A LOW-MASS MAIN-SEQUENCE STAR AND ACCRETION DISK IN THE VERY FAINT X-RAY TRANSIENT M15 X-3

For this analysis, we reduced and analyzed the 2012 Advanced CCD Imaging Spectrometer (ACIS-S) ${\text{}}{Chandra}$ observation of M15, taken in subarray mode with an effective frame time of 0.9 s. We also reduced and analyzed three unpublished archival observations from the ${\text{}}{Chandra}$ data archive (see Table 1). These observations were reduced using CALDB version 4.5.9 and the 2012 August time-dependent gain file (acisD2012-08-01t_gainN0006.fits). All four observations were taken using ACIS-S in faint mode with no grating. Each observation was reprocessed with the chandra_repro script, to generate a new level 2 events file. Using CIAO 4.6, the spectrum of M15 X-3 was obtained for each observation via the specextract script. In all observations, M15 X-3 can be clearly detected as a source 21'' from the core of M15 (see Figure 1). Two other bright X-ray sources, the LMXBs AC-211 and M15 X-2 (White & Angelini 2001), are also nearby in the ${\text{}}{Chandra}$ frame. The source region used was a 2.4'' circular region, surrounded by an annulus with 6'' radius that provided a representative background region. Background regions of varying sizes were tested, however the choice of background region did not affect our results. The spectra were grouped to give a minimum of 25 counts per bin in the 0.5–10 keV range, so that Gaussian statistics were valid. Each spectrum was then fit using XSPEC 12.8.0. Initial fitting indicated that M15 X-3 was in a consistent spectral state across all four observations, motivating the choice to fit the spectra simultaneously, first with model parameters tied and then untied from each other. To perform a check for variability in M15 X-3, background subtracted light curves were generated using the dmextract command. Testing for variability using the CIAO tool glvary,⁴ which looks for deviations among optimally sized bins (Gregory & Loredo 1992), indicated evidence of variability within all 4 observations (probability of nonvariability always less than 10⁻⁸), as previously noted in other observations of M15 X-3 (Heinke et al. 2009).

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Table 1.  Summary of ${\text{}}{Chandra}$ Observations of M15 X-3

ObsID	Date	Exposure	Instrument	Lx
		(ks)		(1033 erg s−1)
9584	2007 Sep 05	2.15	HRC-I	${6}_{-2}^{+2}$
11029	2009 Aug 26	34.18	ACIS-S	${8.3}_{-0.4}^{+0.4}$
11886	2009 Aug 28	13.62	ACIS-S	${9.9}_{-0.7}^{+0.7}$
11030	2009 Sep 23	49.22	ACIS-S	${9.2}_{-0.4}^{+0.4}$
13420	2011 May 30	1.45	HRC-I	${5}_{-1}^{+2}$
13710	2012 Sep 18	4.88	ACIS-S	${10}_{-1}^{+1}$

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There are five HRC-I observations of M15 X-3 in the ${\text{}}{Chandra}$ archive. The first three, obtained in 2001, found M15 X-3 in a quiescent state, barely detectable when using special processing (Heinke et al. 2009), and we do not re-analyze them here. We analyze the 2007 and 2011 HRC-I observations, which catch M15 X-3 in a relatively bright state (see details in Table 1). We reprocessed the events files (with CALDB 4.5.9 and the 2012 August time-dependent gainfile) using the chandra_repro script, and extracted source counts from the position of M15 X-3, and background counts from an annular region. We infer the unabsorbed 0.5–10 keV luminosities using PIMMS and a power law with photon index Γ = 1.4 and N_H = 4.6 × 10²⁰ cm⁻². The short length of these observations (Table 1) makes the data unsuitable for timing analyses.

Previously, M15 X-3 was found to be well-described using a power law with photon index 1.51 ± 0.14, absorbed through a column density fixed at the cluster value of 4.6 × 10²⁰ cm⁻² (Heinke et al. 2009). Very faint X-ray transients are often described with a hard power law, or a hard power law with a softer thermal component (Armas Padilla et al. 2013a). Even for a relatively faint source such as M15 X-3, the count rate of ≈0.06 counts s⁻¹ (or 0.18 per frame, for the three long archival ACIS-S observations), makes pileup a potential issue. Therefore, pileup is accounted for by convolving spectral models with the XSPEC model for pileup in ${\text{}}{Chandra}$ observations (Davis 2001). The dependence of fits upon the grade correction parameter α was considered in the fitting process.⁵

For this analysis we fit all four observations simultaneously. These separate fits all suggested that M15 X-3 was in the same state with L_x ≈ 8 × 10³³ erg s⁻¹ (χ² = 211.73 for 188 degrees of freedom (dof)). This motivated the choice to fit all four ACIS observations of M15 X-3 simultaneously, with the power-law index of all four observations tied together. In this case, only the power-law component normalization, which corresponded to the measured flux of the source, was allowed to be independent. Fitting to all four of the ${\text{}}{Chandra}$ observations gives an acceptable fit (χ²/dof = 214.13/191, implying a null hypothesis probability of 0.12, see Table 2 SPL 1) with an absorbed power law photon index of 1.42 ± 0.03. Allowing the indices to vary does not significantly improve the fit (an F-test gives an F-statistic of 0.71, and probability 0.55 for obtaining such an improvement by chance), suggesting that the power-law index does not change. The fit was not significantly affected by the value of the grade correction parameter α, which was unconstrained in the fit, due to the relatively small degree of pileup. We therefore report results with α fixed to 0.5, as empirically suggested (Davis 2001), as none of the results presented here qualitatively change for different choices of α.

