Search for Very High-energy Emission from the Millisecond Pulsar PSR J0218+4232

PSR J0218+4232 (hereafter J0218) is a millisecond pulsar (MSP) with a period of 2.3 ms in a 2 day orbit with a ∼ 0.2 M_⊙ white dwarf companion (Bassa et al. 2003). It was serendipitously discovered as a steep spectrum, highly polarized source in a low-frequency radio study of an unrelated supernova (Navarro et al. 1995). Its broad radio peak, with a large unpulsed component (∼50%) makes it unusual, suggesting that it may be an aligned rotator, a pulsar in which the magnetic field is aligned with the axis of rotation. This was later supported by subsequent polarimetry studies (Stairs et al. 1999). With a characteristic age τ < 0.5 Gyr and a spin-down power of 2.4 × 10³⁵ erg s⁻¹, it is one of the youngest and most energetic MSPs known. It has an extremely strong magnetic field at the light cylinder (B_LC ∼ 3.2 × 10⁵ G), only slightly weaker than young Crab-like pulsars, but several orders of magnitude weaker at the neutron star surface (Saito et al. 1997). Like the Crab Pulsar, J0218 also displays giant pulses (Joshi et al. 2004; Knight et al. 2006), something very uncommon among MSPs. Its distance, previously estimated to be greater than 6 kpc, potentially making it the most luminous MSP known (Du et al. 2014), has since been revised downward to ∼3 kpc (Verbiest & Lorimer 2014). Table 1 provides a summary of the key properties of J0218.

J0218 was first detected as a steady source of X-rays and γ-rays using ROSAT and EGRET, respectively (Verbunt et al. 1996). It was later shown to display nonthermal pulsed X-ray emission (Kuiper et al. 1998). Like in radio, J0218 has a large unpulsed component in X-rays. It has been detected by most of the X-ray missions, including BeppoSAX (Mineo et al. 2000), Chandra (Kuiper et al. 2002), XMM-Newton (Webb et al. 2004), RXTE (Kuiper et al. 2004), and most recently NICER (Deneva et al. 2019). Nonthermal hard X-ray emission was detected with NuSTAR up to ∼70 keV (Gotthelf & Bogdanov 2017), with no evidence for a spectral break above these energies, although it must turn over somewhere between 100 keV and 100 MeV, to be consistent with the γ-ray emission detected at GeV energies (Gotthelf & Bogdanov 2017).

Table 1. Timing Ephemeris for PSR J0218+4232, Obtained with the Nançay Radio Telescope

Timing and binary parameters
R.A., α (J2000.0)...	${02}^{{\rm{h}}}\,{18}^{{\rm{m}}}\,06\mathop{.}\limits^{}35863{\left(1\right)}^{{\rm{s}}}$
Decl., δ (J2000.0)...	$+42^\circ 32^{\prime} 17\mathop{.}\limits^{}3722(2)^{\prime\prime}$
Frequency, F0 (Hz)...	430.46105998103612106(6)
1st frequency derivative, F1, (Hz s−1)	−1.434128(1) × 10−14
PMRA ($\dot{\alpha }/\cos \delta ,\mathrm{mas}\,{\mathrm{yr}}^{-1}$)	5.32(3)
PMDEC ($\dot{\delta },\mathrm{mas}\,{\mathrm{yr}}^{-1}$)	−3.68(6)
PEPOCH (MJD)	56000
POSEPOCH (MJD)	56000
DMEPOCH (MJD)	56000
DM (cm−3 pc)	61.2374(7)
DM1 (cm−3 pc s−1)	−0.0004(2)
BINARY MODEL	ELL1
PB (d)	2.0288460845(6)
A1 (lt-s)	1.9844348(2)
TASC (MJD)	49148.5799767(2)
EPS1	5.0(2) × 10−6
EPS2	4.9(2) × 10−6
START (MJD)	53579.2
FINISH (MJD)	58960.5
UNITS	TDB (barycentric dynamical time)
EPHEM	DE436
Derived parameters
Period, P (ms)	2.32309053
1st period derivative, $\dot{P}$ (s s−1)	7.739 × 10−20
Characteristic age, τc (yr)	4.8 × 108
Spin-down power, $\dot{E}$ (erg s−1)	2.4 × 1035
Surface B-field strength, BS (G)	4.3 × 108
Light-cylinder B-field, BLC (G)	3.1 × 105
Distance, d (kpc)	${3.15}_{-0.60}^{+0.85}$
ON-pulse region	(0.34–0.98)
OFF-pulse region	[0–0.34)∪(0.98,1]

Note. We used the DE436 solar system ephemeris, with time units in barycentric dynamic time (TDB) and the ELL1 binary model for low eccentricity orbits, where EPS1 and EPS2 represent the first and second Laplace–Lagrange parameters (Lange et al. 2001). We refer the reader to the Tempo2 manual (Hobbs et al. 2006) for a detailed definition of all parameters included in our timing model.

