THE FIRST FERMI LARGE AREA TELESCOPE CATALOG OF GAMMA-RAY PULSARS

We have conducted two distinct pulsation searches of Fermi-LAT data. One search uses the ephemerides of known pulsars, obtained from radio and X-ray observations. The other method searches for periodicity in the arrival times of gamma rays coming from the direction of neutron star candidates ("blind period searches"). Both search strategies have advantages. The former is sensitive to lower gamma-ray fluxes, and the comparison of phase-aligned pulse profiles at different wavelengths is a powerful diagnostic of beam geometry. The blind period search allows for the discovery of new pulsars with selection biases different from those of radio searches, such as, for example, favoring pulsars with a broader range of inclinations between the rotation and magnetic axes.

For each gamma-ray event (index i), the topocentric gamma-ray arrival time recorded by the LAT is transferred to times at the solar-system barycenter t_i by correcting for the position of Fermi in the solar-system frame. The rotation phase ϕ_i(t_i) of the neutron star is calculated from a timing model, such as a truncated Taylor series expansion,

$$\begin{equation} \phi _i(t_i) = \phi _0 + \sum _{j = 0}^{j = N} { {f_{j} \times (t_i - T_0)^{j+1}}\over {(j+1)!}}. \end{equation} \tag{ 1 }$$

Here, T₀ is the reference epoch of the pulsar ephemeris and ϕ₀ is the pulsar phase at t = T₀. The coefficients f_j are the rotation frequency derivatives of order j. The rotation period is P = 1/f₀. Different timing models, described in detail in Edwards et al. (2006), can take into account various physical effects. Most germane to the present work is accurate ϕ_i(t_i) computations, even in the presence of the rotational instabilities of the neutron star called "timing noise." "Phase-folding" a light curve, or pulse profile, means filling a histogram with the fractional part of the ϕ_i values. An ephemeris includes the pulsar coordinates necessary for barycentering, the f_j and T₀ values, and may include parameters describing the pulsar proper motion, glitch epochs, and more. The radio dispersion measure (DM) is used to extrapolate the radio pulse arrival time to infinite frequency, and uncertainties in the DM translate to an uncertainty in the phase offset between the radio and gamma-ray peaks.

2.1.1. Pulsars with Known Rotation Ephemerides

The ATNF database (version 1.36; Manchester et al. 2005) lists 1826 pulsars, and more have been discovered and await publication (Figure 1). The LAT observes them continuously during the all-sky survey. Phase-folding the gamma rays coming from the positions of all of these pulsars (consistent with the energy-dependent LAT PSF) requires only modest computational resources. However, the best candidates for gamma-ray emission are the pulsars with high $\dot{E}$, which often have substantial timing noise. Ephemerides accurate enough to allow phase-folding into, at a minimum, 25 bin phase histograms can degrade within days to months. The challenge is to have contemporaneous ephemerides.

We have obtained 762 contemporaneous pulsar ephemerides from radio observatories, and five from X-ray telescopes, in two distinct groups. The first consists of 218 pulsars with high spin-down power ($\dot{E}>10^{34}$ erg s⁻¹) timed regularly as part of a campaign by a consortium of astronomers for the Fermi mission, as described in Smith et al. (2008). With one exception (PSR J1124−5916, which is a faint radio source with large timing noise), all of the 218 targets of the campaign have been monitored since shortly before Fermi launch. Some results from the timing campaign can be found in Weltevrede et al. (2010a).

The second group is a sample of 544 pulsars from nearly the entire $P-\dot{P}$ plane (Figure 2) being timed for other purposes for which ephemerides were shared with the LAT team. These pulsars reduce possible bias of the LAT pulsar searches created by our focus on the high $\dot{E}$ sample requiring frequent monitoring. Gamma-ray pulsations from six radio pulsars with $\dot{E} < 10^{34}$ erg s⁻¹ were discovered in this manner, all of which are MSPs.

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Table 1 lists properties of the 46 detected gamma-ray pulsars. Five of the 46 pulsars, all MSPs, are in binary systems. The period first derivative $\dot{P}$ is corrected for the kinematic Shklovskii effect (Shklovskii 1970): $\dot{P}=\dot{P}_{\rm obs}-\mu ^2P_{\rm obs}d/c$, where μ is the pulsar proper motion and d is the distance. The correction is small except for a few MSPs (Abdo et al. 2009b). The characteristic age is $\tau _{\rm c} = P/2\dot{P}$ and the spin-down luminosity is

$$\begin{equation} \dot{E} = 4\pi ^2I\dot{P}P^{-3}, \end{equation} \tag{ 2 }$$

taking the neutron star's moment of inertia I to be 10⁴⁵g cm². The magnetic field at the light cylinder (radius R_LC = cP/2π) is

$$\begin{equation} B_{\rm LC} = \left(\frac{3I8\pi ^4\dot{P}}{c^3P^5} \right)^{1/2} \approx 2.94 \times 10^{8}(\dot{P}P^{-5})^{1/2}\, {\rm G}. \end{equation} \tag{ 3 }$$

