DISCOVERY OF THE OPTICAL COUNTERPARTS TO FOUR ENERGETIC FERMI MILLISECOND PULSARS

The Fermi Gamma-Ray Space Telescope (hereafter simply Fermi) has discovered hundreds of unidentified point sources whose positional uncertainties are small enough to enable deep targeted pulsar searches with the world's largest radio telescopes. These searches have resulted in the discovery of 43 energetic millisecond pulsars (MSPs) to date (Cognard et al. 2011; Hessels et al. 2011; Kerr et al. 2012; Ransom et al. 2011; Ray et al. 2012). Intriguingly, many of these are "black widow" and "redback" systems, of which previously only a handful were known—Fermi has so far increased their number to over 20 (Roberts 2013).

Black widows are energetic MSPs (spin-down luminosity $\dot{E} \sim 10^{34\hbox{--}35}$ erg s⁻¹) with very low mass (few 0.01 M_☉) companions in compact, few-hour orbits. The original black widow, PSR B1957+20 (Fruchter et al. 1988b), and other members of the class derive their name from the fact that their strong relativistic wind ablates the surface of the companion star, which might have been significantly more massive in the past (Fruchter et al. 1988b). Redbacks, on the other hand, have similar pulsars but more massive companions, of ∼0.2 M_☉. As yet, is it not clear whether redbacks are the close progeny of accreting X-ray systems, nor whether they can evolve into black widow systems. The evidence that the prototype redback PSR J1023+0038 accreted matter not long ago is suggestive of the former at least (Archibald et al. 2009).

Because of their "cannibalistic" behavior, it has been proposed that some black widows eventually completely ablate their companion (Kluzniak et al. 1988; van den Heuvel & van Paradijs 1988). If this is the case, it could at least partially resolve the conundrum of the existence of isolated MSPs, since these sources clearly appear to have evolved in binary systems despite the fact that they are now isolated. It appears, however, that at least for the original black widow pulsar, PSR B1957+20, the evaporation is too slow (Eichler & Levinson 1988; Eichler & Gedalin 1995). Hence one has to wonder whether the known black widows are somewhat special (and less extreme than the isolated millisecond progenitors) or whether the scenario is simply wrong. Recent work suggested that evolution in a triple system might provide a viable formation channel to explain a small fraction of the isolated MSPs and peculiar binary pulsars (Portegies Zwart et al. 2011; Freire et al. 2011). Nonetheless, the ablation mechanism still appears as the most plausible general scenario for creating isolated MSPs.

Black widows and redbacks usually display extended eclipses at radio wavelengths, which are accompanied by rapid variations of the dispersion measure (DM; Stappers et al. 1996). At X-ray energies, persistent emission is often visible in addition to pulsations, which likely results from the shocked relativistic wind colliding with the companion (Stappers et al. 2003; Tavani 1993). Black widows and redbacks were thought to be a relatively small fraction of the total MSP population, but it has become clear that previous wide-field pulsar surveys have missed many sources (e.g., because of eclipses). In contrast, since energetic MSPs are efficient γ-ray emitters, targeted radio searches of γ-ray sources are biased toward finding them, especially since repeated searches of the same source have a better chance of catching it out of eclipse. About one-third of the pulsars discovered by targeting γ-ray sources have been black widows or redbacks (Ray et al. 2012). In the light of these discoveries, black widows might after all offer a viable channel for the formation of at least some of the isolated MSPs.

Optical and near-infrared observations of black widows and redbacks are an important probe of the state of the companion and the energetics of the system (see, e.g., Fruchter et al. 1988a; van Paradijs et al. 1988; Stappers et al. 2001). The optical light is dominated by the companion—the pulsar contributes a negligible amount—and shows significant flux and color variations. Mostly, these reflect strong irradiation of the hemisphere facing the pulsar, but superposed on this are ellipsoidal variations due to the tidal distortion. Combined, these allow one to constrain the inclination and other physical parameters (Stappers et al. 2001; Reynolds et al. 2007). Combined with phase-resolved spectroscopy, this can be used to determine the component masses (e.g., van Kerkwijk et al. 2011; Romani & Shaw 2011; Romani et al. 2012). Using this technique, van Kerkwijk et al. (2011) inferred that PSR B1957+20 is likely very massive, ∼2.40 ± 0.12 M_☉. Similarly, Romani et al. (2012) find evidence that PSR J1311−3430 is also heavyweight—perhaps as much as ∼2.7 M_☉. These large masses suggest that mass transfer was relatively effective, in contrast to what is inferred for other pulsar binaries (e.g., Lin et al. 2011; Antoniadis et al. 2012), posing both an interesting quandary as to why this might be the case and an opportunity to probe the upper mass limit of neutron stars.