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The γ-ray emission of J0218 has often been confused with that of the blazar 3C 66A because of their close proximity (0.97'' separation, see Figure 1), the poor angular resolution of many γ-ray instruments, and the varying intensity of the blazar. The Second EGRET (2EG) Catalog (Thompson et al. 1995) contained a source (2EG J0220+4228) that was more often associated with 3C 66A. In the third EGRET (3EG) catalog (Hartman et al. 1999), the source 3EG J0222+4253 was identified with 3C 66A based on its >1 GeV emission; however, its low-energy flux (100–300 MeV) was found to be dominated by the pulsar, rather than the blazar, leading to the conclusion that both were likely contributing to the EGRET source (Kuiper et al. 2000; Guillemot et al. 2007). Indeed, Kuiper et al. (2000) reported marginal evidence (∼3.5σ) for the detection of pulsed γ-ray emission from J0218, making it potentially the first MSP detected at these energies.

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With the launch of the Large Area Telescope (LAT; Atwood et al. 2009) on board the Fermi satellite, the picture became significantly clearer. Next to 3C 66A, a strong source (>19σ), 1FGL J0218.1+4232, was reported in the First Fermi-LAT Catalog, based on 11 months of data (Abdo et al. 2010a), and a firm detection of GeV γ-ray pulsations confirmed the identification with J0218 (Abdo et al. 2009, 2010b). The LAT detection, however, showed a broad single-peaked γ-ray light curve, bearing little similarity to the EGRET one reported in Kuiper et al. (2000), or indeed the double-peaked X-ray pulse profile (Kuiper et al. 2002; Webb et al. 2004). Despite the fact that the First Fermi-LAT Catalog of sources above 10 GeV (1FHL; Ackermann et al. 2013) contained no source associated with J0218, this pulsar was identified as having hints of pulsed emission above 10 GeV (Ackermann et al. 2013), and the Third Catalog of Hard Fermi-LAT Sources (3FHL; Ajello et al. 2017) contained a source associated with J0218 (3FHL J0218.3+4230), which was shown to have >10 GeV (and even >25 GeV) pulsations (Saz Parkinson et al. 2017), despite the limited statistics due to the small effective area of the LAT at such high energies, compared to ground-based γ-ray telescopes. ⁵⁴ These preliminary LAT results provided a strong motivation for observing this pulsar at even higher energies, using ground-based γ-ray telescopes.

At very high energies (VHE >100 GeV), J0218 has been a target for the Major Atmospheric Gamma Imaging Cherenkov (MAGIC) telescope, starting in 2004, when it was observed for 13 hr during the commissioning phase (single-telescope MAGIC-1 observations), in large part due to it being in the same field of view as 3C 66A (Oña-Wilhemi 2005). It was subsequently observed for 20 hr, between 2006 October and 2007 January yielding no significant detection, and a 3σ flux upper limit (UL) of < 9.4 × 10⁻¹² cm⁻² s⁻¹, above 140 GeV (Anderhub et al. 2010). Since then, the performance of the MAGIC telescopes has significantly improved (Aleksić et al. 2016a).

In this paper, we report results from an analysis of 11.5 yr of Fermi-LAT data, together with ∼90 hr of data from new MAGIC stereoscopic observations of J0218, collected from 2018 November to 2019 November using the low-energy threshold Sum-Trigger-II system (Dazzi et al. 2021). The structure of the paper is as follows: Section 2.1 describes the Fermi-LAT data analysis. Section 2.2 describes our MAGIC observations and analysis. Section 3 presents our results, both based on LAT data (Section 3.1) and the MAGIC observations (Section 3.2). Section 4 discusses our theoretical modeling. Finally, in Section 5 we discuss the main conclusions of our work.

