THE ACCRETION RATE DEPENDENCE OF BURST OSCILLATION AMPLITUDE

First we compute for each burst the burst start time (t₀) and the background count rate. We estimate the background count rate using the count rate in the range 20–5 s prior to the approximate burst start time given in one of the databases. t₀ is then defined as the time where the count rate equals 1.5 times the estimated background count rate. This ensures that all the burst start times are defined by the same criterion. Next, the true background count rate (${C}_{{\rm{B}}}$) is defined as the average count rate in the range 17–1 s preceding t₀. Muno et al. (2004) used the 16 s directly prior to the burst to compute the background. A time buffer of one second is kept between the burst start time and the range from which the background is calculated to ensure that the background is not overestimated in bursts with a slow rise.

Second, the first 16 s of the burst, $[{t}_{0}-({t}_{0}+16.0\,{\rm{s}})]$, is divided into non-overlapping time bins with 5000 counts each. We divided the bursts into time bins with an equal amount of counts to make the error bars on each measurement similar (Watts et al. 2005). Note that in a previous analysis by Muno et al. (2004) equal time bins were used, so that error bars later in the burst were larger (see discussion in Section 5.3). The number of time bins in our analysis thus depends on the strength of the burst and the underlying background. We use non-overlapping time bins to ensure that each bin is independent of the others. This makes it easier to compute the number of trials to obtain a signal (see Section 3.2.4).

In each time bin we look for signals within 5 Hz of the known oscillation frequency. For each time bin we set up 10 frequency bins (${\nu }_{{\rm{o}}}\pm 5\,\mathrm{Hz}$), to obtain a frequency resolution of 1 Hz (equal to the resolution in Muno et al. (2004)). We thus create for each burst a two-dimensional grid of time-frequency bins in which we attempt to detect oscillation signals (see Figure 2 for a visualization of the grid).

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We compute for each time bin the signal power for each of the 10 trial frequencies. We obtain the measured power for a signal with trial frequency ν by calculating the Z² statistic (see Buccheri et al. 1983; Strohmayer & Markwardt 1999). The Z² statistic is similar to a fast Fourier transform in the sense that it decomposes a signal into the sines and cosines that it consists of (see Equation (2)). This results in a power spectrum in which the power of the signal is plotted as function of frequency. The difference with a fast Fourier transform is that Z² statistics does not require that the arrival times of the counts are binned. The Z² statistic is defined as:

$$\begin{eqnarray}&&{Z}_{n}^{2}=\displaystyle \frac{2}{{N}_{\gamma }}\displaystyle \sum _{k=1}^{n}\left[{\left(\displaystyle \sum _{j=1}^{{N}_{\gamma }}\cos k\nu {t}_{j}\right)}^{2}+{\left(\displaystyle \sum _{j=1}^{{N}_{\gamma }}\sin k\nu {t}_{j}\right)}^{2}\right]\end{eqnarray} \tag{ 2 }$$

where Z² is the measured power of the signal, n is the number of harmonics, N_γ is the number of counts in the time bin, and t_j the arrival time of the jth count relative to some reference time. We only look for the first harmonic of each signal, so n = 1. By definition of the time bins ${N}_{\gamma }=5000$.

Using this statistic, we obtain a power spectrum for each time bin in which the power of the oscillation signals is plotted as function of the 10 trial frequencies. From such a spectrum one can easily determine at which frequency the signal is strongest. However, when frequency drifts between Fourier bins, the computed amplitude drops artificially (van der Klis 1989). We assume that the frequency of the oscillation is constant within each time bin and thus do not take into account frequency drifts within each bin.

For each individual time bin of a burst, we select the frequency bin with the largest measured power and determine whether or not the signal is considered a detection. We assume a Poisson noise process, for which powers in the absence of a signal are distributed as ${\chi }^{2}$ with two degrees of freedom. This assumption is reasonable at high frequencies, but not at low frequencies where the red noise contribution due to the burst light curve envelope becomes significant. For this reason, we do not look for oscillation signals below 50 Hz in the two sources without previously detected oscillations. Based on the assumption for noise distribution, we can then determine the chance that any measured power is produced by noise alone. We can then set a threshold for the measured power above which we define a signal to be significant. We choose to set the detection criterion such that the chance that a signal was produced by noise is less than 1% when taking into account the number of trials for each burst $(N)$. The number of trials in defined as the total number of time-frequency bins in which one looks for a signal; where $N={N}_{{\rm{t}}}\ast {N}_{\nu }$ with ${N}_{{\rm{t}}}$ the number of time bins and N_ν the number of frequency bins.