While this kind of light curve modeling is also possible in other types of binary neutron star systems (see, e.g., Muñoz-Darias et al. 2009; Wang et al. 2013 for low-mass X-ray binaries), black widows and redbacks are much cleaner systems. Indeed, the only source of optical light is the companion since there is no accretion disk or jet.¹⁴ Also, the irradiation is due to relativistic photons/particles which penetrate several optical depths inside the companion's photosphere. As a result, the atmosphere remains in quasi-equilibrium and this avoids the formation of prominent emission line features. Finally, because the neutron star is a pulsar, the radio timing provides a set of accurate orbital parameters.

We have searched for the optical counterpart to four of the new black widow/redback systems, using Gemini data complemented by New Technology Telescope (NTT) and archival Swift data. In this paper, we present the optical discovery of these companions (Section 2) and use their light curves to constrain the system parameters by modeling them using the synthesis code Icarus (Section 3). We find that the temperature increase on the dayside of the irradiated pulsar companions typically corresponds to a conversion efficiency ∼15% of the incident energy available from the host pulsars' rotational spin-down (Section 4).

We were awarded 7.67 hr on Gemini North (program GN-2010B-Q-77) to search for the counterparts of four energetic binary MSPs with the GMOS-N instrument (Hook et al. 2004): PSRs J1810+1744, J0023+0923, J2215+5135, and J2256−1024. The first three pulsars were found in a targeted Robert C. Byrd Green Bank Telescope (GBT) search of unidentified Fermi point sources¹⁵ (P. Bangale et al., in preparation; J. W. T. Hessels et al., in preparation; Hessels et al. 2011) while the last one was discovered during the 350 MHz GBT pulsar drift scan survey¹⁶ (Boyles et al. 2013; Lynch et al. 2013; I. H. Stairs et al., in preparation). Table 1 presents the main properties of the targets, and a detailed discussion about each source follows in Section 4. Our observing program had two main goals: detect the pulsar companions and, if they are detected, identify variability at the orbital period. Our observing strategy consisted of collecting data at four different epochs, with each observing session consisting of a 320 s i-band, 620 s g-band, and 320 s i-band exposure sequence. Since we observed under non-photometric conditions (cloud cover: 90%, image quality: 85%, sky background: 80%, water vapor: any), resolution was not an issue and we binned the EEV CCD detector by a factor 2 × 2 in order to reduce the readout time to 35 s per exposure (fast mode) while still properly sampling the point-spread function. Using this strategy, we were expecting to find interepoch variability and, given the short orbital periods of these binaries, maybe even intraepoch variations (for the i band).

