Deep Chandra Survey of the Small Magellanic Cloud. III. Formation Efficiency of High-mass X-Ray Binaries

X-ray binaries (XRBs) are our main tool for studying the populations of compact objects in galaxies, and the formation and evolution of intermediate and higher mass binary stellar systems. Systematic studies of nearby galaxies have provided initial estimates of the formation rate of XRBs as a function of the star formation rate (SFR) and stellar mass (M⋆) of their host galaxies (Lehmer et al. 2010; Mineo et al. 2012). A strong dependence of the XRB formation rate on age and metallicity has been predicted (Dray 2006; Linden et al. 2010; Fragos et al. 2013); however, we are only now starting to probe the details of this connection (Shtykovskiy & Gilfanov 2005; Antoniou et al. 2010; Williams et al. 2013; Antoniou & Zezas 2016). The Small Magellanic Cloud (SMC) is the ideal target to study the dependence of the formation efficiency of XRBs on the age of their parent stellar population. It is our second nearest (D = 61.9 ± 0.6 kpc; de Grijs & Bono 2015; 1' ∼ 18 pc) star-forming galaxy, and offers a clear picture of its spatially resolved star formation (SF) history (Harris & Zaritsky 2004, hereafter HZ04). Furthermore, it has low metallicity (Luck et al. 1998; Antoniou & Zezas 2016 and references therein), and hosts one of the largest populations of high-mass X-ray binaries (HMXBs) known in a galaxy, rivaling that of the Milky Way (e.g., Coe & Kirk 2015; Haberl & Sturm 2016, hereafter HS16).

To study the HMXB populations in the SMC in detail, and in particular their connection with their parent stellar population, we performed a deep Chandra X-ray Visionary Project (XVP) survey of selected SMC regions (PI A. Zezas) chosen to sample stellar populations of different ages. The first results on the spectral and timing properties of pulsars detected in the survey fields are presented in Hong et al. (2016, 2017). In this work, we present our measurement of the formation efficiency of the SMC HMXB populations as a function of their age, the most detailed such measurement so far. In Section 2 we describe briefly the Chandra XVP SMC survey, and the source sample used, while in Section 4 we discuss the SF history of the regions studied, and we link the SF in each region with the XRB populations. In Section 6 we estimate the formation efficiency of HMXBs, and present the HMXB delay time distribution (DTD). In Section 7 we discuss these results and compare the different HMXB formation efficiency indicators. The most important findings are summarized in Section 8.

Throughout this work, we adopt a distance modulus of (m − M)_V = 18.96 ± 0.02 mag (de Grijs & Bono 2015), R_V = 2.74 ± 0.13 (Gordon et al. 2003), and E(B − V) (=A_V/R_V) = 0.09 ± 0.02 mag (Udalski et al. 1999), thus the extinction A_V is estimated to be 0.25 mag, and in turn¹¹ A_I = 0.12 mag, and E(V − I) = 0.13 mag.

The SMC has been surveyed extensively in the X-ray band with Einstein (e.g., Seward & Mitchell 1981), ROSAT (e.g., Haberl et al. 2000; Sasaki et al. 2000), RXTE (e.g., Laycock et al. 2005; Corbet et al. 2009), ASCA (e.g., Yokogawa et al. 2003), and XMM-Newton (e.g., Haberl et al. 2012; Sturm et al. 2013), with the latter yielding the most extensive survey of the galaxy down to luminosities of 5 × 10³³ erg s⁻¹ (Haberl et al. 2012). These observations yielded a very rich population of HMXBs (120 sources; Haberl & Sturm 2016), while the spectroscopic and/or photometric properties of their vast majority identify them as Be/X-ray binaries (Be-XRBs; e.g., Coe & Kirk 2015). To reach well within the regime of the X-ray emission of quiescent XRBs (∼10³² erg s⁻¹) and obtain as complete a picture of its HMXB populations as possible, we have been awarded a Chandra XVP Program to perform a comprehensive survey of 11 fields, to a depth of 100 ks exposure, selected to represent young (<100 Myr) stellar populations of different ages. These observations were performed from 2012 December to 2014 February, utilizing the ACIS-I imaging mode.

In addition, we also analyzed three archival observations reaching the same 100 ks depth. Two of these fields (PI A. Zezas; observed in 2006) overlap partially with fields from the XVP survey, and the third is centered on NGC 346 (PI M. Corcoran; observed in 2001). Although analyses of these data have been presented elsewhere (Laycock et al. 2010, and Nazé et al. 2002, respectively), for consistency we opted to reanalyze them. In Figure 1 we present a Magellanic Cloud Emission Line Survey (MCELS) Hα image of the SMC showing the observed fields, color coded for the age of their stellar population derived using data from HZ04 (for details, see Section 4).

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Source detection employed CIAO 4.5 (Fruscione et al. 2006) wavdetect in four bands (broad 0.5–7.0 keV, soft 0.5–1.2 keV, medium 1.2–2.0 keV, and hard 2.0–7.0 keV) on all individual ObsIDs and the merged data set for each field. All subsequent data analysis steps (source photometry, screening, spectral fitting, and timing analysis) were performed with ACIS Extract (AE Version 2014may23; Broos et al. 2010, 2012). This yielded 2,393 sources down to a limiting flux of 2.6 × 10⁻¹⁶ erg cm⁻² s⁻¹ in the full (0.5–8.0 keV) band (∼50% complete at 7.94 × 10⁻¹⁶  erg cm⁻² s⁻¹). Further details on the survey, data analysis, and the complete source catalog¹² will be presented in V. Antoniou et al. (2020, in preparation).