Table 2.  Summary of Fits to ${\text{}}{Chandra}$ Observations of M15 X-3

Model		SPL 1	SPL 2	BB + PL	DISKBB + PL	NSATMOS + PL	NSA + PL	BPLb
NH	(1020 cm−2)	[4.6]	${9}_{-1}^{+2}$	${6}_{-4}^{+4}$	${4}_{-4}^{+5}$	${10}_{-2}^{+2}$	${5}_{-5}^{+5}$	${6}_{-3}^{+2}$
Γ1		${1.42}_{-0.03}^{+0.03}$	${1.57}_{-0.07}^{+0.07}$	${1.5}_{-0.2}^{+0.2}$	${1.3}_{-0.5}^{+0.5}$	${1.57}_{-0.07}^{+0.04}$	${1.5}_{-0.3}^{+0.3}$	${1.3}_{-0.2}^{+0.1}$
Γ2		⋯	⋯	⋯	⋯	⋯	⋯	${1.9}_{-0.2}^{+0.2}$
Ebreak	(keV)	⋯	⋯	⋯	⋯	⋯	⋯	${2.7}_{-0.6}^{+0.4}$
kTthermal	(keV)	⋯	⋯	${0.7}_{-0.2}^{+0.3}$	${1.2}_{-0.3}^{+1.0}$	${0.05}_{-0.04}^{+0.03}$ a	${0.6}_{-0.2}^{+0.3}$ a	⋯
Rthermal	(km)	⋯	⋯	${0.2}_{-0.1}^{+0.1}$	${0.7}_{-0.5}^{+0.3}$	[10]	${0.3}_{-0.2}^{+0.4}$	⋯
LX, MJD = 55069	(1034 erg s−1)	${0.83}_{-0.04}^{+0.04}$	${0.83}_{-0.04}^{+0.04}$	${0.77}_{-0.05}^{+0.05}$	${0.74}_{-0.06}^{+0.06}$	${0.83}_{-0.04}^{+0.04}$	${0.76}_{-0.05}^{+0.05}$	${0.75}_{-0.05}^{+0.05}$
PL fraction, MJD = 55069		[1]	[1]	${0.88}_{-0.09}^{+0.07}$	${0.7}_{-0.3}^{+0.2}$	${0.99}_{-0.01}^{+0.01}$	${0.84}_{-0.05}^{+0.04}$	[1]
LX, MJD = 55071	(1034 erg s−1)	${0.99}_{-0.07}^{+0.07}$	${0.98}_{-0.07}^{+0.07}$	${0.93}_{-0.08}^{+0.08}$	${0.91}_{-0.09}^{+0.09}$	${0.98}_{-0.07}^{+0.07}$	${0.92}_{-0.08}^{+0.08}$	${0.90}_{-0.07}^{+0.08}$
PL fraction, MJD = 55071		[1]	[1]	${0.90}_{-0.08}^{+0.06}$	${0.8}_{-0.2}^{+0.2}$	${0.99}_{-0.01}^{+0.01}$	${0.87}_{-0.04}^{+0.03}$	[1]
LX, MJD = 55097	(1034 erg s−1)	${0.92}_{-0.04}^{+0.04}$	${0.92}_{-0.04}^{+0.04}$	${0.87}_{-0.05}^{+0.05}$	${0.85}_{-0.07}^{+0.06}$	${0.92}_{-0.04}^{+0.04}$	${0.86}_{-0.06}^{+0.06}$	${0.84}_{-0.03}^{+0.06}$
PL fraction, MJD = 55097		[1]	[1]	${0.89}_{-0.09}^{+0.07}$	${0.7}_{-0.2}^{+0.2}$	${0.99}_{-0.01}^{+0.01}$	${0.86}_{-0.03}^{+0.03}$	[1]
LX, MJD = 56188	(1034 erg s−1)	${1.0}_{-0.1}^{+0.1}$	${1.0}_{-0.1}^{+0.1}$	${1.0}_{-0.1}^{+0.1}$	${1.0}_{-0.1}^{+0.1}$	${1.1}_{-0.1}^{+0.1}$	${1.0}_{-0.1}^{+0.1}$	${1.0}_{-0.1}^{+0.1}$
PL fraction, MJD = 56188		[1]	[1]	${0.91}_{-0.08}^{+0.06}$	${0.8}_{-0.2}^{+0.2}$	${0.99}_{-0.01}^{+0.01}$	${0.88}_{-0.04}^{+0.03}$	[1]
${\chi }^{2}/\mathrm{dof}$		214.13/191	195.08/190	187.52/188	188.04/188	195.03/189	187.99/188	185.19/188
n.h.p.		0.12	0.38	0.50	0.49	0.37	0.49	0.54

Notes. M15 X-3 best spectral fit parameters, all ACIS-S data. Values in square brackets are fixed as part of the model choice. In this table SPL 1 is the single power law with column density fixed, while SPL 2 is the same but with column density free.