Table 1. Measured and Inferred Source Parameters

Quantity	J0023+0923a	J2256−1024b	J1810+1744a	J2215+5135a	B1957+20c	J1023+0038c
Measured
Right Ascensiond	$00^{{\rm h}}23^{{\rm m}}16\buildrel{\mathrm{s}}\over{.}89(2)$	$22^{{\rm h}}56^{{\rm m}}56\buildrel{\mathrm{s}}\over{.}39(1)$	$18^{{\rm h}}10^{{\rm m}}37\buildrel{\mathrm{s}}\over{.}28(1)$	$22^{{\rm h}}15^{{\rm m}}32\buildrel{\mathrm{s}}\over{.}68(1)$	$19^{{\rm h}}59^{{\rm m}}36\buildrel{\mathrm{s}}\over{.}77$	$10^{{\rm h}}23^{{\rm m}}47\buildrel{\mathrm{s}}\over{.}69$
Declinationd	09°23'2418(20)	−10°24'3437(12)	17°44'3738(7)	51°35'3645(10)	20°48'1512	00°38'4115
Gal. longitude (deg)	111.38	59.23	44.64	99.87	59.20	243.49
Gal. latitude (deg)	−52.85	−58.29	16.81	−4.16	−4.70	45.78
fγ, 0.1–100 GeV	10.7 ± 1.5	10.1 ± 1.4	25.5 ± 2.1	10.9 ± 1.6	16.7 ± 1.9	5.4 ± 1.0
(10−12 erg s−1 cm−2)
Pspin (ms)	3.1	2.3	1.7	2.6	1.61	1.69
Porb (h)	3.3312	5.1092	3.5561	4.1401	9.1672	4.7543
Tasc.node (MJD (TDB))	55186.11343	54853.22391	55130.04813	55186.16449	48196.06352	54801.97065
x (lt-s)	0.035	0.083	0.095	0.47	0.089	0.343
DM (pc cm−3)	14	14	40	69	29.12	14.33
AVe	0.37	0.14	0.43	1.15	⋅⋅⋅	⋅⋅⋅
Inferred
dDM (kpc)f	0.69	0.65	2.00	3.01	2.49	0.62
$M_{\rm c}^{\rm min}$ (M☉)g	0.017	0.030	0.045	0.213	0.022	0.138
$L_{\rm sd,\dot{P}}$ (1034 erg s−1)h	1.51	3.95	3.97	5.29	16.0	9.82
Lγ (1033 erg s−1)i	1.28	1.21	3.05	1.30	2.00	0.65
a (R☉)j	1.27	1.69	1.33	1.53	2.49	1.65
RL (R☉)k	0.13	0.22	0.19	0.36	0.29	0.34
Light curves (Minimum/Maximum/Quadrature)l
i band	24.3/21.7/22.7	24.3/20.8/22.1	22.3/19.5/20.3	19.5/18.6/18.9	⋅⋅⋅	⋅⋅⋅
g band	28.0/23.4/25.0	28.0/22.2/26.8	23.2/19.2/20.2	19.9/18.1/18.7	⋅⋅⋅	⋅⋅⋅
Light curve fitting
Inclination (deg)	58 ± 14	68 ± 11	48 ± 7	66 ± 16	⋅⋅⋅	⋅⋅⋅
Filling factor	0.30 ± 0.30m	0.40 ± 0.20	0.80 ± 0.30	0.99 ± 0.03	⋅⋅⋅	⋅⋅⋅
Tnight (K)	2900 ± 700	2450 ± 350	∼4600n	4800 ± 450	∼2500	∼5600
Tday (K)	4800 ± 2000	4200 ± 700	≳ 8000n	6200 ± 500	∼5800	∼6650
Tirr (K)	4600	4100	7800n	5550	5750	5580
irr	0.17	0.07	0.60n	0.15	0.15	0.09

Notes. ^aTiming data (P_spin, P_orb, x, DM, and $L_{\rm sd,\dot{P}}$ from J. W. T. Hessels et al. (in preparation). ^bTiming data (P_spin, P_orb, x, DM, and $L_{\rm sd,\dot{P}}$ from I. H. Stairs et al. (in preparation). ^cPSRs B1957+20 and J1023+0038 are shown for comparison. Data from the ATNF Pulsar Catalogue (Manchester et al. 2005; http://www.atnf.csiro.au/research/pulsar/psrcat). ^dPositions derived from optical astrometry. Uncertainties are dominated by the catalog systematics of 007 in UCAC3 (J0023+0923, J1810+1744, and J2215+5135) and 01 in SDSS DR7 (J2256−1024). ^eTotal reddening along the line of sight (see Section 3). ^fDistance based on the dispersion-measure NE2001 model (Cordes & Lazio 2002). ^gMinimum companion mass assuming a 1.4 M_☉ pulsar and a 90° orbital inclination. ^hSpin-down luminosity inferred from P_spin and its derivative, and a moment of inertia of 10⁴⁵ g cm². ⁱγ-ray luminosity estimated from the γ-ray flux and dispersion-measure distance. ^jOrbital separation $a = x (1+1.4/M_{\rm c}^{\rm min})$. ^kCompanion Roche radius $R_L = 0.46 a [M_{\rm c}^{\rm min}/(M_{\rm c}^{\rm min}+1.4)]^{1/3}$. ^lThe reported magnitudes are inferred from the light curve modeling presented in the text. ^mThe probability distribution for this parameter is highly non-Gaussian, and hence the value must be taken with caution. The median value is 0.40, the mode at ∼0.15 and the distribution extends with a heavy tail all the way to unity. ⁿAs explained in Section 3.3, we believe that our analytic estimate is more robust than the numerical values for the case of PSR J1810+17. Hence, we present the analytic results in this table.

