we consider the variations in luminosity from all remaining bursts, and determine the average luminosity and its error. This average light curve with error band is used for our sensitivity study. Using an average is justified because observationally precise burst light curves can also be obtained by averaging long sequences of bursts (Galloway et al. 2004).

We also investigate the sensitivity of the steady-state burst ashes to nuclear reaction rate variations. These steady-state burst ashes of a burst sequence are determined in the following way. The deepest zones at the end of our multi-burst simulation sequence are special because they represent the ashes of the first bursts before a steady-state burst behavior was achieved. Similarly to the procedure adopted for the light curves, we therefore remove ashes from the first two bursts from our analysis to prevent anomalous zones from contributing. Too-shallow zones also have to be excluded because they will continue to be modified by subsequent bursts and therefore do not represent the steady-state ashes. Clearly this includes zones that still contain hydrogen. Deep unburned helium, however, can also be burned during heating from bursts occurring in shallower regions (Woosley et al. 2004). This can be seen in Figure 2, where the helium mass fraction successively decreases with increasing depth. As the helium mass fraction becomes lower, the modifications in the ashes induced by burning of residual helium become less important. We therefore exclude shallow zones with helium mass fractions above ∼3.0 × 10⁻⁵ from the calculation of our final composition. For a typical burst model used with the reaction rate variations, these constraints result in an averaging of the final composition over a depth range of M_acc ∼ (3–6) × 10²¹ g (see Figure 2). We chose the time (rounded to the nearest hundred time steps, because one in every 100 steps is stored) at which the light curve is minimum after the 12th burst as our point for determining this composition.

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Figure 3 shows the resulting isobaric ash composition. The abundance peak around A = 40 stems from the impedance of the nuclear reaction sequence caused by the Z = 20 shell closure at the ${}^{40}{\rm{Ca}}$ waiting point, whereas the abundance peaks around A = 60–64 are produced by the ${}^{60}{\rm{Zn}}$, ${}^{64}{\rm{Ge}}$, and ${}^{68}{\rm{Se}}$ rp-process waiting points, which have particularly long half-lives despite being located at the proton drip line (Schatz et al. 1998). The Ni–Cu and Zn–Ga cycles further enhance abundances in this mass region (van Wormer et al. 1994). The α-chain isotopes (A = 12, 16, 20, 24, 28, 32) are further enhanced by late helium burning once hydrogen is consumed. The time- and zone-integrated reaction flow during a typical X-ray burst is shown in Figure 13 to illustrate the type of reactions that occur.

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The one-zone model ONEZONE is similar to the model of Koike et al. (1999) and has been used previously to estimate the reaction sequence of the rp-process (Schatz et al. 2001). It evolves thermodynamic conditions and composition in a single zone, neglecting gradients in temperature, density, and composition, as well as radiative transport and convection. ONEZONE assumes nuclear burning at constant pressure P. This is justified, because mass accretion during the burst is negligible and only little material is ejected. As the burning happens in a thin layer on the surface, the influence of the neutron star is fully characterized through the surface gravity g (neglecting magnetic fields and rotation). The evolution of the temperature starting at burst ignition is calculated for a time step dt using ${dT}={c}_{P}^{-1}({\epsilon }_{{\rm{nuc}}}+{\epsilon }_{{\rm{cool}}}+{\epsilon }_{\nu }){dt}$ with the specific heat capacity at constant pressure c_P, the positive specific nuclear energy generation rate _nuc, the negative specific surface cooling rate _cool, and the negative specific neutrino loss rate _ν. Here

$$\begin{eqnarray}&&{\epsilon }_{{\rm{nuc}}}=\displaystyle \frac{{\sum }_{i}{{dY}}_{i}{{\rm{\Delta }}}_{i}}{{dt}}\end{eqnarray} \tag{ 1 }$$

is calculated from the changes in abundance dY using atomic mass excesses Δ, assuming that positrons emitted in β⁺ decays are annihilated instantly. For X-ray burst conditions, _ν is entirely due to neutrinos emitted in β⁺ decays. The respective energy losses are taken from Fuller et al. (1982) and Pruet & Fuller (2003). _cool is approximated as (Fujimoto et al. 1981)

$$\begin{eqnarray}&&{\epsilon }_{{\rm{cool}}}=\displaystyle \frac{{{acT}}^{4}}{3\kappa {y}^{2}}\end{eqnarray} \tag{ 2 }$$

where the opacity κ is calculated according to Schatz et al. (1999). a is the radiation constant, c the speed of light, and y = P/g the column density. The change in mass density dρ during a time step dt is calculated from P and dT using an equation of state that includes the pressure of the radiation and of a partially degenerate electron gas (Paczynski 1983). The model is coupled to an implicitly solved nuclear reaction network that includes 688 nuclei from hydrogen to tellurium, which for a given T and ρ calculates dY. We used the same reaction rates as in the multi-zone model.

ONEZONE needs to be coupled with a model predicting the ignition conditions. In the past, a full 1D steady-state hydrogen and helium atmosphere model has been used that evolves the composition as a function of depth during the accretion process until a thermal instability criterion is fulfilled (Cumming & Bildsten 2000). Pressure and composition at that location are then used as initial conditions for ONEZONE. The ignition temperature is not a critical parameter because of the steep temperature rise at the beginning of an X-ray burst. The initial temperature is chosen to be just high enough to initiate a thermonuclear runaway in ONEZONE on a timescale that is sufficiently short to not modify the composition further by any steady-state burning. The results for burst timescale, major reaction sequence, and final composition have been shown to agree reasonably well with the first burst calculated with Kepler (Woosley et al. 2004) and other multi-zone models (Fisker et al. 2008; José et al. 2010). The details of the light curve cannot be predicted accurately, as expected for a one-zone model without radiation transport. As the energy generation and the reaction network are coupled self-consistently, however, ONEZONE can be used to explore the sensitivities of energy generation and the X-ray burst light curve to changes in nuclear reaction rate, provided realistic ignition conditions are chosen.

Our goal here is for the single-zone model to most closely resemble a typical burst of the multi-zone model calculation, not the first burst. We therefore extract the ignition conditions for ONEZONE from the baseline calculation of our multi-zone burst model shortly before a typical burst is ignited. The ignition time is defined by significant breakout from the CNO cycle via the ¹⁵O(α, γ) reaction, and is identified from a peak in the ${}^{15}{\rm{O}}$ abundance. The conditions for ONEZONE model ignition are taken at that time from the zone for which the ONEZONE results for light curve and final composition most closely resemble the results from the multi-zone model for total luminosity and steady-state composition. The chosen ignition conditions were a temperature of 0.386 GK, a pressure of 1.73 × 10²² erg cm⁻³, and hydrogen and helium mass fractions of 0.51 and 0.39, respectively.

With these ignition conditions, the peak temperature in ONEZONE agrees with the peak temperature in the hottest zone of the multi-zone model (1.2 GK). Figures 3 and 4 show that ONEZONE does predict the burst light curve and ashes composition reasonably well. The main difference in the light curve is the sudden drop in luminosity defining the end of the burst in the single-zone model. This is mainly the result of the absence of radiation transport modeling in the one-zone model. As far as the burst ashes are concerned, ONEZONE predicts quite well the main features of the composition, including the A ≈ 40 peak and subsequent abundance drop, and the main components of the ashes at A = 56, 60, 64, 68, 72, and 76. The main difference is the additional helium burning in the multi-zone model, which is induced by heating of the ashes in subsequent bursts and is not included in the one-zone approach. Compared to the single-zone results, this helium burning leads to the depletion of helium and other A < 24 nuclei, which serve as seeds for α-capture reactions, and the buildup of heavier α-chain nuclides at A = 28 and 32. A less important difference is the enhancement in the multi-zone calculation of the only weakly produced mass chains with A > 60 owing to the broader range of conditions for rp-process freeze-out in multiple zones. The ONEZONE reaction flow shown in Figure 12 is indeed similar to the reaction flow in the multi-zone model during an X-ray burst (see Figure 13).

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Based on the similarities between ONEZONE and the multi-zone model we expect that reaction rate sensitivities in the αp- and rp-processes during a burst can be reasonably well approximated in ONEZONE, with the caveat that zones of lower temperature and density contribute to some extent in the multi-zone model. On the other hand, reaction rate sensitivities related to interburst burning, for example, in the CNO region, and reaction rate sensitivities related to deep helium burning triggered by subsequent bursts, such as sensitivity to α-capture reactions, are not expected to be present in ONEZONE because those burning regimes are not included in the single-zone simulation.

A total of 1931 (p, γ), (α, p), and (α, γ) reactions, together with their respective inverse reactions, were varied individually in the single-zone burst model by factors of 100 up and down. The reactions selected were those at or near the time-integrated reaction flow sequence of the baseline single-zone model (see Figure 12).