2.1. Fermi-LAT Data Analysis

For the Fermi-LAT analysis, we used 11.5 yr of Pass 8 data (Atwood et al. 2013; Bruel et al. 2018) (specifically, P8R3_SOURCE_V2), from 2008 August 4 (MJD 54682.7) to 2020 February 10 (MJD 58890). ⁵⁵ We used Fermipy (Wood et al. 2017) to select Source class (evclass 128), Front and Back converting events (evtype 3) with an energy range from 100 MeV to 870 GeV, and from a square region of 15° × 15°, centered on the position of 4FGL J0218.1+4232 (R.A. = 34.5344'', decl = 42.5459''). A maximum Earth zenith angle of 90° is imposed, helping to eliminate contamination from the Earth's limb. We further ensured that the selection only included events at times where the LAT was in normal science configuration and taking good data.

Figure 1 shows the Fermi-LAT >1 GeV all-sky γ-ray intensity map, highlighting the region around J0218. As discussed in Section 1, the blazar 3C 66A, located 0.97'' away, complicates the analysis of J0218. The bright γ-ray counterpart of 3C 66A (4FGL J0222.6+4302, see Figure 1) has a >1 GeV flux of 1.6 × 10⁻⁸ cm⁻² s⁻¹ (>160σ) in 4FGL, compared to 5.2 × 10⁻⁹ cm⁻² s⁻¹ (73.5σ) for J0218. Because blazars are typically variable sources, we considered excluding certain flaring periods of 3C 66A. Unfortunately, the Fermi-LAT light curves of J0218 and 3C 66A, over the entire time interval (11.5 yr) and energy (100 MeV to 870 GeV) range (Figure 2) reveal that such a strategy would not be possible as the blazar has been quite active throughout the entire 11.5 yr period of our observations. Figure 3 shows the zoomed-in 1 yr period covered by our MAGIC observations, illustrating how 3C 66A is brighter than J0218 at GeV energies. We note that there is a report of quasi-periodic variability in the optical light curve of 3C 66A with a period of ∼3 yr, although these have not been confirmed in the γ-ray data (Otero-Santos et al. 2020).

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A binned likelihood analysis was performed, utilizing spatial bins of 0.1° × 0.1° and 8 logarithmically spaced bins per decade of energy. The initial region model was seeded from the 8 yr Fermi-LAT Fourth Source Catalog (4FGL; Abdollahi et al. 2020) by including sources lying within a square area of 40° × 40° centered on 4FGL J0218.1+4232. Because the LAT only measures the energy of photons with finite resolution (<10% in the 1–100 GeV range, but ∼20% at 100 MeV, and ∼28% at 30 MeV) ⁵⁶ a correction for this energy dispersion was enabled for all sources except the isotropic diffuse emission, as recommended by the Fermi Science Support Center. ⁵⁷ The fitting of source spectra within the region model was carried out in an iterative process, with sources being removed from the model if they fell below a test statistic (TS) of 10. Here, TS is defined as TS = −2ln(${{ \mathcal L }}_{0}/{{ \mathcal L }}_{1}$), where ${{ \mathcal L }}_{1}$ is the probability that a model that includes the source of interest matches the data, and ${{ \mathcal L }}_{0}$ is the probability that the same model without the source of interest matches the data. ⁵⁸ The final iteration of the fit was performed with Minuit (James & Roos 1975). In addition to the spectral parameters of 4FGL J0218.1+4232, the normalization and index of the Galactic diffuse emission (modeled with a power-law spectrum), the normalization of the isotropic diffuse emission, and the normalizations of background sources with a TS of at least 100 were allowed to vary.

We modeled the J0218 spectrum using a power law with an exponential cutoff ⁵⁹ , $\displaystyle \frac{{dN}}{{dE}}={N}_{0}{\left(\displaystyle \frac{E}{{E}_{0}}\right)}^{\gamma }\exp (-{{aE}}^{b})$. We set the index (b) to a fixed sub-exponential value of 2/3, as this source is too faint for it to be determined by maximum likelihood estimation, and 2/3 approximates the values of other, brighter pulsars (Abdollahi et al. 2020).

After obtaining our best region model, we examined events within 5° of J0218, and used the gtsrcprob Fermitool to assign them a probability (weight) of originating from either J0218 or 3C 66A relative to other sources in the model. Finally, we used Tempo2 (Hobbs et al. 2006) with the fermi plug-in (Ray et al. 2011) to assign the pulsar rotational phases ϕ_i , according to our pulsar ephemeris obtained with the Nançay radio telescope, given in Table 1.