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We reduced the data following standard procedures, implemented using custom Python scripts. We used standard Gemini nightly calibration data to remove the bias and flat field our science frames. We registered our frames astrometrically relative to the UCAC3 catalog for PSRs J1810+1744, J0023+0923, and J2215+5135, and relative to Sloan Digital Sky Survey (SDSS) for PSR J2256−1024 (the only source that falls within the coverage of SDSS Data Release 7 (DR7)). For all but PSR J0023+0923, a few tens of reference stars fall within the central frame of the CCD detector and hence our calibration yielded positional uncertainties dominated by systematic uncertainties, of 007 and 01 in UCAC3 and SDSS DR7, respectively. For PSR J0023+0923, only three UCAC3 stars fall on the central CCD chip. In this case, we calibrated the image that looked the cleanest using the UCAC3 catalog, and then calibrated the other images in the same band relative to the reference one using a list of bright stars found in all images. The measured optical positions are reported in Table 1 and agree with the radio position derived from the timing.

We performed aperture photometry using an extraction radius of 5 pixels (i.e., 0.73 arcsec at the plate scale of 0.146 arcsec per binned pixel), and sky inner and outer annuli of 10 and 15 pixels, respectively. We calibrated our photometry against bright, non-saturated stars appearing in the SDSS DR7 catalog for PSR J2256−1024, and USNO-B1 in the case of the three other targets. For USNO-B1, we converted the catalog magnitudes from the photographic B, R, and I magnitudes to the ugriz system using the transformation of Jordi et al. (2006), i − I = (0.247 ± 0.003)(R − I) + (0.329 ± 0.002), for the i band, and from Lupton (2005), B − g = 0.3130(g − r) + 0.2271 and R − r = −0.1837(g − r) − 0.0971, for the g band. In the case of PSR J1810+1744, there is a neighboring star that might contaminate our aperture photometry. For this reason, we performed point-spread function photometry using a Moffat profile f(r)∝(1 + r²/σ²)^−β, with β = 3 and σ optimized using a set of bright, non-saturated stars for each frame. Reference stars were fitted individually while the immediate vicinity of PSR J1810+1744 was simultaneously fitted for the source and the two other stars located east and northeast from it.

Our complete photometric results for the pulsar companions are available in Tables 2–5, as well as for a set of comparison field stars in Table 6. Our photometric errors were calculated by adding in quadrature the sky background, the photon counting noise, and the intraband relative zero point. The zero-point calibration errors correspond to the standard deviation of the mean of the zero-point calibration for each band to the reference catalog stars, which were added in quadrature to the catalog systematic calibration. In the case of PSR J0023+0923, very few catalog stars overlap with our field and hence the band calibration is poorer than for the other systems analyzed here. The SDSS systematic calibration error is 0.02 mag (for PSR J2256−1024), while that of USNO-B1.0 is 0.3 mag (for PSRs J1810+1744, J0023+0923, and J2215+5135).