As expected, the number of individual reaction rate variations that affect the burst light curve significantly (Category 1) is rather small. Figures 5 and 6 show the largest resulting variations in the light curve. A large change in light curve is produced by variations in seven (p, γ) reactions, seven (α, p) reactions, and the ¹⁵O(α, γ)¹⁹Ne reaction (see also Table 1 and Figure 12). By far the largest change is produced by varying the ⁵⁹Cu(p, γ)⁶⁰Zn and ⁵⁹Cu(p, α)⁵⁶Ni rates, because a low ⁵⁹Cu(p, γ) rate or a high ⁵⁹Cu(p, α)⁵⁶Ni rate leads to the formation of a stronger NiCu cycle (van Wormer et al. 1994) that strongly limits synthesis of heavier nuclei. The critical quantity determining the strength of the NiCu cycle is the ratio of the (p, α) to (p, γ) reaction rates at ⁵⁹Cu. The ¹⁵O(α, γ)¹⁹Ne reaction rate has a strong impact on the total luminosity and leads to a strongly increased peak energy release when lowered. Variations of four additional Category 2 reactions listed in Table 1 cause smaller, but still significant changes in the light curve. An additional nine reactions do have some noticeable impact on the light curve, but rate variations of much more than a factor of 100 will be needed for a significant change. We included the top 28 reactions in the multi-zone variations.

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Table 1.  Reactions that Impact the Burst Light Curve in the Single-zone X-Ray Burst Model

Rank	Reaction	Typea	Sensitivityb	Category
1	${}^{56}{\rm{Ni}}$(α, p)${}^{59}{\rm{Cu}}$	U	12.5	1
2	${}^{59}{\rm{Cu}}$(p, γ)${}^{60}{\rm{Zn}}$	D	12.1	1
3	${}^{15}{\rm{O}}$(α, γ)${}^{19}{\rm{Ne}}$	D	7.9	1
4	${}^{30}{\rm{S}}$(α, p)${}^{33}{\rm{Cl}}$	U	7.8	1
5	${}^{26}{\rm{Si}}$(α, p)${}^{29}{\rm{P}}$	U	5.3	1
6	${}^{61}{\rm{Ga}}$(p, γ)${}^{62}{\rm{Ge}}$	D	5.0	1
7	${}^{23}{\rm{Al}}$(p, γ)${}^{24}{\rm{Si}}$	U	4.8	1
8	${}^{27}{\rm{P}}$(p, γ)${}^{28}{\rm{S}}$	D	4.4	1
9	${}^{63}{\rm{Ga}}$(p, γ)${}^{64}{\rm{Ge}}$	D	3.8	1
10	${}^{60}{\rm{Zn}}$(α, p)${}^{63}{\rm{Ga}}$	U	3.6	1
11	${}^{22}{\rm{Mg}}$(α, p)${}^{25}{\rm{Al}}$	D	3.5	1
12	${}^{56}{\rm{Ni}}$(p, γ)${}^{57}{\rm{Cu}}$	D	3.4	1
13	${}^{29}{\rm{S}}$(α, p)${}^{32}{\rm{Cl}}$	U	2.8	1
14	${}^{28}{\rm{S}}$(α, p)${}^{31}{\rm{Cl}}$	U	2.7	1
15	${}^{31}{\rm{Cl}}$(p, γ)${}^{32}{\rm{Ar}}$	U	2.7	1
16	${}^{35}{\rm{K}}$(p, γ)${}^{36}{\rm{Ca}}$	U	2.5	2
17	${}^{18}{\rm{Ne}}$(α, p)${}^{21}{\rm{Na}}$	D	2.3	2
18	${}^{25}{\rm{Si}}$(α, p)${}^{28}{\rm{P}}$	U	1.9	2
19	${}^{57}{\rm{Cu}}$(p, γ)${}^{58}{\rm{Zn}}$	D	1.7	2
20	${}^{34}{\rm{Ar}}$(α, p)${}^{37}{\rm{K}}$	U	1.6	3
21	${}^{24}{\rm{Si}}$(α, p)${}^{27}{\rm{P}}$	U	1.4	3
22	${}^{22}{\rm{Mg}}$(p, γ)${}^{23}{\rm{Al}}$	D	1.1	3
23	${}^{65}{\rm{As}}$(p, γ)${}^{66}{\rm{Se}}$	U	1.0	3
24	${}^{14}{\rm{O}}$(α, p)${}^{17}{\rm{F}}$	U	1.0	3
25	${}^{40}{\rm{Sc}}$(p, γ)${}^{41}{\rm{Ti}}$	D	0.9	3
26	${}^{34}{\rm{Ar}}$(p, γ)${}^{35}{\rm{K}}$	D	0.8	3
27	${}^{47}{\rm{Mn}}$(p, γ)${}^{48}{\rm{Fe}}$	D	0.8	3
28	${}^{39}{\rm{Ca}}$(p, γ)${}^{40}{\rm{Sc}}$	D	0.8	3

Notes.

^aUp (U) or down (D) variation that has the largest impact. ^b ${M}_{{LC}}^{(i)}$ in units of 10¹⁷ erg g⁻¹ s⁻¹.

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The composition of the burst ashes is affected by a much larger number of reactions. Table 4 and Figure 12 list reactions for which a change by a factor of 100 (either up or down) in the rate leads to a change by at least a factor of 2 in the mass fraction of a mass chain with significant (>10⁻⁴) mass fraction. The maximum ratio listed in Table 4 gives the largest change in the mass fraction of a mass chain, calculated for each mass chain as $\max ({X}_{{\rm{initial}}},{10}^{-4})/\max ({X}_{{\rm{final}}},{10}^{-4})$ or, if less than 1, its inverse. Also listed are the mass chains with changes of a factor of 2 or more and 10 or more, respectively. For 21 reactions, changes in the rate by a factor of 100 lead to changes in mass fraction in the composition of the burst ashes by a factor of 10 or more (as defined above). For an additional 75 reactions changes in abundance range between factors of 2 and 9. Whereas most reactions affect only a small number of final mass chains in a significant way, typically mass numbers close to the nuclei involved in the reaction, there are a few reactions that affect the final composition broadly. As expected, these tend to be the same reactions that have a significant impact on the burst light curve (see Table 1). In some cases, such as ¹⁵O(α, γ)¹⁹Ne, the broad compositional changes induced by a reaction that strongly affects the light curve are less than a factor of 2 and therefore do not appear in Table 4.

Table 2.  Reactions that Impact the Burst Light Curve in the Multi-zone X-ray Burst Model

Rank	Reaction	Typea	Sensitivityb	Category
1	${}^{15}{\rm{O}}$(α, γ)${}^{19}{\rm{Ne}}$	D	16	1
2	${}^{56}{\rm{Ni}}$(α, p)${}^{59}{\rm{Cu}}$	U	6.4	1
3	${}^{59}{\rm{Cu}}$(p, γ)${}^{60}{\rm{Zn}}$	D	5.1	1
4	${}^{61}{\rm{Ga}}$(p, γ)${}^{62}{\rm{Ge}}$	D	3.7	1
5	${}^{22}{\rm{Mg}}$(α, p)${}^{25}{\rm{Al}}$	D	2.3	1
6	${}^{14}{\rm{O}}$(α, p)${}^{17}{\rm{F}}$	D	5.8	1
7	${}^{23}{\rm{Al}}$(p, γ)${}^{24}{\rm{Si}}$	D	4.6	1
8	${}^{18}{\rm{Ne}}$(α, p)${}^{21}{\rm{Na}}$	U	1.8	1
9	${}^{63}{\rm{Ga}}$(p, γ)${}^{64}{\rm{Ge}}$	D	1.4	2
10	${}^{19}{\rm{F}}$(p, α)${}^{16}{\rm{O}}$	U	1.3	2
11	${}^{12}{\rm{C}}$(α, γ)${}^{16}{\rm{O}}$	U	2.1	2
12	${}^{26}{\rm{Si}}$(α, p)${}^{29}{\rm{P}}$	U	1.8	2
13	${}^{17}{\rm{F}}$(α, p)${}^{20}{\rm{Ne}}$	U	3.5	2
14	${}^{24}{\rm{Mg}}$(α, γ)${}^{28}{\rm{Si}}$	U	1.2	2
15	${}^{57}{\rm{Cu}}$(p, γ)${}^{58}{\rm{Zn}}$	D	1.3	2
16	${}^{60}{\rm{Zn}}$(α, p)${}^{63}{\rm{Ga}}$	U	1.1	2
17	${}^{17}{\rm{F}}$(p, γ)${}^{18}{\rm{Ne}}$	U	1.7	2
18	${}^{40}{\rm{Sc}}$(p, γ)${}^{41}{\rm{Ti}}$	D	1.1	2
19	${}^{48}{\rm{Cr}}$(p, γ)${}^{49}{\rm{Mn}}$	D	1.2	2

Notes.

^aUp (U) or down (D) variation that has the largest impact. ^b ${M}_{{LC}}^{(i)}$ in units of 10³⁸ erg s⁻¹.

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In total, 84 reactions were selected to be explored further in the multi-zone model analysis. This includes all reactions that were found to affect the burst light curve significantly, but only a subset of the reactions that affect the final composition (see reactions marked in Table 4 and in Figure 13).