We calculated the source spectrum and flux points for the theoretical modeling described in Section 4 by utilizing the aforementioned region model. Three energy bins spanning 12.38–28.99 GeV were combined in order to produce a flux point instead of a UL. To extract the overall spectrum of J0218, the normalization of the isotropic and Galactic diffuse emission components were allowed to vary while Minuit fit the spectral parameters (summarized in Table 2). To generate flux points, the index of the Galactic diffuse emission was also allowed to vary, along with the normalizations of background sources with a TS of at least 500 or which lie within 5° of 4FGL J0218.1+4232. The spectrum of 4FGL J0218.1+4232 is replaced by a power law with an index of −2, and Minuit is used to fit the normalization of this modified spectrum within each energy bin. The result is interpreted as either a flux point or a UL, depending on the significance with which the power-law source was detected.

Table 2. Gamma-ray Spectral Parameters for the Total Emission from PSR J0218+4232

Parameter	Value
N0 (ph cm−2 s−1 MeV−1)	(2.07 ± 0.03) × 10−11
γ	−1.76 ± 0.01
E0 (MeV)	821.6 (fixed)
a	(6.19751 ± 0.00007) × 10−3
b	0.6667 (fixed)
Photon flux (photons cm−2 s−1)	(7.67 ± 0.15) × 10−8
Energy flux (MeV cm−2 s−1)	(3.05 ± 0.04) × 10−5

Note. Photon and energy flux cover the entire 100 MeV to 870 GeV energy range.

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2.2. MAGIC Observations and Data Analysis

3.1. Fermi-LAT Results

Figure 4 shows the LAT spectrum obtained in our analysis. Table 2 reports the best-fit spectral parameters and Table 3 gives the spectral values, including ULs. Note that the spectrum falls steeply at energies above 10 GeV with only ULs reported for energies above ∼29 GeV. These results are consistent with previous LAT results reported in 4FGL (Abdollahi et al. 2020) and 3FHL (Ajello et al. 2017), the latter of which reported an index of Γ = − 4.5, when fitting the >10 GeV data with a simple power law.

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Table 3. Fermi-LAT and MAGIC Spectral Points and ULs

	Fermi-LAT	MAGIC		Fermi-LAT	MAGIC
E	E2 dN/(dEdAdt)	E2 dN/(dEdAdt)	E	E2 dN/(dEdAdt)	E2 dN/(dEdAdt)
(GeV)	(TeV cm−2 s−1)	(TeV cm−2 s−1)	(GeV)	(TeV cm−2 s−1)	(TeV cm−2 s−1)
0.12	(7.24 ± 1.58) × 10−12	⋯	33.40	<3.79 × 10−13	⋯
0.15	(7.69 ± 1.02) × 10−12	⋯	44.35	<4.71 × 10−13	⋯
0.20	(8.11 ± 0.73) × 10−12	⋯	45.93	⋯	<8.32 × 10−12
0.27	(7.77 ± 0.57) × 10−12	⋯	58.88	<5.82 × 10−13	⋯
0.36	(8.25 ± 0.48) × 10−12	⋯	78.18	<7.69 × 10−13	⋯
0.48	(8.15 ± 0.42) × 10−12	⋯	81.68	⋯	<2.36 × 10−12
0.63	(8.34 ± 0.38) × 10−12	⋯	103.80	<1.33 × 10−12	⋯
0.84	(7.50 ± 0.36) × 10−12	⋯	137.80	<1.38 × 10−12	⋯
1.11	(8.22 ± 0.37) × 10−12	⋯	145.25	⋯	<6.57 × 10−13
1.48	(6.79 ± 0.36) × 10−12	⋯	183.00	<1.80 × 10−12	⋯
1.96	(5.34 ± 0.35) × 10−12	⋯	243.00	<2.40 × 10−12	⋯
2.61	(5.37 ± 0.38) × 10−12	⋯	258.30	⋯	<3.50 × 10−13
3.46	(4.05 ± 0.38) × 10−12	⋯	322.60	<3.19 × 10−12	⋯
4.59	(3.07 ± 0.37) × 10−12	⋯	428.30	<4.29 × 10−12	⋯
6.10	(2.90 ± 0.40) × 10−12	⋯	459.34	⋯	<1.94 × 10−13
8.10	(1.28 ± 0.30) × 10−12	⋯	568.70	<5.83 × 10−12	⋯
10.75	(9.24 ± 3.01) × 10−13	⋯	755.00	<8.10 × 10−12	⋯
18.95	(4.66 ± 1.71) × 10−13	⋯	816.84	⋯	<3.30 × 10−13
25.83	⋯	<9.12 × 10−11	1452.58	⋯	<1.06 × 10−13
			2583.09	⋯	<1.05 × 10−13

Note. Centers of energy bins are reported. Fermi-LAT data utilizes 32 logarithmically spaced bins between 100 MeV and 870 GeV. Three bins, spanning 12.38–28.99 GeV, were combined in order to produce a flux point instead of a UL. As such, a total of 30 bins are reported for the Fermi-LAT. MAGIC utilizes 14 logarithmically spaced bins between 20 GeV and 63 TeV. Note that we did not obtain ULs for the last five MAGIC bins (i.e., E > 3.56 TeV) because they have zero counts and such limits would be considered too unreliable.