^aIndicate a model limit. ^bWe adopt the broken power-law model as our best fit.

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Examining the single power-law fit suggests the presence of curvature in the residuals above 1 keV. This curvature motivated the choice to attempt other fits. First, the column density N_H was allowed to vary, since high column density is a potential source of curvature. All subsequent fits permit N_H to vary. In the case of the single power law, this strongly improved the fit (χ²/dof = 195.08/190; SPL 2 in Table 2) with a new column density of ${9}_{-1}^{+2}\times {10}^{20}\;{\mathrm{cm}}^{-2}$, though the residuals of the fit (Figure 2) still suggested curvature. Since other VFXTs are described acceptably by a spectra composed of a power law with a thermal component (Armas Padilla et al. 2013a), several thermal components (a low magnetic field neutron star hydrogen atmosphere, NSATMOS, Heinke et al. 2009; a multicolor disk blackbody, DISKBB, e.g., Mitsuda et al. 1984; and a single-temperature blackbody, BBODYRAD) were tested to see if they improved the fit.

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Adding an NSATMOS component (assuming emission from the entire neutron surface, i.e., normalization fixed to 1.0; NSATMOS + PL in Table 2) does not substantially improve the fit (Δχ² is only 0.04 for 1 fewer dof), and the fit prefers a temperature of ${0.05}_{-0.04}^{+0.03}$ keV. The 90% confidence upper limit on the temperature permits this component to contribute only 2% (∼2 × 10³² erg s⁻¹) of the 0.5–10 keV unabsorbed L_X, and the lower limit is pegged to the lower temperature boundary of the model. Adding a disk blackbody component (DISKBB + PL in Table 2) does improve the fit (χ²/dof = 188.04/188, F-statistic of 3.52 and probability 3.2 × 10⁻² of this improvement being due to chance). However, the disk blackbody model can be physically excluded, since the inferred inner disk radius (R_inner cos θ = 0.7 km) is much smaller than the neutron star radius, making an accretion disk origin untenable except at very high inclination.

Addition of a blackbody model can also improve the fit (χ²/dof = 187.52/188, F-statistic of 3.79 and probability 2.4 × 10⁻² of this improvement compared to the variable N_H single power-law model being due to chance). The best-fit blackbody parameters are kT = ${0.7}_{-0.07}^{+0.06}$ keV, with an emitting radius R = ${0.2}_{-0.1}^{+0.1}$ km (see Table 2, BB + PL). This could possibly be interpreted as a hot spot on the neutron star surface, where infalling material is channeled by the magnetic field to the poles. However, the inferred emitting region is smaller than is expected for polar caps on a neutron star surface spinning at millisecond periods (1.4–2.6 km for 3–10 ms periods, using ${R}_{\mathrm{pc}}={(2\pi {R}_{\mathrm{NS}}/({cP}))}^{1/2}{R}_{\mathrm{NS}}$; Lyne & Graham-Smith 2006). A longer spin period (e.g., 0.5 s for 0.2 km) would predict a polar cap of the appropriate size.

Alternatively, it is well-known (e.g., Rajagopal & Romani 1996; Zavlin et al. 1996) that a hydrogen atmosphere model gives inferred radii a few times larger than blackbody fits to the same data. Indeed, blackbody fits to the thermal spectra of old millisecond pulsars find inferred polar cap radii of order 0.1–0.2 km (Bogdanov et al. 2006; Forestell et al. 2014). Using hydrogen atmosphere models, emitting radii of ∼1 km, more typical of NS polar caps, are found (e.g., Becker et al. 2002; Zavlin et al. 2002; Bogdanov et al. 2006). We therefore try fitting the thermal component with the NSATMOS model again, now permitting radiation from only a small fraction of the stellar surface, by freeing the normalization parameter. This result of this fit gave an aphysical normalization (corresponding to the fraction of the neutron star emitting) ≫1. Thus, fitting was repeated with an alternate neutron star atmosphere NSA model (Zavlin et al. 1996), with magnetic field set to 0 G, which is a reasonable approximation for the 10⁸⁻⁹ G regime (NSA + PL in Table 2). The resulting fit is again an improvement over the single power-law fit (χ²/dof = 187.99/188, F-statistic of 3.55 and probability 3.1 × 10⁻² of this improvement compared to the variable N_H single power-law model being due to chance). The best-fit temperature is ${0.6}_{-0.2}^{+0.3}$ keV and the inferred radius of the emitting region is ${0.3}_{-0.2}^{+0.4}$ km, with the temperature upper error limit pegged to the upper boundary of the model.

