ESTIMATING THE GeV EMISSION OF MILLISECOND PULSARS IN DWARF SPHEROIDAL GALAXIES

Neutron stars receive a "kick" in the supernova events in which they originate. Predicted kick-velocity distributions vary widely in the literature: Hooper et al. (2013) and Cordes & Chernoff (1997, for long lived MSPs) find 10–50 km s⁻¹, Hobbs et al. (2005) and Toscano et al. (1999) find ∼85 ± 13 km s⁻¹, while Lyne et al. (1998) find a higher velocity range ∼130 ± 30 km s⁻¹. If unbound, MSPs would leave these systems in ${t}_{\mathrm{esc}}\sim {10}^{8}(R/1\,\mathrm{kpc})(10\,\mathrm{km}\ {{\rm{s}}}^{-1}/{v}_{\mathrm{kick}})$ years, i.e., much shorter than the typical stellar ages of dSphs. The kick-velocity estimates above are larger than the typical stellar velocity dispersion of dSphs, ∼10 km s⁻¹ in dSphs (McConnachie 2012), but are comparable to the escape velocity. In particular, the enclosed masses of classical dwarfs within 600 pc from the center are (2–7) × 10⁷ M_⊙ (Walker et al. 2007), corresponding to an escape velocity ${v}_{\mathrm{esc}}=\sqrt{2{{GM}}_{600}/{r}_{600}}\sim 30$ km s⁻¹ ∼ v_kick. This suggests that a fraction of MSPs should be retained within the virial radii of dSphs, which are assumed to be at kiloparsec distances with a contained mass of ∼10^8–9 M_⊙ (see also the calculations of Dehnen & King 2006).³

An MSP orbiting at 600 pc from the center of a dSph located 80 kpc away would span an apparent 04 angular size, which is within the 95% confinement radius of LAT Pass8 FRONT events at 2 GeV.⁴ Depending on the distance to a given dSph and the photon event class used for the analysis, a slight signal extension might be present.

Star formation in most MW dSphs ended several Gyr ago (e.g., Brown et al. 2012; Weisz et al. 2014), whereas the MW disk remains active. The formation of LMXBs, and accordingly MSPs, is thought to peak most strongly in the first 0.5 Gyr after star formation (Fragos et al. 2008), suggesting that MSPs in dSphs are potentially older and less luminous (Hooper & Linden 2016) than the average Galactic MSP.

However, other factors might increase the MSP population in dSphs. Neutron star production could be enhanced in low-metallicity systems relative to the Galactic field by ∼20% (Ivanova et al. 2008). Also, two dSphs host their own globular clusters: five in Fornax (de Boer & Fraser 2016) and one in Eridanus II (Crnojević et al. 2016). The abundance of MSPs might be higher in these systems due to more frequent stellar encounters.

Our pulsar sample consists primarily of MSPs in the Second LAT Pulsar Catalog (2PC; Abdo et al. 2013), although all LAT-detected MSPs were considered.⁵ Pulsars residing within globular clusters, as well as those lacking distance measurements, were both excluded, leaving 66 MSPs in our sample.

The luminosities of Galactic MSPs often have large uncertainties because accurate distance measurement is difficult. Distance measurements based on dispersion measure suggest that for roughly 75% of the directions in the sky the accuracy is no better than a factor of 1.5–2 (Schnitzeler 2012). We compare luminosities calculated with distances from 2PC and from the Australian Telescope National Facility Pulsar Catalog (ATNF; Manchester et al. 2005)⁶ in Figure 1.

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As a first step toward constructing the Galactic MSP LF we perform a Monte Carlo (MC) incompleteness correction and apply it to our MSP sample. In our model, we assume an exponential spatial distribution for MSPs in the MW disk:

$$\begin{eqnarray}&&\rho (R,z)\propto {e}^{(-R/{R}_{0})}{e}^{(-| z| /{z}_{0})},\end{eqnarray} \tag{ 1 }$$

where R is the radial distance from the Galactic center and z is the vertical scale height above the Galactic plane. While other authors model the radial distribution of Galactic MSPs using a Gaussian density profile (Faucher-Giguère & Loeb 2010), Grégoire & Knödlseder (2013) find that their results are fairly insensitive to the selected radial law. We make no attempt to model a special population of MSPs in the Galactic bulge; our population of interest is field MSPs.