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3.1.1. Search for High-energy Pulsation in LAT data

To test for possible pulsed emission above 10 GeV, an analysis analogous to what was carried out in the First Fermi-LAT Catalog of >10 GeV sources (1FHL; Ackermann et al. 2013) and in Saz Parkinson et al. (2017) was performed. We defined a low-energy probability density function (PDF), which we refer to as PDF_LE, based on the best estimate fit of the 1–10 GeV events (see Figure 5, top panel). For the high-energy PDF, we considered the family of distributions given by PDF_HE(ϕ) = (1 − x) + x · PDF_LE(ϕ), with 0 ≤ x ≤ 1. We maximized the unbinned likelihood function derived from this PDF, with respect to x, obtaining ${ \mathcal L }(\hat{x})$, and comparing it to the null hypothesis, for x = 0, that there is no pulsation (i.e., PDF_HE(ϕ) = 1). By construction, ${ \mathcal L }(0)=1$, so TS = −2ln(${ \mathcal L }(0)/{ \mathcal L }(\hat{x})$ can be simplified to TS = 2 ln ${ \mathcal L }(\hat{x})$. We converted the measured TS value into a tail probability (or p-value), by assuming, following Wilks' theorem (Wilks 1938), that for the null hypothesis the TS follows a χ² distribution with 1 degree of freedom.

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Using the LAT data set described in Section 2.1, we first selected events in the 1–10 GeV energy range (with a probability >50% of coming from the pulsar, as obtained via the Fermi Science Tool ⁶⁰ gtsrcprob) and used this histogram to generate a low-energy template. We used the nonparametric SOPIE (Sequential Off-Pulse Interval Estimation) R package (Schutte & Swanepoel 2016), to obtain a smooth kernel density estimator of our histogram (which we defined as our low-energy template), and also derived an estimate of the off-pulse interval of the light curve, using the median value of the results obtained from four different goodness-of-fit tests: Kolmogorov–Smirnov, Cramér–von Mises, Anderson–Darling, and Rayleigh test statistics. Figure 5 (top panel) shows our results, including the 1–10 GeV histogram, along with the resulting low-energy template, and the estimated off-pulse interval, all calculated using SOPIE.

To test for emission at higher energies, we looked at the >10 GeV events arriving within the 95% containment radius of the PSF (0.5°/0.8° for front/back converting events) and performed a likelihood test to determine whether they are likely to come from a similar distribution function, as represented by the lower energy template. We set a threshold p-value of 0.05 to claim evidence for emission at a specific energy. We carried out the same test with events of energies greater than 25 GeV. Figure 5 (bottom panel) shows the distribution of 58 (17) events above 10 (25) GeV, leading to a p-value of 1e-4 (0.01), thus showing evidence for emission above 10, and marginally, above 25 GeV. We also tested for possible emission above 30 GeV but found that, despite the presence of 10 events above this energy, their distribution in phase yielded a p-value that was not significant (p > 0.05).

3.2. MAGIC Results

We analyzed our MAGIC data to search for possible pulsed and unpulsed γ-ray emission above 20 GeV. The sky map is shown in Figure 6, where no emission is observed from J0218. The high emission spot observed in the image is the blazar 3C 66A, which is significantly detected as a by-product of the observations centered on J0218.

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We used the Tempo2 package (Hobbs et al. 2006) to assign the rotational phase to each event using the same ephemeris as in our LAT analysis described in Section 3.1 given in Table 1.