Finally, we tried a broken power-law model, where the spectrum is described by a power-law index of Γ₁ = 1.3^+0.1_−0.2 up to an energy of ${2.7}_{-0.6}^{+0.4}$ keV, where it becomes a power law index of ${{\rm{\Gamma }}}_{2}={1.9}_{-0.2}^{+0.2}$. This model also gives a significantly better fit than the single power-law (${\rm{\Delta }}{\chi }^{2}$ = 9.89 over the single power-law, F-statistic 5.02, probability 7.5 × 10⁻³ of this improvement compared to the variable N_H single power-law model being due to chance). Releasing the power-law indices or break energies between observations did not improve this fit. The broken power-law model would presumably imply a bremsstrahlung origin of the emission, with the break energy suggesting the bremsstrahlung temperature (Chakrabarty et al. 2014), so it is reasonable to search for a separate emission contribution from the neutron star surface in this case. However, as above, adding a NSATMOS component (with emitting size fixed to the whole neutron star) gave no improvement, and adding a BBODYRAD component gave very little improvement (Δχ² = 0.01). Although it is an empirical fit only, we adopt the broken power-law model as our best fit as it gave the best improvement compared to the variable N_H single power-law model.

The 2012 Hubble Space Telescope (${\text{}}{HST}$) observation, taken in one orbit with the WFC3 camera using the UVIS channel in subarray mode, one day before a ${\text{}}{Chandra}$ observation, consists of exposures in WFC3 B (F438W, total exposure length 1360 s), broad V (F606W, total exposure length 188 s), and I (F814W, total exposure length 332 s). The exposures were obtained using the standard four-point box dither pattern, with integer plus subpixel dithers, to allow for bad pixels and increase the effective angular resolution. A 1k × 1k subarray was chosen to minimize the overhead time and thus allow the necessary exposures to fit into a single orbit. In programming the observations, care was taken to constrain the roll angle in order to prevent diffraction spikes from a nearby bright giant from falling on the M15 X-3 counterpart. The frames were combined with correction for cosmic rays using the IRAF/STSDAS Multidrizzle task with 2× oversampling, giving an effective pixel size of 0.02''. Photometry on the drizzle-combined frames was carried out using the DAOPHOT/ALLSTAR software package, since the field of M15 rapidly becomes crowded near the core. Point-spread functions (PSFs) were constructed using 100 candidate PSF stars per filter. Photometric calibration to the VEGAMAG system was performed by doing aperture photometry on the PSF stars within a 0.1'' diameter aperture, finding the aperture correction to an infinite radius aperture from Sirianni et al. (2005), calculating the offset between the ALLSTAR photometry and the aperture photometry, and applying the calibrations from the HST WFC3 calibration webpage.⁶

As shown in Figure 3, the optical counterpart of M15 X-3 is clearly detected in all 3 filters. Its magnitudes in this epoch, as well as in previous usable ${\text{}}{HST}$ epochs, are given in Table 3. CMDs of the cluster were constructed in B − V and V − I, as shown in Figure 4, where the counterpart is bluer than the main sequence and becomes progressively bluer as one moves to bluer filters.

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To determine the nature of the M15 X-3 companion using its optical colors, the system is modeled using Pysynphot (STScI development Team 2013). Previous observations of M15 X-3 in the X-ray and optical bands have suggested that the system is likely to be a neutron star with a low-mass main sequence companion (Heinke et al. 2009). M15 X-3 appears to be too X-ray luminous to be a WD accretor (Verbunt et al. 1997) and its X-ray/optical flux ratio rules this out conclusively (Heinke et al. 2009). The optical spectrum of the M15 X-3 system is therefore modeled with two components: a low-mass main sequence companion and an accretion disk whose emission is approximated by a power law.

The range of power-law indices used to model a power-law accretion disk (and its expected optical spectrum) depends on two limiting behaviors. For small frequencies, F_ν approaches a Rayleigh–Jeans description where F_ν ∝ ν². For a conventional disk, the spectrum of the outer part of the disk behaves approximately as F_ν ∝ ν^1/3 (Frank et al. 2002). In the wavelength space utilized by Pysynphot for spectral modeling, this corresponds to a photon index Γ_λ = −4 in the Rayleigh–Jeans limit, and Γ_λ = −7/3 for a conventional disk. In fitting the power law, Γ_λ indices between −4 and −2 were considered. To have an index larger than −2 (which corresponds to the peak of the spectrum) would imply that the measured WFC3 bandpasses lie close to the Wien limit of the spectrum. Observations of quiescent disks in other accreting NS and black hole systems (e.g., Froning et al. 2011; Hynes & Robinson 2012; Wang et al. 2013) indicate that the disks have typical inner temperatures between 5000 and 13,000 K. Compared to Aql X-1 in quiescence, M15 X-3 is more luminous in both the X-rays (by a factor of 20; Narayan et al. 1997) and UV (by a factor of ∼4; Hynes & Robinson 2012). Assuming they have a similar temperature-radius relation, we conclude that M15 X-3 is likely hotter than Aql X-1. Since Aql X-1's disk temperature was 12,400 ± 1400 K (Hynes & Robinson 2012), it is unlikely that the disk temperature of M15 X-3 is low enough to place the Wien tail into our WFC3 bandpasses.