The probability (Prob) that a measured signal with noise chance δ was produced by noise for N trials is given by:

$$\begin{eqnarray}&&\mathrm{Prob}=N\delta {(1-\delta )}^{N-1}.\end{eqnarray} \tag{ 3 }$$

Based on the detection criterion for each burst, we can define three criteria, similar to Muno et al. (2004), for which we determine a measured power to be a significant detection (see Figure 2 for a visualization of each criterion).

• 1.  
  The chance that a measured power Z_m was produced by noise is less than 7 × 10⁻⁵ in a single trial ($\delta \leqslant 7\times {10}^{-5}$), assuming that a burst will on average consist of 16 individual time bins, such that $N=16\ast 10$. This corresponds for 1% chance overall to a measured power criterion ${Z}_{m}^{2}\geqslant 19.4$.
• 2.  
  A signal occurring in the first second of a burst has a single trial chance probability $\delta \leqslant {10}^{-3}$. This probability results in a measured power limit ${Z}_{m}^{2}\geqslant 13.8$. This detection criterion was introduced by Muno et al. (2004) as well. At the burst onset, the difference in brightness between burning and non-burning material is largest, and therefore oscillation signals would be expected to be largest in the burst rise (first second).
• 3.  
  A signal distributed over two adjacent time-frequency bins has a combined single trial noise chance probability ${\delta }_{1}\ast {\delta }_{2}\leqslant 1.3\times {10}^{-6}$. We check this using the fact that this is similar to a measured power limit of the averaged signal in these two adjacent bins of ${\bar{Z}}_{m}^{2}\geqslant 13.8$. There is a significant chance that a signal does not peak exactly in one time-frequency bin, but is spread over multiple bins instead. Therefore, we select in each time bin the signal with the largest measured power and compute the noise chance of the signal that is spread over the selected time-frequency bin and one of three directly adjacent bins: the same time bin and one of two the adjacent frequency bins, or the same frequency bin and the next time bin. The chance that both bins consist of noise alone is given by the product of the noise chance probabilities of the two individual bins (${\mathrm{Prob}}_{\mathrm{1,2}}={\mathrm{Prob}}_{1}({N}_{1},{\delta }_{1})\ast {\mathrm{Prob}}_{2}({N}_{2},{\delta }_{2})$).To meet the detection criterion of the burst, the single trial probabilities of the two bins (${\delta }_{1}$ and ${\delta }_{2}$) must satisfy the equation for ${\mathrm{Prob}}_{\mathrm{1,2}}$. Using an approximation for ${\mathrm{Prob}}_{\mathrm{1,2}}$ given by Equation (4) (taking into account that N₂ is reduced due to the fact that the second bin has to be selected from one of the three bins surrounding the first bin) yields the solution ${\delta }_{1}{\delta }_{2}=1.3\times {10}^{-6}$ that adjacent bins must satisfy to meet the threshold burst probability ${\mathrm{Prob}}_{\mathrm{1,2}}={10}^{-2}$.

$$\begin{eqnarray}&&{\mathrm{Prob}}_{\mathrm{1,2}}\approx 3{N}_{{\rm{t}}}^{2}{N}_{\nu }{\delta }_{1}{\delta }_{2}.\end{eqnarray} \tag{ 4 }$$

Each of the detection criteria satisfies that, on average, an oscillation signal detected from a single burst has a 1% chance of being a false detection. If one considers each of the three detection criteria as individual trials, the noise probability would increase to a 3% chance that a detected oscillation is actually a false detection.

