Using X-ray time-delays to get close to black holes and neutron stars

Using X-ray time-delays to get close to black holes

and neutron stars

Bart van Baal, 10615989

February 26, 2017

Report Bachelor Project Physics and Astronomy, 15 EC, conducted between 24-03-2016 and 26-02-2017

Supervisor: P. Uttley

Second Assessor: M. van der Klis Version: Final

University of Amsterdam, FNWI Anton Pannekoek Institute

Abstract

This work takes a look at the compact objects found in X-ray binary systems and uses their timing delays in order to see if there is a dif-ference in the power variability of a black hole system, GX339-4, and a neutron star system, Aquila X-1. By sorting many observations of each system into different groups, depending on the state they are in, a comparison between these two systems can be made.The light curves in two different energy bands, 3 - 6 keV and 7 - 13 keV, are taken, and these were used to create a phase lag diagram in the frequency range of 0.0078 - 54.78 Hz. The results from these diagrams show that there are indeed differences in the two systems, but not all of them can be explained by the fact that a neutron star surface is an extra source of seed photons, as a difference has been found for much higher time scales, which cannot be explained even if the size of the system is assumed to be 100RG.

Samenvatting

In dit onderzoek is gekeken naar verschillen tussen een zwart gat en een neutronenster, aan de hand van uitgestraalde lichtdeeltjes van deze twee compacte systemen. Onderstaande figuur 1 is een visualisatie van een dergelijk systeem, met een begeleidende ster en het (centrale) compacte object.

Figure 1: De begeleidende ster verliest langzaam materie aan zijn compacte partner.

De materie die vanaf de begeleidende ster op het compacte object valt, komt niet meteen op het zwarte gat of de neutronenster terecht, maar komt eerst in een accretieschijf terecht waarin het om dit compacte object heendraait. Uit deze schijf kunnen lichtdeeltjes worden uitges-traald, die wij kunnen waarnemen. Door invloeden van een zogeheten

corona (een soort hele hete wolk) om het centrale object heen kun-nen deze deeltjes extra energie krijgen. De verschillende energiewaar-den van de uitgezonenergiewaar-den lichtdeeltjes kunnen op Aarde worenergiewaar-den geme-ten, waardoor te achterhalen is of de deeltjes direct uit de accreti-eschijf komen of eerst een interactie hadden met deze corona. Aan de hand van wiskundige technieken (Fouriertransformaties) kan dan bepaald worden of er verschillen zijn tussen een neutronenster en een zwart gat. Deze verschillen zijn te verwachten omdat een neutronen-ster zelf ook lichtdeeltjes uitzendt, terwijl een zwart gat dat niet doet. De verwachting is dat dit ervoor zorgt dat er uit een neutronenster-systeem relatief meer lichtdeeltjes komen met hoge energie, en het is gebleken dat dit inderdaad klopt. Er zijn twee verschillen gevonden, de eerste op relatief korte tijdschalen (ongeveer 1 seconde), welke waren verwacht. Daarnaast blijkt dat ook op veel langere tijdschalen er ver-schillen voorkomen, maar deze verver-schillen kunnen niet verklaard wor-den aan de hand van dit onderzoek.

Contents

1

Introduction

Black holes and neutron stars are the most compact objects found in space. Sometimes these can have companion stars, forming a system known as an X-ray binary, which allows for accretion disks to form and the emission of X-rays. There are however multiple components to such a system, which can change over time. A rough sketch of the situation can be seen below in Figure 2. The central object can be either a neutron star or a black hole. This central object is surrounded by something called a corona, but it is unsure what exactly the corona is or what geometry it takes. What is certain however, is that the corona is some form of plasma, made up from highly relativistic electrons. Regardless of the corona’s exact composition, it is sure that the corona itself has an effect in creating a power-law emission. The accretion disk is not constant through time, and can get closer and further away from the central object, which, when observed, leads to different states. If the accretion disk comes closer to the central object, the system is in a ”soft” state, while if the disk is further out it is called the ”hard” state. According to Heil et al. (2015a), important clues about the geometry of the ”coronal” X-ray power emitting regions could be discovered if inclination-dependant differences can be found for hard and intermediate states.

Figure 3: The different components cause different types of emission with different energies in the photons being radiated outwards.