The selected 84 reactions were varied individually in the multi-zone model Kepler, for a total of 168 calculations, each with a full sequence of typically 13–15 bursts. The rate variation factors are listed in Table 3. A factor of less than 100 was chosen in cases where the reaction rate uncertainty is likely much smaller. These are mostly reaction rates calculated with the Hauser–Feshbach approach that are closer to stability, where statistical methods should be well applicable (reduction to a factor of 10), (α, p) reactions where limited experimental data hint at a total uncertainty span of a factor of 100 (for example, Deibel et al. 2011; Parikh et al. 2013), and, in a few cases, reaction rates where some experimental information is available. We do note that with a few exceptions none of the reactions is based solely on direct experimental measurements. The goal was not to determine realistic uncertainties but to err on the side of a larger variation while still minimizing cases where a reaction rate is flagged as important even though it is sufficiently well known. In cases where reaction rates are calculated using shell-model information and are dominated by contributions from a few resonances, we calculated upper and lower limits by varying resonance energies by the experimental uncertainty or, when not known experimentally, by 200 keV (see Table 3). During this study a compilation of reaction rate uncertainties was published for a small subset of the reactions of interest here (Iliadis et al. 2010). In cases where the new rate uncertainties did not agree within an order of magnitude and, if available, the larger variation showed a significant impact on light curve and composition, we reran the multi-zone model calculation with a new estimate of the variation factor based on the data in Iliadis et al. (2010) for a 99.7% confidence range (see Table 3). This 3σ confidence range was chosen to be conservative because we want to make sure we are not missing an important sensitivity.

Table 3.  Rate Variations in the Multi-zone Model Calculations

Reaction	Variationa	Reaction	Variationa	Reaction	Variationa
3α	x1.2 (2)	${}^{12}{\rm{C}}$(α, γ)${}^{16}{\rm{O}}$	2.0 (3)	${}^{12}{\rm{C}}$(p, γ)${}^{13}{\rm{N}}$	1.1 (1)
${}^{13}{\rm{N}}$(p, γ)${}^{14}{\rm{O}}$	10	${}^{14}{\rm{O}}$(α, p)${}^{17}{\rm{F}}$	10	${}^{15}{\rm{O}}$(α, γ)${}^{19}{\rm{Ne}}$	10
${}^{16}{\rm{O}}$(α, γ)${}^{20}{\rm{Ne}}$	1.80 (1)	${}^{16}{\rm{O}}$(α, p)${}^{19}{\rm{F}}$	10	${}^{17}{\rm{F}}$(α, p)${}^{20}{\rm{Ne}}$	10
${}^{17}{\rm{F}}$(p, γ)${}^{18}{\rm{Ne}}$	6.33b	${}^{18}{\rm{F}}$(α, p)${}^{21}{\rm{Ne}}$	100	${}^{19}{\rm{F}}$(α, p)${}^{22}{\rm{Ne}}$	10
${}^{18}{\rm{Ne}}$(α, p)${}^{21}{\rm{Na}}$	30 (4)	${}^{19}{\rm{Ne}}$(α, p)${}^{22}{\rm{Na}}$	10	${}^{19}{\rm{Ne}}$(p, γ)${}^{20}{\rm{Na}}$	100
${}^{20}{\rm{Ne}}$(α, γ)${}^{24}{\rm{Mg}}$	1.40 (1)	${}^{22}{\rm{Na}}$(α, p)${}^{25}{\rm{Mg}}$	10	${}^{22}{\rm{Na}}$(p, γ)${}^{23}{\rm{Mg}}$	2 (1)
${}^{22}{\rm{Mg}}$(α, p)${}^{25}{\rm{Al}}$	10	${}^{23}{\rm{Mg}}$(α, p)${}^{26}{\rm{Al}}$	10	${}^{24}{\rm{Mg}}$(α, γ)${}^{28}{\rm{Si}}$	10
${}^{23}{\rm{Al}}$(p, γ)${}^{24}{\rm{Si}}$	30–100c	${}^{26}{\rm{Al}}$(α, p)${}^{29}{\rm{Si}}$	10	${}^{26}{\rm{Al}}$(p, γ)${}^{27}{\rm{Si}}$	2b
${}^{24}{\rm{Si}}$(α, p)${}^{27}{\rm{P}}$	10	${}^{25}{\rm{Si}}$(α, p)${}^{28}{\rm{P}}$	10	${}^{26}{\rm{Si}}$(α, p)${}^{29}{\rm{P}}$	10
${}^{27}{\rm{Si}}$(p, γ)${}^{28}{\rm{P}}$	3b	${}^{27}{\rm{P}}$(p, γ)${}^{28}{\rm{S}}$	2–3c	${}^{29}{\rm{P}}$(p, γ)${}^{30}{\rm{S}}$	10
${}^{28}{\rm{S}}$(α, p)${}^{31}{\rm{Cl}}$	10	${}^{29}{\rm{S}}$(α, p)${}^{32}{\rm{Cl}}$	10	${}^{30}{\rm{S}}$(α, p)${}^{33}{\rm{Cl}}$	10
${}^{31}{\rm{S}}$(p, γ)${}^{32}{\rm{Cl}}$	6b	${}^{31}{\rm{Cl}}$(p, γ)${}^{32}{\rm{Ar}}$	2–3c	${}^{34}{\rm{Ar}}$(α, p)${}^{37}{\rm{K}}$	10
${}^{35}{\rm{Ar}}$(p, γ)${}^{36}{\rm{K}}$	100	${}^{35}{\rm{K}}$(p, γ)${}^{36}{\rm{Ca}}$	3–10c	${}^{36}{\rm{K}}$(p, γ)${}^{37}{\rm{Ca}}$	10
${}^{39}{\rm{Ca}}$(p, γ)${}^{40}{\rm{Sc}}$	3b	${}^{40}{\rm{Ca}}$(p, γ)${}^{41}{\rm{Sc}}$	1.40b	${}^{40}{\rm{Sc}}$(p, γ)${}^{41}{\rm{Ti}}$	100
${}^{45}{\rm{V}}$(p, γ)${}^{46}{\rm{Cr}}$	100	${}^{47}{\rm{Cr}}$(p, γ)${}^{48}{\rm{Mn}}$	10	${}^{48}{\rm{Cr}}$(p, γ)${}^{49}{\rm{Mn}}$	100
${}^{49}{\rm{Cr}}$(p, γ)${}^{50}{\rm{Mn}}$	10	${}^{47}{\rm{Mn}}$(p, γ)${}^{48}{\rm{Fe}}$	100	${}^{51}{\rm{Mn}}$(p, γ)${}^{52}{\rm{Fe}}$	100
${}^{52}{\rm{Fe}}$(p, γ)${}^{53}{\rm{Co}}$	100	${}^{53}{\rm{Fe}}$(p, γ)${}^{54}{\rm{Co}}$	100	${}^{54}{\rm{Fe}}$(p, γ)${}^{55}{\rm{Co}}$	10
${}^{54}{\rm{Co}}$(p, γ)${}^{55}{\rm{Ni}}$	10	${}^{56}{\rm{Ni}}$(α, p)${}^{59}{\rm{Cu}}$	100	${}^{56}{\rm{Ni}}$(p,γ)${}^{57}{\rm{Cu}}$	5c
${}^{57}{\rm{Cu}}$(p, γ)${}^{58}{\rm{Zn}}$	100d	${}^{59}{\rm{Cu}}$(p, γ)${}^{60}{\rm{Zn}}$	100	${}^{60}{\rm{Cu}}$(p, γ)${}^{61}{\rm{Zn}}$	10
${}^{60}{\rm{Zn}}$(α, p)${}^{63}{\rm{Ga}}$	100	${}^{61}{\rm{Zn}}$(p, γ)${}^{62}{\rm{Ga}}$	100	${}^{62}{\rm{Zn}}$(p, γ)${}^{63}{\rm{Ga}}$	10
${}^{61}{\rm{Ga}}$(p, γ)${}^{62}{\rm{Ge}}$	100	${}^{63}{\rm{Ga}}$(p, γ)${}^{64}{\rm{Ge}}$	10	${}^{61}{\rm{Ge}}$(p, γ)${}^{62}{\rm{As}}$	100
${}^{65}{\rm{Ge}}$(p, γ)${}^{66}{\rm{As}}$	100	${}^{66}{\rm{Ge}}$(p, γ)${}^{67}{\rm{As}}$	100	${}^{65}{\rm{As}}$(p, γ)${}^{66}{\rm{Se}}$	100
${}^{67}{\rm{As}}$(p, γ)${}^{68}{\rm{Se}}$	100	${}^{69}{\rm{Se}}$(p, γ)${}^{70}{\rm{Br}}$	100	${}^{70}{\rm{Br}}$(p, γ)${}^{71}{\rm{Kr}}$	100
${}^{71}{\rm{Br}}$(p, γ)${}^{72}{\rm{Kr}}$	100	${}^{72}{\rm{Br}}$(p, γ)${}^{73}{\rm{Kr}}$	10	${}^{73}{\rm{Kr}}$(p, γ)${}^{74}{\rm{Rb}}$	10
${}^{74}{\rm{Rb}}$(p, γ)${}^{75}{\rm{Sr}}$	10	${}^{75}{\rm{Rb}}$(p, γ)${}^{76}{\rm{Sr}}$	100	${}^{76}{\rm{Rb}}$(p, γ)${}^{77}{\rm{Sr}}$	10
${}^{79}{\rm{Y}}$(p, γ)${}^{80}{\rm{Zr}}$	10	${}^{83}{\rm{Zr}}$(p, γ)${}^{84}{\rm{Nb}}$	10	${}^{83}{\rm{Nb}}$(p, γ)${}^{84}{\rm{Mo}}$	10
${}^{84}{\rm{Nb}}$(p, γ)${}^{85}{\rm{Mo}}$	10	${}^{85}{\rm{Mo}}$(p, γ)${}^{86}{\rm{Tc}}$	100	${}^{86}{\rm{Mo}}$(p, γ)${}^{87}{\rm{Tc}}$	100
${}^{89}{\rm{Tc}}$(p, γ)${}^{90}{\rm{Ru}}$	10	${}^{92}{\rm{Rh}}$(p, γ)${}^{93}{\rm{Pd}}$	10	${}^{93}{\rm{Pd}}$(p,γ)${}^{94}{\rm{Ag}}$	10

Notes.