For each time bin we determine four different measured powers, all related to the frequency bin with the largest measured power: we determine the single bin measured power, which can either be in the first second (criterion 2) or at any other timespan in the analyzed part of the burst (criterion 1), and three double bin measured powers (criterion 3). However, each measured power consists of two components: the signal power and the noise power. To obtain the signal power and its uncertainty, we correct the measured power for the noise component. We do this using the method outlined in Section 2 of Watts et al. (2005) to compute the signal power of a given measured power and the number of harmonics.

The signal power is derived using the probability distribution p_n of measured signals Z_m for given signal power Z_s:

$$\begin{eqnarray}{p}_{n}({Z}_{m}| {Z}_{s}) & = & \displaystyle \frac{1}{2}\exp \left[-\displaystyle \frac{({Z}_{m}+{Z}_{s})}{2}\right]{\left(\displaystyle \frac{{Z}_{m}}{{Z}_{s}}\right)}^{(n-1)/2}\\ & & \times {I}_{n-1}(\sqrt{{Z}_{m}{Z}_{s}})\end{eqnarray} \tag{ 5 }$$

where n is the number of harmonics (we always use n = 1), and I is a first kind modified Bessel function. The computational procedure provides a signal power and 1-σ errors.

The oscillation amplitude of the signal in each time bin is computed from the signal power. As mentioned, there are five possibilities to pass the detection criteria: one from the first criterion, one from the second and three from the third. Per time bin we select from the five options the signal with the largest (averaged) measured power that passed the detection criteria. From the signal power of this oscillation, we compute the fractional rms amplitude of the oscillation (${A}_{\mathrm{rms}}$) using Equation (6).

$$\begin{eqnarray}&&{A}_{\mathrm{rms}}=\sqrt{\displaystyle \frac{{Z}_{s}^{2}}{{N}_{\gamma }}}\left(\displaystyle \frac{{N}_{\gamma }}{{N}_{\gamma }-B}\right).\end{eqnarray} \tag{ 6 }$$

The second term in Equation (6) is the factor that corrects for the background emission, where N_γ is the number of counts, and B is the estimated number of background counts in the investigated time bin (${N}_{\gamma }=5000$ and $B={C}_{B}\ast {\rm{\Delta }}t$, with ${\rm{\Delta }}t$ the time width of the bin(s) over which the signal is considered). We calculate the 1-σ error on the amplitude using linear error propagation of the independent parameters, for which the standard deviations of N_γ and B are calculated as the square root of the considered parameter.

If none of the detection criteria are passed, an upper limit on the oscillation amplitude is determined with Equation (6) using the strongest (average) signal power (Z²_s) from those that did not pass the detection criteria (non-significant signals). This definition results in upper limits that are defined in the exact same manner as the detections, such that the results from the detections can be compared to those of the non-significant signals. Note that Muno et al. (2004) defined the upper limit as the largest amplitude that could be obtained from the first five seconds after the start of the burst decay, because they argue that most detections are found in this phase. However, this upper limit is not necessarily based on the largest measured power found throughout the whole burst, and their non-detections are thus defined differently from the detections.

From the oscillation signals detected in a burst, we select the amplitude of signal with the largest signal power to compare with the results from other bursts (see Figure 3). We thus select one specific time-frequency bin for each individual burst. If no oscillation signals are found throughout an entire burst, we select the upper limit found for the signal with the largest non-significant signal power.

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In Figure 4 the results of our analysis are plotted for each of the six oscillation sources. We plot from each burst the amplitude of the strongest oscillation signal as a function of S_Z, which indicates the accretion state. Most of the detected oscillations were found to pass detection criterion 1 (149 out of 185 bursts with oscillations). We do not show any errors on S_Z, because these are not yet available from the MINBAR database.

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Our results are consistent with Figure 2 from Muno et al. (2004). There are no deviations in the general distribution of the amplitudes of the detections as a function of S_Z. Extension of the data sample seems to cause a larger scatter of the amplitudes rather than a confinement toward any fittable trend. A more specific comparison with Muno et al. (2004) shows that there does appear to be a small offset in the strongest signal of individual bursts; our amplitudes and upper limits tend to be slightly higher by 0.02–0.03. However, we do not detect a constant offset. We will discuss this difference in more detail in Section 5.3.