Given the broad pulse shape in the high-energy band, the use of the off-pulse region to estimate the background would lead to large uncertainties due to its smaller phase extent of 0.36 compared to the on-pulse width of 0.64. Therefore, we chose three source-free reflected-region backgrounds located at the same distance from the field-of-view (FoV) center, which we expect to have the same acceptance as the region containing the source (Berge et al. 2007). The ULs to the differential flux were obtained by following the Rolke & López (2001) method under the assumption of a Gaussian systematic uncertainty in the detection efficiency, with a standard deviation of 30% systematic uncertainty in the flux level. Hereafter, the ULs will be given at 95% confidence level. We assumed a spectral index Γ = −4.5 obtained from the power-law fit to the high-energy (>10 GeV) Fermi-LAT data, as reported in the Third Hard Source Catalog (3FHL; Ajello et al. 2017).

Figure 4 shows our MAGIC ULs, indicated with blue arrows, along with the green points and ULs from the Fermi-LAT analysis. The numerical values are reported in Table 3.

Given the soft spectrum observed by Fermi-LAT, we concentrated our search for pulsed signal in the lowest energy decade starting at threshold energy for the MAGIC observations. Hence, the phase-folded light curve of J0218 (see Figure 7) is computed in the energy range from 20–200 GeV. We performed the same unbinned likelihood test described in Section 3.1, to determine whether they are likely to come from the distribution function represented by the lower energy template (see Figure 5, top panel), obtaining no evidence for pulsation (p-value ≫0.05). In addition, we chose to carry out a standard pulsation search, looking at ON and OFF events. The on-pulse region was selected as the phase interval between 0.34 and 0.98, and shown as the gold area in Figure 7, as defined by our Fermi-LAT analysis (see the top panel of Figure 5 and Table 1). The same source-free reflected-region backgrounds, shown with the gray horizontal band in Figure 7, were used for calculating the significance of the excess events using Equation 17 of Li & Ma (1983), and no significant (0.057σ) pulsation was found. Moreover, we applied region-independent signal tests (de Jager et al. 1989) (χ² and H test), also with null results (5.54, for 11 degrees of freedom, and 0.05σ, respectively).

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We modeled the broadband spectrum of J0218 from UV to VHE γ-rays (14 orders of magnitude in energy) using a numerical force-free magnetosphere model for the global magnetic field, computing the individual trajectories of particles injected at the neutron star surface. Two populations of particles are injected: primary electrons/positrons along field lines that connect to the current sheet and are accelerated by an assumed parallel electric field distribution, and secondary electrons/positrons from polar cap (PC) pair cascades along field lines where there is no accelerating electric field. The dynamics and radiation of the particles are followed from the neutron star surface to a distance of 2 light cylinder radii (2R_lc ) and radiated photons are stored in energy-dependent sky maps of observer angle versus rotation phase (Harding & Kalapotharakos 2015; Harding et al. 2018).

All particles radiate by synchro-curvature (SC) and inverse Compton (IC) emission. The pitch angles for SC are maintained through cyclotron resonant absorption of radio photons emitted above the PC (Harding et al. 2008). The SC also assumes the radius of curvature of the particle trajectory in the inertial observers' frame. The IC requires that trajectories be followed twice, once to store the SC radiation emissivity and another to compute the local photon densities from the stored emissivity and radiate IC (Harding & Kalapotharakos 2015; Harding et al. 2018).

The main assumptions of the model are the parallel electric field (E_∣∣) distribution, the source of pairs, pair multiplicity and their injection distribution on the PC, and the mechanism for generating pitch angle. Apart from this, the model requires the observed parameters of the pulsar (P and $\dot{P}$). The magnetic and electric field distribution assumptions are based on results of particle-in-cell simulations showing that pulsars producing high pair multiplicity have near-force-free magnetospheres and that the highest parallel electric fields are in the current sheet. Increasing the pair multiplicity increases the pair SC (mostly synchrotron radiation (SR)) and the pair IC (mostly synchrotron self-Compton (SSC)). Increasing the E parallel increases the SC of primaries, increases the high-energy (GeV) cutoff, and increases the IC (at 10 TeV). For J0218, we assumed a magnetic inclination angle α = 45°, a viewing angle ζ = 65°, and a pair multiplicity of M₊ = 1 × 10⁵. Figure 8 shows the model predictions, including the various individual emission components.

