Chapter 1 Introduction
3.1 Simulating the outbursts at periastron passage
55
Chapter 3
Simulated Populations of High Mass X-ray Binaries
To our knowledge, the only instances of capture-recapture applied in astronomy is by Laycock (2017) in a time-domain context and Romine et al. (2016) in a multi- wavelength context to determine the completeness of the population of protostars in the MYStIX survey. However, in a series of five papers on flare stars in the Pleiades, Ambartsumyan et al. (1970, 1971, 1972, 1973) and Mirzoyan et al. (1977) used similar techniques for capture probability and population size estimation without an explicit mention of capture-recapture.
This chapter reproduces results similar to those encountered in the Laycock (2017) paper. This chapter uses simulations to characterise population estimators based on astronomical parameters and constraints such as observational cadence and the rate of detection in time and as a function of the number of observations. 25 The effect of increased brightness threshold is also explored w.r.t. rate of convergence to the true population size as a function of capture occasion observation k. I explain the steps performed and the motivations for doing so to arrive at those results.
located within the SMC (Liu, van Paradijs, and Van den Heuvel, 2005, 2007; Haberl and Sturm, 2016, and references therein). For these reasons their prevalence was used as a motivation to simulate a sample of BeXRB systems that is representative of the HMXB population that one may encounter by using the SMC as an example. The features of BeXRB systems include Type I outbursts which are periodic outbursts at periastron (see §1.2.1). The outbursts in the BeXRB systems can thus be modelled using characteristic orbital periods (Charles and Coe, 2006; Laycock, 2017).
Several scenarios of an HMXB population were considered by Laycock to probe esti- mator performance of the population size by considering the interaction of parameters such as the ‘sampling window’ (which we will refer to as cadence from hereon) and the recurrence of outbursts of HMXB. Laycock (2017) created six model distributions from pulsar spin periods, described in Table 3.1, which I have also used in the same methodology to simulate X-ray lightcurves of HMXBs. These models represent instances of possible BeXRB spin period distributions, empirically linked to the orbital period through the relation in Eq. 1.1. Each of the models (A through F) were characterised either by a Gaussian (type G) or a uniform (type U) distribution in Ppulse. A sample of N = 100 pulse periods was randomly drawn from each distribution and the peri- ods transformed to a corresponding orbital period using the empirical relationship in Eq. 1.1. The spin period distributions were transformed to skew log-normal orbital period distributions, except for population model B, which was of Type U and subse- quently transformed to a triangular distribution. The distributions are shown in Figure 3.1 with peaks ranging between 130 and 300 days.
Table 3.1: Model A to F used for simulating populations of HMXBs with comparison orbital period distributions, as defined in Laycock (2017). Type G refers to a Gaussian distribution with P1 as the mean and P2 the standard deviation, where as Type U refers to a Uniform distribution with P1 as the minimum and P2 the maximum bounds. T indicates the median orbital period
for each model. Negative values for spin periods are excluded.
Model Type P1 P2 T
s s d
A G 150 50 132
B U 1 1000 <311
C G 0 200 16
D G 200 100 150
E G 200 50 150
F G 0 100 16
3.1. Simulating the outbursts at periastron passage 57
Figure 3.1: Simulated period distribution functions for the pulsar spin period (top) and the N = 100 drawn samples that were transformed to BeXRB binary
orbital periods (bottom).
Each of the 100 binary orbital periods was used to simulate a BeXRB X-ray lightcurve modelled by a Gaussian-shaped outburst profile which is assumed to reoccur at each periastron passage. The width of the Gaussian-shaped outburst is described by the full-width half-maximum (FWHM) (where FWHM ≈ 2.355σ where σ standard devia- tion of the Gaussian). The standard deviation σ was set at a constant 10% of each system’s orbital period (which fixes the duty cycle of the sources and hence, the capture probability):
τoutburst =FWHM≈2.355σ= 0.2355Porbit (3.1)
A relative flux scale was used for simplicity, with all lightcurves at a zero quiescent flux. This assumes that all systems are located at the same distance and have little variation in luminosity. The amplitude of each outburst was randomly scaled between 0 and 1 from a uniform distribution. The randomness simulates the varying degree to which BeXRB systems outburst based on various factors such as accretion rate (mass transfer rate) and disk size. It allowed for those odd occasions of ‘missed’ outbursts in BeXRB systems where the geometry of the system varies significantly compared to the regular periastron outbursts (Charles and Coe, 2006) and the outburst is not seen when expected. Each lightcurve signal in the simulation can be described as a sum of Gaussian functions in the time domain by the following equation:
~ 200
200 400 600 800 1000
FiJliise (seconds)
40
~
20o , _ _ _ ~ - - ~ - - ~ - - ~ - - ~ - - ~ - - ~ - - - <
0 50 100 150 200
Porbit (days)
250 300 350 400
Model A Model B
Model C Model D Model E Model F
Signal form : X∞
i
1 σ√
2π
e
−12(
t−σµi)
2; µ
i≥ t
0where µi is the time at the peak flux of the ith outburst, t0 the simulation start time, and σ = 0.1Porbit =τoutburst/2.355 the standard deviation related to the outburst cycle in Eq. 3.1.
Figure 3.2: Example of a small population of simulated Model A HMXB lightcurves that show periodic outbursts at periastron passage with randomised amplitudes. The lightcurves are plotted for the first 400 days of the simulation.
Models A to F were simulated over a timescale of 4000 days (∼ 11 years). The lightcurves are shown for a few sources in Figure 3.2.