Table 1. Measured and Intrinsic Parameters of LAT-detected Pulsars

PSR	Type,	l	b	P	$\dot{P}$	Age τc	$\dot{E}$	BLC	S1400
	Ref.	(°)	(°)	(ms)	(10−15)	(kyr)	(1034 erg s−1)	(kG)	(mJy)
J0007+7303	ga,b	119.7	10.5	316	361	14	45.2	3.1	1 < 0.1
J0030+0451	mc,d	113.1	−57.6	4.9	10 × 10−6	7.7 × 106	0.3	17.8	0.6
J0205+6449	re	130.7	3.1	65.7	194	5	2700	115.9	0.04
J0218+4232	mbd	139.5	−17.5	2.3	77 × 10−6	0.5 × 106	24	313.1	0.9
J0248+6021	rf	136.9	0.7	217	55.1	63	21	3.1	9
J0357+32	gb	162.7	−16.0	444	12.0	590	0.5	0.2	...
J0437−4715	mbd	253.4	−42.0	5.8	14 × 10−6	6.6 × 106	0.3	13.7	140
J0534+2200	rh	184.6	−5.8	33.1	423	1	46100	950.0	14
J0613−0200	mbd	210.4	−9.3	3.1	9 × 10−6	5.3 × 106	1.3	54.3	1.4
J0631+1036	ri	201.2	0.5	288	105	44	17.3	2.1	0.8
J0633+0632	gb	205.0	−1.0	297	79.5	59	11.9	1.7	2 < 0.2
J0633+1746	gh	195.1	4.3	237	11.0	340	3.3	1.1	<1
J0659+1414	ri	201.1	8.3	385	55.0	110	3.8	0.7	3.7
J0742−2822	ri	243.8	−2.4	167	16.8	160	14.3	3.3	15
J0751+1807	mbd	202.7	21.1	3.5	6 × 10−6	8.0 × 106	0.6	32.3	3.2
J0835−4510	rk	263.6	−2.8	89.3	124	11	688	43.4	1100
J1028−5819	rl	285.1	−0.5	91.4	16.1	90	83.2	14.6	0.36
J1048−5832	rm	287.4	0.6	124	96.3	20	201	16.8	6.5
J1057−5226	rn	286.0	6.6	197	5.8	540	3.0	1.3	11
J1124−5916	r	292.0	1.8	135	747	3	1190	37.3	0.08
J1418−6058	gb	313.3	0.1	111	170	10	495	29.4	2,3 < 0.06
J1420−6048	ri	313.5	0.2	68.2	83.2	13	1000	69.1	0.9
J1459−60	gb	317.9	−1.8	103	25.5	64	91.9	13.6	2 < 0.2
J1509−5850	ri	320.0	−0.6	88.9	9.2	150	51.5	11.8	0.15
J1614−2230	mbd	352.5	20.3	3.2	4 × 10−6	1.2 × 106	0.5	33.7	...
J1709−4429	rn	343.1	−2.7	102	93.0	18	341	26.4	7.3
J1718−3825	ri	349.0	−0.4	74.7	13.2	90	125	21.9	1.3
J1732−31	gb	356.2	0.9	197	26.1	120	13.6	2.7	2 < 0.2
J1741−2054	gb,t	6.4	4.6	414	16.9	390	0.9	0.3	2 0.16
J1744−1134	md	14.8	9.2	4.1	7 × 10−6	9 × 106	0.4	24.0	3
J1747−2958	ro	359.3	−0.8	98.8	61.3	26	251	23.5	0.08
J1809−2332	gb	7.4	−2.0	147	34.4	68	43.0	6.5	2,3 < 0.06
J1813−1246	gb	17.2	2.4	48.1	17.6	43	626	76.2	2 < 0.2
J1826−1256	gb	18.5	−0.4	110	121	14	358	25.2	2,3 < 0.06
J1833−1034	ro	21.5	−0.9	61.9	202	5	3370	137.3	0.07
J1836+5925	gb	88.9	25.0	173	1.5	1800	1.2	0.9	4 < 0.007
J1907+06	gb,r	40.2	−0.9	107	87.3	19	284	23.2	<0.02
J1952+3252	rn	68.8	2.8	39.5	5.8	110	374	71.6	1
J1958+2846	gb	65.9	−0.2	290	222	21	35.8	3.0	...
J2021+3651	rp	75.2	0.1	104	95.6	17	338	26.0	0.1
J2021+4026	gb,s	78.2	2.1	265	54.8	77	11.6	1.9	...
J2032+4127	gb,t	80.2	1.0	143	19.6	120	26.3	5.3	2  0.24
J2043+2740	rq	70.6	−9.2	96.1	1.3	1200	5.6	3.6	5  3
J2124−3358	md	10.9	−45.4	4.9	12 × 10−6	0.6 × 106	0.4	18.8	1.6
J2229+6114	rm	106.6	2.9	51.6	78.3	11	2250	134.5	0.25
J2238+59	gb	106.5	0.5	163	98.6	26	90.3	8.6	...

Notes. The first two columns are pulsar names and types: "r" for radio-selected, "g" for gamma-ray-selected, "m" for MSPs, and "b" for binary pulsars. The third and fourth columns are Galactic coordinates for each pulsar. The fifth and sixth columns list the period (P) and its first derivative ($\dot{P}$), corrected for the Shklovskii effect (see the text). Following are the characteristic age τ_c (Column 7), the spin-down luminosity $\dot{E}$ (Column 8), and the magnetic field at the light cylinder B_LC (Column 9). The last column is the radio flux density at 1400 MHz, or an upper limit when one is available. These values are taken from the ATNF database (Manchester et al. 2005) except for the noted entries where (1) Halpern et al. 2004; (2) Camilo et al. 2009; (3) Roberts et al. 2002; (4) Halpern et al. 2007; (5) Ray et al. 1996. Note that PSR J1509−5850 should not be confused with PSR B1509−58 observed by CGRO. References. Fermi-LAT publications specific to these pulsars: (a) Abdo et al. 2008; (b) Abdo et al. 2009c; (c) Abdo et al. 2009n; (d) Abdo et al. 2009b; (e) Abdo et al. 2009d; (f) I. Cognard et al. 2010, in preparation (PSRs J0248+6021 and J2240+5832); (h) Abdo et al. 2010b; (i) Weltevrede et al. 2010b; (j) Abdo et al. 2009a; (k) Abdo et al. 2009g; (l) Abdo et al. 2009e; (m) Abdo et al. 2009f; (n) A. A. Abdo et al. 2010, in preparation (Three EGRET Pulsars); (o) F. Camilo et al. 2010, in preparation (PSRs J1833−1034 and J1747−2958); (p) Abdo et al. 2009m; (q) A. Noutsos et al. 2010, in preparation (PSR J2043+2740); (r) Abdo et al. 2010a; (s) A. A. Abdo et al. 2010, in preparation (γ-Cygni); (t) Camilo et al. 2009.