To identify the HMXBs in the complete source catalog, we cross-correlated the locations of the 2,393 X-ray sources with the OGLE-III catalog of stars in the SMC (Udalski et al. 2008). We used a cross-correlation radius based on the relative error of the X-ray source position estimated from Equation (5) of Hong et al. (2005). We set a conservative minimum radius of 1'' based on the minimum combined positional uncertainty of the X-ray and optical catalogs, and the standard Chandra boresight error (08 rms; 90% confidence¹³ ) Following Antoniou et al. (2010) and Antoniou & Zezas (2016), we classified as HMXBs X-ray sources with optical counterparts within the OB-star locus of the (V, V − I) color–magnitude diagram (CMD). This locus is based on the location of massive stars from the spectroscopic census of the SMC (Bonanos et al. 2010). To account for the well-known effect that Be-XRBs (the most numerous subclass of SMC HMXBs) appear redder than OB stars due to the circumstellar disk of their Oe/Be star companions (e.g., Antoniou et al. 2009a, 2009b; Bonanos et al. 2010), we extended the locus to redder colors based on the location of all 120 known HMXBs from HS16 on the same CMD. We define the "extended" OB-star locus (hereafter referred to simply as the OB-star locus) to lie within V ≤ 18 mag, I ≤ 18 mag, and −0.4 ≤ V − I ≤ 0.6 mag (Figure 2). In total, we have identified 3,938 OGLE-III matches for the 2,393 final Chandra XVP sources (V. Antoniou et al. 2020, in preparation).

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To estimate the chance coincidence probability of identifying spurious matches from the OGLE-III catalog as the optical counterparts of the HMXBs, we performed extensive Monte Carlo simulations following Antoniou et al. (2009b), and Antoniou & Zezas (2016). These results indicate that for a search radius of 1'', about 10% of the bright blue (V ≤ 16 mag and −0.4 ≤ (V − I) ≤ 0.6 mag) matches are spurious associations, with this probability increasing to 17% and 79% for 16 < V ≤ 17 mag, and 17 < V ≤ 18 mag, respectively (V. Antoniou et al. 2020, in preparation). These results show that the brightest (and subsequently bluest, for objects of similar brightness) match for sources with multiple matches is the most likely optical counterpart (see Antoniou et al. 2009b).

Using the above criteria, the HMXB sample consists of 127 candidate sources, identified in this work on the basis of their association with an early-type star and/or a known HMXB. Since XRBs are highly variable sources, in order to obtain a more complete picture of their population, they were supplemented by all additional HMXBs identified by HS16 (likely and confirmed sources) that fall within the XVP area, but are neither detected in our survey (nine sources) nor matched with any of the 127 sources identified in the XVP data (five sources). Based on the X-ray colors reported in HS16 we identified four sources with colors inconsistent with HMXBs:

• 1.  
  X-ray source CXOU J011302.24-724142.1 having X-ray colors typical of a foreground star. The identified counterpart is a Gaia DR2 source (4687253738933765632; Gaia Collaboration et al. 2018) with a parallax and proper motion also suggestive of a nearby star.
• 2.  
  X-ray source CXOU J005108.94-732144.7 being a spurious source identified as a knot in the known bright SNR IKT6.
• 3.  
  X-ray source CXOU J005905.52-721035.1 being a YSO-4, i.e., an HAeBe Young Stellar Object (Ruffle et al. 2015).
• 4.  
  X-ray source CXOU J010043.11-721133.6 being consistent with the anomalous X-ray pulsar (AXP) CXOU J010043.1-721134 with P_spin = 8.020392(9) s (Lamb et al. 2002, 2003; McGarry et al. 2005). No convincing optical identification was found by Durant & van Kerkwijk (2008), while the counterpart identified in this work has a ∼19% probability of being a spurious match (based on its photometry, and extensive Monte Carlo simulations; Antoniou et al. 2009b).

Based on the above findings, we exclude these four sources from the sample of the 127 candidate HMXBs identified in the Chandra XVP observations. Thus, our final list of candidate HMXBs consists of 137 sources. In more detail, from the comparison of the Chandra XVP detected sources with those listed in HS16, we found that