To account for systematic uncertainties both in the spatial distribution of Galactic MSPs, as well as the effective selection function of the LAT pulsar survey, we repeat the following MC procedure many times with different input parameter sets. First, following the MSP population model of Grégoire & Knödlseder (2013), we draw radial and vertical scale lengths from log-normal distributions defined by ${R}_{0}={3}_{-1}^{+3}$ kpc and ${z}_{0}={0.6}_{-0.3}^{+0.6}$ kpc, respectively (see their Table 3). Using these scale lengths, we generate 10⁷ MSPs at random locations consistent with the spatial distribution of Equation (1) and assuming azimuthal symmetry in Galactocentric coordinates. We then determine the apparent Galactic coordinates (l, b) of each MSP as viewed from the Sun's position at (R, ϕ, z) = (8.5 kpc, 0 rad, 20 pc).

Each MSP (l, b) coordinate is mapped to an effective flux detection threshold using the sensitivity curve in Figure 17 of 2PC (Abdo et al. 2013), which is expressed as a function of Galactic latitude. This direction-dependent flux threshold partially accounts for variations in the intensity of diffuse Galactic emission. We model systematic uncertainty in the 2PC selection function by drawing a direction-dependent flux threshold curve between the 10% and 90% percentile sensitivity range for each realization of the MSP population (sampled from a uniform distribution). The survey completeness is evaluated as the number of detectable MSPs at a given luminosity divided by the total number of simulated MSPs.

After calculating the detection efficiency for 10⁵ sets of spatial parameters and flux detection thresholds, we arrive at the completeness function shown in Figure 1.

We derive the MSP γ-ray LF using a Markov Chain Monte Carlo (MCMC) assuming a broken-power-law parameterization. During the fit, the median incompleteness correction (Section 3.2) is applied to the MSPs in each luminosity bin of our sample. After determining the LF posterior, we normalize to unit stellar mass, assuming an MW stellar mass of 7 × 10¹⁰ M_☉ (Malhotra et al. 1996). The resultant LF is shown in Figure 2.

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To estimate the systematic uncertainty, we re-fit the LF pairing luminosities calculated from 2PC and ATNF distances with the upper and lower limits of the incompleteness correction. These four bracketing cases establish the upper and lower systematic envelope on the LF. At high luminosity, uncertainty on the LF is driven by the small number of extremely luminous Galactic MSPs, since the LAT census of such sources is expected to be largely complete. Systematic uncertainty dominates at low luminosity, due in part to the large and imperfectly known incompleteness correction that must be applied.

Our model predicts that the main contribution comes from MSPs with luminosities ∼10³³ erg s⁻¹. Both our incompleteness correction and LF are in reasonable agreement with Hooper & Mohlabeng (2016); their best-fit LF for field MSPs peaks around 3 × 10³³ erg s⁻¹ in ${L}_{\gamma }^{2}{dN}/{{dL}}_{\gamma }.$

For each of the 30 dSphs in our sample, we scale the LF of Galactic MSPs (Section 3.3) to the dSph stellar mass. The expectation value for the cumulative MSP luminosity is found by integrating the LF, dN/dL_γ:

$$\begin{eqnarray}&&\langle {L}_{\gamma ,\mathrm{MSP}}\rangle =\displaystyle \frac{{M}_{* ,\mathrm{dSph}}}{{M}_{* ,\mathrm{MW}}}{\int }_{{L}_{\min }}^{{L}_{\max }}{L}_{\gamma }\displaystyle \frac{{dN}}{{{dL}}_{\gamma }}{{dL}}_{\gamma },\end{eqnarray} \tag{ 2 }$$

where L_min = 10³¹ erg s⁻¹ and L_max = 10³⁶ erg s⁻¹.

dSphs have small enough stellar populations that the number of luminous MSPs in a particular dSph may exhibit substantial "shot noise." To model this effect, we evaluate the cumulative MSP flux including Poisson fluctuations in the number of MSPs within multiple luminosity intervals. We repeat this process 10⁴ times for each dSph to construct a PDF of the expected flux. As shown in Figures 3 and 4, the spread in cumulative MSP flux for a low stellar mass dSph, such as Segue 1, spans several orders of magnitude, whereas high stellar mass dSphs, such as Fornax and Sculptor, have sharply peaked flux PDFs. The cumulative MSP flux in an ultra-faint dSph is likely dominated by a handful of luminous sources, but the relative impact of such statistical variations diminishes with increasing stellar mass.