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Table 2 lists which observatories provided ephemerides for the gamma-ray pulsars. An "L" indicates that the pulsar was timed using LAT gamma rays, as described in the following section.

Table 2. Pulsation Detection Significances for LAT-detected Pulsars

PSR	Z22 Value	H Value	MaxROI (°)	ObsID
J0007+7303	2072.1	2371.8	1.0	L
J0030+0451	121.1	362.7	1.0	N
J0205+6449	90.9	206.0	1.0	G, J
J0218+4232	24.7	22.5	1.0	N, W
J0248+6021	57.5	75.1	0.5	N
J0357+32	422.7	450.7	1.0	L
J0437−4715	126.9	153.6	1.0	P
J0534+2200	4397.8	15285.0	1.0	N, J
J0613−0200	93.6	139.9	1.0	N
J0631+1036	48.6	44.8	1.0	N, J
J0633+0632	230.2	573.3	1.0	L
J0633+1746	10053.6	20346.4	1.0	L
J0659+1414	80.5	99.0	1.0	N, J
J0742−2822	38.9	44.9	1.0	N, J
J0751+1807	29.7	26.5	1.0	N
J0835−4510	26903.9	74716.7	1.0	P
J1028−5819	291.5	915.9	0.5	P
J1048−5832	208.5	634.0	1.0	P
J1057−5226	1668.9	1772.4	1.0	P
J1124−5916	93.5	179.9	1.0	L
J1418−6058	230.1	343.7	1.0	L
J1420−6048	104.7	114.4	1.0	P
J1459−60	148.2	159.3	1.0	L
J1509−5850	71.6	73.3	0.5	P
J1614−2230	36.2	69.5	0.5	G
J1709−4429	4680.1	5612.1	1.0	P
J1718−3825	111.9	109.8	0.5	N, P
J1732−31	141.2	279.6	1.0	L
J1741−2054	332.6	355.9	1.0	L
J1744−1134	28.4	38.1	1.0	N
J1747−2958	47.2	69.0	0.5	G
J1809−2332	589.3	1562.5	1.0	L
J1813−1246	140.0	162.0	1.0	L
J1826−1256	442.4	979.0	1.0	L
J1833−1034	35.2	87.6	1.0	G
J1836+5925	349.2	385.3	1.0	L
J1907+06	257.1	521.0	1.0	L
J1952+3252	464.8	1008.8	1.0	J, N
J1958+2846	146.9	233.1	1.0	L
J2021+3651	1433.5	4603.7	1.0	G, A
J2021+4026	222.0	275.8	1.0	L
J2032+4127	224.9	485.9	0.5	L
J2043+2740	28.2	38.2	1.0	N, J
J2124−3358	77.8	80.9	1.0	N
J2229+6114	1026.0	1237.4	1.0	G, J
J2238+59	135.8	373.0	1.0	L

Notes. Columns 2 and 3 list the Z²₂ (Buccheri et al. 1983) and H (de Jager et al. 1989) periodicity test values for E>0.3 GeV, respectively. Detection of gamma-ray pulsations are claimed when the significance of the periodicity test exceeds 5σ, with the exceptions of J0218+4232, J0751+1807, J1744−1134, and J2043+2740 as described in Section 2.1.3. A significance greater than 5σ corresponds to Z²₂>36 and H>42; greater than 7σ corresponds to Z²₂>61; and greater than 10σ corresponds to Z²₂>114 (see Section 2.1.3). Column 4 gives the maximum angular radius (maxROI) around the pulsar position within which gamma-ray events were searched for pulsations. The final column indicates the observatories that provided ephemerides (see Section 2.1.1 for details): A, Arecibo telescope; G, Green Bank Telescope; J, Lovell telescope at Jodrell Bank; L, Large Area Telescope; N, Nançay Radio Telescope; P, Parkes Radio Telescope; W, Westerbork Synthesis Radio Telescope.

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"P" is the Parkes radio telescope (Manchester 2008; Weltevrede et al. 2010a). The majority of the Parkes observations were carried out at intervals of four to six weeks using the 20 cm Multibeam receiver (Staveley-Smith et al. 1996) with a 256 MHz band centered at 1369 MHz. At six-month intervals, observations were also made at frequencies near 0.7 and 3.1 GHz with bandwidths of 32 and 1024 MHz, respectively. The required frequency resolution to avoid dispersive smearing across the band was provided by a digital spectrometer system.

"N" is the Nançay radio telescope (Theureau et al. 2005). Nançay observations are carried out every three weeks on average. The recent version of the BON back-end is a GPU-based coherent dedispersor allowing the processing of a 128 MHz bandwidth over two complex polarizations (Cognard et al. 2009). The majority of the data are collected at 1398 MHz, while for MSPs in particular, observations are duplicated at 2048 MHz to allow DM monitoring.

"J" is the 76 m Lovell radio telescope at Jodrell Bank in the United Kingdom. Jodrell Bank observations (Hobbs et al. 2004) were carried out at typical intervals of between two and 10 days in a 64 MHz band centered on 1404 MHz, using an analog filterbank to provide the frequency resolution required to remove interstellar dispersive broadening. Occasionally, observations are also carried out in a band centered on 610 MHz to monitor the interstellar dispersion delay.