Figure 4 shows that most of the detected oscillations are found at high accretion rate; ${S}_{{\rm{Z}}}\gt 1.7$ (see also Table 2). However, at lower accretion rates, oscillations can also be detected. There does not seem to be a limit on S_Z below which oscillations can no longer be detected, since for 4U 1636-536 and 4U 1728-34 detections are found in the bursts observed at the lowest S_Z. Furthermore, all sources show similar behavior in oscillation amplitude as a function of ${S}_{{\rm{Z}}};$ this holds both for detected oscillations (see Figure 5) as well as for the amplitude upper limits of the non-detections (Figure 6). In Figure 5 (Figure 6) we plot the results of the detections (non-detections) of the individual sources on a background formed by the results of the detections (non-detections) from the other five investigated oscillation sources. The distribution of results from individual sources are consistent with the distribution of the general population of the other sources.

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Table 2.  Detectability for High and Low S_Z

Source	Bursts with Detected Oscillations
	${S}_{{\rm{Z}}}\leqslant 1.7$	${S}_{{\rm{Z}}}\gt 1.7$
4U 1608-52	0/27	8/20
4U 1636-536	7/141	73/196
4U 1702-429	0/6	35/43
4U 1728-34	4/76	36/55
KS 1731-26	1/22	5/5
Aql X-1	1/50	7/21

Note. Fraction of the bursts in which oscillations have been detected for high (${S}_{{\rm{Z}}}\gt 1.7$) and low (${S}_{{\rm{Z}}}\leqslant 1.7$) accretion rates. Note that we omit bursts for which the S_Z value is unknown.

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There is a trend in fractional rms amplitude as a function of S_Z. The detections found at low accretion rate (${S}_{{\rm{Z}}}\leqslant 1.7$) generally have low fractional rms amplitudes, ${A}_{\mathrm{rms}}\leqslant 0.10$. The amplitude upper limits of the non-detections are equally low. At higher accretion rate, the signals are found to have amplitudes over a broad spread;  $0.05\leqslant {A}_{\mathrm{rms}}\leqslant 0.20$. A significant fraction of oscillation signals has a large amplitude: $\sim 1/3$ of the oscillations have an amplitude ${A}_{\mathrm{rms}}\gt 0.10$ at ${S}_{{\rm{Z}}}\gt 1.7$. The only exception is Aql X-1; in this source all six bursts with detected oscillation signals at ${S}_{{\rm{Z}}}\gt 1.7$ are found to have amplitudes that satisfy ${A}_{\mathrm{rms}}\leqslant 0.10$. However, it should be noted that the number statistic is rather low.

The distribution in amplitude as a function of S_Z is emphasized in Figure 7, which shows for each source (horizontal panels) histograms of the distribution of amplitudes for different S_Z ranges (vertical panels). This figure includes the distribution of amplitudes of oscillations in combination with amplitude limits of non-detections. 4U 1636-536 and 4U 1728-34 have the largest burst sample and therefore the most reliable results. In the range $1.0\leqslant {S}_{{\rm{Z}}}\leqslant 1.5$ the amplitude distribution is strongly peaked around ${A}_{\mathrm{rms}}=0.05$. On the other hand, at higher accretion rate ($2.0\leqslant {S}_{{\rm{Z}}}\leqslant 2.5$) the peak of the distribution is still at 0.05, but the amplitude distribution is significantly broader due to the contribution of amplitudes of detected oscillations. Note that we tried but were unable to find a simple functional fit to the distributions.

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The proposed burst oscillation models differ in expected oscillation amplitudes and burst phases (rise, peak, or tail) during which oscillations can be detected. To be able to explain the results in the light of oscillation models, we also analyzed during which burst phase the detections are found using the obtained analysis figures of the individual bursts. We focus on the burst phase during which detections are found in bursts at both low accretion rates (${S}_{{\rm{Z}}}\leqslant 1.7$) and in bursts at high accretion rates. At high accretion rate we specifically focus on oscillations signals with rms amplitudes that exceed ${A}_{\mathrm{rms}}=0.10$, to find indications of what mechanism might cause such high amplitudes. The S_Z limit is based on the results of sources 4U 1636-536 and 4U 1728-34, as there seem to be more oscillations, and those with higher amplitudes above this limit. Also, this limit is consistent with earlier studies by Galloway et al. (2008) who stressed that more detections were found for ${S}_{{\rm{Z}}}\geqslant 1.75$.