As can be seen in Figure 3, from Gilfanov (2010), the different components of an X-ray binary cause different types of emission. The disk itself sends out seed photons which can scatter directly towards an observer, or which can first enter the corona where they undergo Compton scattering, causing them to acquire more energy before they also scatter towards an observer. Finally, it is also possible for such an upscattered photon to fall back on the accretion disk and to be reflected from there. As Figure 4, also from Gilfanov (2010) shows, these different types of emission all result in different

energies for the photons we observe. Each type of emission peaks at a dif-ferent point in the spectrum, where all three can be combined to get the total line. This means that if the two systems are observed in different en-ergy bands, it should become clear if one of them is being influenced by an extra component when comparing the systems. The reason to assume that these differences would exist comes from the difference between a black hole and neutron star binary: the neutron star itself can also send out photons, while the black hole cannot. Theoretically, this means that if a black hole and neutron star system are compared, differences should be found as there should be an extra power-law component coming from the neutron star sur-face. The goal of this research is to discover if this is also true in practice, by comparing the variable power output of two systems, the neutron star Aquila X-1 and the black hole GX339-4, when they were in comparable states. Gilfanov et al. (2003) have already shown that there is variability on second to millisecond time scales in neutron star systems, which is caused primarily by variations of the luminosity of the boundary layer.

The seed photons reaching the observer directly will at most have a couple of keV energy, while the comptonized photons can get up to more than 100 keV. As a result of this, if an X-ray binary is observed in different energy ranges, it is possible to get information about the state the system is in. This is because if the system is in the hard state, where the disk is further away from the central object and the corona, there would possibly be less Comptonized photons reaching us, when compared to a system in the soft state. For this reason, two different energy bands have been determined at 3 to 6 keV and 7 to 13 keV, in which measurements will be made to observe for potential differences.

Since the disk of an X-ray binary does not stay constant over time, many different observations of one object are used in order to allow filtering for comparable states between the black hole system and the neutron star sys-tem. These observations are clustered into bins, according to the state of the accretion disk. For more information on accretion disks, see Done et al. (2007). The bins will be covered in section 2.

As is explained in Uttley & Casella (2014), when measuring different energy bands there is an implicit correlation, which means that the variability has some coherence. This correlation points at either a direct, but causal,

con-nection between the mechanisms which cause the photon variations, or that both bands are causally linked to some underlying process. The earliest pro-posals (Payne (1980), Kazanas & Hua (1999)) for the origin of these lags in black hole X-ray binary systems (BHXRBs) focussed on Compton scatter-ing. Photons with a higher energy have undergone more of these scatterings, spending more time in the region which is responsible for these scatterings, causing these higher energy photons to lag behind the photons which have undergone less scatterings. However, this means that this region responsi-ble for the upscattering would have to be very large (about a light-second across), and following Nowak et al. (1999) these models were abandoned on energetics grounds, as it would be unfeasable for a BHXRB to sustain such a large area with high temperatures. Reig et al. (2003) suggested a more likely upscattering model, where the photons undergo their scattering in the jet of a black hole.

1.1 Spectral timing

As discussed in Wijnands & van der Klis (1999), power spectra of different X-ray sources can be hard to distinguish, as they all take on roughly the same shapes. However, if X-ray light curves are made for these sources it turns out that they are somewhat like an unpredictable, stochastic, noise process. An example of such a noise processs is Poisson noise. However, the noise processes observed in these light curves are inherently unpredictable to some degree. They are not completely unpredictable though, as there is an underlying function which depends on the frequency, which influences these processes. This underlying function is called the Power Spectral Density P (f ), which describes the average variance per unit frequency of a signal at a given frequency f . Different PSD noise processes (P (f ) ∝ fα) can be observed: a red noise PSD (α ≤ −2), a f licker noise PSD (α ≈ −1) and a white noise PSD (α = 0). An example of a red noise process is a random walk, while a f licker noise has a limiting time-scale and below the related frequency the PSD typically flattens at a rapid pace. This is a characteristic bend which can be found in active galactic nuclei and BHXRBs both, and which happens at a frequency corresponding to a characteristic time-scale which scales linearly with black hole mass, and perhaps inversely

with the accretion rate (Uttley et al. 2005). At even lower frequencies the PSD could become flatter again, becoming a white noise; meaning that the total variance is finite and the light curve has no long-term memory on the time-scales corresponding to these frequencies. The aforementioned example of Poisson noise is a white noise process. It should be noted that the type of PSD depends on the frequency and thus that if a black hole PSD is like a white noise process at a certain frequency, this does not mean all PSD functions are like a white noice process for this black hole.