• 1.  
  Out of the 120 high-confidence sources reported in HS16 as HMXBs, only 65 fall within the 14 Chandra fields used in this work. The remaining 55 HS16 HMXBs have not been covered by the Chandra XVP survey.
• 2.  
  Out of the 65 HS16 HMXBs in the Chandra fields, 46 have an XVP match, i.e., 29% of the HS16 HMXBs covered by the Chandra survey do not have an XVP counterpart. This fraction is an indication of the number of HMXBs active at any given time.
• 3.  
  Forty-one of these 46 HS16 sources have an XVP match associated with at least one OB star.
• 4.  
  Five HS16 sources (12, 77, 87, 90, 117) have an XVP match, but are not associated with an OB star using the conservative minimum radius of this work.
• 5.  
  From the remaining 19 HS16 HMXBs (out of the 65) that are covered by the Chandra XVP survey but have not been detected in that, we exclude from further consideration 5 sources that have very large positional uncertainties (>100''). Four are RXTE sources (HS16, #5, 17, 28, 32), and one is an INTEGRAL source (HS16, #8). The lack of more accurate positions for these sources does not allow us to associate them with any individual fields and hence their parent stellar populations. The remaining 14 sources supplement the 123 HMXBs identified in the Chandra XVP survey and comprise our final source sample of 141 candidate and confirmed HMXBs used in this work.
• 6.  
  Five of these 14 HS16 sources have at least one OB counterpart in the OGLE-III catalog.
• 7.  
  The minimum and maximum positional uncertainty listed for the 14 HS16 sources without an XVP match is 06 and 400, respectively, while regarding their spatial distribution, there are two HS16 sources within each of the DF05, DF07, DF09, DF11, and DF01_A fields, one in DF06, and 3 in DF02_A.

The optical properties of the most likely counterpart of the 123 candidate HMXBs from this work are listed in Table 1, while those of the 14 HS16 sources that supplemented our XVP sample are listed in Table 2. In particular, in Table 1 we list the HMXB ID (Column 1), along with the Chandra XVP field ID (Column 2), and the source ID within that particular field (Column 3). The Chandra catalog name appears on Column 4, followed by the R.A. and decl. (J2000.0) coordinates of each source (Columns 5 and 6), and their associated relative error (Column 7). Given that for the majority of the fields we did not find an adequate number of matches with the 2MASS catalog that would allow us to correct the absolute astrometry, we did not attempt to correct the absolute astrometry of the fields used in this work. Nonetheless, the rms error of the positions of sources in areas of the sky covered by multiple pointings is consistent with the typical absolute astrometric error of Chandra used in our cross-correlation.

Subsequently, in Table 1 we present the optical properties of the identified OGLE-III OB matches. The field ID, subfield ID and database number are listed in Columns 8–10, the R.A. and decl. (J2000.0) coordinates in Columns 11–12, and the V, I and V − I photometry in Columns 13, 15 and 17, followed by their errors in Columns 14, 16 and 18, respectively. The offset between the X-ray and optical source positions is listed in arcseconds in Column 19.

For the 14 sources that supplemented the 127 candidate HMXBs identified in the XVP survey the information provided in Columns 1 and 3–14 in Table 2 is similar to that listed for the 123 candidate HMXBs from this work in Columns 1 and 8–19 of Table 1, respectively. In Table 2 we also present the HMXB ID (Column 2) and the comments on the individual sources (Column 15) from the work of HS16. Regarding the optical properties of the additional sources we did not use the optical photometry presented in HS16 (which is based on the MCPS catalog); instead for consistency with the analysis of the Chandra sources we adopted their counterparts in the OGLE-III survey.

As part of the Magellanic Clouds Photometric Survey (MCPS; HZ04), the spatially resolved SF history of the SMC with a scale of 12' × 12' (216.2 pc × 216.2 pc) is publicly available. The SF history in each of the Chandra fields we adopt is the total SF history of the 12' × 12' MCPS subregions they encompass, weighted by the fraction of the area of each MCPS subregion covered by the Chandra field.

We note that for the purpose of measuring the formation rate of HMXBs with respect to their parent stellar populations, when an HMXB falls within two or more overlapping Chandra fields, we associate it with the field that has a peak of SF at a time consistent with its age (indicated in Column 4 of Table 3). This is necessary because we are measuring the SF history of each Chandra field. Also, for simplicity we approximate each SF episode as a sequence of Gaussian events; generally 1–3 Gaussians are adequate to reproduce the evolution of the SFR during a SF event. In Table 3 we list the SFR surface density for each burst of SF (Column 10), the area of each Chandra field (Column 11; the intersection area of overlapping fields is assigned to only one of those fields), the burst SF (Column 12), and the stellar mass formed during the SF episode associated with each HMXB population (Column 13). The latter is calculated by integrating the SF history during the period of the SF episode of interest in each relevant field.

Table 3.  HMXB Populations and Formation Efficiency Indicators for the Major SF Bursts of Each Chandra Field