We discriminate between oscillations detected during the burst rise, peak, and tail. First we determine the maximum count rate of the burst using 0.25 s time resolution. The boundaries of the peak phase are then defined as the first and last time bin that exceed 90% of the maximum count rate. The burst rise phase starts at the burst start time and ends at the beginning of the peak phase, and similarly, the tail starts at the end of the peak phase and ends at the last analyzed time bin. This method is similar to the one described in Galloway et al. (2008). In cases where the selected 5000-count bin falls on both sides of one of the boundaries, we set the burst phase of the oscillation signal equal to the phase in which the signal was measured over the largest timespan.

4U 1636-536 has a significantly larger burst sample than all other sources, and is therefore the only source with a significant number of detections at different accretion rates. Determining the dependence of oscillation detectability on burst phase will therefore provide results with the highest significance for this source, but we present the results from all sources for completeness (see Table 3). Note that, in Table 3, we only present the burst phase results of the selected time bin (the one with the largest signal power) in each burst and do not consider oscillation signals with smaller signal powers detected in other burst phases.

Table 3.  Burst Phase Analysis

Source	${S}_{{\rm{Z}}}\leqslant 1.7$			${S}_{{\rm{Z}}}\gt 1.7$
				${A}_{\mathrm{rms}}\leqslant 0.10$			${A}_{\mathrm{rms}}\gt 0.10$
	R	P	T	R	P	T	R	P	T
4U 1608-52	0	0	0	2	1	3	1	0	1
4U 1636-536	4	3	0	10	3	38	17	1	4
4U 1702-429	0	0	0	5	5	17	0	0	8
4U 1728-34	0	1	3	4	7	10	7	3	5
KS 1731-26	0	1	0	3	0	1	0	0	1
Aql X-1	0	1	0	2	1	4	0	0	0

Note. This table shows how many signals were detected during either the rising phase (R), peak (P), and tail (T) of the burst. For each burst with oscillations, we only determine the phase of the strongest signal. We discriminate bursts observed at high (${S}_{{\rm{Z}}}\gt 1.7$) and low accretion rate (${S}_{{\rm{Z}}}\leqslant 1.7$). In bursts observed at high accretion rate, we also distinguish oscillation signals with large amplitude (${A}_{\mathrm{rms}}\gt 0.10$) from low-amplitude oscillations (${A}_{\mathrm{rms}}\leqslant 0.10$).

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Oscillations have been detected in 81 of the 339 observed bursts from 4U 1636-536 in our sample. Seven of these bursts occurred while the source was in a low accretion state and all of the detections in these bursts were found either during the rising phase or peak of the burst. Moreover, considering all detected oscillations throughout each of these bursts, we stress that none of these seven bursts was found to have oscillation signals in the tail of the burst. From the 22 bursts observed at ${S}_{{\rm{Z}}}\gt 1.7$ that were found to have oscillation signals with ${A}_{\mathrm{rms}}\gt 0.10$, 18 were observed during the rising phase or peak of the burst.

Overall, for 4U 1636-536 two trends seem to be present: at low accretion rate (${S}_{{\rm{Z}}}\leqslant 1.7$), oscillation signals are only found during the rising phase of the bursts or during the peak rather than the tail, and at high accretion rate most of the high amplitude oscillations (${A}_{\mathrm{rms}}\gt 0.10$) are detected in the rising phase. This second trend seems to be present in the bursts of 4U 1728-34 as well (see Table 3). In general most oscillations are detected at high accretion rate and have low amplitudes. These signals are most often found during the tail of the burst (in all sources). This trend is consistent with the results found by Galloway et al. (2008).