The PSD can be estimated: to do this one takes the modulus-squared of the discrete Fourier transform (DFT) of the light curve. The DFT X of light curve x, which consists of fluxes which are measured in N contiguous time bins, each with width ∆t, is given by:

Xn= N −1

X

k=0

xk exp (2πink/N ) (1)

In Equation 1, xk is the kth value of the light curve, and Xn is the DFT at each Fourier frequency fn= n/(N ∆t), with n = 1, 2, ..., N/2. The modulus-squared of the DFT is also called the periodogram, which is given by the product of Xn and the complex conjugate Xn∗:

|Xn|2= Xn∗Xn (2)

Equation 2 can be further normalised so it has the same units as the PSD:

Pn=

2∆t

< x >2_N|Xn|

2 ₍₃₎

In Equation 3 < x > is the mean flux of the light curve, causing the nor-malised periodogram Pn to be expressed in units of fractional variance per Hz (Belloni & Hasinger 1990), leading to the name of ”rms-squared” nor-malization. So if the PSD is now integrated according to this normalisation over a certain frequency range and the square root is also taken, the frac-tional rms variability (also called Fvar) gives the contributions by variations over this frequency range.

It is important to note that for a noise process the observed periodogram is a random realisation of the underlying PSD (Uttley et al. 2014, Ch. 2.1.1),

with values drawn from a scewed distribution. This means that the ”nois-iness” of an observed periodogram is in a way intrinsic to the signal, i.e. the underlying variability process. As the underlying PSD of this process is the interesting quantity, the periodogram is usually binned up (in both frequency and by using periodograms measured from multiple, independant light curve segments) to obtain an estimate of the PSD, so that the PSD of a frequency bin νj, averaged over M segments and with K frequencies is given by: ¯ P (νj) = 1 KM i+K−1 X n=i M X m=1 Pn,m (4)

In Equation 4 ¯P (νj) is the estimate of the PSD, obtained from the average of the periodogram in the νjth bin, and Pn,m is the value of one sample of the periodogram, taken from the mth segment and with frequency fn, contained within the frequency bin νj. This bin contains frequencies in the range from fi tofi+K−1. It should be noted that Poisson noise leads to a flattening of the PSD at high frequencies, with a normalisation depending on the observed count rate, which can be fixed by substracting this constant Pnoise from the PSD.

If two light curves are used over different energy bands, a Fourier cross spectrum can be taken between these two. Assuming the two light curves to be x(t) and y(t) with DFTs Xn and Yn, this cross spectrum is defined as:

CXY,n= Xn∗Yn (5)

This cross spectrum can be used to find the frequency-dependent phase lag between the two bands, by using the complex conjugate of Xn = AX,n ∗ exp (iψn). Here AX,n is the absolute magnitude (or amplitude) of the Fourier transform and ψn is the phase of the signal at the frequency fn. This phase signal is a random value between π and −π for a noise process. Assuming y(t) to be linearly correlated but with an extra phase-shift φn at this same frequency, this leads to Equation 5 becoming:

CXY,n= AX,nAY,n exp (iφn) (6) Equation 6 shows that the phase of the cross spectrum is causing the phase

lag between the two light curves. A more thorough explanation is given by Uttley et al. (2014), including on how one goes from phase lag to time-lag. This cross-spectral approach can be used to measure the lag-frequency spec-trum: the phase lag (or time-lag) between two broad energy bands, which can be plotted as a function of the Fourier frequency. It is expected that the power-law component in the black hole system behaves differently than the one in the neutron star system, as the surface of the neutron star is another source for seed photons. These surface seed photons are different: they ef-fectively cool the corona, while the disk photons heats up the corona. In a black hole system, only the heating effect is present which produces hard lag. However, as the surface seed photons in a neutron star should cool the corona, they suppress the lag because the corona is heated and cooled at the same time, lowering the lag.