^aReaction rates were each multiplied and divided by the factor given. References refer to work that was used in the estimate of the variation factor. ^bAdjusted with data from Iliadis et al. (2010). ^cValues approximate. Variation calculated using resonance uncertainty (see text) and depends on temperature. ^dRecent experiment reduced the uncertainty significantly (Langer et al. 2014).

References. (1) Angulo et al. (1999), (2) Herwig et al. (2006), (3) Buchmann & Barnes (2006), (4) Matic et al. (2009).

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The reaction rate variations that were found to have a significant impact on the burst light curve are listed in order of significance in Table 2 and Figure 13. There are eight reaction rate variations that affect the light curve strongly (Category 1) (Figure 7). An additional 11 reaction rate variations lead to smaller but still significant changes (Category 2, see Figure 8 for examples). We also provide the value for ${M}_{{LC}}^{(i)}$ as defined in Equation (3). Note that while ${M}_{{LC}}^{(i)}$ tends to be large for reaction rate variations with a strong impact on the light curve, it does not always provide a reliable quantitative measure for ranking the significance of the impact of the light curve. ${M}_{{LC}}^{(i)}$ weights changes near peak luminosity more strongly (which may be relevant for some model applications but not for others), and it can become artificially large for small shifts in burst rise or decline, as is the case, for example, for the ${}^{17}{\rm{F}}$(α, p)${}^{20}{\rm{Ne}}$ rate variation.

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Table 5 and Figure 13 summarize the significant changes in composition (change by at least a factor of 2 for a mass chain with mass fraction >10⁻⁴, see above) induced by the reaction rate variations explored in the multi-zone model. A total of 47 rate variations produce changes of more than a factor of 2 in at least one mass chain, whereas 14 result in changes of a factor of 10 or more. For the multi-zone calculations, changes are defined as the ratio between the up and the down variations. Note that this definition differs from what was used for the single-zone model analysis, where changes were defined relative to a baseline calculation. This addresses ambiguities in the definition of a baseline in the multi-zone model that are a result of our limited burst statistics, combined with rate-dependent changes of recurrence time and burst-to-burst variations. Figures 9 and 10 show a few examples of changes in composition.

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The multi-zone model calculations also offer the opportunity to explore the impact of changes in rate on the burst recurrence time. Only two reaction rate variations produce significant changes beyond average burst-to-burst interval variations: the ¹⁵O(α, γ) reaction rate variation leads to an 11% change, and the ¹⁴O(α, p) reaction rate variation leads to a 7% change. In both cases, only a decrease of the rate has a significant impact and leads to a shortening of the recurrence time (see discussion below).

The reactions that we find to have a significant influence on the burst light curve of the one-zone model are summarized in Figure 12. The breakout reactions of the hot CNO cycle, ¹⁵O(α, γ) and ¹⁸Ne(α, p), affect the onset of the burst light curve, which sets the initial conditions for the later stages. ¹⁴O(α, p) does not lead to breakout of the CNO cycle but opens in the early stages of the burst a rapid pathway from ¹⁴O to ¹⁵O that accelerates the hot CNO cycle.

A second set of reactions that are important for the light curve are the (α, p) reactions in the αp-process. In this model they occur on target nuclei ²²Mg, ^24–26Si, ^28–30S, and ³⁴Ar. As the temperature during the burst rise increases, these nuclei serve as waiting points until it is hot enough for the (α, p) reactions to turn on. In addition, for ²²Mg, ²⁶Si, ³⁰S, and ³⁴Ar, a proton capture pathway can compete with the (α, p) reaction. Therefore, proton capture rates in this pathway also become important. Typically the waiting point is in equilibrium with the following isotone because of the low proton capture Q-value and the strong inverse (γ, p) reaction that enables the much slower (α, p) reaction to compete with the proton capture sequence. In this case, proton capture on the $Z+1$ isotone determines the strength of the proton capture branch—for ²²Mg, ²⁶Si, ³⁰S, and ³⁴Ar these are the proton captures on ²³Al, ²⁷P, ³¹Cl, and ³⁵K, respectively. In the case of ²²Mg and ³⁴Ar the proton capture on the waiting point itself also plays a role, indicating that (p, γ)–(γ, p) equilibrium is not always established.

Also important for the light curve are reactions related to the Ni–Cu and Zn–Ga cycles (van Wormer et al. 1994). Of key importance here is the branching into the cycle at ⁵⁹Cu and ⁶³Ga, which is determined by the competition of the proton capture rate and the (p, α) rate. Note the (p, α) reactions are listed as (α, p) reactions.

The proton capture rates in the rp-process are mostly unimportant for determining the burst light curve. This is not surprising, because the process is characterized by proton capture being more rapid than β⁺ decays, leaving the β⁺ decay rates, which we did not consider in this study, as the critical parameter that determines energy generation. Nevertheless there are exceptions at particular bottlenecks. These are either associated with shell structure—³⁹Ca and ⁵⁶Ni are isotopes where the rp-process crosses the Z = 20 and Z = 28 proton shells—or with the rp-process waiting points ⁶⁰Zn and ⁶⁴Ge, where half-lives are unusually long and proton capture Q-values are low enough to slow down proton captures. As proton capture Q-values tend to be low, (p, γ)–(γ, p) equilibrium tends to play an important role at these waiting points, making the proton capture rate on the Z + 1 isotones ⁴⁰Sc, ⁵⁷Cu, ⁶¹Ga, and ⁶⁵As the important rate. In the cases of ³⁹Ca and ⁵⁶Ni, proton capture on the waiting point is also important.

Figure 13 and Table 2 summarize the reactions that have been found to influence the light curve in the multi-zone model. The number of important reactions is somewhat smaller than in the single-zone model, largely because we did use smaller variations in several cases. Additional multi-zone calculations demonstrate that the following reactions would appear in Table 2 when varied by the same factor of 100 as in the single-zone calculations: ²²Mg(p, γ)²³Si, ²⁹S(α, p)³²Cl, ³⁰S(α, p)³³Cl, ³¹Cl(α, p)³⁴Ar (though with a very small effect), ³⁴Ar(α, p)³⁷K, ³⁹Ca(p, γ)⁴⁰Sc, ³⁵K(p, γ)³⁶Ca, and ⁵⁶Ni(p, γ)⁵⁷Cu. Taking this into account, the qualitative agreement between the single-zone model and the multi-zone model is quite reasonable, validating the overall approach. Nevertheless, the quantitative impact of the various rate variations can be quite different owing to the different conditions in different zones. Only six out of the 28 reactions that impact the light curve in the single-zone model have no impact at all in the multi-zone model. Four of these are (α, p) reactions on the very neutron-deficient nuclei ²⁴Si, ²⁵Si, and ²⁸S. This may be due to contributions from zones in the multi-zone model where lower peak temperatures lead to a somewhat less pronounced and less extended αp-process. This may also explain the unimportance of ³⁴Ar(p, γ)³⁵K in the multi-zone model and the fact that in general the (α, p) reactions on heavier nuclei are less important in the multi-zone model than they are in the single-zone model. A detailed analysis of the origin of the differences between single-zone and multi-zone model sensitivities is beyond the scope of this work and would require considerable additional computational effort.

There are also a number of reaction variations that do not impact the light curve in the single-zone model, but have a significant impact on the prediction of the light curve in the multi-zone model. These reactions fall mostly into two categories: reactions on ¹⁷F and ¹⁹F that affect the CNO cycles, and the helium-burning reactions ¹²C(α, γ)¹⁶O and ²⁴Mg(α, γ)²⁸Si. The former group likely affects burning between bursts and in shallower zones where the rp-process is mostly absent or less dominant. The latter group affects the helium-burning phase once hydrogen is exhausted, and in particular the additional helium burning that occurs when a layer of burst ashes is reheated from the ignition of a subsequent burst above it. Both burning regimes are neglected in the single-zone approximation. We were able to identify these reactions in the multi-zone model because we chose to vary them for other reasons than the impact of the light curve in the single-zone model. We cannot exclude the possibility, however, that there are a few additional reactions not listed here that do not appear to be important in the single-zone model but would affect the predictions of the multi-zone model.