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We have also used an SC model where all unknowns are reduced to just a few parameters that represent the observed spectrum (see Torres 2018; Torres et al. 2019, for details). The model follows particle trajectories in a generic region of a pulsar magnetosphere threaded by an accelerating parallel electric field, (E_∣∣). The region is located around the light cylinder, and particles are assumed to enter it at x_in with a (sizeable) pitch angle α. The model parameterizes the magnetic field by a power law $B(x)={B}_{s}{\left({R}_{s}/x\right)}^{b}$ (see the discussion in Viganò et al. 2015a), where x is the distance along the field line, b is referred to as the magnetic gradient, B_s is the surface magnetic field, and R_s is the pulsar radius. Given (E_∣∣, b) as free parameters, and the period and period derivative $(P,\dot{P})$, the model solves the equations of motion that balance acceleration and losses by SC radiation (see Cheng & Zhang 1996; Viganò et al. 2015b), computing the total emission. The model assumes that the distribution of particles emitting toward us can be parameterized as ${{dN}}_{e}/{dx}\propto {e}^{-(x-{x}_{\mathrm{in}})/{x}_{0}}$ where the inverse of x₀/R_lc (R_lc is the light cylinder radius) is referred to as the contrast.

Figure 9 shows the results of the model with best-fit parameters, $\mathrm{log}({E}_{| | }/V\,{m}^{-1})=10.92$, $\mathrm{log}({x}_{0}/{R}_{\mathrm{lc}})=-4.20$ and b = 3.70. The agreement between the model description and the broadband data is acceptable (the fractional residual errors are of the order ∼10%), despite the significant increase in the precision of each spectral measurement and the number of data points.

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The detection by ground-based Cherenkov telescopes of pulsed emission from the Crab pulsar (Aliu et al. 2008; VERITAS Collaboration et al. 2011), most recently detected up to TeV energies (Ansoldi et al. 2016), followed by the detection of pulsations from Vela (H.E.S.S. Collaboration et al. 2018), also up to TeV energies (Harding et al. 2018), has led to a flurry of activity by pulsar experts to develop self-consistent models able to explain the detected emission over such a broad range of energies.

In addition, the development of the Sum-Trigger-II system in MAGIC has significantly improved the sensitivity of the telescopes below 100 GeV, something that has enabled the detection of Geminga between 15 and 75 GeV (MAGIC Collaboration et al. 2020), making this the third γ-ray pulsar (and first middle-aged one) detected with ground-based telescopes. ⁶¹

J0218 is part of a small but diverse population of MSPs with a well-characterized broadband nonthermal energy distribution. Several of these MSPs have an inverted spectrum in X-rays (where E² dF/dE decreases with energy), quite different from that of J0218 as we have reported here, see Figure 3 of Coti Zelati et al. (2020). We also note that the fitted magnetic gradient b for J0218 and other MSPs within the SC model (Torres 2018) is larger than for normal pulsars. This is perhaps the result of the larger B_lc of MSPs compared to typical pulsars, due to the smaller size of R_lc. This needs to be taken into account when making predictions for their observability at lower energies based only on the γ-ray data. Fits to the γ-ray data alone are mostly insensitive to the value of the magnetic gradient, and assuming a lower b could lead to incorrect predictions that an MSP is undetectable in the X-ray band.

In accordance with previous studies, we also find here that the relevant scales for the production of the pulsar's spectrum (given by x₀) is small in comparison with the light cylinder radius.

This is true in general for MSPs, for which the light cylinder is already orders of magnitude smaller than in normal pulsars, i.e., the x₀/R_lc values imply a relevant region of emission ≪ 1 km.

Instead of subtracting the background events from the off-pulse region, we applied reflected-region background subtraction approach for the MAGIC analysis due to the large on-pulse interval of the LAT phaseogram. No evidence of emission (either pulsed or unpulsed) is apparent in the MAGIC data, and the measured MAGIC ULs are well above our two theoretical model predictions for VHE emission. The curvature radiation component from particles accelerated mostly in the current sheet is expected to fall to flux levels too low at VHE energies for detection by MAGIC and the ICS components from both pairs (mostly SSC) and accelerated primaries are predicted to be at even lower flux levels.

Most models for γ-ray emission from pulsars do not predict high levels of ICS and SSC emission for MSPs. In the model we used here, described by Harding & Kalapotharakos (2015) and Harding et al. (2018), for example, the pairs that come from the PC cascade and MSP surface magnetic fields are so low that the photons need to have much higher energies to produce pairs by one-photon magnetic pair production than do normal pulsars. The MSP pair spectra are thus shifted to much higher energies (typically γ ∼ 10⁴–10⁷) (Harding & Muslimov 2011). This will produce higher energy SR near the light cylinder. Since VHE emission is most likely ICS or SSC, and both particles and photons have higher energies, the VHE emission will be Klein–Nishina limited and therefore suppressed. This is also a problem for outer gap (OG) models since the latest models have pairs also produced near the PC since otherwise, MSPs cannot sustain OGs (Harding 2021). Observationally, we see that the SR spectra seem to extend to higher energy in MSPs (at least the energetic ones that have nonthermal emission). So the SR photons and the particles that produce them must be at higher energy. The Cherenkov Telescope Array is expected to have significantly better sensitivity than MAGIC in the 10–100 GeV range, and this and other pulsars will thus be prime targets for observation (Burtovoi et al. 2017). On the other hand, pulsars like J0218 are also good sources for MeV telescopes, such as the All-sky Medium Energy Gamma-ray Observatory (McEnery et al. 2019), that can detect the predicted SR peaks around 1–10 MeV.