The power-law component is processed with Synphot's built-in power-law model, while the low-mass companion is modeled using stellar models from the Castelli and Kurucz library of stellar atmospheres included in the Synphot package. The advantage of this library is that it permits the modeling of main-sequence stars at the low metallicity characteristic of M15. As inputs, the Castelli and Kurucz models require the following: effective temperature T_eff, logarithm of surface gravity $\mathrm{log}(g)$, and metallicity [M/H]. To obtain these input parameters for a main-sequence star of a particular mass, results were taken from Baraffe et al. (1997) model calculations for low-mass stars at [M/H] = −2.0, which is reasonable given that M15 has [Fe/H] = −2.34 (Carretta et al. 2009). The nature of the companion and disk is determined by fitting the B, V, and I magnitudes predicted by (Synphot) synthetic photometry for a particular choice of low-mass companion and accretion disk against the measured WFC3 magnitudes for the M15 X-3 system using a standard chi-square test. A grid-search for the best chi-square value is performed over the parameter space for each free component—mass in the case of the companion, and power-law index in the case of the accretion disk. Four cases were considered: A low-mass main-sequence companion only, a power law accretion disk only (with free index), a companion and power-law disk with a free index, and a companion and power-law disk with index fixed to −7/3. In the case of a low-mass companion + free index power-law disk, testing only at the mass values given in Baraffe et al. (1997) gives a best-fit companion mass of 0.5 ± 0.1 M_⊙, with a power-law index Γ_λ = −3.28. To estimate a more precise mass, linear interpolation of model parameters was performed on the table of masses in the range 0.200−0.700 M_⊙.

As shown in Table 4, neither a power-law disk description nor a companion alone is sufficient to explain the observed optical colors of M15 X-3. The best fit is obtained by having a combined spectrum, where the accretion disk of M15 X-3 and its low-mass companion both contribute to the optical flux. Allowing the mass of the companion as well as both power-law parameters (Γ_λ and the normalization) to vary gives an acceptable fit with an inferred companion mass of ${0.515}_{-0.270}^{+0.005}$ M_⊙. However, with this many free parameters, the dof reduce to zero, overconstraining the model and limiting its utility. Such a fit is only useful for constraining the model parameters—in this case, the fit shows the stellar component is not consistent with a low stellar mass. If the index is fixed at the conventional accretion disk value of −7/3 an acceptable fit is also obtained, with an inferred companion mass of ${0.440}_{-0.060}^{+0.035}$ M_⊙. The inferred spectrum of this combined system is plotted in Figure 5. This choice of index describes the accretion disk as the largest contributor to the system's optical flux in B and V, while in I the flux contribution of the disk and companion become approximately equal. This naturally explains why M15 X-3's optical counterpart appears much bluer than the main sequence, especially in bluer filters, as it is the accretion disk which dominates emission at bluer wavelengths. This fit has χ² = 0.44 for 1 dof. With this small dof, this low χ² corresponds to a null hypothesis probability of 0.49 (close to the null hypothesis probability of 0.5 for a reduced χ² = 1 for large dof), which indicates that there is no need to consider additional systematic errors when constraining our fit. Note that the fits prefer a power-law photon index of −4, consistent with a relatively high temperature for the accretion disk.

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Table 4.  Summary of Fits to ${\text{}}{HST}$ Observations of M15 X-3

Model	Γλ	Fdisk/Ftotal						Inferred Mcomp	χ2/dof
		U1994	B2002	V2002	B2012	V2012	I2012
Companion only	⋯	[0]	[0]	[0]	[0]	[0]	[0]	0.540 ± 0.005	33.78/2
PL only	$-{2.00}_{-0.06}^{+0.00*}$	[1]	[1]	[1]	[1]	[1]	[1]	⋯	8.87/1
Fixed PL + Companion	$[-\displaystyle \frac{7}{3}]$	${0.98}_{-0.01}^{+0.01}$	${0.92}_{-0.04}^{+0.04}$	${0.82}_{-0.07}^{+0.07}$	${0.87}_{-0.06}^{+0.07}$	${0.68}_{-0.13}^{+0.13}$	${0.53}_{-0.15}^{+0.15}$	${0.440}_{-0.060}^{+0.035}$	0.44/1
Free PL + Companion	$-{4.00}_{-0.00*}^{+2.00*}$	${0.94}_{-0.01}^{+0.05}$	${0.85}_{-0.03}^{+0.14}$	${0.62}_{-0.02}^{+0.34}$	${0.71}_{-0.07}^{+0.28}$	${0.35}_{-0.03}^{+0.58}$	${0.16}_{-0.01}^{+0.71}$	${0.515}_{-0.270}^{+0.005}$	0.18/0

Note. Summary of fits to 2012 ${\text{}}{HST}$ data of M15 X-3 optical counterpart using PYSYNPHOT modeling. Values in square brackets were fixed because of the model choice, while uncertainties marked with a single asterisk indicate that the uncertainty is at the hard limit of the model. Note that adding a power law with free index to the model adds two model parameters (PL normalization, PL index), while a power law with a fixed index adds only one model parameter (PL normalization). A low-mass main-sequence companion adds only one free parameter to the model (mass of the companion).