Figure 12 and Table 4 summarize the reactions that affect the composition of the burst ashes in the single-zone model significantly (by more than a factor of 2 for mass chains with a mass fraction of more than 10⁻⁴). Most reactions along the reaction path affect the composition. An exception are reactions between Ca and Ni, where the rp-process splits into multiple parallel paths. There, only reactions on the path closest to stability are important for the final composition, as this is where the longest β⁺ decay half-lives will be located that define the bottlenecks that determine the composition in this mass region.

Table 4.  Reactions that Impact the Composition in the Single-zone X-Ray Burst Model

Count	Reaction	Max. Ratiob	Affected Mass Numbers with Mass Fraction >10−4
			max	>×10 change	×2 < change < ×10
1	${}^{12}{\rm{C}}$(p, γ)${}^{13}{\rm{N}}$	6	16		16
2	${}^{15}{\rm{O}}$(α, γ)${}^{19}{\rm{Ne}}$ a	4	15		15
3	${}^{16}{\rm{O}}$(α, γ)${}^{20}{\rm{Ne}}$ a	7	16		16, 20–21
4	${}^{17}{\rm{F}}$(α, p)${}^{20}{\rm{Ne}}$ a	2	21		21
5	${}^{17}{\rm{F}}$(p, γ)${}^{18}{\rm{Ne}}$ a	3	19		18–19, 21
6	${}^{18}{\rm{F}}$(α, p)${}^{21}{\rm{Ne}}$ a	2	23		18, 21, 23
7	${}^{18}{\rm{Ne}}$(α, p)${}^{21}{\rm{Na}}$ a	7	19		18–19, 21, 24, 57
8	${}^{20}{\rm{Ne}}$(α, γ)${}^{24}{\rm{Mg}}$ a	6	20		20, 24
9	${}^{22}{\rm{Na}}$(α, p)${}^{25}{\rm{Mg}}$ a	4	27		16, 21, 25, 27
10	${}^{22}{\rm{Na}}$(p, γ)${}^{23}{\rm{Mg}}$ a	4	23		23
11	${}^{22}{\rm{Mg}}$(α, p)${}^{25}{\rm{Al}}$ a	20	22	22	21, 23–27, 57
12	${}^{23}{\rm{Mg}}$(p, γ)${}^{24}{\rm{Al}}$	2	23		23
13	${}^{23}{\rm{Al}}$(p, γ)${}^{24}{\rm{Si}}$ a	4	57		16, 19, 21, 24, 26–28, 46, 57, 70, 74, 81–82
14	${}^{24}{\rm{Al}}$(p, γ)${}^{25}{\rm{Si}}$	2	24		24
15	${}^{26}{\rm{Al}}$(p, γ)${}^{27}{\rm{Si}}$ a	6	27		27
16	${}^{26}{\rm{Si}}$(α, p)${}^{29}{\rm{P}}$ a	3	18		18–21, 46, 57, 70, 73–75, 78, 82, 86
17	${}^{27}{\rm{Si}}$(p, γ)${}^{28}{\rm{P}}$ a	3	27		27
18	${}^{28}{\rm{Si}}$(p, γ)${}^{29}{\rm{P}}$	2	28		28
19	${}^{27}{\rm{P}}$(p, γ)${}^{28}{\rm{S}}$ a	2	26		26
20	${}^{28}{\rm{P}}$(p, γ)${}^{29}{\rm{S}}$	2	28		28
21	${}^{29}{\rm{P}}$(p, γ)${}^{30}{\rm{S}}$ a	4	29		29
22	${}^{30}{\rm{P}}$(p, γ)${}^{31}{\rm{S}}$	3	31		31
23	${}^{28}{\rm{S}}$(α, p)${}^{31}{\rm{Cl}}$ a	2	19		18–19
24	${}^{30}{\rm{S}}$(α, p)${}^{33}{\rm{Cl}}$ a	3	18		18–21, 46, 57, 70–71, 73–75, 77–78, 81–83, 86
25	${}^{31}{\rm{S}}$(p, γ)${}^{32}{\rm{Cl}}$ a	6	31		31, 33
26	${}^{32}{\rm{Cl}}$(p, γ)${}^{33}{\rm{Ar}}$	5	32		32–33
27	${}^{33}{\rm{Cl}}$(p, γ)${}^{34}{\rm{Ar}}$	4	33		33
28	${}^{35}{\rm{Ar}}$(p, γ)${}^{36}{\rm{K}}$ a	7	35		35–37
29	${}^{36}{\rm{Ar}}$(p, γ)${}^{37}{\rm{K}}$	4	36		36
30	${}^{36}{\rm{K}}$(p, γ)${}^{37}{\rm{Ca}}$ a	5	36		36
31	${}^{37}{\rm{K}}$(p, γ)${}^{38}{\rm{Ca}}$	6	37		37
32	${}^{39}{\rm{K}}$(p, γ)${}^{40}{\rm{Ca}}$	3	40		40
33	${}^{40}{\rm{Ca}}$(p, γ)${}^{41}{\rm{Sc}}$ a	6	40		40, 42–46
34	${}^{41}{\rm{Sc}}$(p, γ)${}^{42}{\rm{Ti}}$	10	41	41	42
35	${}^{42}{\rm{Sc}}$(p, γ)${}^{43}{\rm{Ti}}$	3	42		42–43
36	${}^{43}{\rm{Ti}}$(p, γ)${}^{44}{\rm{V}}$	8	43		43–45
37	${}^{44}{\rm{Ti}}$(p, γ)${}^{45}{\rm{V}}$	2	44		44
38	${}^{44}{\rm{V}}$(p, γ)${}^{45}{\rm{Cr}}$	7	44		44–45
39	${}^{45}{\rm{V}}$(p, γ)${}^{46}{\rm{Cr}}$ a	6	45		45–46
40	${}^{46}{\rm{V}}$(p, γ)${}^{47}{\rm{Cr}}$	2	46		46
41	${}^{47}{\rm{Cr}}$(p, γ)${}^{48}{\rm{Mn}}$ a	8	47		47
42	${}^{48}{\rm{Cr}}$(p, γ)${}^{49}{\rm{Mn}}$ a	3	48		48
43	${}^{48}{\rm{Mn}}$(p, γ)${}^{49}{\rm{Fe}}$	10	48	48
44	${}^{49}{\rm{Mn}}$(p, γ)${}^{50}{\rm{Fe}}$	7	49		49
45	${}^{50}{\rm{Mn}}$(p, γ)${}^{51}{\rm{Fe}}$	3	50		50–51
46	${}^{51}{\rm{Mn}}$(p, γ)${}^{52}{\rm{Fe}}$ a	2	51		51
47	${}^{51}{\rm{Fe}}$(p, γ)${}^{52}{\rm{Co}}$	5	51		51
48	${}^{52}{\rm{Fe}}$(p, γ)${}^{53}{\rm{Co}}$ a	7	52		52
49	${}^{53}{\rm{Fe}}$(p, γ)${}^{54}{\rm{Co}}$ a	3	53		53
50	${}^{54}{\rm{Fe}}$(p, γ)${}^{55}{\rm{Co}}$ a	3	54		54
51	${}^{52}{\rm{Co}}$(p, γ)${}^{53}{\rm{Ni}}$	5	52		52
52	${}^{53}{\rm{Co}}$(p, γ)${}^{54}{\rm{Ni}}$	10	53	53
53	${}^{54}{\rm{Co}}$(p,γ)${}^{55}{\rm{Ni}}$ a	10	54	54
54	${}^{55}{\rm{Co}}$(p, γ)${}^{56}{\rm{Ni}}$ a	10	55	55	56
55	${}^{56}{\rm{Ni}}$(α, p)${}^{59}{\rm{Cu}}$ a	7	24		12, 16, 21–22, 24–34, 36–39, 42, 44, 46, 50, 52, 54–57, 61, 63, 65–67, 69–71, 73–75, 77–78, 80–85
56	${}^{56}{\rm{Ni}}$(p, γ)${}^{57}{\rm{Cu}}$ a	10	56	56	18–19, 21, 57–59
57	${}^{57}{\rm{Ni}}$(p, γ)${}^{58}{\rm{Cu}}$	5	57		57
58	${}^{57}{\rm{Cu}}$(p, γ)${}^{58}{\rm{Zn}}$ a	6	57		56–58
59	${}^{58}{\rm{Cu}}$(p, γ)${}^{59}{\rm{Zn}}$	30	58	58
60	${}^{59}{\rm{Cu}}$(p, γ)${}^{60}{\rm{Zn}}$ a	10	59	59	12, 16, 21–22, 24–34, 36–39, 42, 44, 46, 50, 52, 54–57, 61, 65–67, 69–71, 73–75, 77–78, 80–85
61	${}^{60}{\rm{Cu}}$(p, γ)${}^{61}{\rm{Zn}}$ a	3	61		61, 63
62	${}^{60}{\rm{Zn}}$(α, p)${}^{63}{\rm{Ga}}$ a	2	57		21, 24, 27, 57
63	${}^{61}{\rm{Zn}}$(p, γ)${}^{62}{\rm{Ga}}$ a	10	61	61	63
64	${}^{62}{\rm{Zn}}$(p, γ)${}^{63}{\rm{Ga}}$ a	5	62		62
65	${}^{61}{\rm{Ga}}$(p, γ)${}^{62}{\rm{Ge}}$ a	8	60		12, 18–19, 60–68
66	${}^{63}{\rm{Ga}}$(p, γ)${}^{64}{\rm{Ge}}$ a	10	63	63	21, 24, 27, 57
67	${}^{64}{\rm{Ga}}$(p, γ)${}^{65}{\rm{Ge}}$	6	65		65–67
68	${}^{65}{\rm{Ge}}$(p, γ)${}^{66}{\rm{As}}$ a	30	65	65	66–67
69	${}^{66}{\rm{Ge}}$(p, γ)${}^{67}{\rm{As}}$ a	20	66	66	67
70	${}^{66}{\rm{As}}$(p, γ)${}^{67}{\rm{Se}}$	5	66		66
71	${}^{67}{\rm{As}}$(p, γ)${}^{68}{\rm{Se}}$ a	30	67	67
72	${}^{68}{\rm{As}}$(p, γ)${}^{69}{\rm{Se}}$	5	69		69–71
73	${}^{69}{\rm{Se}}$(p,γ)${}^{70}{\rm{Br}}$ a	20	69	69	70–71
74	${}^{70}{\rm{Se}}$(p, γ)${}^{71}{\rm{Br}}$	10	70	70	71
75	${}^{70}{\rm{Br}}$(p, γ)${}^{71}{\rm{Kr}}$ a	3	70		70
76	${}^{71}{\rm{Br}}$(p, γ)${}^{72}{\rm{Kr}}$ a	10	71	71
77	${}^{72}{\rm{Br}}$(p, γ)${}^{73}{\rm{Kr}}$ a	5	73		73–75
78	${}^{73}{\rm{Kr}}$(p, γ)${}^{74}{\rm{Rb}}$ a	20	73	73	74–78
79	${}^{74}{\rm{Kr}}$(p, γ)${}^{75}{\rm{Rb}}$	10	74	74	75
80	${}^{74}{\rm{Rb}}$(p, γ)${}^{75}{\rm{Sr}}$ a	3	74		74
81	${}^{75}{\rm{Rb}}$(p, γ)${}^{76}{\rm{Sr}}$ a	9	75		75–76
82	${}^{76}{\rm{Rb}}$(p, γ)${}^{77}{\rm{Sr}}$ a	5	77		77–78
83	${}^{77}{\rm{Sr}}$(p,γ)${}^{78}{\rm{Y}}$	10	77	77
84	${}^{78}{\rm{Sr}}$(p, γ)${}^{79}{\rm{Y}}$	6	78		78, 80–85
85	${}^{78}{\rm{Y}}$(p, γ)${}^{79}{\rm{Zr}}$	10	80	80	78–79, 81–92
86	${}^{79}{\rm{Y}}$(p, γ)${}^{80}{\rm{Zr}}$ a	5	80		80
87	${}^{80}{\rm{Y}}$(p, γ)${}^{81}{\rm{Zr}}$	5	81		81–85
88	${}^{81}{\rm{Zr}}$(p, γ)${}^{82}{\rm{Nb}}$	5	81		81
89	${}^{82}{\rm{Zr}}$(p, γ)${}^{83}{\rm{Nb}}$	3	82		82–85
90	${}^{82}{\rm{Nb}}$(p, γ)${}^{83}{\rm{Mo}}$	2	82		82
91	${}^{83}{\rm{Nb}}$(p, γ)${}^{84}{\rm{Mo}}$ a	3	84		83–84
92	${}^{84}{\rm{Nb}}$(p, γ)${}^{85}{\rm{Mo}}$ a	3	85		84–85
93	${}^{85}{\rm{Nb}}$(p, γ)${}^{86}{\rm{Mo}}$	2	85		85
94	${}^{85}{\rm{Mo}}$(p, γ)${}^{86}{\rm{Tc}}$ a	3	85		85
95	${}^{86}{\rm{Mo}}$(p,γ)${}^{87}{\rm{Tc}}$ a	3	86		86
96	${}^{88}{\rm{Tc}}$(p, γ)${}^{89}{\rm{Ru}}$	4	88		88