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Using the same MW MSP population model discussed in Section 3.1, we also estimate the probability of confusion due to the chance alignment of foreground MSPs along the same sightlines to the target dSphs. We begin by randomly drawing luminosities from the best-fit LF (normalized to the stellar mass of the MW). Next, the luminosity values are assigned to random MSP locations within the MW according to the fiducial spatial model described in Section 3.2. We then compute the cumulative flux within a 10 radius (comparable to the LAT PSF for 1 GeV γ rays) around the location of each dwarf in our sample. The resulting foreground MSP flux distributions for four representative dSphs are shown in Figures 3 and 4.

For each of the 30 dSphs in our sample, we compute the expectation value and uncertainty on the cumulative MSP flux by adding in quadrature:

• 1.  
  Statistical uncertainty due to the finite number of LAT-detected Galactic MSPs;
• 2.  
  Systematic uncertainty in the luminosities of Galactic MSPs due to distance uncertainties;
• 3.  
  Systematic uncertainty in the detection efficiency of Galactic MSPs due to their unknown spatial distribution and the effective flux threshold of the LAT pulsar catalog; and
• 4.  
  Uncertainty in the stellar mass of the host dSph.

We then compute flux upper limits taking into account Poisson fluctuations in the number of luminous MSPs (using the best-fit LF). While stellar mass uncertainty is dominant for select dSphs, we find that systematic uncertainty, together with Poisson fluctuations, have the greatest impact on the predicted signals from both ultra-faint and classical dSPhs. The results of these calculations, summarized in Table 1, suggest that high stellar mass dSphs are likely to host modest MSP populations. Consistent with our initial assumptions, we find the mean number of predicted MSPs to be higher than a few in all dSphs where LMXBs or LMXB candidates have been detected. However, even for the largest classical dSph, Fornax, the predicted MSP flux >500 MeV is ${6.0}_{-4.4}^{+9.7}\times {10}^{-12}$ ph cm⁻² s⁻¹, which is about an order of magnitude below the typical flux upper limits obtained at high Galactic latitudes after six years of the LAT survey, ∼10⁻¹⁰ ph cm⁻² s⁻¹ (Ackermann et al. 2015). Perhaps more importantly, Draco and Ursa Minor, two classical dSphs with large and well-constrained J-factors (e.g., Geringer-Sameth et al. 2015), have estimated MSP fluxes another order of magnitude fainter than Fornax. The MSP emission of ultra-faint dSphs, such as Segue 1, is typically below 10⁻¹³ ph cm⁻² s⁻¹ and likely beyond the reach of γ-ray telescopes in the foreseeable future.

To more easily compare γ-ray signals from MSPs and DM annihilation, in Figure 5 we show the expected MSP flux versus J-factor for the host dSph. The potential for confusion is greatest between models with similar γ-ray spectra, so we consider a DM model that produces a γ-ray energy flux peaking at ∼1 GeV, namely, $\chi \chi \to b\bar{b}$ with a DM particle mass of 25 GeV. This DM model is also compatible with DM interpretations of the Galactic Center excess (e.g., Calore et al. 2015 and references therein). In addition to being below the current LAT sensitivity threshold, we predict that MSP emission in each of the dSphs is also below the annihilation signal expected from this DM model for an annihilation cross section at the thermal relic value $\langle \sigma v\rangle =3\times {10}^{-26}\,{\mathrm{cm}}^{3}\,{{\rm{s}}}^{-1}$.

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Figure 5 also demonstrates that the GeV emission from MSPs and from DM annihilation follow different scaling relations across the dSph population. The targets with the largest J-factors are mainly nearby ultra-faint dSphs with M_* < 10⁴ M_⊙, whereas we argue that dSphs with M_* ∼ 10⁷ M_⊙ are most likely to host detectable γ-ray MSP populations. This distinction offers another means to distinguish a putative DM signal from conventional astrophysical sources (e.g., using a joint-likelihood analysis).

We find that Galactic foreground MSP emission is potentially a non-negligible source of confusion for a small fraction of dSphs, even at the current LAT sensitivity. However, for a typical target, there is no foreground MSP with luminosity exceeding ${L}_{0.1\mbox{--}100\mathrm{GeV}}={10}^{31}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ in ≈90% of model realizations. For dSphs with M_* > 10⁴ M_⊙, the contribution from MSPs within the dSph is predicted to exceed the foreground in most cases (e.g., Fornax, Sculptor, Draco). Otherwise, the MSP emission internal the dSph is likely to be sub-dominant (e.g., Segue 1). Several LAT analyses have already employed empirical background modeling methods to partially account for sub-threshold point sources coincident with dSphs (e.g., Geringer-Sameth & Koushiappas 2011; Ackermann et al. 2014; Carlson et al. 2015).