Table 6. Photometry of PSR J0023+09's Field

R.A.	Decl.	Fg	$\delta _{F_g}$	Fi	$\delta _{F_i}$	g Mag.a	g Mag. Errorb	i Mag.a	i Mag. Errorb
(°)	(°)	(μJy)	(μJy)	(μJy)	(μJy)
5.8186	9.33167	17.866	0.461	37.066	14.874	20.770	0.028	19.977	0.436
5.8316	9.32724	43.987	5.481	531.848	275.488	19.791	0.135	17.085	0.562
5.8362	9.32149	13.635	0.354	169.842	69.976	21.063	0.028	18.325	0.447
5.8478	9.30288	26.704	0.680	202.056	88.088	20.333	0.028	18.136	0.473

Notes. ^aAB magnitudes in the Lupton system, $m = m_{_0} - 2.5 \log b^\prime - (2.5 \log e) \sinh ^{-1} (f/2b^\prime)$, using softening parameters b' = 0.059 and 0.233 μJy for the g and i band, respectively, and zero-point $m_{_0} = -48.6$. ^bThe flux and magnitude errors represent the formal uncertainties. One should also add in quadrature the zero-point calibration errors, which are 0.319 and 0.757 mag in g and i band, respectively.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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The field of PSR J2215+5135 was also serendipitously observed by the UVOT instrument on board Swift while it was monitoring a nearby gamma-ray burst. We performed photometric reduction of the publicly available data,¹⁷ which were all obtained in the uvw1 band (Poole et al. 2008; λ_c = 260 nm, Δλ = 69.3 nm). For the UVOT data, we used a 5 pixel aperture (i.e., 5 arcsec at 1.004 arcsec pixel⁻¹) and sky inner and outer annuli of 15 and 30 pixels, respectively. We took the photometric zero point from Breeveld et al. (2011), zpuvw1 = 18.95 ± 0.03 (AB system), to convert our count rate to magnitude. Note that the aperture correction is negligible for our aperture size.¹⁸

We also obtained some exploratory exposures during an observing run at the NTT using the tri-band ULTRACAM imager (Dhillon et al. 2007) and managed to obtain an additional z-band and g-band images for PSR J2256−1024. The data processing was performed using the ULTRACAM data reduction pipeline.¹⁹

We found optical counterparts to all four pulsar binaries (see Figure 1). The association of the optical counterparts with the irradiated pulsar companions was confirmed in all cases by variability at the known orbital periods, as can be seen in the light curves shown in Figure 2.

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As discussed in Section 1, the light curves can be used to constrain several parameters of the system. The orbital modulation is caused by a combination of irradiation, which produces a maximum at the superior conjunction of the companion, and ellipsoidal variations, which yield maxima at quadrature. From the colors at superior and inferior conjunction of the companion, one can constrain the dayside and nightside temperatures and thus irradiation. For poorly sampled data like ours, the nightside temperature cannot be constrained very well, since it is degenerate with the orbital inclination. Good coverage is required to distinguish between the high-inclination case of seeing only the brighter, unirradiated nightside at inferior conjunction of the companion, or the low-inclination case where one observes a dimmer nightside combined with a sliver of the irradiated side. However, for strongly irradiated systems, where the dayside is much hotter than the nightside, the irradiated temperature can be determined relatively securely from just the colors near superior conjunction of the companion. The amplitude of ellipsoidal variations directly depends on the filling factor of the companion. Since the amplitude is small relative to irradiation, one must possess good quality sampling near quadrature in order to constrain the filling factor accurately. However, the physical size of the star is also directly related to its apparent luminosity given its distance, and thus any independent information on the latter, such as from the DM, can be used to constrain the filling factor. For small filling factors,²⁰ below ∼0.55, the scaling between filling factor and distance is linear (as expected), but for larger filling factors its dependence becomes sub-linear, as the ellipsoidal variations and gravity darkening start to counteract the irradiation at maximum light (superior conjunction), leading to a ∼20% reduction relative to the linear extrapolation when the companion fills its Roche lobe. Finally, we note that constraining parameters such as inclination and filling factor is generally limited by the quality of the sampling and the uncertainty on individual data points, whereas the determination of the temperatures are mostly affected by the absolute calibration of each band.