Notes.

^aReaction also varied in the multi-zone model. ^bAbundance ratio relative to the baseline calculation.

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The most abundant mass chains that dominate the composition of the burst ashes as calculated by the single-zone model (mass fractions $\gt {10}^{-2}$) are A = 12, 56, 60, 64, 68, 72, and 76. Only a small number of reactions affect these mass chains significantly—these are proton captures on ⁵⁶Ni, ⁶¹Ga, ⁶⁵Ge, ⁶⁷As, ⁵⁷Cu, ⁵⁵Co, and ⁵⁹Cu together with the ⁵⁶Ni(α, p) reaction. An additional 38 reactions affect mass chains with mass fractions >10⁻³.

It was not feasible to vary all the important reactions identified in the single-zone model in individual multi-zone model runs. Nevertheless, by varying a subset of rates we do find 47 reactions that affect the burst ashes significantly (Figure 13 and Table 5). We also find 37 reactions that do not affect the composition of the ashes significantly and that are also indicated in Figure 13. The most important reaction rate uncertainties among the subset of 47 relevant reaction rates will be the ones that affect the most abundant mass chains and the ones that affect mass chains that are of particular interest because of their impact on ocean and crust physics. The most abundant mass chains in the multi-zone model (>10⁻²) are A = 28, 32, 56, 60, 64, 68, and 72. Overall these abundances are quite robust, with no reaction causing a change by more than a factor of 10. The set of reactions affecting the A = 56, 60, 64, 68, and 72 mass chains by more than a factor of 2 is very small. ⁵⁶Ni(α, p)⁵⁹Cu affects A = 56, ⁶¹Ga(p, γ)⁶²Ge has a broad impact on A = 60, 64, and 68, and both, ⁶⁹Se(p, γ)⁷⁰Br and ⁷¹Br(p, γ)⁷²Kr affect A = 72. The A = 28 mass chain is affected by ¹²C(α, γ)¹⁶O, ¹⁵O(α, γ)¹⁹Ne, ¹⁶O(α, p)¹⁹F, ¹⁸Ne(α, p)²¹Na, ²²Mg(α, p)²⁵Al, ²³Mg(α, p)²⁶Al, and ²⁴Mg(α, γ)²⁸Si, whereas A = 32 is affected only by ¹⁵O(α, γ)¹⁹Ne. Reactions such as ¹²C(α, γ)¹⁶O, ¹⁶O(α, p)¹⁹F, and ²⁴Mg(α, γ)²⁸S are not expected to occur in the presence of hydrogen, because proton capture rates on these target isotopes are much faster. This indicates late-time helium burning as the source of these abundance peaks, which also explains their absence from the single-zone model.

Table 5.  Reactions that Impact the Composition in the Multi-zone X-Ray Burst Model