Age	Field	SF			HMXBs	HMXBs	OBs	OBs	SF	Area	SF	M⋆	HMXB Formation Efficiency
Bin	ID	Burst				Error		Error	Burst		Burst		SFR	OBs	M⋆
ID		ID	Age	Span					Rate		Rate
			(Myr)	(Myr)					(10−6 M⊙ yr−6 arcmin−2)	(arcmin2)	(10−3 M⊙ yr−1)	(105M⊙)	(10−1 M⊙/yr)−1	(10−4)	(10−6M⊙−1)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)	(15)	(16)
1	DF11	1	7	4	1	1.0	2466	49.7	${348}_{-159}^{+139}$	330	${115}_{-53}^{+46}$	${3.6}_{-1.7}^{+3.0}$	0.9 ± 0.9	4.1 ± 4.1	${2.8}_{-2.8}^{+3.6}$
	NGC 346	1	5	2	1	1.0	2190	46.8	${192}_{-187}^{+137}$	293	${56}_{-55}^{+40}$	${0.7}_{-0.7}^{+0.5}$	${1.8}_{-1.8}^{+2.2}$	4.6 ± 4.6	${14.3}_{-14.3}^{+17.6}$
2	DF01	1	11	6	0	0.0	523	22.9	${109}_{-30}^{+31}$	343	${37}_{-10}^{+11}$	${2.1}_{-0.6}^{+1.0}$	0.0 ± 0.0	0.0 ± 0.0	0.0 ± 0.0
	DF02	1	11	6	0	0.0	257	16.0	${243}_{-41}^{+42}$	328	${80}_{-14}^{+14}$	${4.5}_{-1.2}^{+2.1}$	0.0 ± 0.0	0.0 ± 0.0	0.0 ± 0.0
	DF03	1	11	6	5	2.2	687	26.2	${220}_{-37}^{+31}$	330	${73}_{-12}^{+10}$	${3.9}_{-0.8}^{+1.0}$	${6.9}_{-3.3}^{+3.2}$	72.8 ± 32.7	${12.8}_{-6.3}^{+6.6}$
	DF04	1	11	6	4	2.0	250	15.8	${175}_{-25}^{+26}$	324	${57}_{-8}^{+8}$	${3.2}_{-0.6}^{+1.1}$	${7.0}_{-3.6}^{+3.6}$	160.0 ± 80.6	${12.5}_{-6.7}^{+7.6}$
3	DF05	1	34	43	5	2.2	1631	40.4	${94}_{-16}^{+19}$	324	${30}_{-5}^{+6}$	${14.5}_{-3.0}^{+3.8}$	${16.7}_{-8.0}^{+8.2}$	30.7 ± 13.7	${3.4}_{-1.7}^{+1.8}$
	DF06	1	34	36	11	3.3	3003	54.8	${112}_{-21}^{+27}$	331	${37}_{-7}^{+9}$	${16.0}_{-4.1}^{+5.8}$	${29.7}_{-10.6}^{+11.5}$	36.6 ± 11.1	${6.9}_{-2.7}^{+3.2}$
	DF07	1	34	34	11	3.3	3138	56.0	${136}_{-55}^{+33}$	309	${42}_{-17}^{+10}$	${18.3}_{-8.3}^{+7.3}$	${26.2}_{-13.2}^{+10.1}$	35.1 ± 10.6	${6.0}_{-3.3}^{+3.0}$
4	DF08	1	42	43	24	4.9	3981	63.1	${116}_{-15}^{+15}$	328	${38}_{-5}^{+5}$	${17.8}_{-2.4}^{+3.1}$	${63.2}_{-15.3}^{+15.3}$	60.3 ± 12.3	${13.5}_{-3.3}^{+3.6}$
	DF09	1	42	28	7	2.6	1957	44.2	${112}_{-20}^{+19}$	298	${33}_{-6}^{+6}$	${9.4}_{-2.1}^{+3.4}$	${21.2}_{-8.9}^{+8.9}$	35.8 ± 13.5	${7.4}_{-3.3}^{+3.9}$
	DF11	2	42	22	16	4.0	2466	49.7	${161}_{-31}^{+28}$	330	${53}_{-10}^{+9}$	${16.3}_{-4.9}^{+7.4}$	${30.2}_{-9.5}^{+9.1}$	64.9 ± 16.3	${9.8}_{-3.8}^{+5.1}$
	DF01_A	1	42	30	18	4.2	2551	50.5	${151}_{-13}^{+13}$	245	${37}_{-3}^{+3}$	${12.8}_{-1.6}^{+2.0}$	${48.6}_{-12.1}^{+12.1}$	70.6 ± 16.7	${14.1}_{-3.8}^{+4.0}$
	DF02_A	1	42	41	26	5.1	3111	55.8	${92}_{-19}^{+20}$	268	${25}_{-5}^{+5}$	${12.8}_{-4.6}^{+6.0}$	${104.0}_{-29.1}^{+29.1}$	83.6 ± 16.5	${20.3}_{-8.3}^{+10.3}$
	NGC 346	2	42	30	5	2.2	2190	46.8	${125}_{-14}^{+16}$	293	${37}_{-4}^{+5}$	${11.2}_{-2.1}^{+2.9}$	${13.5}_{-6.2}^{+6.3}$	22.8 ± 10.2	${4.5}_{-2.2}^{+2.3}$
5	DF10	1	67	29	2	1.4	790	28.1	${8}_{-7}^{+20}$	345	${3}_{-3}^{+7}$	${7.2}_{-2.2}^{+3.4}$	${66.7}_{-66.7}^{+162.5}$	25.3 ± 17.9	${2.8}_{-2.1}^{+2.4}$
6	DF02_A	2	266	436	1	1.0	3111	55.8	${31}_{-6}^{+7}$	268	${8}_{-2}^{+2}$	${67.5}_{-7.3}^{+8.8}$	${12.5}_{-12.5}^{+12.9}$	3.2 ± 3.2	0.1 ± 0.1

Note. Column (1) Age bin ID (Figure 3); Column (2) Chandra field ID (Figure 1); Columns (3)–(5) ID, age and time-span (FWHM) of the dominant SF episode; Columns (6)–(7) Number and number error of HMXBs in each field associated with the respective SF episode; Columns (8)–(9) Number of OB stars in each Chandra field, and error of the total number of OB stars in each Chandra field; Column (10) Peak SFR of this episode in units 10⁻⁶ M_⊙ yr⁻¹ arcmin⁻² (errors are based on the upper and lower SFR ranges reported by HZ04); Column (11) Area of each Chandra field; Column (12) Peak SFR of this episode in units 10⁻³ M_⊙ yr⁻¹; Column (13) Total stellar mass (M⋆) produced in the SF episode (based on the integration of the SFR time evolution); Columns (14)–(16) HMXB formation efficiency based on the ratio of N(HMXBs) (Column 6) to the SFR (Column 12), the N(OBs) (Column 8), and the stellar mass (Column 13) produced during the SF burst they are associated with (see Section 4).