The Fermi-LAT Collaboration acknowledges generous ongoing support from a number of agencies and institutes that have supported both the development and the operation of the LAT as well as scientific data analysis. These include the National Aeronautics and Space Administration and the Department of Energy in the United States, the Commissariat à l'Energie Atomique and the Centre National de la Recherche Scientifique/Institut National de Physique Nucléaire et de Physique des Particules in France, the Agenzia Spaziale Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), High Energy Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency (JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research Council and the Swedish National Space Board in Sweden. This research was partially carried out using the HKU Information Technology Services research computing facilities that are supported in part by the Hong Kong UGC Special Equipment Grant (SEG HKU09). P.S.P. and C.Y.N. are supported at HKU by a grant from the Big Data Project Fund (BDPF) and a GRF grant (Project 17307618) from the Hong Kong government. A.K.H. acknowledges resources supporting the theoretical modeling provided by the NASA High-End Computing Program through the NASA Center for Climate Simulation. D.F.T. acknowledges support from the Spanish grants PGC2018-095512-B-I00 and SGR2017-1383, as well as from the Chinese Academy of Sciences Presidential Fellowship Initiative 2021VMA0001. The Nançay Radio Observatory is operated by the Paris Observatory, associated with the French Centre National de la Recherche Scientifique (CNRS). We acknowledge financial support from the "Programme National Hautes Energies" (PNHE) of CNRS/INSU, France.

We would like to thank the Instituto de Astrofísica de Canarias for the excellent working conditions at the Observatorio del Roque de los Muchachos in La Palma. The financial support of the German BMBF, MPG and HGF; the Italian INFN and INAF; the Swiss National Fund SNF; the ERDF under the Spanish Ministerio de Ciencia e Innovación (MICINN) (FPA2017-87859-P, FPA2017-85668-P, FPA2017-82729-C6-5-R, FPA2017-90566-REDC, PID2019-104114RB-C31, PID2019-104114RB-C32, PID2019-105510GB-C31,PID2019-107847RB-C41, PID2019-107847RB-C42, PID2019-107988GB-C22); the Indian Department of Atomic Energy; the Japanese ICRR, the University of Tokyo, JSPS, and MEXT; the Bulgarian Ministry of Education and Science, National RI Roadmap Project DO1-268/16.12.2019 and the Academy of Finland grant No. 320045 is gratefully acknowledged. This work was also supported by the Spanish Centro de Excelencia "Severo Ochoa" SEV-2016-0588 and CEX2019-000920-S, and "María de Maeztu" CEX2019-000918-M, the Unidad de Excelencia "María de Maeztu" MDM-2015-0509-18-2 and the "la Caixa" Foundation (fellowship LCF/BQ/PI18/11630012) and by the CERCA program of the Generalitat de Catalunya; by the Croatian Science Foundation (HrZZ) Project IP-2016-06-9782 and the University of Rijeka Project 13.12.1.3.02; by the DFG Collaborative Research Centers SFB823/C4 and SFB876/C3; the Polish National Research Centre grant UMO-2016/22/M/ST9/00382; and by the Brazilian MCTIC, CNPq and FAPERJ. We thank Matthew Kerr and Philippe Bruel for their careful reading of the manuscript and useful comments and suggestions, which greatly improved the paper. We thank Slavko Bogdanov for providing the X-ray data points used in this work. Finally, we sincerely thank the journal referee for taking the time and effort to provide a detailed review of our paper, providing some very helpful feedback, which helped us improve the final version of the article.

Facilities: Fermi-LAT - , MAGIC - , ADS. -

Software: Fermipy (Wood et al. 2017), Minuit (James & Roos 1975), MARS (Zanin et al. 2013), R (R Core Team 2013), SOPIE (Schutte & Swanepoel 2016), Tempo2 (Hobbs et al. 2006).