"G" is the 100 m NRAO Green Bank Telescope (GBT). PSR J1833–1034 was observed monthly at 0.8 GHz with a bandwidth of 48 MHz using the BCPM filter bank (Backer et al. 1997). The other pulsars monitored at the GBT were observed every two weeks at a frequency of 2 GHz across a 600 MHz band with the Spigot spectrometer (Kaplan et al. 2005). Individual integration times ranged between 5 minutes and 1 hr.

Arecibo ("A") observations of the very faint PSRs J1930+1852 and J2021+3651 are carried out every two weeks, with the L-wide receiver (1100–1730 MHz). The back-ends used are the Wideband Arecibo Pulsar Processor (WAPP) correlators (Dowd et al. 2000), each with a 100 MHz wide band. The antenna voltages are three-level digitized and then autocorrelated with a total of 512 lags, accumulated every 128 μs, and written to disk as 16-bit sums. Processing includes Fourier transforms to obtain power spectra, which are then dedispersed and phase-folded. The average pulse profiles are cross-correlated with a low-noise template profile to obtain topocentric times of arrival.

"W" is the Westerbork Synthesis Radio Telescope, with which observations were made approximately monthly at central frequencies of 328, 382, and 1380 MHz with bandwidths of 10 MHz at the lower frequencies and 80 MHz at the higher frequencies. The PuMa pulsar back-end (Vote et al. 2002) was used to record all the observations. Folding and dedispersion were performed offline.

The rms of the radio timing residuals for most of the solutions used in this paper is <0.5% of a rotation period, but ranges as high as 1.2% for five pulsars. This is adequate for the 50- or 25-bin phase histograms used in this paper. The ephemerides used for this catalog will be available on the Fermi Science Support Center data servers.⁷¹

2.1.2. Pulsars Discovered in Blind Periodicity Searches

2.1.3. Light Curves

The pulsar spectra were fitted with an exponentially cutoff power-law model of the form

$$\begin{equation} \frac{d N}{d E} = K E_{\rm GeV}^{-\Gamma } \exp \left(- \frac{E}{E_{\rm cutoff}} \right), \end{equation} \tag{ 4 }$$

in which the three parameters are the photon index at low energy Γ, the cutoff energy E_cutoff, and a normalization factor K (in units of ph cm⁻² s⁻¹ MeV⁻¹), in keeping with the observed spectral shape of bright pulsars (Abdo et al. 2009g). The energy at which the normalization factor K is defined is arbitrary. We chose 1 GeV because it is, for most pulsars, close to the energy at which the relative uncertainty on the differential flux is minimal.

We wish to extract the spectra down to 100 MeV in order to constrain the power-law part of the spectrum, and to measure the flux above 100 MeV directly. Because the spatial resolution of Fermi is not very good at low energies (∼5° at 100 MeV), we need to account for all neighboring sources and the diffuse emission together with each pulsar. This was done using the framework used for the LAT Bright Source List (Abdo et al. 2009i). A six-month source list was generated in the same way as the three-month source list described in Abdo et al. (2009i), but covering the extended period of time used for the pulsar analysis. We have added the source Cyg X-3 (Abdo et al. 2009l), although it was not detected automatically as a separate source, because it is very close to PSR J2032+4127, and impacts the spectral fit of the pulsar. Cyg X-3 was fit with a simple power law as were all other non-pulsar sources in the list.

We used a Galactic diffuse model designated 54_77Xvarh7S calculated using GALPROP,⁷³ an evolution of that used in Abdo et al. (2009i). A similar model, gll_iem_v02, is publicly available (see footnote ⁷¹).

We kept events with E>100 MeV belonging to the diffuse event class, which has the tightest cosmic-ray background rejection (Atwood et al. 2009). To avoid contamination by gamma rays produced by cosmic-ray interactions in the Earth's atmosphere, we select time intervals when the entire ROI, of radius 10° around the source, has a zenith angle <105°. We extracted events in a circle of radius 10° around each pulsar, and included all sources up to 17° into the model (sources outside the extraction region can contribute at low energy). Sources further away than 3° from the pulsar were assigned fixed spectra, taken from the all-sky analysis. Spectral parameters for the pulsar and sources within 3° of it were left free for the analysis.

The fit was performed by maximizing unbinned likelihood (direction and energy of each event is considered) as described in Abdo et al. (2009i) and using the minuit fitting engine.⁷⁴ The uncertainties on the parameters were estimated from the quadratic development of the log(likelihood) surface around the best fit. In addition to the index Γ and the cutoff energy E_cutoff which are explicit parameters of the fit, the important physical quantities are the photon flux F₁₀₀ (in units of ph cm⁻² s⁻¹) and the energy flux G₁₀₀ (in units of erg cm⁻² s⁻¹),

$$\begin{eqnarray} &&F_{100} = \int _{\rm 100 \, MeV}^{\rm 100 \, GeV} \frac{d N}{d E} d E,\\ &&G_{100} = \int _{\rm 100 \, MeV}^{\rm 100 \, GeV} E \frac{d N}{d E} d E. \end{eqnarray} \tag{ 5 }$$

These derived quantities are obtained from the primary fit parameters. Their statistical uncertainties are obtained using their derivatives with respect to the primary parameters and the covariance matrix obtained from the fitting process.

For a number of pulsars, an exponentially cutoff power-law spectral model is not significantly better than a simple power law. We identified these by computing TS_cutoff = 2Δlog(likelihood) (comparable to a χ² distribution with one degree of freedom) between the models with and without the cutoff. Pulsars with TS_cutoff < 10 have poorly measured cutoff energies. TS_cutoff is reported in Table 4.

The above analysis yields a fit to the overall spectrum, including both the pulsar and any unpulsed emission, such as from a PWN. To do better we split the data into on-pulse and off-pulse samples and modeled the off-pulse spectrum by a simple power law. The off-pulse window used for this background estimation is defined in the last column of Table 3.