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Since our goal is to measure the formation rate of HMXBs as a function of the age of their parent stellar population, we first need to constrain the ages of the HMXBs and associate them with individual SF episodes responsible for the birth of their progenitors. Ages are derived from optical counterpart positions on the (V, V − I) CMD with respect to the PARSEC isochrones (v1.2S; Bressan et al. 2012) generated by CMD¹⁴ 2.8 for Z = 0.004 (Figure 2). We note that due to their circumstellar disks, the Be stars are intrinsically redder than B-type stars of the same spectral type. By comparing the (V − I)_o colors of B and Be stars from the census of Bonanos et al. (2010), we have found that early [B0–B2), and mid [B2–B4) spectral-type Be stars (i.e., within the typical range of companions of known SMC HMXBs; e.g., McBride et al. 2008; Antoniou et al. 2009a; Maravelias et al. 2014) have median values of reddening-corrected (V − I)_o colors ∼0.3 and ∼0.2 mag redder, respectively, and V-band absolute magnitudes ${M}_{{V}_{o}}$ ∼0.2 and ∼0.1 mag fainter, respectively, than B stars of the same spectral type. This systematic displacement (also obvious from the background points in Figure 2) implies that a Be system appears redder and slightly fainter than a B star due to its equatorial disk.

Furthermore, each optical counterpart is associated with a SF episode taking into account the fact that stars can be associated with a SF event that overlaps with the age range of isochrones consistent with its location on the CMD. For example, the SF history of Chandra field DF11 shows two prominent peaks at 7 Myr and 42 Myr (indicated in Column 4 of Tables 3 and 4). Out of the 17 HMXBs of DF11 only one X-ray source has an optical counterpart with a location on the OGLE-III (V, V − I) CMD consistent with the SF burst at 7 Myr (best described by the 6.3 Myr isochrone of Figure 2), while the remaining 16 have ages consistent the peak in the SF history at 42 Myr. These 16 counterparts appear to be best described by isochrones with ages from 79.5 Myr (cyan) to 158.5 Myr (yellow), but as described above this is merely a circumstellar reddening effect. The optical counterparts of these sources have mid- to late-B spectral types (based on their magnitudes and colors) indicating much younger ages, thus their photometry needs to be corrected by at least the amount indicated with the dark green arrow in Figure 2. Although this correction is estimated for B[2, 4) spectral types, i.e., for spectral types perhaps a bit earlier than that of some of these 16 counterparts, it is the best approximation available due to the fact that the sample of late (i.e., later than B4) Be-type stars in the SMC is so small that we cannot derive a reliable reddening correction for the latest B-type stars. Assuming this reddening correction, we find ages younger than ∼63–79 Myr (magenta and cyan isochrones, respectively), which associates them with the SF burst at ∼42 Myr (that has a span of ∼22 Myr; age bin #4 in Table 3).

Table 4.  Combined HMXB Populations and Average Formation Efficiency

Age	SF Burst		HMXB Formation Efficiency
Bin	Age	Span	SFR	OB stars	M⋆
ID	(Myr)		(10−2 M⊙/yr)−1	(10−3)	(10−6M⊙−1)
(1)	(2)	(3)	(4)	(5)	(6)
1	6	3	0.12 ± 0.10	0.43 ± 0.30	${4.7}_{-0.10}^{+0.12}$
2	11	6	0.36 ± 0.13	5.2 ± 1.8	${6.6}_{-0.13}^{+0.14}$
3	34	38	${2.5}_{-0.64}^{+0.58}$	3.5 ± 0.67	${5.5}_{-0.69}^{+0.70}$
4	42	32	4.3 ± 0.52	5.9 ± 0.60	${12}_{-0.61}^{+0.74}$
5	67	30	${6.7}_{-6.7}^{+16}$	2.5 ± 1.8	${2.8}_{-2.8}^{+5.7}$
6	266	436	1.3 ± 1.3	0.32 ± 0.32	${0.15}_{-0.15}^{+1.3}$

Note. Column (1) Age bin ID (similar to Column 1 of Table 3); Column (2) Average age (using the values of Column 4 in Table 3) for the stellar populations in a given SF episode; Column (3) Time-span (FWHM) of the dominant SF episode; Columns (4)–(6) Average HMXB formation efficiency based on the ratio of N(HMXBs) to the SFR, the N(OBs), and the stellar mass $M\star$ (using the values of Columns (15)–(17) in Table 3) produced during the SF burst they are associated with (see Section 4).