Since there are significantly more detections of oscillations found at higher accretion rate, we decided to look at the detectability of the oscillations as a function of accretion rate. We combined the results from all sources, divided the S_Z range [0.6–2.8] into non-overlapping bins with 0.1 width, and determined in each bin what fraction of the burst contained detected oscillation signals. The oscillation detectability is defined as the number of bursts with oscillations in one S_Z bin, divided by the total number of bursts in the considered bin. We used the combined results rather than those of individual sources to increase the number of bursts per bin and with that the significance of each point. Combining the results is justified by the observation that the behavior of each of the oscillation sources was found to be similar as a function of accretion rate (Section 4.1.1).

Figure 8 shows the oscillation detectability as function of S_Z. For ${S}_{{\rm{Z}}}\lt 1.7$ the oscillation detectability is smaller than 10%. Moreover, for ${S}_{{\rm{Z}}}\lt 1$ no oscillations were detected. However, because of the low number of bursts in this range (note that in Figure 8 the number of bursts in a specific S_Z range is indicated by the size of the purple circles), these points cannot be considered to be statistically significant. For S_Z values larger than 1.7, the oscillation detectability significantly increases. A steep increase of oscillation detectability as a function of accretion rate can be observed for $1.7\leqslant {S}_{{\rm{Z}}}\lt 2.5$. The oscillation detectability goes up to 85% for $2.5\leqslant {S}_{{\rm{Z}}}\lt 2.6$. For the highest accretion rates (${S}_{{\rm{Z}}}\gt 2.6$) the detectability is found to decrease, but so does the significance.

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From 4U 1705-44 a total of 70 bursts were analyzed after the burst selection process (Section 2.3). In the frequency range ν = 1800–1810 Hz we measured four pass signals. This range was selected as most positive and therefore we present the results obtained from this frequency band (see Table 4 for the three frequency bands with the best results). The largest signal power among the four pass signals is ${Z}_{{\rm{s}}}^{2}=22.7$.

Table 4.  Frequency Analysis for 4U 1705-44

Rank	ν Band	Pass	Max ${Z}_{{\rm{s}}}^{2}$	Significant
	(Hz)
1	1800–1810	4	22.7	No
2	270–280	3	20.8	No
3	1820–1830	3	13.7	No

Note. None of the signals meets the detection criteria and we thus do not detect oscillations for any frequency

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Figure 9 shows the results of the fractional rms amplitude as a function of S_Z (upper left panel), as well as an overplot of the results on a background of the results from all oscillation sources (lower left panel). From the upper panel one can observe that very few bursts are observed at the highest accretion rates. The lower panel shows that the distribution in fractional rms amplitude as a function of accretion rate of the observed signals seems to be similar to those of the oscillation sources.

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We analyzed 22 bursts of 4U 1746-37 that passed the burst selection, and found in all frequency bands for which the analysis was carried out no more than one pass signal. We present the results from the frequency range ν = 1880–1890 Hz, because the pass signal in this range has the largest signal power; ${Z}_{{\rm{s}}}^{2}=24.4$ (see Table 5). The upper right panel of Figure 9 shows the fractional rms amplitudes as function of S_Z. A notable observation from this figure is that more than half of the observed bursts at high accretion rate are found to have upper limits larger than ${A}_{\mathrm{rms}}=0.10$, while in all oscillation sources the upper limits are rarely found to be this large. For most bursts of the oscillation sources, the upper limits are lower than the typical amplitudes of detected oscillations at that accretion rate, because in general a weak signal has a very low amplitude. Other than that, the selected signals from the observed bursts of this source show no abnormalities in distribution of amplitudes as function of accretion rate (lower right panel of Figure 9).

Table 5.  Frequency Analysis for 4U 1746-37

Rank	ν Band	Pass	Max ${Z}_{{\rm{s}}}^{2}$	Significant
	(Hz)
1	1170–1180	2	24.4	No
2	1970–1980	1	24.3	No
3	1240–1250	1	22.8	No

Note. None of the signals meets the detection criteria and we thus do not detect oscillations for any frequency

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To determine whether or not the two new sources are anomalous compared to the oscillation sources, we tried to obtain the probability of finding a distribution of pass signals as low as we did. Initially we tried to do this by fitting the distribution of amplitudes as function of S_Z observed for the oscillation sources. However, we were unable to find a simple function that would adequately fit the data (primarily due to the broad spread in amplitudes at high accretion rate). Then we fitted the distributions of amplitudes in different S_Z bins (Figure 7) with several functions (see Section 4.1.1).