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Given the relative blueness of the optical counterpart of M15 X-3 compared to other members of the cluster, it is reasonable to ask whether its optical colors may be substantially altered by X-ray irradiation of the main sequence companion of M15-3 by the primary. To determine whether X-ray irradiation is important, we compare the amount of X-ray light received by the companion star (measured near-simultaneously) with the total flux from the star. We begin by assuming a non-irradiated companion, M₂ = 0.44 M_☉ and a 1.4 M_☉ primary that radiates ≈10³⁴ erg s⁻¹ of X-rays. With a mass ratio q = M₂/M₁ one can write the binary separation as (following Frank et al. 2002):

$$\begin{eqnarray}&&a=3.5\times {10}^{10}{({M}_{1})}^{\displaystyle \frac{1}{3}}{(1+q)}^{\displaystyle \frac{1}{3}}{({P}_{\mathrm{hr}})}^{\displaystyle \frac{2}{3}}{\rm{cm}}.\end{eqnarray} \tag{ 1 }$$

For a lower MS star filling its Roche Lobe, the period can be approximated by

$$\begin{eqnarray}&&{M}_{2}\approx 0.11{P}_{\mathrm{hr}},\end{eqnarray} \tag{ 2 }$$

which means that the orbital period of the system is approximately 4 hr (Frank et al. 2002). Substituting this relation in for M₂ gives the following:

$$\begin{eqnarray}&&a\simeq 3.5\times {10}^{10}{M}_{1}{(1+q)}^{\displaystyle \frac{1}{3}}{\left(\displaystyle \frac{{M}_{2}}{0.11}\right)}^{\displaystyle \frac{2}{3}}{\rm{cm}}.\end{eqnarray} \tag{ 3 }$$

For M15 X-3, this gives a ∼ 1.3 × 10¹¹ cm. This means that the X-ray flux received by the companion at its surface (assuming isotropic emission) is

$$\begin{eqnarray}&&{F}_{x}\approx \displaystyle \frac{{L}_{x}}{4\pi {a}^{2}}=3.5\times {10}^{10}\;\mathrm{erg}\;{\mathrm{cm}}^{-2}\;{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 4 }$$

By contrast, the total flux of X-3's companion can be estimated by taking a luminosity of ${L}_{\textFiveStartOpen }\approx 0.05{L}_{\odot }$ (Baraffe et al. 1997) and ${R}_{\textFiveStartOpen }\approx 0.44{R}_{\odot }$ (Frank et al. 2002) to get the companion's flux at its surface to be:

$$\begin{eqnarray}&&{F}_{\textFiveStartOpen }\approx \displaystyle \frac{{L}_{\textFiveStartOpen }}{4\pi {R}_{\textFiveStartOpen }^{2}}=1.6\times {10}^{10}\;\mathrm{erg}\;{\mathrm{cm}}^{-2}\;{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 5 }$$

Since the received X-ray flux is roughly a factor of 2 larger than the flux emitted by the companion itself, X-ray irradiation is likely to be a contributing factor to the optical color of this system. This means that it is likely that the companion of M15 X-3 is less massive than the fitted mass of ${0.440}_{-0.060}^{+0.035}$ M_⊙. To constrain the mass of the companion, we assumed that the companion is irradiated by its primary with L_primary = 10³⁴ erg s⁻¹ and re-radiates these X-rays away as a blackbody of temperature

$$\begin{eqnarray}&&{T}_{\mathrm{irradiation}}={\left(\displaystyle \frac{\sigma }{{F}_{\textFiveStartOpen }+{F}_{x}}\right)}^{\displaystyle \frac{1}{4}}.\end{eqnarray} \tag{ 6 }$$

We assume that the emission is dominated by the irradiated half of the star, and thus that:

$$\begin{eqnarray}&&{\left(\displaystyle \frac{R}{{R}_{\odot }}\right)}^{2}=\left(\displaystyle \frac{L}{{L}_{\odot }}\right){\left(\displaystyle \frac{T}{{T}_{\odot }}\right)}^{-4}.\end{eqnarray} \tag{ 7 }$$

Modeling the star as a blackbody with temperature T and radius R, the acceptability of various fits was explored over the parameter space in Figure 6. When compared with a maximally irradiated main-sequence star, at the 3σ level we can still rule out the possibility of a 0.15 R_☉ or smaller companion. This rules out a brown dwarf, white dwarf, or planetary companion. At the 2σ level, the radius of the companion lies between 0.19 and 0.41 R_☉, which suggests a ∼2–4 hr orbital period.