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Following this procedure, we associate each of the 137 most likely optical counterparts of the identified HMXBs with a SF episode. These results are summarized in Table 3 where we list the number of sources from each field grouped in 6 age bins. The age bins are given in Column 1. In Column 2, we list the Chandra field ID, while in Columns 3–5 we present the number of the major SF burst, its age and time-span (identified as the FWHM of an approximate Gaussian event; see Section 4). In Columns 6 and 8 we give the number of HMXBs and OB stars, along with their related number errors in Columns 7 and 9, respectively.

The underlying assumption in this analysis is that the HMXBs do not have significant displacements from their birthplaces, so the stellar populations in their neighborhood do probe their parent stellar populations. This assumption is supported by (a) the strong association of HMXBs with star-forming regions in the SMC (e.g., Shtykovskiy & Gilfanov 2007; Antoniou et al. 2009b); and (b) the indications that HMXBs in the SMC have similar (or smaller) kick velocities compared to HMXBs in our Galaxy (e.g., Coe 2005; Antoniou et al. 2009b, 2010; Knigge et al. 2011). The latter is also consistent with the similar distribution of orbital separations and eccentricities in SMC and Galactic HMXBs (Maravelias et al. 2014). Previous works estimated typical displacements of less than ∼250 pc assuming a travel time of ∼20–30 Myr (Antoniou & Zezas 2016; see Politakis et al. 2019), which is less than the size of the Chandra fields. This is in line with population synthesis models which show that the vast majority of the HMXBs produced assuming a flat (continuous) star formation history have travel times <10 Myr, which when convolved with the distribution of their kick velocities result in displacements of less that ∼250 pc (Andrews et al. 2018).

We derive three different metrics of the age-dependent formation efficiency of HMXBs in Table 3, the number of HMXBs in different ages with respect to the (a) SFR of their parent stellar population (Column 14); (b) number of OB stars, N(OBs), in their respective Chandra field (Column 15); and (c) stellar mass formed during the SF episode they are associated with (Column 16).

Then, we group together Chandra fields that have similar ages (as indicated by the different group of fields shown in Table 3; e.g., DF05, DF06, and DF07, all show a prominent peak in their SF histories at ∼34 Myr), and we present the mean values in each age bin in Table 4. The age dependence of these three different tracers of the HMXB formation rate is shown in three different panels in Figure 3. The error bars in the x-axis indicate the average age range of the stellar populations in each age bin.

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We find that the N(HMXBs)/SFR ratio (top left panel, black squares) increases rapidly up to ∼40–60 Myr, and then gradually decreases for older stellar populations. This result is consistent with previous lower age-resolution small-scale studies (involving shallow Chandra and XMM-Newton observations), which show an increased formation efficiency of HMXBs at ages between 30 and 60 Myr (of fields across the SMC Bar) compared to younger stellar populations (SMC Wing) (Shtykovskiy & Gilfanov 2005; Antoniou et al. 2009b). On the other hand, the N(HMXBs)/N(OBs) ratio (top right panel; blue circles) shows a steep increase (by about an order of magnitude) from ∼6 to 10 Myr, then a flattening up to ∼60 Myr, followed by a drop (by about an order of magnitude again) for ages older than ∼60 Myr. The N(HMXBs)/M⋆ ratio instead (bottom panel; red triangles) remains flat up to ∼60 Myr, but it decreases for later ages as well.

An alternative formulation of the time taken for the formation of a class of astronomical objects (in this case, HMXBs) from the SF event that gives rise to its progenitor stellar systems is described by the DTD. The DTD is defined as the production rate of objects as a function of time after an hypothetical brief SF burst. Badenes et al. (2015, hereafter B15) described a method to recover the DTD from an object catalog and a SF history map, and applied it to LMC planetary nebulae. Here, we apply the same method to the SMC HMXB catalog described in Section 3. The only difference with respect to the B15 analysis is that the Chandra fields we used to derive the HMXB catalog do not cover a contiguous or uniform part of the SMC, and in many cases there is only partial overlap between a given Chandra field and a specific MCPS subregion. For this reason, we multiplied the SFR of each MCPS subregion in the SF history map of HZ04 by a weight between 0 and 1, which represents the fraction of the surface area of the subregion covered by Chandra (as was also done in Section 4).

The resulting DTD is presented in Figure 4. We have used the temporal binning that offers the best compromise between DTD resolution and detection significance, given N(HMXBs) and the native resolution of the SF history map. We detect significant signal in the DTD of HMXB progenitors for stellar ages 21–53 Myr, and 53–84 Myr. Stellar populations in this age range generate ∼2 × 10⁻⁵ HMXBs per unit stellar mass. This formation efficiency, Ψ T_HMXB, is the product of the specific HMXB formation rate, Ψ [HMXBs yr⁻¹ M_⊙⁻¹], and mean HMXB lifetime, T_HMXB [yr] —see Equations (1)–(3) and Section 2 in B15. For stellar populations younger than 21 Myr, we obtain a shallow 2σ upper limit to the HMXB formation efficiency of ∼2.3 × 10⁻⁵ M_⊙⁻¹. For stellar populations older than 84 Myr, we obtain a much lower upper limit of 2.5 × 10⁻⁷  M_⊙⁻¹. This indicates that there must be a maximum delay time for HMXB formation of less than 84 Myr, but longer than 53 Myr, given the significant detection in this bin.