Count	Reaction	Max. Ratioa	Affected Mass Numbers with Mass Fraction >10−4
			max	>×10 change	×2 < change < ×10
1	${}^{8}{\rm{Be}}$(α, γ)${}^{12}{\rm{C}}$	3	98		30, 93–99
2	${}^{12}{\rm{C}}$(α, γ)${}^{16}{\rm{O}}$	2	28		28
3	${}^{14}{\rm{O}}$(α, p)${}^{17}{\rm{F}}$	4	29		29
4	${}^{15}{\rm{O}}$(α, γ)${}^{19}{\rm{Ne}}$	30	29	24, 29	12, 20, 26–28, 30–32, 36, 91–92, 95–98
5	${}^{16}{\rm{O}}$(α, p)${}^{19}{\rm{F}}$	3	28		28
6	${}^{17}{\rm{F}}$(α, p)${}^{20}{\rm{Ne}}$	3	98		90, 93–99
7	${}^{18}{\rm{Ne}}$(α, p)${}^{21}{\rm{Na}}$	4	30		28, 30, 98–99
8	${}^{22}{\rm{Mg}}$(α, p)${}^{25}{\rm{Al}}$	4	30		28–30
9	${}^{23}{\rm{Mg}}$(α, p)${}^{26}{\rm{Al}}$	2	28		28
10	${}^{24}{\rm{Mg}}$(α, γ)${}^{28}{\rm{Si}}$	20	24	24	28–30
11	${}^{23}{\rm{Al}}$(p, γ)${}^{24}{\rm{Si}}$	2	75		75, 79
12	${}^{26}{\rm{Al}}$(α, p)${}^{29}{\rm{Si}}$	4	26		26, 29
13	${}^{29}{\rm{P}}$(p, γ)${}^{30}{\rm{S}}$	3	98		90, 92–99
14	${}^{40}{\rm{Sc}}$(p, γ)${}^{41}{\rm{Ti}}$	3	29		29
15	${}^{45}{\rm{V}}$(p, γ)${}^{46}{\rm{Cr}}$	10	45	45	46
16	${}^{47}{\rm{Cr}}$(p, γ)${}^{48}{\rm{Mn}}$	2	47		47
17	${}^{48}{\rm{Cr}}$(p, γ)${}^{49}{\rm{Mn}}$	30	48	48	51–53
18	${}^{49}{\rm{Cr}}$(p, γ)${}^{50}{\rm{Mn}}$	5	49		49
19	${}^{47}{\rm{Mn}}$(p, γ)${}^{48}{\rm{Fe}}$	2	98		97–99
20	${}^{51}{\rm{Mn}}$(p, γ)${}^{52}{\rm{Fe}}$	30	51	51	52
21	${}^{52}{\rm{Fe}}$(p, γ)${}^{53}{\rm{Co}}$	40	52	52	53–55, 57
22	${}^{53}{\rm{Fe}}$(p, γ)${}^{54}{\rm{Co}}$	40	53	53	57–58
23	${}^{54}{\rm{Fe}}$(p, γ)${}^{55}{\rm{Co}}$	6	54		54
24	${}^{56}{\rm{Ni}}$(α, p)${}^{59}{\rm{Cu}}$	5	29		12, 29–30, 56, 75, 78–79, 82
25	${}^{59}{\rm{Cu}}$(p, γ)${}^{60}{\rm{Zn}}$	200	59	59	12, 29–30, 75, 78–79
26	${}^{60}{\rm{Cu}}$(p, γ)${}^{61}{\rm{Zn}}$	2	98		93–98
27	${}^{60}{\rm{Zn}}$(α, p)${}^{63}{\rm{Ga}}$	2	98		97–98
28	${}^{61}{\rm{Zn}}$(p, γ)${}^{62}{\rm{Ga}}$	9	61		61–63
29	${}^{62}{\rm{Zn}}$(p, γ)${}^{63}{\rm{Ga}}$	7	62		62, 90–99
30	${}^{61}{\rm{Ga}}$(p, γ)${}^{62}{\rm{Ge}}$	20	60	60–61	12, 30, 62–81
31	${}^{63}{\rm{Ga}}$(p, γ)${}^{64}{\rm{Ge}}$	3	63		63
32	${}^{65}{\rm{Ge}}$(p, γ)${}^{66}{\rm{As}}$	10	65	65	67
33	${}^{66}{\rm{Ge}}$(p, γ)${}^{67}{\rm{As}}$	20	66	66	67
34	${}^{67}{\rm{As}}$(p, γ)${}^{68}{\rm{Se}}$	100	67	67	93–95, 97–98
35	${}^{69}{\rm{Se}}$(p, γ)${}^{70}{\rm{Br}}$	7	69		69, 71–75
36	${}^{71}{\rm{Br}}$(p, γ)${}^{72}{\rm{Kr}}$	90	71	71	72, 98
37	${}^{72}{\rm{Br}}$(p, γ)${}^{73}{\rm{Kr}}$	2	98		74, 93–94, 96–98
38	${}^{73}{\rm{Kr}}$(p,γ)${}^{74}{\rm{Rb}}$	4	73		73, 75
39	${}^{75}{\rm{Rb}}$(p, γ)${}^{76}{\rm{Sr}}$	40	75	75
40	${}^{79}{\rm{Y}}$(p, γ)${}^{80}{\rm{Zr}}$	6	79		79
41	${}^{83}{\rm{Zr}}$(p, γ)${}^{84}{\rm{Nb}}$	3	83		83
42	${}^{83}{\rm{Nb}}$(p, γ)${}^{84}{\rm{Mo}}$	2	83		83
43	${}^{84}{\rm{Nb}}$(p, γ)${}^{85}{\rm{Mo}}$	3	84		84
44	${}^{85}{\rm{Mo}}$(p, γ)${}^{86}{\rm{Tc}}$	3	85		85, 90–98
45	${}^{86}{\rm{Mo}}$(p, γ)${}^{87}{\rm{Tc}}$	7	86		86, 90–92
46	${}^{89}{\rm{Tc}}$(p, γ)${}^{90}{\rm{Ru}}$	3	89		89
47	${}^{92}{\rm{Rh}}$(p, γ)${}^{93}{\rm{Pd}}$	3	98		91, 93–98

Note.

^aAbundance ratio between up and down variations of the reaction rate.

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Among the mass chains that affect ocean and crust physics, A = 12 stands out. ¹²C produced by X-ray bursts and reignited deeper in the ocean of the neutron star is the leading model to explain the occasionally observed superbursts. The small mass fraction of ¹²C produced in current X-ray burst models (0.4% in our model) is a major issue in this context because superburst models require ¹²C mass fractions in the 20% range (Cumming et al. 2006) to achieve ignition and explain burst energetics. We find that only four reactions—¹⁵O(α, γ)¹⁹Ne, ⁵⁶Ni(α, p)⁵⁹Cu, ⁵⁹Cu(p, γ)⁶⁰Zn, and ⁶¹Ga(p, γ)⁶²Ge—affect ¹²C production by more than a factor of 2, and none affects it by more than a factor of 10, as would be required to explain the origin of superbursts. With the caveat that we studied only a limited set of reactions, this may indicate that the problem of the low ¹²C production in X-ray bursts is not caused by nuclear physics uncertainties.

It has recently been found that the presence of nuclei with odd mass numbers may lead to strong crust cooling via electron capture–β-decay Urca cycles (Schatz et al. 2014). The cooling rates are directly proportional to the abundance of the respective mass chain. Reaction rate uncertainties that affect odd-mass chains will therefore be of particular importance. The most abundant odd-mass chains in the burst ashes are A = 61, 63, 65, and 69, which have mass fractions of close to 10⁻² or larger. In our subset of reaction rate variations we identify proton captures on ⁶¹Zn, ⁶¹Ga, ⁶³Ga, ⁶⁵Ge, ⁶⁶Ge, ⁶⁷As, and ⁶⁹Se as important for determining the amount of these nuclides in the burst ashes.

Only three reaction variations were found to affect burst recurrence times significantly (beyond average burst-to-burst interval variations): ¹⁵O(α, γ)¹⁹Ne, ¹⁴O(α, p)¹⁷F, and the 3α reaction. Variation of the 3α reaction rate has a small effect because the rate is relatively well known. A change in the rate of 20% up or down results in a change in recurrence time of about 4%. As expected, an increase of the 3α reaction rate leads to faster burst ignition and a shorter recurrence time, indicating that indeed as predicted the 3α reaction is the main burst ignition mechanism (Strohmayer & Bildsten 2006). On the other hand, only a decrease of the ¹⁴O(α, p)¹⁷F and ¹⁵O(α, γ)¹⁹Ne reactions has a strong impact, and, somewhat counterintuitively, leads to faster burst ignition and a decrease in recurrence time by 7% and 11%, respectively.

A possible explanation is the effect of these reactions on the operation of the hot CNO cycle prior to burst ignition (see also the discussion in Fisker et al. 2006). Energy generation by the hot CNO cycle is given by 5.8 × 10¹⁵Z_CNO erg g⁻¹ s⁻¹ with Z_CNO being the mass fraction of CNO nuclei (Strohmayer & Bildsten 2006). There exists a positive feedback loop between the hot CNO cycle and the 3α reaction. As the hot CNO cycle increases the ⁴He abundance, the 3α reaction rate, which depends strongly on the ⁴He abundance, will increase. This will increase the amount of CNO nuclei Z in the hot CNO cycle, further increasing ⁴He production. As the energy generation in the hot CNO cycle increases with Z, this may significantly affect burst ignition. The ¹⁵O(α, γ)¹⁹Ne reaction may then act as a "valve," removing material from the CNO cycle, damping the ⁴He abundance feedback loop, slowing down energy generation, and delaying burst ignition. Thus, a smaller ¹⁵O(α, γ)¹⁹Ne reaction increases energy generation between bursts via the CNO cycle, decreases recurrence times, and for a given accretion rate pushes models closer to the boundary to stable burning. This interpretation is supported by the trends in peak luminosity and burst duration. Figure 11 shows recurrence time, peak luminosity, and burst timescale τ for additional variations in the multi-zone model of the ¹⁵O(α, γ)¹⁹Ne reaction rate. A smaller ¹⁵O(α, γ)¹⁹Ne reaction rate leads to shorter recurrence times, higher peak brightness, and shorter bursts, just as one would expect for a larger ⁴He/H ratio at burst ignition, which would be the result of a stronger CNO cycle in between bursts. We also confirm previous results (Fisker et al. 2006) that a lower ¹⁵O(α, γ)¹⁹Ne reaction rate leads to an increase in ¹²C production (Figure 11). It has been shown previously that stable burning of hydrogen and helium can lead to the production of large amounts of ¹²C (Schatz et al. 2003; Stevens et al. 2014). Increased ¹²C production would therefore be expected for an increase in stable burning between bursts caused by a reduced ¹⁵O(α, γ)¹⁹Ne reaction rate.