To get first-order estimates of the system parameters, we modeled the light curves of these irradiated binaries using Icarus²¹ (Breton et al. 2011) along with BTSettl atmosphere models (Allard et al. 2003, 2007, 2011) available through the Phoenix Web simulator.²² The free parameters in the model are the mass ratio q, the orbital inclination i, the distance d, and reddening A_V to the system, the companion's filling factor f = R/R_L (where R is the volume-averaged radius and R_L is the volume-averaged radius of the Roche lobe), and its dayside and nightside temperatures T_day and T_night. We assumed that the companion is tidally locked to the pulsar and added Gaussian priors on the distance and reddening. We set the mean of the distance priors to the values inferred from DM using the NE2001 model (Cordes & Lazio 2002), and, since our sources are well outside the dust layer, we estimated the reddening as being equal to the total along each line of sight as inferred by Schlegel et al. (1998; see values in Table 1). In both cases, we took standard deviations corresponding to 25% relative uncertainties. Also, since the mass ratio is unconstrained without spectroscopic information, we imposed that the mass of the pulsar lies in the range 1–3 M_☉. For the model fitting, we used the uncertainties from Tables 2 to 5 on individual data point, and a band-to-band uncertainty corresponding to our band calibration errors to which we added in quadrature an additional 0.1 mag to account for systematics in the zero-point calculation inherent to the atmosphere models (this is important only for PSR J2256−1024 for which the photometry could be accurately tied to SDSS). In the following subsections, we discuss the main results of the light curve modeling (see Figure 2 for best-fit light curves).

3.1. PSR J0023+0923

3.2. PSR J2256−1024

3.3. PSR J1810+1744

3.4. PSR J2215+5135

As can be seen in Figure 2, the model light curves generally agree well with our data. While our light curves are not sufficiently well sampled to produce strong constraints on the system parameters, they allow us to address two interesting aspects: the efficiency of the irradiation and the extent to which the companion fills its Roche lobe.

To estimate the efficiency of the irradiation, i.e., the effective fraction _irr of the spin-down luminosity incident on the companion that is absorbed and re-radiated, we assume that the irradiating flux is thermalized and re-radiated locally, i.e., that it simply adds to the intrinsic flux wherever it impinges. Then, from the hottest point, one can derive a characteristic "irradiation temperature" $T_{\rm irr}^4 = T_{\rm day}^4 - T_{\rm night}^4$, which is related to the pulsar's spin-down luminosity, L_sd, as $\epsilon _{\rm irr} L_{\rm sd} = 4\pi a^2 \sigma T_{\rm irr}^4$, where a is the orbital separation and σ is the Stefan–Boltzmann constant. Note that we implicitly assume that the pulsar spin-down energy is isotropically radiated.

In Table 1, we list our inferred irradiation temperatures as well as the implied irradiation efficiencies. With the exception of PSR J1810+1744, which we will discuss below, we find that the typical values of the irradiation efficiency of the new systems presented in this paper are consistent with those of PSRs B1957+20 and J1023+0038. Moreover, the other known irradiated pulsar systems (PSRs J2051−0827, J1311−3430, J2339−0533; Stappers et al. 2001; Romani et al. 2012; Romani & Shaw 2011) also display similar efficiencies (≳ 30%, ∼30%, and ∼15%, respectively).²⁴ We conclude that the typical irradiation efficiency factor in these systems lies in the range 10%–30%, with 15% being a representative figure. Note that the intrinsic spread in values might be smaller if one compensates for the fact that the orbital separation is poorly known given the uncertainties in the inclination, the pulsar mass, and its moment inertia.

The energetics derived from the irradiation of the companions are consistent with the idea that the relativistic wind, which is powered by the rotational spin down of the neutron star, is the major driver of the heating mechanism. The case of PSR J1810+1744 is, however, puzzling. Its rather large inferred irradiation efficiency not only departs from the other known irradiated pulsar systems but it also implies an input energy larger than the nominal spin-down luminosity (though our analytic estimate of the lower limit of ∼0.60 is below unity). One cause may be that our assumption of an isotropic wind is not justified. For instance, if the pulsar is aligned with the orbit and emits its wind preferentially in the equatorial plane (as is the case for, e.g., the Crab and Vela pulsars), the irradiation efficiency would be reduced. From our Gemini data, there is a clear indication that both g- and i-band light curves are not symmetric, with the companion being brighter after its inferior conjunction (phases ∼0.0–0.15) than before (phases ∼0.85–1.0). This could be an indication of non-isotropic heating or heat redistribution at the surface of the star. More detailed light curves of this system would help resolve these issues.