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In section Section 6 we presented an analysis of the formation efficiency of HMXBs in the SMC based on a set of deep Chandra observations of this galaxy. We calculate this based on three different indicators, (N(HMXBs)/SFR, N(HMXBs)/N(OBs), and N(HMXBs)/M⋆, all as a function of the age of the major associated SF burst), as well at the delay function formulation. We find that there is an increase in the formation rate for ages ≳10–20 Myr and up to 40–60 Myr followed by a decline at older ages. The three HMXB formation efficiency indicators presented in Figure 3 serve different purposes.

N(HMXBs)/N(OBs) is observationally driven, and takes into account the present-day numbers of OB stars. However, it does not take into account the donor star rejuvenation due to the first mass transfer from the initially more massive star that subsequently explodes as a SN and leaves behind a neutron star (or a black hole). Because of this rejuvenation, the system will live longer than single stars of similar mass formed during the same SF episode. This discrepancy is smaller for SF episodes of similar or longer duration compared to the lifetime of HMXB systems. Nonetheless, the N(HMXBs)/N(OBs) ratio is an indicator that can be calculated directly for any nearby galaxy with resolved stellar populations, without the need to derive their SF history. Therefore, it serves as a useful proxy of the relative formation rate of HMXBs that can be applied to large samples of galaxies. In this work, we measured a peak formation efficiency N(HMXBs)/N(OBs) of (5.9 ± 0.60) × 10⁻³ at 42 Myr, and an average formation efficiency in the 30–40 Myr age range of (4.7 ± 0.90) × 10⁻³.

N(HMXBs)/SFR is based on the SF episode of the parent stellar population. It takes into account the SF event that created the binaries we observe today, but not the duration of the SF burst. In this work, we derived a peak formation efficiency N(HMXB)/SFR of (430 ± 52)(M_⊙/yr)⁻¹ at 42 Myr, and an average formation efficiency in the 30–40 Myr age range of ${339}_{-83}^{+78}$ ${({M}_{\odot }/\mathrm{yr})}^{-1}$. These are in good agreement with previous estimates of the average formation efficiency in the broad ∼20–60 Myr age range. We also find a factor of 12 rise in the peak formation efficiency with respect to younger populations (∼10 Myr) and a factor of 3 decline in older epochs (∼260 Myr). The different behavior of the N(HMXBs)/SFR with respect to the N(HMXBs)/N(OBs) indicator could be the result of the age dependence of N(OBs), i.e., as the stellar populations age a smaller number of OB stars is expected to be present. From a simple stellar lifetime argument folded through the IMF, the number of OB stars will be reduced with time, while the rejuvenation of the donor star would result in a longer lifetime of the binary systems. While N(HMXBs)/SFR is considered a more accurate representation of the formation efficiency of young accreting binaries than N(HMXBs)/N(OBs), it is similarly problematic for providing observational constraints in sophisticated population synthesis models (e.g., Andrews et al. 2018).

More suitable is the ratio of N(HMXBs) to the total stellar mass produced in the relevant SF burst (N(HMXB)/M⋆), as this takes into account the SF burst duration (the integral of the SFR as a function of time). This is the fundamental relation that we were aiming to derive from this Chandra XVP program, and the one that best resembles the delay function of the HMXBs. The time evolution of the HMXB formation rate normalized by the total stellar mass of their parent stellar populations is shown in the bottom panel (red points) of Figure 3). We find an increase up to ages of ∼30–40 Myr, followed by a sharp decline at older ages, similar to the behavior of the other two indicators. We measure a peak formation efficiency N(HMXB)/M⋆ of $({12}_{-0.61}^{+0.74})\times {10}^{-6}$ M_⊙⁻¹ at 42 Myr, and an average formation efficiency in the 30–40 Myr age range of $({8.74}_{-0.92}^{+1.0})\times {10}^{-6}$ M_⊙⁻¹. The overall evolution of the N(HMXB)/M⋆ formation efficiency (Figure 3) is consistent within the errors with the DTD (Figure 4). Moreover, our results are in good qualitative agreement with the simulated stellar mass normalized total X-ray luminosity output of a galaxy as a function of age of Fragos et al. (2013), who find an increase at ages ≳20 Myr and a decrease at ages ≳80 Myr. This effect becomes more prominent in metallicities like those of the SMC (∼1/5 Z_⊙; Luck et al. 1998; Antoniou & Zezas 2016, and references therein).

The Chandra fields along the SMC Wing (DF01–DF04 in Figure 1) produce only a small number of HMXBs based on the surveys conducted so far (McGowan et al. 2008; this work). We attribute this deficiency on the strong but very recent (<10 Myr) star formation of the fields in this area (Antoniou et al. 2010) compared to the SMC Bar regions (typically ∼25–60 Myr). Although this deficit might indicate an elusive, young, population of HMXBs, such as highly absorbed HMXBs (e.g., Walter et al. 2015), based on XRB evolution models we would not expect a large number of XRBs at such young ages as only the few, most massive systems would have produced compact objects (Belczynski et al. 2008). Because of the large mass of the progenitors of these systems and the low metallicity of the SMC, we would expect these systems to be predominantly black hole XRBs (Antoniou et al. 2010).