2

Analysis

In order to compare a black hole system to a neutron star system, many different observations are needed for both systems, in order to compare their different states. The chosen systems are GX339-4, a black hole system, and Aquila X-1, a neutron star system. Next the observations were compared by looking at the state the system was in during the observation, for which power colours, defined by Heil et al. (2015b), were used. As stated by Heil et al. (2015b), different X-ray binary systems evolve along a very consistent path around the power colour-colour diagram. This evolution follows a cycle for black hole systems, where both the energy and power spectra change within the hard state with an increase in the X-ray flux from the source. The energy spectra become softer while the blackbody-like emssion from the component associated with the accretion disk becomes stronger. In addition to this the variability becomes concentrated in a narrow frequency range, ranging from 0.1-10 Hz, close to the peak flux of the outburst. This state is classified as the hard-intermediate state (Heil et al. 2015b). These reduced time scales have been interpreted as evidence that the inner edge of the accretion disk is moving closer to the black hole. With the disk getting closer to the central object, the power-law becomes steeper, as stated by Done et al. (2007), which is due to the increase of disk photon flux

causing the corona to cool. Eventually the broad-band variability dissapears which leaves quasi-periodic features to dominate. This marks the transition to a soft-intermediate state. Dunn et al. (2010) found little evidence for differences in the energy spectra of these two states, despite the changes of broadband variability. While in the soft state, the source flux decreases and after some time the object will transition back to a hard state, restarting the cycle. As is stated by Heil et al. (2015b), there are complications with the methods of classifying timing behavior, which led to the use of energy spectral ”colours”, to create a ”colour based” approach of classifying the power spectra. This was done by defining power colours, which are the ratios of integrated power variance over different Fourier frequency ranges. These power colours can be measured as long as the rms-variance (see Equation 3) can be measured across all set frequency bands. This means that this method will be effective, even in observations where there is a relative high amount of noise compared to the signal.

Figure 5: The selected observations are marked for both systems. The selection criteria where: pc1 between 3 and 10, pc2 between 0.4 and 1 for Aquila X-1 and pc1 between 6 and 10, with pc2 between 0.8 and 1.2 for GX339-4.

The different observations were obtained by the Rossi X-ray Timing Ex-plorer (RXTE), but not all observations on the two objects, Aquila X-1 and GX339-4, were used. The power colours were used in order to obtain a small selection of similar states for both objects, which can be seen in Figure 5. In this figure, all observatios have been plotted, with the selected observations in bold. The selection criteria were based on the ratios of variance values between different frequency bands, which were set to 0.0039 - 0.031 Hz, 0.031 - 0.25 Hz, 0.25 - 2.0 Hz and 2.0 - 16.0 Hz. Ratios are used because the rms-Intensity diagrams in figure 4 of Heil et al. (2012) show that sources can have similar amounts of fractional rms over a broad frequency band, even when they are in different states. Observations were used for Aquila X-1 if the ratio between the third and first frequency bands was between 3 and 10 and if the ratio between the second and fourth was between 0.4 and 1. The observations for GX339-4 were used if this first ratio was between 6 and 10, and the second ratio was between 0.8 and 1.2. The observations which fit the criterea were then binned according to where they fit in the diagram. The binning can be found in Figure 6, where each colour/shape combination matches one group. The first ratio here (C/A in the figure) is again defined as the ratio between the 0.0039 - 0.031 Hz and 0.25 - 2.0 Hz bands, while the second ratio (B/D in the figure) is defined as the ratio between the 0.031 - 0.25 Hz and 2.0 - 16.0 Hz bands. The states on the left side of these two figures correspond to the hard states, where the accretion disk is further out from the central object, while the states on the right correspond to the soft states, where the accretion disk is closer to the central object. As can be observed, there are more bins for GX339-4 on the hard states side and there is one extra bin for Aquila X-1 on the soft state side, but only bins where both GX339-4 and Aquila X-1 have data have been used, as this work has only focussed on making the comparison between these two systems when they were in comparable states.