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M15 X-3's unusual X-ray behavior for a globular cluster X-ray source, and position blue-ward of the main sequence, suggests that we check whether it can be explained as a foreground or background source. If M15 X-3 were a foreground star, we would expect it to be redder, rather than bluer, than the main sequence. We use the empirical relations from Baldi et al. (2002) to estimate the probability of finding a background active galactic nucleus at or above the flux of M15 X-3 projected within the half-mass radius of M15. For M15 X-3, we use the bright state flux of S ≈ 5.6 × 10⁻¹³ erg (s⁻¹ cm⁻²), which gives an expected 0.6 sources per square degree, or 5 × 10⁻⁴ sources expected in the half-mass radius of M15, roughly the region where M15 X-3 was discovered. (Note that the likelihood of finding a source at the distance of M15 X-3 from the cluster center is a factor of 9 smaller.) This suggests that M15 X-3 is unlikely to be a foreground or background source.

We can quantitatively consider the probability of the optical counterpart being an unaffiliated interloper by considering the local density of optical sources that could be potential counterparts. We consider a region of 5'' around M15 X-3, as there is a strong density gradient of sources with distance from the center of the cluster. Within this region, there are 3 objects that have V magnitudes in the range 19 < V < 23 that lie as far to the blue side of M15's main sequence as M15 X-3's proposed counterpart, including the counterpart itself. This gives a chance probability of one of these objects falling within the M15 X-3 error circle (0.05''—the offset between the ${\text{}}{Chandra}$ and ${\text{}}{HST}$ relative astrometry, determined using the optical and X-ray detections of the nearby XRB AC 211 Heinke et al. 2009) of $3*{(0.05\prime\prime /5\prime\prime )}^{2}\approx 3\times {10}^{-4}$. Standard counting errors would give ${3}_{-1.6}^{+2.9}$ objects at 1σ, and ${3}_{-2.8}^{+9.7}$ at 3σ. At the upper end, the probability of a non-associated optical counterpart that is a member of the cluster is ≈0.13% even at 3σ. The remaining possibility for the optical emission of M15 X-3 is that the reddish companion is a cluster member unassociated with M15 X-3 itself. However, if the B flux is dominated by M15 X-3, and the I flux is dominated by an unassociated main-sequence star, the offset between these two objects could not be more than about 1 WFC3 pixel = 0.04'', as the counterpart is well-aligned in the B and I frames. This gives a chance probability of an unassociated red star happening to align with M15 X-3's blue counterpart of ≈3%, considering stars within one magnitude of the proposed counterpart in I.

As discussed in Heinke et al. (2009), we can immediately rule out a white dwarf accretor on the basis of long-term variability and the X-ray to optical flux ratio. Radio observations of M15 X-3 argue against the idea that we view it edge-on, permitting most of its intrinsic luminosity to be obscured by the accretion disk, because the radio emission, which scales with the intrinsic X-ray luminosity, should still be visible (e.g., Bower et al. 2005). Strader et al. (2012) report no source at M15 X-3's location, with an rms sensitivity of 2.1 μJy/beam. A non-detection implies that M15 X-3's radio brightness cannot be higher than 10.5 μJy at 5σ. For a black hole at 10³⁶ erg s⁻¹, with X-rays obscured by a high inclination accretion disk, we would expect a radio flux of roughly 10³⁰ erg s⁻¹. At 10.3 kpc, this would give roughly 1 mJy at 6 GHz (Gallo et al. 2012). For a neutron star, we would expect closer to 20 μJy at 6 GHz (Migliari & Fender 2006).

M15 X-3 appears to be the faintest VFXT for which we have an X-ray spectrum of very high quality; for instance, the targets discussed in Armas Padilla et al. (2013a) have L_X(0.5–10 keV) of 5–10 × 10³⁴ erg s⁻¹, versus ∼1 × 10³⁴ erg s⁻¹ here. The X-ray spectrum is somewhat atypical, and can be well-fit by either a broken power law or a power law plus a thermal component. The thermal component, if fit by a blackbody, produces an inferred radius of ${0.2}_{-0.1}^{+0.1}$ km (or ${0.3}_{-0.2}^{+0.4}$ km if fit with an NSA model), which is significantly smaller than the inferred blackbody radii (∼5 km) of the thermal components in the persistent VFXTs discussed by Armas Padilla et al. (2013a). It is unclear if the possible thermal component here has the same physical origin as in the other transients.

The X-ray spectrum of material falling onto a neutron star is predicted to include a thermal component (from the lower atmosphere) plus a very hot surface layer that effectively Comptonizes the outgoing radiation (Deufel et al. 2001). This model may provide a first step toward understanding the spectra of several observed neutron star transients at low luminosity (e.g., Armas Padilla et al. 2013a; Bahramian et al. 2014; Chakrabarty et al. 2014). In the case of M15 X-3, the total luminosity and small inferred radius of the thermal component are consistent with such a high proton energy flux (∼10²⁴ erg cm⁻² s⁻¹) that the Deufel et al. (2001) model predicts an entirely non-thermal spectrum. To reproduce the approximately 10% thermal component in M15 X-3, one might invoke a central high proton flux hotspot that is smaller than the radius of the thermal component we measure, where the thermal component arises from the surrounding area being heated either by conduction or by a lower proton energy flux. It would be of great interest to obtain models such as Deufel's in a convenient form for the spectral fitting of X-ray transients.⁷