In a second step, we re-fitted the on-pulse emission to the exponentially cutoff power law as before, with the off-pulse emission (scaled to the on-pulse phase interval) added to the model and fixed to the off-pulse result. In many cases, the off-pulse emission was not significant at the 5σ or even 3σ level, but we kept the formal best fit anyway, in order to not bias the pulsed emission upwards. The results summarized in Table 4 come from this on-pulse analysis.

Using an off-pulse pure power law is not ideal for the Crab or any other PWN with synchrotron and inverse Compton components within the Fermi energy range. Judging from the Crab pulsar, using a simple power law to model the off-pulse emission mainly affects the value of the cutoff energy. The analysis specific to the Crab, with a model adapted to the pulsar synchrotron component low energies and to the high-energy nebular component, yields E_cutoff ∼ 6 GeV (Abdo et al. 2010b). This is the value listed in Table 4. The cutoff value obtained with the simplified model applied to most pulsars in this paper is higher (> 10 GeV). The photon and energy fluxes given by the two analyses are within 10% of each other. Additional exceptions in Table 4 are for PSRs J1836+5925 and J2021+4026. The off-pulse phase definition for these pulsars is unclear, so the spectral parameters reported in the table are from the initial, phase-averaged spectral analysis.

We have checked whether our imperfect knowledge of the Galactic diffuse emission may impact the pulsar parameters by applying the same analysis with a different diffuse model, as was done in Abdo et al. (2009i). The phase-averaged emission is affected. Seven (relatively faint) pulsars see their flux move up or down by more than a factor of 1.5. On the other hand, the pulsed flux is much more robust, because the off-pulse component absorbs part of the background difference, and the source-to-background ratio is better after on-pulse phase selection. Only two pulsars see their pulsed flux move up or down by more than a factor of 1.2, and none shift by more than a factor of 1.4 when changing the diffuse model. Overall, the systematic uncertainties due to the diffuse model on the fluxes F₁₀₀ and G₁₀₀, on the photon index Γ, and on E_cutoff scale with the statistical uncertainty. Adding the statistical and systematic errors in quadrature amounts, to a good approximation, to multiplying the statistical errors on F₁₀₀, G₁₀₀, and Γ by 1.2. The uncertainties listed in Table 4 and plotted in the figures include this correction. The increase in the uncertainty on E_cutoff due to the diffuse model is <5% and is neglected.

Systematic uncertainties on the LAT effective area are of order 5% near 1 GeV, 10% below 0.1 GeV, and 20% above 10 GeV. To propagate their effect on the spectral parameters in Equation (4), we modify the instrument response functions to bracket the nominal values, and repeat the likelihood calculations. This is reported in detail for most of the individual LAT pulsars already referenced. The bias values reported in Abdo et al. (2009b) well describe our current knowledge of the effect of the uncertainties in the instrument response functions on the spectral parameters: they are δΓ = (+0.3, − 0.1), δE_cutoff = (+20%, − 10%), δF₁₀₀ = (+30%, − 10%), and δG₁₀₀ = (+20%, − 10%). The bias on the integral energy flux is somewhat less than that of the integral photon flux, due to the weighting by photons in the energy range where the effective area uncertainties are smallest. We do not sum these uncertainties in quadrature with the others, since a change in instrument response will tend to shift all spectral parameters similarly.

The pulsar spectra were also evaluated using an unfolding method (D'Agostini 1995; Mazziotta 2009) that takes into account the energy dispersion introduced by the instrument response function and does not assume any model for the spectral shapes. "Unfolding" is essentially a deconvolution of the observed data from the instrument response functions. For each pulsar we selected photons within 68% of the PSF with a minimum radius of 035 and a maximum of 5°.

The observed pulsed spectrum was built by selecting the events in the on-pulse phase interval and subtracting the events in the off-pulse interval, properly scaled for the phase ratio. The instrument response function, expressed as a smearing matrix, was evaluated using the LAT Geant4-based⁷⁵ Monte Carlo simulation package called Gleam (Boinee et al. 2003), taking into account the pointing history of the source.

The true pulsar energy spectra were then reconstructed from the observed ones using an iterative procedure based on Bayes' theorem (Mazziotta 2009). Typically, convergence is reached after a few iterations. When the procedure has converged, both statistical and systematic errors on the observed energy distribution can be easily propagated to the unfolded spectra. The results obtained from the unfolding analysis were consistent within errors with the likelihood analysis results.

Converting measured pulsar fluxes to radiated power requires reliable distance estimates. Annual trigonometric parallax measurements are the most reliable, but are generally only available for a few relatively nearby pulsars.

The most commonly used technique to obtain radio pulsar distances exploits the pulse delay as a function of wavelength by free electrons along the path to Earth. A distance can be computed from the DM coupled to an electron density distribution model. We use the NE2001 model (Cordes & Lazio 2002) unless noted otherwise. It assumes uniform electron densities in and between the Galactic spiral arms, with smooth transitions between zones, and spheres of greater density for specific regions such as the Gum nebula, or surrounding Vela. Specific lines of sight can traverse unmodeled regions of over- (or under-) density, as, for example, along the tangents of the spiral arms, causing significant discrepancies between the true pulsar distances and those inferred from the electron-column density.

A third method, kinematic, associates the pulsar with objects whose distance can be measured from the Doppler shift of absorption or emission lines in the neutral hydrogen (H i) spectrum, together with a rotation curve of the Galaxy. It breaks down where the velocity gradients are very small or where the distance–velocity relation has double values. The associations are often uncertain, and these distance measurements can be controversial.

In a small number of cases, the distance is evaluated either from X-ray measurements of the absorbing column at low energies (below 1 keV), or from consideration of the X-ray flux assuming some standard parameters for the neutron star.