Radio eclipses are observed in three out of four of these systems (J. W. T. Hessels et al., in preparation; I. H. Stairs et al., in preparation) and large increases in the DM at the ingress and egress in similar systems indicates total intrabinary electron column densities of N_e ∼ 10¹⁶ cm⁻² (see, e.g., Fruchter et al. 1988b). Given that eclipses are coincident with the inferior conjunction of the companions, plasma must certainly be surrounding them and hence some form of mass loss from the companion is required. In the case of a nearly Roche lobe-filling star, material is loosely bound to the surface and can be peeled off easily when exposed to a relativistic pulsar wind. Our work suggests that, however, some of the irradiated pulsar companions that we have studied (PSRs J0023+0923 and J2256−1024) are not close to filling their Roche lobe.

As mentioned in the previous section, a first possibility is that the distances inferred from the DMs of these pulsars are underestimated. It is now well established that for pulsars located far off the Galactic plane, the measured parallactic distances tend to be larger than the DM distances (see, e.g., Gaensler et al. 2008; Roberts 2011). Roberts (2011) shows that based on 13 sources with measured parallax and Galactic latitude larger than 10°, d_DM/d_parallax = 0.66 ± 0.26. Since the filling factor is correlated with the distance, we ran another set of fits using priors on the distance rescaled using the above conversion factor and error. As a result, we found that the typical filling factor for PSRs J0023+0923, J1810+1744, and J2256−1024 did not change significantly.²⁵ This comes from the fact that the dayside temperature is not very precisely constrained due to the large systematic uncertainties in the absolute calibration of the bands. Consequently, the larger distance priors tend to increase the dayside temperatures rather than changing the filling factors as one would expect. We also ran another set of fits, this time by holding the filling factor of the companions to unity and removing the distance priors in order to see how much further these systems would need to be located in order to match the observed fluxes. We found that the DM distances would need to be off by a factor 8.5, 1.2, and 2.3 for the three above sources, respectively.

While it is not excluded that our distance estimates are wrong, it appears unlikely that PSRs J0023+0923 and J2256−1024's companions are Roche lobe filling. Whether nearly Roche lobe-filling stars are required in order to explain the radio eclipses of the pulsars is uncertain since neither the mechanism supplying particles to the plasma nor the role of the pulsar in triggering it are understood. Caution should be taken before drawing definitive conclusions and precise distance constraints from parallactic measurements would help shed light on this. Better-quality multi-color light curves will improve the measurement of the orbital inclination and address the contribution of ellipsoidal variations, hence also help to constrain the filling factor. It is worth mentioning that the second black widow system to be found, PSR J2051−0827, also displays puzzling light curves. Previous work highlighted an ambiguous behavior either indicating a filling factor near unity or closer to 50% (Stappers et al. 2001).

A common feature of black widow and redback systems is the presence of non-secular orbital period derivatives in the radio timing (see, e.g., Arzoumanian et al. 1994; Lazaridis et al. 2011; Archibald et al. 2009). It has been suggested that gravitational quadrupole coupling of the companion with the orbit might explain the orbital variability (Applegate & Shaham 1994). The dissipated tidal energy would drive convection, which would power a dynamo-induced magnetic field and provide a significant source luminosity. While such a mechanism would be compatible with the amplitude of the orbital variations of PSR B1957+20 and the luminosity of its companion (Applegate & Shaham 1994), it would require some fine tuning—namely, a ∼50% filling factor—in order to work for PSR J2051−0827 (Lazaridis et al. 2011). These newly discovered irradiated pulsar systems could therefore provide extended leverage to test the gravitational quadrupole theory, since the Roche lobe underfilling systems are predicted to display smaller orbital variability.