For each of these bins, all the observations were stacked on top of each other, in order to obtain the power variability plots for the two used energy bands of 3 - 6 keV and 7 -13 keV. Next a cross spectrum (see Equation 5) was taken between the two energy bands, for each of the created bins, in order to obtain the different phase lag diagrams for each bin. In this cross spectrum, the 3 - 6 keV band takes the role of Xn while the 7 - 13

Figure 6: The different bins for the two systems with the black hole system on the left and the neutron star on the right.All 9 bins span 0.301 along a logarithmic conversion of C/A.

keV band is used as Yn. The practical steps taken here are to start with the observations and create the light curves for the energy bands, and then follow up by splitting each of these light curves into small segments of ₁₂₈1 seconds. For each of the segments in both energy bands the PSD is obtained, and from there the cross spectrum. These are then taken together, to form an average. Next the PSD and cross spectrum are binned again over the frequency, to create the required frequency bins. These bins were chosen so that they all have the same width in log-frequency, starting at 0.0078 Hz, up to 54.78 Hz, spread over 53 different bins. The phase of these binned cross spectrums is now used to calculate the phase lag for each frequency bin. These steps are also described in Uttley et al. (2014, Ch. 2.2.1).

3

Results

Comparing the two systems has to be done in two energy bands to allow for the creation phase lag plots. The first energy band used was 3 - 6 keV, with its power spectral density plots for all five bins in Figure 7. Each of the bins has been plotted separately, so the differences between the black hole system and the neutron star system can be observed as they go from hard states towards the soft states. The second band which was used was spanning energy range of 7 - 13 keV, with the respective power spectral

density plots for each bin plotted in Figure 8.

What can be observed from these energy bands is that for the 3 - 6 keV band, the two systems follow roughly the same shape while the systems are in harder states, as can be seen in figures 7a, 7b and 7c, but GX339-4 is more variable at the same frequencies when comparing it to Aquila X-1. However, at the lowest frequencies (sub 0.1 Hz) the neutron star starts to behave differently for the softer states, first appearing weakly in Figure 7c but most prominently in 7d and 7e. Another difference which also becomes visible at the highest frequencies is that the neutron star now has a higher variability than the black hole, which has lower variability in its power output for the high end of the frequency range. If the differences in these lowest frequencies are ignored for a moment, it somewhat appears as if the variability of the power output for the neutron star has shifted slightly along the frequencies, as it starts showing variability at somewhat higher frequencies and does not start the strong drop at the end of the frequency range as early as the black hole does. When moving on to the other energy band of 7 - 13 keV, again starting at the harder states the two systems appear to follow the same shape, as can be observed in 8. If the spike found in the neutron star curve in 8c at roughly 0.5 Hz is ignored, again the first three bins (figures 8a, 8b and 8c) seem to have little to no difference in variability shape, but again for the softer states in 8d and 8e it becomes clear that the neutron star has a completely different variable output at the lowest frequencies (below 0.1 Hz). Again, the soft states show that the neutron star has variable power output, while the black hole in this energy band again has almost no variability for these lower frequencies, as can be observed from the last two bins in Figures 8d and e. Somewhat different from the first energy band, in the 7 - 13 keV band the ”shift” along the frequency no longer appears in the hard states, and only the delayed drop of variability happens for the soft states.

The results for these two bands can be combined together, by using Equation 5 where Xnis the 3 - 6 keV band, and Ynis the 7 - 13 keV band. This results in a phase lag plot, with the results which can be found below in Figure 9.

(a) The power plot for the first bin from 3 - 6 kev.

(b) The power plot for the second bin from 3 - 6 kev.

(c) The power plot for the third bin from 3 - 6 kev.

(d) The power plot for the fourth bin from 3 - 6 kev.

(e) The power plot for the fifth bin from 3 - 6 kev.

Figure 7: The variability power plots for GX339-4 and Aquila X-1 in the energy range of 3 - 6 keV, in the frequency range of 0.01 Hz to 50 Hz.

(a) The power plot for the first bin from 7 - 13 kev.

(b) The power plot for the second bin from 7 - 13 kev.

(c) The power plot for the third bin from 7 - 13 kev.

(d) The power plot for the fourth bin from 7 - 13 kev.

(e) The power plot for the fifth bin from 7 - 13 kev.

Figure 8: The variability power plots for GX339-4 and Aquila X-1 in the energy range of 7 - 13 keV, in the frequency range of 0.01 Hz to 50 Hz.

(a) The phase lag for the first bin. (b) The phase lag for the second bin.

(c) The phase lag for the third bin. (d) The phase lag for the fourth bin.

(e) The phase lag for the fifth bin.

Figure 9: The phase lag plots for GX339-4 and Aquila X-1, in the frequency range of 0.01 Hz to 50 Hz. The cycles go from π to −π.