Table 5 presents the best-known distances of 37 pulsars detected by Fermi, the methods used to obtain them, and the references. For distances obtained from the NE2001 model and the DM, the reference indicates the DM measurement. We assume a minimum DM distance uncertainty of 30%. When distances from different methods disagree and no method is more convincing than the other, a range is given, and 30% uncertainties on the upper and lower values are used. For the remaining nine Fermi-discovered pulsars no distance estimates have been established so far. Here follow comments for some of the distance values reported in Table 5.

PSR J0205+6449. The pulsar is in the PWN 3C 58. NE2001 gives 4.5 kpc for DM=141 cm⁻³ pc in this direction (Camilo et al. 2002d). Using H i absorption and emission lines from the PWN yields from 2.6 kpc (Green & Gull 1982) to 3.2 kpc (Roberts et al. 1993). The lower V-band reddening (Fesen et al. 1988, 2008) compared to the Galactic-disk edge (Schlegel et al. 1998) suggests that the PWN is in the range 3–4 kpc. Table 5 quotes the distance range found by Green & Gull (1982) and Roberts et al. (1993).

PSR J0218+4232. The DM measurements from Navarro et al. (1995) together with NE2001 yield 2.7 ± 0.8 kpc. Comparing the pulsar characteristic age with the cooling models of its white-dwarf companion gives a distance range of 2.5–4 kpc (Bassa et al. 2003).

PSR J0248+6021. The DM of 376 cm⁻³ pc (I. Cognard et al. 2010, in preparation (PSRs J0248+6021 and J2240+5832)) puts this pulsar beyond the edge of the Galaxy for this line of sight. The line of sight, however, borders the giant H ii region W5 in the Perseus Arm. We bracket the pulsar distance as being between W5 (2 kpc) and the Galaxy edge (9 kpc).

PSR J0631+1036. The DM =125.3 cm⁻³ pc (Zepka et al. 1996) is large for a source in the direction of the Galactic anticenter. The dark cloud LDN 1605, part of the active star-forming region 3 Mon, is in the line of sight. The pulsar could be inside the cloud, at ∼0.75 kpc. Ionized material in the cloud could cause NE2001 to overestimate the distance.

PSR J1124−5916. It lies toward the Carina arm where NE2001 biases are acute. The DM distance is 5.7 kpc (Camilo et al. 2002c). The kinematic distance of the associated SNR (G292.0+1.8) indicates a lower limit of 6.2 ± 0.9 kpc (Gaensler & Wallace 2003). The value in Table 5 is derived by Gonzalez & Safi-Harb (2003) linking the X-ray absorption column with the extinction along the pulsar direction.

PSR J1418−6058. This pulsar is likely associated with the PWN G313.3+0.1, near the Kookaburra complex. A nearby H ii region is at 13.4 kpc (Caswell & Haynes 1987) but could easily be in the background. Such a large distance implies an unreasonably large gamma-ray efficiency. Table 5 lists a crude estimate of the distance range with the lower limit (Yadigaroglu & Romani 1997) taking the pulsar to be related to one of the near objects (Clust 3, Cl Lunga 2 or SNR G312.4−0.4), and the higher limit (Ng et al. 2005) determined by applying the relation found by Possenti et al. (2002) and the correlation between pulsar X-ray photon index and luminosity given by Gotthelf (2003).

PSR J1709−4429. The NE2001 DM distance is 2.3 ± 0.7 kpc (Taylor et al. 1993). Kinematic distances give upper and lower limits of 3.2 ± 0.4 kpc and 2.4 ± 0.6 kpc, respectively (Koribalski et al. 1995). The X-ray flux from the neutron star detected by Chandra (Romani et al. 2005) and XMM-Newton (McGowan et al. 2004) is compatible with a distance of 1.4–2.0 kpc. We assume the range 1.4–3.6 kpc.

PSR J1747−2958. The pulsar is associated with the PWN G359.23−0.82. H i measurements yield a distance upper limit of 5.5 kpc (Uchida et al. 1992), but the DM (101 pc cm⁻³) suggests 2.0 ± 0.6 kpc (Camilo et al. 2002b). The X-ray absorbing column detected by Chandra is between 4 and 5 kpc, while the closer value of 2 kpc would imply that an otherwise unknown molecular cloud lies in front of the pulsar (Gaensler et al. 2004). A range of 2–5 kpc is used in our analysis.

PSR J2021+3651. The DM distance of ∼12 kpc implies a high gamma-ray conversion efficiency (Roberts et al. 2002; Abdo et al. 2009m). The open cluster Berkeley 87 near the line of sight could be responsible for an electron column density higher than modeled by NE2001. The distance in Table 5 comes from a Chandra X-ray observation of the pulsar and its surrounding nebula (Van Etten et al. 2008). A similar range (1.3–4.1 kpc) was obtained for the X-ray flux detected from the associated PWN.

PSR J2032+4127. The DM value (115 pc cm⁻³) gives an NE2001 distance of 3.6 kpc. If the pulsar belongs to the star cluster Cyg OB2, it would be located at approximately 1.6 kpc (Camilo et al. 2009). In this text we use a range of 1.6–3.6 kpc for this source.

PSR J2229+6114. The distance derived from the X-ray absorption is ∼3 kpc (Halpern et al. 2001b), between the values from the DM (6.5 kpc; Halpern et al. 2001a) and from the kinematic method (0.8 kpc; Kothes et al. 2001).

Figure 3 shows a polar view of the distribution of known pulsars over the Galactic plane. When two different distances are listed, we plot the closer one.

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Figure 1 shows the pulsars projected on the sky. A Gaussian fit to the Galactic latitude distribution for those with |b| < 10° and having distance estimates yields a standard deviation of σ_b = 3.5 ± 0.8 degrees. The distances range from d = 0.25 to 5.6 kpc, and we can use the most distant to place an upper limit on the scale height, obtaining h < dsin 35 = 340 pc, close to the typical scale height for all radio pulsars (the value used in Lyne et al. 1998 is 450 pc).