Our view of the binary pulsar population is currently shifting toward a new paradigm. Until the launch of Fermi, the bulk (∼90%) of the known population in the Galactic field consisted of pulsar–white dwarf systems (Lorimer & Kramer 2004), while the remaining pulsar binaries had neutron star, main sequence, very low mass star, or planet companions. Only about four of the known binary pulsars in the Galactic field were irradiated systems like those presented here. As of today, the number of irradiated systems has increased to over 20 members and candidates, which implies that they now account for about 10% of the binary pulsar population outside of globular clusters (based on the ATNF catalog; Manchester et al. 2005). It is clear that a large selection bias against finding these binaries in classical radio surveys existed until high-energy missions were added in the picture—and yet they still remain challenging to find. New radio pulsar surveys, benefiting from multibeam receivers and larger bandwidth, are also contributing to finding irradiated systems in blind searches, as was the case for PSR J2256−1024. Black widows and redbacks therefore constitute a fundamental component of the pulsar ecosystem, a component that dominates among the fastest-spinning MSPs (Hessels 2008).

The work presented here shows that the spin-down luminosity of pulsars is a good indicator of the level of irradiation sustained by their companions. We found that these systems display a rather universal irradiation efficiency _irr ∼ 10%–30% for reprocessing the incoming energy flux from the pulsar's spin-down into heat on the companion's surface. As a result, one may easily estimate the brightness and amplitude of the optical light curves due to irradiation in these pulsar binaries, provided an orbital separation, if the pulsar spin-down luminosity is known from timing or, alternatively, from the γ-ray luminosity.

The typical reflection albedo of stars with temperatures in the range 2000–10000 K is between 0.5 and 1.0 (Claret 2001) for atmospheres that are convective and in radiative equilibrium, respectively. Given the above 15% irradiation efficiency, the above albedos imply that 10%–30% of the energy from the spin-down luminosity would actually reach the companion. It is worth noting that the temperature of PSR J1810+1744 suggests that its outer envelope might be radiative, as opposed to convective in the other systems presented here. If so, the observed irradiation efficiency would be a factor two larger because of the difference in the stellar albedo and this could partly explain why it appears unusual. If further studies of these systems find bow shock nebulae (like for PSR B1957+20), it would allow an independent measurement of the energy loss by the pulsar, which would make for an interesting comparison with that inferred from the irradiated companion (see, e.g., van Kerkwijk & Ingle 2008).

The possibility that some of these pulsar companions do not fill their Roche lobe leads one to ponder the underlying cause for radio eclipses in these systems. What is the mechanism responsible for replenishing the plasma responsible for the eclipses? Other missing pieces of the evolutionary puzzle are: Were these companion stars closer to filling their Roche lobe in the past or have they contracted thermally since the mass transfer episode has terminated? Can black widows and redbacks completely destroy their companion and eventually become the isolated MSPs that we observe? Or, are we simply observing the systems that are incapable of destroying their companions on a relatively short timescale?

The fact that some of these irradiated systems are relatively bright also offers an interesting opportunity for spectroscopic follow-up. Radial velocity curves should allow for the measurement of the component masses, and test whether these neutron stars typically are more massive, as found for the single system studied so far. Further, detailed photometry work will also allow one to investigate the possible asymmetry in the light curves of PSR J1810+1744 and the reason for its anomalously large dayside temperature.

J.W.T.H. is a Veni Fellow of the Netherlands Foundation for Scientific Research (NWO). Pulsar research at UBC and UofT is supported by NSERC Discovery Grants. This work was partially supported by the NASA Fermi Guest Observer Program. Based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Science and Technology Facilities Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência, Tecnologia e Inovação (Brazil), and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina). The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.

Facilities: Fermi - Fermi Gamma-Ray Space Telescope (formerly GLAST), GBT - Green Bank Telescope, Gemini:Gillett - Gillett Gemini North Telescope, NTT - New Techology Telescope, Swift - Swift Gamma-Ray Burst Mission

Table 6 contains the field photometries of PSR J0023+09, PSR J2256−10, PSR J1810+17, and PSR J2215+51.