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The ¹⁴O(α, p)¹⁷F reaction may further support this mechanism by opening a pathway to bypass the ¹⁴O β⁺ decay in the hot CNO cycle via the reaction sequence ¹⁴O(α, p)¹⁷F(p, γ)¹⁸Ne(β⁺)¹⁸F(p, α)¹⁵O. This will increase the ¹⁵O abundance relative to the ¹⁴O abundance in the hot CNO cycle. An increase of the ¹⁴O(α, p)¹⁷F reaction rate will therefore also lead to more efficient breakout of the CNO cycle via ¹⁵O(α, γ)¹⁹Ne.

This effect can occur only if conditions for breakout via ¹⁵O(α, γ) are reached prior to burst ignition for a long enough time for the hot CNO cycle to operate so that breakout occurs in parallel to the hot CNO cycle. If, on the other hand, the time between the onset of the ¹⁵O(α, γ)¹⁹Ne reaction and ignition is short, one would expect the opposite behavior, with a larger ¹⁵O(α, γ)¹⁹Ne reaction leading to ignition sooner. This may explain why different burst models show different sensitivities to ¹⁵O(α, γ)¹⁹Ne reaction rate variations (see below).

Our results can be compared with previous studies of the sensitivities of X-ray burst models to reaction rate variations. We emphasize that we do not expect the results to be the same for different models (see discussion below). Nevertheless, such a comparison is useful because experimentalists or nuclear theorists may prefer to focus on reaction rates that serve multiple model needs.

Only very few and extremely limited fully self-consistent one-zone or 1D multi-zone model-based sensitivity studies, which take into account the changes in astrophysical conditions induced by changes in rate, have been carried out so far, however. Thielemann et al. (2001) varied proton capture rates on ${}^{27}{\rm{Si}}$, ${}^{31}{\rm{S}}$, ${}^{35}{\rm{Ar}}$, and ${}^{38}{\rm{Ca}}$ and found a strong influence on the burst light curve in their AGILE-based 1D multi-zone model. We find that these reactions do not play a significant role in either our single-zone or multi-zone burst model. This agrees with post-processing studies (Iliadis et al. 1999) though the post-processing approach cannot be used to draw firm conclusions (see Section 1). The same multi-zone X-ray burst model has been used by Fisker et al. (2004) to investigate the impact of variations in the ³⁰S(α, p)³³Cl and ³⁴Ar(α, p)³⁷K reaction rates on a double-peak structure in their burst light curve. We also find small changes in the burst light curve when these reactions are varied, though the details of the variations of burst shape are different. In particular our chosen burst model does not exhibit the double-peak structure that Fisker et al. (2004) obtain.

The impact of rate variations of the ¹⁵O(α, γ)¹⁹Ne reaction has been investigated in a number of multi-zone models (Fisker et al. 2006, 2007; Davids et al. 2011; Keek et al. 2014). Fisker et al. (2007) used the AGILE model and carried out detailed calculations for eight accretion rates with nine variations of the ¹⁵O(α, γ)¹⁹Ne rate each, ranging from a reduction by a factor of 1000 to an increase by a factor of 10 of their recommended rate. They found that for very low ¹⁵O(α, γ)¹⁹Ne reaction rates burst rise times become longer and longer, and eventually burst activity is replaced by an oscillatory burning behavior that resembles a stable burning regime. For an accretion rate and recurrence time that are similar to our study (10% of the Eddington accretion rate and 200 minutes, respectively) they find that this transition occurs for a rate reduction by a factor of 30 or more. To explore this effect we carried out additional variations of the ¹⁵O(α, γ)¹⁹Ne rate with our multi-zone model, spanning a change in reaction rate of ×0.01 to ×100. Even for our lowest ¹⁵O(α, γ)¹⁹Ne rate we do not find the oscillatory behavior found by Fisker et al. (2006). We do find, however, a decrease in recurrence time (see Figure 11) with decreasing rate that confirms the effect of the ¹⁵O(α, γ)¹⁹Ne rate on the hot CNO cycle that Fisker et al. (2006) propose to explain their transition to stable burning. Therefore, whereas we confirm that a lower ¹⁵O(α, γ)¹⁹Ne rate pushes the model toward larger energy generation between bursts and toward stable burning, the effect in our model is not strong enough to actually trigger the transition to stable burning, in contrast to the model of Fisker et al. (2006).

Davids et al. (2011) used the multi-zone X-ray burst code SHIVA to carry out calculations with the same ¹⁵O(α, γ)¹⁹Ne rate variation used in this work. They find the opposite behavior, with a decreasing ¹⁵O(α, γ)¹⁹Ne rate leading to longer recurrence times. This may indicate that in their model ¹⁵O(α, γ)¹⁹Ne does not play a significant role during the hot CNO cycle and rather contributes on short timescales to burst ignition with an additional energy boost. The different behavior is not necessarily surprising because Davids et al. (2011) explore a somewhat different burst regime with longer recurrence times of around 300 minutes (compared to 175 minutes and 200 minutes in this work and Fisker et al. 2006, respectively). In addition, they follow only four bursts and the results may be affected by additional uncertainties from intrinsic burst-to-burst variations.

The only previous large-scale sensitivity study for X-ray bursts was carried out in the post-processing approximation by Parikh et al. (2008). In this approach, a fixed temperature and density profile from a single-zone X-ray burst model is used to calculate energy generation and final composition using a nucleosynthesis network code. As temperature and density are fixed, changes in temperature and density induced by changes in reaction rate are neglected. This is a major limitation for X-ray burst studies, where the same reactions that create the burst ashes are the sole source of energy and entirely determine the evolution of temperature and density. In general, therefore, changes in reaction rates in X-ray burst models will lead to changes in temperature and density and associated changes in reaction pathways. Nevertheless post-processing studies are fast and efficient, and can be used to identify qualitatively likely candidates for important reactions by identifying changes in energy generation and composition. It is not generally possible to make reliable predictions of the quantitative impact of a change in rate, however. Similarly to our study, Parikh et al. (2008) find that a large number of reactions tend to affect the burst composition. Detailed comparisons are not useful, because Parikh et al. (2008) post-processed models with very different temperature and density profiles. They also use a different criterion of importance (rates must affect more than three mass chains) than our study. Parikh et al. (2008) do identify a set of 17 reactions that lead to significant changes in energy generation and composition. Indeed eight of the 17 reactions have an impact on the burst light curve in either our single-zone or our multi-zone study. These are ${}^{56}{\rm{Ni}}$(α, p)${}^{59}{\rm{Cu}}$, ${}^{59}{\rm{Cu}}$(p, γ)${}^{60}{\rm{Zn}}$, ${}^{15}{\rm{O}}$(α, γ)${}^{19}{\rm{Ne}}$, ${}^{23}{\rm{Al}}$(p, γ)${}^{24}{\rm{Si}}$, ${}^{30}{\rm{S}}$(α, p)${}^{33}{\rm{Cl}}$, ${}^{22}{\rm{Mg}}$(α, p)${}^{25}{\rm{Al}}$, ${}^{31}{\rm{Cl}}$(p, γ)${}^{32}{\rm{Ar}}$, and ${}^{18}{\rm{Ne}}$(α, p)${}^{21}{\rm{Na}}$. Many rates that we find impact our light curves strongly, however, are not listed in Parikh et al. (2008). Again, this is not surprising as different models are used. For example, we find a strong impact of the ¹⁴O(α, p)¹⁷F reaction in our multi-zone model, which is not expected to appear prominently in a single-zone post-processing study because it mainly affects the pre-burst phase and ignition conditions (see Hu et al. 2014 for a post-processing analysis). An example of a rate that we find to be unimportant even though it appears prominently in Parikh et al. (2008) is ¹⁰³Sn(α, p). The burst model S01 used for post-processing by Parikh et al. (2008) has different astrophysical conditions that lead to a more extended main rp-process path that reaches ¹⁰³Sn, whereas in our model the main rp-process ends at lower masses and only a very weak reaction flow reaches the A = 103 mass region.

This comparison reinforces an important point. Sensitivity studies by design are specific to a particular model and identify specifically the important nuclear physics in that model. This is of particular importance for X-ray bursts, where a broad range of astrophysical parameters occurs in nature and leads to variations in bursts and nuclear processes. Various models reported in the literature typically choose different astrophysical parameters, which may all be reasonable and reflect different types of systems and sources. Differences among sensitivity studies are therefore expected and reflect the broad needs of nuclear physics data. This is also true for models that use the same input parameters but different astrophysical approximations, for example descriptions for convection or diffusion. The goal of a sensitivity study is to enable the nuclear physics uncertainties of a particular model to be addressed, so that this particular model can be validated against observations. With reliable nuclear physics this validation process can then be used to adjust astrophysical parameters (to effectively extract these from observations) or to guide improvements of the astrophysical approximations.