The time-resolved HMXB formation efficiency with respect to the stellar mass presented in Figure 3 is in good agreement with the general trend estimated by Shtykovskiy & Gilfanov (2007), who find a peak at similar ages (∼40 Myr; albeit with coarser time resolution). We attribute differences in the absolute value of the formation efficiency between the two works to the fact that Shtykovskiy & Gilfanov (2007) consider only massive stars (M > 8 M_⊙) in their calculation of stellar mass that was used to normalize the number of HMXBs.

Our results are also in agreement with studies of the formation efficiency of massive Oe/Be stars in the Magellanic Clouds (e.g., Martayan et al. 2006, 2007b; Bonanos et al. 2009, 2010), and the Milky Way (McSwain & Gies 2005). These works show a peak at ages of ∼20–50 Myr (Iqbal & Keller 2013), matching the age of maximum production of HMXBs at least at the metallicity of the SMC. This similarity could indicate that (a) the Be stars, the donor stars of Be-XRBs (the predominant HMXB population in the SMC), are the result of binary evolution (e.g., Porter & Rivinius 2003, and references therein), and/or (b) the larger mass-loss rates of Be stars through their equatorial winds (in comparison to the much weaker spherical stellar winds) lead to an enhanced population of active XRBs (see Antoniou et al. 2010). However, only detailed population synthesis models accounting for the complex orbital evolution and mass transfer in eccentric binaries (e.g., Dray 2006) can distinguish between these possibilities.

Finally, a first assessment of the overall XRB formation rate in the LMC, which has two SF episodes at similar ages as the SMC (12.6 and 42 Myr) but with different intensities, indicates that the formation efficiency of its overall XRB population is ∼17 times lower than in the SMC (Antoniou & Zezas 2016). This could be the result of a metallicity effect (e.g., Be stars form more efficiently at lower metallicities as shown by Martayan et al. 2007a and Iqbal & Keller 2013). Furthermore, Dray (2006) finds that at the ∼1/5 Z_⊙ metallicity of the SMC, population synthesis models predict 3 times larger populations of HMXBs than in the Milky Way. However, only a more systematic study of the formation efficiency of XRBs in the higher metallicity LMC galaxy will show how this truly depends on the metallicity.

We have investigated the formation efficiency of HMXBs in the low SMC metallicity for the first time as a function of the age of their parent stellar population. We have used the different formation efficiency indicators N(HMXBs)/SFR, N(HMXBs)/N(OBs), and N(HMXBs)/M⋆, all as a function of the age of the major associated SF burst. In all cases, we find an increase in the formation efficiency up to an age of ∼40–60 Myr, and a gradual decrease thereafter. In this work, we derive a peak formation efficiency N(HMXB)/SFR of (430 ± 52) (M_⊙/yr)⁻¹ at 42 Myr, and an average formation efficiency of ${339}_{-83}^{+78}$ ${({M}_{\odot }/{\rm{yr}})}^{-1}$ in the 30–40 Myr age range, in good agreement with previous estimates of the average formation efficiency in the broad ∼20–60 Myr age range. This peak in the formation efficiency of the SMC HMXBs is 12 and 3 times higher than at earlier (∼10 Myr) and later epochs (∼260 Myr), respectively, and it is in excellent agreement with previous studies that have examined it on Be stars in both the Magellanic Clouds and the Milky Way. We also measure a peak formation efficiency N(HMXBs)/N(OBs) of (5.9 ± 0.60) × 10⁻³ and N(HMXB)/M⋆ of $({12}_{-0.61}^{+0.74})\,\times {10}^{-6}$ M_⊙⁻¹ at 42 Myr. Finally, in the 30–40 Myr age range, we derive an average formation efficiency N(HMXBs)/N(OBs) of (4.7 ± 0.90) × 10⁻³ and N(HMXB)/M⋆ of $({8.74}_{-0.92}^{+1.0})\,\times {10}^{-6}$ M_⊙⁻¹.

We are grateful to Pat Broos for all his advice, support, and assistance throughout this work, and we thank K. D. Kuntz for useful comments that have improved the quality of the paper. V.A. acknowledges financial support from NASA/Chandra grant GO3-14051X, NASA/ADAP grant NNX10AH47G, and the Office of the Provost at Texas Tech University. A.Z. acknowledges financial support from NASA/ADAP grant NNX12AN05G and funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n. 617001. This project has also received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie RISE action, grant agreement No 691164 (ASTROSTAT). J.J.D., T.J.G., and P.P.P. were funded by NASA contract NAS8-03060 to the Chandra X-ray Center. M.S. acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) through the Heisenberg grants SA 2131/3-1, 5-1, 12-1. P.F.W. acknowledges financial support from the NSF through grant AST-1714281. The OGLE project has received funding from the National Science Centre, Poland, grant MAESTRO 2014/14/A/ST9/00121 to AU. We thank the CXC Director, Belinda Wilkes, for advice and support, and for funding the publication of this work.

Software: CIAO (Fruscione et al. 2006), ACIS Extract (AE Version 2014may23; Broos et al. 2010, 2012), TOPCAT (Taylor 2005), PARSEC isochrones (v1.2S; Bressan et al. 2012).