Subsequently we applied a more robust method to obtain a value for the probability of the amplitudes found for the two sources without previously detected oscillations. Since 4U 1636-536 has the most bursts and the largest amount of bursts with oscillation signals, we set this source as a model distribution. Most detections are found at high accretion rate (${S}_{{\rm{Z}}}\gt 1.7$) and the amplitudes of the oscillations are higher. Larger fractional rms amplitudes indicate stronger signals, and we thus expect to find the most, and the most significant, results at high accretion rate. We determine how likely it is that the signals that passed any of the detection criteria, observed in bursts at high accretion rate of the two non-oscillating sources, have amplitudes as low as they do. We use the distribution formed by the detected oscillations in the bursts from 4U 1636-536 to determine this probability. First, we determine how many pass signals (${N}_{\mathrm{obs}}$) are found for each of the two non-oscillating sources at ${S}_{{\rm{Z}}}\gt 1.7$ and the largest amplitude (${A}_{\max }$) among these pass signals. Then we randomly pick ${N}_{\mathrm{obs}}$ amplitudes of the detected signals with ${S}_{{\rm{Z}}}\gt 1.7$ of the model distribution, and determine whether or not the condition that all ${N}_{\mathrm{obs}}$ amplitudes satisfy ${A}_{\mathrm{rms}}\leqslant {A}_{\max }$ is met. We repeat the sampling process 10,000 times and determine how often the condition is satisfied. We find that for 4U 1705-44 there is a 15% chance of finding oscillation amplitudes as low as those observed. For 4U 1746-37 this chance is 95%. We conclude that neither source is anomalous given the limitations of current data.

Surface mode patterns cannot give amplitudes as high as hotspot models, since in the latter the bright spot can be restricted, physics permitting, to a smaller region of the star (either as it spreads from the ignition point or due to some mechanism like magnetic confinement). Obtaining amplitudes ${A}_{\mathrm{rms}}\sim 0.1$ with any of the suggested models is certainly feasible, but obtaining amplitudes as high as ${A}_{\mathrm{rms}}\sim 0.2$ with surface mode models will be more challenging. For canonical cooling wake models, amplitudes are predicted to be low, ${A}_{\mathrm{rms}}\sim 0.03$, while for asymmetric cooling wake models, amplitudes can be as large as ${A}_{\mathrm{rms}}=0.10\mbox{--}0.20$, depending on the ignition latitude (Mahmoodifar & Strohmayer 2016). An overview of the maximum amplitudes predicted for each model is given in Table 6, along with other relevant phenomena predicted for each model.

Table 6.  Summary of Model Predictions

	Hotspot	Surface Waves	Cooling Wakes
${A}_{\mathrm{rms}}$ max.	0.2	0.1	0.2a
Burst phaseb	R Pc  Tc	T	T
SZ dependence	- Ignition latitude might	- Signals might only be able
	depend on SZ	to grow to detectable size at
		high SZ
Ignition latitude (${\phi }_{\mathrm{ign}}$) dependence	- If stationary, max. ${A}_{\mathrm{rms}}$ lower		- Max. ${A}_{\mathrm{rms}}$ decreases with
	for high ${\phi }_{\mathrm{ign}}$		increasing ${\phi }_{\mathrm{ign}}$
	- Coriolis force confinement
	more effective at high ${\phi }_{\mathrm{ign}}$,
	reducing low ${\phi }_{\mathrm{ign}}$ amplitudes

Notes. Empty boxes indicate that, at present, there are no clear model predictions.

^aA_rms up to 0.2 can only be obtained for asymmetric cooling wake models. ^bR—rising phase, P—peak of the burst, T—burst tail. ^cA hotspot can only produce oscillations during the peak and tail of the burst if the hotspot is confined.