The light-curve peak separations Δ and the radio lags δ from Table 3 are summarized in Figure 4. As we will discuss in Section 5, outer magnetosphere emission models predict correlations between these parameters. Figure 5 shows B_LC versus the characteristic age (τ_c). The magnetic fields at the light cylinder for the detected MSPs are comparable to those of the other gamma-ray pulsars, suggesting that the emission mechanism for the two families may be similar.

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Table 4 lists L_γ and η, while Figure 6 plots L_γ versus $\dot{E}$. The dashed line indicates $L_\gamma =\dot{E}$, while the dot-dashed line indicates $L_\gamma \propto \dot{E}^{1/2}$, where

$$\begin{equation} L_\gamma \equiv 4\pi d^2 f_\Omega G_{100}. \end{equation} \tag{ 7 }$$

The flux correction factor f_Ω (Watters et al. 2009) is model-dependent and depends on the magnetic inclination and observer angles α and ζ. Both the outer gap and slot gap models predict f_Ω ∼ 1, in contrast to earlier use of f_Ω = 1/4π ≈ 0.08 (in, e.g., Thompson et al. 1994), or f_Ω = 0.5 for MSPs (in, e.g., Fierro et al. 1995). For simplicity, we use f_Ω = 1 throughout the paper, which presumably induces an artificial spread in the quoted L_γ values. However, it is the quadratic distance dependence for L_γ that dominates the uncertainty in L_γ in nearly all cases.

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Gamma rays dominate the total power L_tot radiated by most known high-energy pulsars, that is, L_tot ≈ L_γ. The Crab is a notable exception, with X-ray luminosity L_X ∼ 10L_γ. In Figure 6, we plot both L_γ and L_X + L_γ. L_X for E < 100 MeV is taken from Figure 9 of Kuiper et al. (2001).

Table 6 provides some alternate names and positional associations of the pulsars in this catalog with other astrophysical sources. For the EGRET 3EG, EGR, and GEV and AGILE AGL catalogs, the uncertainties in the localization of the counterparts is worse than for the LAT sources. In these cases, we consider a source is a possible counterpart to a LAT pulsar when the separation between the two positions is less than the quadratic sum of their 95% confidence error radii.

We see that 25 of the 46 pulsars are associated with sources in the 3EG, EGR, and GEV catalogs of EGRET sources, though 19 were seen only as unidentified unpulsed sources. A number of these unidentified EGRET sources had previously been associated with SNRs, PWNe, or other objects (e.g., Walker et al. 2003; De Becker et al. 2005). In all cases, the gamma-ray emission seen with the LAT is dominated by the pulsed emission. Of the 25 EGRET sources, 14 are gamma-ray-selected pulsars and 11 are radio-selected, including two MSPs. All six high-confidence EGRET pulsars (Nolan et al. 1996) are detected, and the three marginal EGRET detections are confirmed as pulsars (Ramanamurthy et al. 1996; Kaspi et al. 2000; Kuiper et al. 2000). The 21 sources without 3EG, EGR, or GEV counterparts include 18 previously detected radio pulsars (six of which are MSPs) and three gamma-ray-selected pulsars.

Not surprisingly, many of the young pulsars have SNR or PWN associations. At least 19 of the 46 pulsars are associated with a PWN and/or SNR (Roberts et al. 2005; Green 2009). We do not test here whether the gamma-ray flux from any of these pulsars includes a non-magnetospheric component, as might be indicated by spatially extended emission or a spectrum at pulse minimum not characteristic of a pulsar. Such studies are underway.

At least 12 of the pulsars are associated with TeV sources, nine of which are also associated with PWNe. Those pulsars with both TeV and PWN associations are typically young, with ages less than 20 kyr.

A widely cited predictor of gamma-ray pulsar detectability is the spin-down flux at Earth ${\dot{E}}/d^2$ (see, e.g., Smith et al. 2008). However, as argued by Arons 2006 (see also Harding & Muslimov 2002), it is natural in many models for the gamma-ray emitting gap to maintain a fixed voltage drop. This implies that L_γ is simply proportional to the particle current (Harding 1981), which gives $L_\gamma \propto {\dot{E}}^{1/2}$, i.e., gamma-ray efficiency increases with decreasing spin-down power down to ${\dot{E}} \sim 10^{34}\hbox{--} 10^{35}$ erg s⁻¹ where the gap saturates at large efficiency. In Figure 12, we show how our detected pulsars rank in ${\dot{E}}^{1/2}/d^2$ against the set of searched pulsars. We see that for both MSPs and young pulsars, the detected objects have among the largest values of this metric. The presence of missing objects among the detected pulsars is interesting, but must be treated with caution, as the detectability metric may be inflated by poor DM distances, or the sensitivity of the pulse search might be anomalously low due to high local background or unfavorable pulsar spectrum or pulse profile. Alternatively, some missing objects may be truly gamma-ray faint for the Earth line of sight. A more complete study of the implications of the pulsar non-detections and upper limits is in progress.

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To study the luminosity evolution in the observed pulsar population, we plot in Figure 6 our present best estimate of L_γ against ${\dot{E}}$, based on the pulsed flux measured for each pulsar. Two important caveats must be emphasized here. First, the inferred luminosities are quadratically sensitive to the often large distance uncertainties. Indeed, for many radio-selected pulsars (green points) we have only DM-based distance estimates. For many gamma-ray-selected pulsars we have only rather tenuous SNR or birth cluster associations with rough distance bounds. Only a handful of pulsars have secure parallax-based distances. Second, we have assumed here uniform phase-averaged beaming across the sky (f_Ω = 1). This is not realized for many emission models, especially for low ${\dot{E}}$ pulsars (Watters et al. 2009).