If M15 X-3 spends most of its time in its "bright" state, with a persistent luminosity of 10³⁴ erg s⁻¹, then we can infer an upper limit on the time-averaged accretion rate by assuming that all of its 0.5–10 keV X-ray luminosity is from accretion. This assumption implies that M15 X-3's time-averaged accretion rate could be at most 9 × 10⁻¹³ M_⊙ yr⁻¹. Under conventional binary evolution, the low X-ray luminosity of M15 X-3 is difficult to explain. One suggestion is that the low accretion rates needed for VFXTs are possible from a brown dwarf or planetary-mass companion. Another option is a system where the accretor is an intermediate-mass (∼1000 M_⊙) black hole with a primordial companion that is very low mass in the current epoch (King & Wijnands 2006). If the companion of M15 X-3 is a brown dwarf or other very low-mass companion, then we would expect the optical emission from the system to be produced only by the accretion disk. At 10.3 kpc, a brown dwarf companion would be well below the detection threshold from the ${\text{}}{HST}$ observations, in contrast with the presence of a second, non-accretion disk component. The poor fit of a power law alone to the BVI values for M15 X-3 suggests that a disk-only model is insufficient to explain the nature of M15 X-3—the accretor's companion is not sufficiently low-mass for this explanation to be tenable. The location of M15 X-3, 21'' from the cluster center, places it well outside the region where an intermediate-mass black hole would be expected to wander (Chatterjee et al. 2002; Gerssen et al. 2002).

It is also plausible for some VFXTs that accretion from the wind of a MS star could explain low accretion rates. In the case of M15 X-3, a companion of M ≲ 0.44 M_☉ is unlikely to be driving wind-fed accretion in this system (Heinke et al. 2015). It has also been argued by in't Zand et al. (2007) that persistently low accretion rates require a very small, completely ionized disk, implying an ultracompact system. Our optical observations of M15 X-3 are inconsistent with an ultracompact system, as we have found evidence for a relatively large (0.15–0.45 R_⊙) donor.

An alternative explanation for the unusually low X-ray persistent X-ray luminosity of M15 X-3 could be that it is an otherwise normal LMXB (with a 2–4 hr orbital period) where the NS magnetic field is sufficiently strong to interfere with accretion, via the propeller effect (in which the rotation of the NS's magnetic field is able to repel inflowing matter; Illarionov & Sunyaev 1975). D'Angelo & Spruit (2012) modeled a leaky propeller and "trapped" disk, leading to continuous low-level accretion; similarly leaky propellers have been studied by e.g., Romanova et al. (2003), Ustyugova et al. (2006), Kulkarni & Romanova (2008). Heinke et al. (2015) argued that all the quasi-persistent VFXTs such as M15 X-3 are in this physical state. This argument is based, on one hand, on the difficulty of explaining continued extremely low-level accretion via disk instability models, and on the other hand the example of transitional millisecond pulsars, which maintain low-level accretion and clearly show evidence of magnetic effects on the accretion flow.

Three transitional millisecond pulsars (which cycle between states of low-level accretion at 10³³−10³⁴ erg s⁻¹, non-accreting radio pulsar states at ∼10³¹ erg s⁻¹, and occasional higher-L_X accreting states) have recently been identified (Archibald et al. 2009; Papitto et al. 2013; Bassa et al. 2014; Bogdanov et al. 2014; Patruno et al. 2014; Roy et al. 2014; Stappers et al. 2014). The recent detection of X-ray pulsations from two of these transitional millisecond pulsars in their L_X ∼ 10³³−10³⁴ erg s⁻¹ state (Archibald et al. 2014; Papitto et al. 2014) strongly indicates that these pulsars are indeed in this leaky propeller state (D'Angelo et al. 2014). The quasi-persistent VFXTs, including M15 X-3, may therefore also be transitional millisecond pulsars (Degenaar et al. 2014; Heinke et al. 2015). At least one transitional millisecond pulsar has been observed to be relatively radio-bright in its X-ray active phase (Deller et al. 2014). Of the known tMSP systems, M15 X-3's X-ray luminosity is closest to XSS J12270-4859, which would imply it has a radio brightness of roughly 5 × 10²⁷ erg s⁻¹ at 5 GHz if it is a similar source. Assuming a spectrally flat source, M15 X-3 would have a 5 GHz flux density of approximately 5.3 μJy at 10.3 kpc, roughly a factor of two too faint to be detected by existing radio observations.

A relatively conservative estimate of M15 X-3's duty cycle is roughly 70%, as seen in Figure 7. However, the quasi-persistent VFXTs with a duty cycle resembling that of M15 X-3 tend to have relatively soft power-law fits, or harder power-law fits only when a thermal component is also present (Armas Padilla et al. 2013a). The X-ray spectrum of M15 X-3 is unlikely to be an intrinsically harder spectrum being suppressed at lower energies because of absorption effects—radio observations suggest the system is not being viewed edge-on, and the column density of M15 X-3 we measure is only ≲10²¹ cm⁻².

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