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What then do the data require us to explain in terms of accretion rate dependence? As the accretion rate rises, the mechanisms operating must permit higher maximum amplitudes, but also allow the possibility of the amplitude remaining low, since we see bursts with amplitudes that never get much above the detection limit even at high accretion rate. What then is the expectation in this regard for the different proposed mechanisms?

For the spreading hotspot model, the most obvious dependence on accretion rate should come if ignition does indeed move off-equator to higher latitudes as the accretion rate increases (Cooper & Narayan 2007). This is due to variations in local accretion rate from rotational effects, which can cause burning to stabilize at the equator before it stabilizes at higher latitudes. This should occur for a relatively narrow band of accretion rates, but recall that it is not straightforward to map S_Z to an absolute accretion rate. The predicted effect on amplitude, however, is not immediately clear. All other things being equal, a stationary hotspot at higher latitudes should give a lower amplitude than a hotspot on the equator (Muno et al. 2002b). However, the speed at which the flame spreads from the ignition point will also have an effect. Coriolis confinement is more effective off-equator (Spitkovsky et al. 2002; Cavecchi et al. 2013, 2015) so an equatorial hotspot can be quickly wiped out as the flame spreads, reducing amplitude. There may even be multiple phases of on- and off-equator ignition as the accretion rate changes, at the transition phases between different burning regimes (Maurer & Watts 2008). The intrinsic dependence of flame speed on accretion rate (as e.g., the properties of the ocean change), and the dependence of stalling mechanisms such as magnetic confinement (Cavecchi et al. 2016) on accretion rate, are not known.

So is the hotspot model consistent with our observations? Let us start with the overall rise in oscillation amplitudes. The very simplest model, where ignition location moves off-equator as accretion rate increases and flame spread does not change, would not explain the large rise in maximum amplitude, since the amplitude should fall. For the spreading hotspot model to fit the data, either flame spread must slow significantly as ignition moves off-equator due to Coriolis effects (as predicted), or ignition is actually moving back on-equator at high accretion rates, due to the presence of multiple burning regimes. The alternative is that some stalling mechanism becomes more effective as the accretion rate rises. What about the spread in amplitudes seen at higher accretion rates? In the simple spreading hotspot model, ignition latitude and flame spread speed from that point should vary monotonically with accretion rate. It is therefore hard to see how such a broad spread is possible if this is the only mechanism in operation, unless ignition location depends on other factors as well.

Predictions for the accretion rate dependence of surface mode amplitudes center around the need to excite the oscillations to detectable amplitudes. Narayan & Cooper (2007) have argued that there is a nuclear burning driven instability mechanism that would operate to pump mode amplitudes primarily in helium-dominated bursts, which occur preferentially at higher accretion rates. If this is correct, surface mode amplitudes should rise as the accretion rate increases. This matches the observation of a general rise in amplitude, although surface mode models would still struggle to explain the very highest amplitudes seen at the highest accretion rates. Given that there are mechanisms that could also suppress surface mode amplitudes again e.g., in bursts where a large convective layer develops (Cooper 2008), surface mode models could also viably explain the lower amplitudes seen in some bursts as well.

Our conclusions at this stage are rather tentative. Hotspot models can in principle give rise to the highest amplitudes seen, and it is possible to explain the overall rise in the envelope of the amplitude distribution with accretion rate provided that the flame spread speed varies strongly with latitude (for simple expectations of the change in ignition latitude with accretion rate). However, this simple model on its own would struggle to explain the fact that at similar accretion rates there are also bursts that do not show high-amplitude oscillations. Perhaps ignition location varies more than expected, or we are seeing the effects of a stalling mechanism, or flame spread speed simply varies much more than indicated by current studies. Alternatively we are seeing a second mechanism coming into play at higher accretion rates, to give the lower-amplitude population. In this scenario the high- and low-amplitude oscillations at high accretion rate would be caused by two different mechanisms. Surface mode models, which can explain amplitudes up to about 0.1, would fit well in this regard since instability mechanisms that might pump them to detectable amplitudes are only expected to kick in as accretion rate rises. We cannot comment on the cooling wake model of Mahmoodifar & Strohmayer (2016) since at this stage there are no concrete predictions for accretion rate dependence against which to compare.