While it is a bit hard to see for the first bin (Figure 9a), the other 4 bins have very readable results. They show that for the second bin in Figure 9b the two systems are very much following each other, which means there is no difference distinguisable. However, for the other three bins differences start appearing: most prominently in 9c but also in 9d and 9e it can be seen that for the lowest frequencies (below rougly 0.03, 0.04 and 0.05 Hz respectively) Aquila X-1 has significant lag compared to GX339-4, which has no lag for 9c and 9d and only a very tiny bit in 9e. There is another difference between the two systems though, which is also only observed in these last three bins. It can be observed in 9c from 1 - 3 Hz, a bit broader from 0.8 - 4 Hz in 9d and even broader again in 9e where it appears from roughly 0.2 Hz up to nearly 10 Hz. In these frequency ranges the black hole system starts to show significant lag but the neutron star shows practically none.

4

Discussion

From the results it can be gathered that there indeed are differences to be found for a black hole system and a neutron star system when both are compared when they were in equal states. Firstly there is a difference to be found in their variable power output: the softer states of the neutron star system start showing a significant difference compared to their black hole counterparts in both the 3 - 6 keV band and the 7 - 13 keV band. It is unclear however where the big difference for the lowest frequencies (below 0.1Hz) comes from, as any difference expected for the systems should not be found in the largest time scales, but rather on short time scales due to the fact that photons travel at the speed of light and the size of the objects observed is limited. Even if the size of the system would be 100RG, photons travelling at the speed of light would take less than a second to traverse it. Another difference which can be explained is something also observed for both energy bands, but which only becomes obvious when comparing their phase lags: the softer states for the black hole system clearly lag behind when compared to their neutron star counterparts, which do not have this lag. As this result is observed at frequencies around 1Hz, this difference can be attributed to the neutron star surface, which means that the surface of the neutron star indeed is another soure of photons, causing a distinguishable

Figure 10: A sketch to show the diffferences between a hard state on the left and a soft state on the right, where the accretion disk is closer to the central object.

difference between a black hole system and a neutron star system. This is also in line with the work of Gilfanov et al. (2003), who found that the variations of the luminosity of the boundary layer of neutron stars cause aperiodic and quasi-periodic variability on these time scales.

The cause that all of these differences only show up for the soft states can also be explained, as in the hard states the accretion disk is further away from the central object, as can be seen in Figure 10. This means the accre-tion disk has less interacaccre-tion with the corona and the central object, while the interaction between them is the underlying cause of these differences. Since the photons coming from the surface of the neutron star effectively cool the corona they reduce the lag caused by the variations of the accre-tion disk. However, for hard states there is not enough interacaccre-tion between the corona and the accretion disk to cause significant lag, and thus the sur-pressing factor for the neutron star only shows up in the later bins, which contain the soft states. Because of this, it is not a big problem that the phase lag plot for the first bin is suffering from large errorbars, which is due to the fact that it only contains three observations for the neutron star, as the remaining four plots give more than enough information to allow for a clear interpretation.

5

Conclusion

In short, the results show that in the harder states both the variable power output and phase lag differences between the two systems are small, due to the corona being less affected by the accretion disk, which is further away in these states. The differences become apparent for the soft states however,

where the systems become more compact. This compactness causes the surface of the neutron star to have a bigger effect on the total system, as photons originating from the surface effectively cool down the corona which results in suppression of the lag which does occur for the soft states of the black hole system.

6

Future research

A recommendation for future research would be to repeat the same steps for different systems, as only one black hole and one neutron star system have been used in this work and it is possible that they were outlier cases, and not following a general trend. Additionally, as can be seen in the phase lag results, there is a deviation in the softer states that is wholly unexpected: the two compared systems are not very large, but the neutron star shows an extra component in the lowest frequencies, which means at longer time scales. This is very unexpected as any lag suppression which is caused by the neutron star surface should be observed around 1Hz and higher frequencies. A follow-up study is recommended in order to explain this, as it goes beyond the scope of this project.

7

Acknowledgements

I would like to express my thanks to Dr. Phil Uttley for sticking with me and giving me helpful explanations whenever I asked, and for giving me the time to finish this project. I also want to thank MSc. David Gardenier for allowing me to use his Python scripts which enabled the data analysis to progress smoothly, and for helping me whenever Phil was not available. Finally I want to thank the team at the Anton Pannekoek Institute for allowing me access on the Taurus computer so I could run David’s program there, and for having this project available.

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