To guide the eye, Figure 6 shows lines for 100% conversion efficiency ($L_\gamma = {\dot{E}}$) and a heuristic constant voltage line L_γ = (10³³ erg s$^{-1} {\dot{E}})^{1/2}$. In view of the large luminosity uncertainties, we must conclude that it is not yet possible to test the details of the luminosity evolution. However, some trends are apparent and individual objects highlight possible complicating factors. For the highest ${\dot{E}}$ pulsars, there does seem to be rough agreement with the ${\dot{E}}^{1/2}$ trend. However, large variance between different distance estimates for the Vela-like PSRs J2021+3651 and J1709−4429 complicate the interpretation. In the range 10³⁵ erg s$^{-1} < {\dot{E}} < 10^{36.5}$ erg s⁻¹, the L_γ seems nearly constant, although the lack of precise distance measurements limits our ability to draw conclusions. For example, the very large nominal DM distance of PSR J0248+6021 would require >100% efficiency, and so is unlikely to be correct. Two other pulsars with apparent high efficiency (J1836+5925 and J2021+4026) are plotted including relatively bright unpulsed emission; this may be magnetospheric, but may also be a surrounding or nearby source. The association distances for the gamma-ray-selected pulsars must additionally be treated with caution. For example, PSR J2021+4026 has a τ_c ∼ 10× larger than the age of the putative associated SNR γ Cygni. Improved distance estimates in this range are the key to probing luminosity evolution.

From 10³⁴ erg s$^{-1} < {\dot{E}} < 10^{35}$ erg s⁻¹ we have several nearby pulsars with reasonably accurate parallax distance estimates. However, we see a wide range of gamma-ray efficiencies. This is the range over which, for both slot gap and outer gap models, the gap is expected to "saturate" and use most of the available potential to maintain the pair cascade. In slot gap models (Muslimov & Harding 2003), the break occurs at about 10³⁵ erg s⁻¹, when the gap is limited by screening of the accelerating field by pairs. The efficiency below this saturation is predicted to be ∼ 10%. In outer gap models (Zhang et al. 2004), the break is predicted to occur at somewhat lower $\dot{E} \sim 10^{34}$ erg s⁻¹. With the present statistics and uncertainties, it is not possible to discriminate between these model predictions except to note that both are consistent with the observed results. In some models, the gap saturation dramatically affects the shape of the beam on the sky and accordingly the flux conversion factor f_Ω; for outer gap models Watters et al. (2009) estimate f_Ω ∼ 0.1–0.15 for Geminga (similar values are obtained for J1836+5925), driving down the rather high inferred luminosity of these pulsars by an order of magnitude. In contrast, another pulsar with an accurate parallax distance, PSR J0659+1414, has an inferred luminosity 30× lower than the ${\dot{E}}^{1/2}$ prediction. Clearly, some parameter in addition to ${\dot{E}}$ controls the observed L_γ. Finally, for <10³⁴ erg s⁻¹ the sample is dominated by the MSPs. These nearby, low-luminosity objects clearly lie below the ${\dot{E}}^{1/2}$ trend, and in fact seem more consistent with $L_\gamma \propto {\dot{E}}$.

Upper limits on radio pulsars with high values of the spin-down flux at Earth or large ${\dot{E}}^{1/2}/d^2$ can help constrain viable efficiency models. In practice, the modest present exposure, the large background in the Galactic plane, and the need to rely on uncertain dispersion-based distance estimates limit the value of such constraints. Still, a few pulsars are already interesting; for example, using the DM-based distance, the sensitivity in Figure 9, and an assumed f_Ω = 1, we find that PSR J1740+1000 shows less than 1/5 of the flux expected from the ${\dot{E}}^{1/2}$ (constant voltage) line in Figure 6. Similarly, PSRs J1357−6429 and J1930+1852 have upper limits just below the expected fluxes. Further, some detected pulsars, e.g., PSRs J0659+1414 and J0205+6449, lie significantly below the constant voltage trend. We expect that as LAT exposure and the significance of such limits increase, we should obtain additional constraints on the factors controlling pulsar detectability.

One likely candidate for the additional factor affecting gamma-ray detectability is beaming. For PSR J1930+1852 (Camilo et al. 2002a), X-ray torus fitting (Ng & Romani 2008) suggests a small viewing angle |ζ| ∼ 33°. In outer gap models this makes it highly unlikely that the pulsar will produce strong emission on the Earth line of sight. Similarly it has been argued that PSR J0659+1414 has a small viewing angle ζ < 20° (Everett & Weisberg 2001; but see Weltevrede & Wright 2009 for a discussion of uncertainties). Again, strong emission from above the null charge surface is not expected for this ζ. One possible interpretation is that we are seeing slot gap or even polar cap emission from this pulsar, which is expected at this ζ. The unusual pulse profile and spectrum of this pulsar may allow us to test this idea of alternate emission zones.

In discussing non-detections, we should also note that the only binary pulsar systems reported in this paper are the radio-timed MSPs. In particular, our blind searches are not, as yet, sensitive to pulsars that are undergoing strong acceleration in binary systems. However, we do expect such objects to exist. Population syntheses (Pfahl et al. 2002) suggest that several percent of the young pulsars are born while retained in massive star binary systems. A few such systems are known in the radio pulsar sample (e.g., the TeV-detected PSR B1259−63); we expect that with the gamma-ray signal immune to dispersion effects an appreciable number of pulsar massive-star binaries will eventually be discovered. Indeed, it is entirely possible that the bright gamma-ray binaries LSI +61° 303 (Abdo et al. 2009h) and LS 5039 (Abdo et al. 2009j) may host pulsed GeV signals that have not yet been found.

With the above caveats about missing binary systems in mind, we can already draw some conclusions about the single gamma-ray pulsar population. For example, there are 17 gamma-ray-selected pulsars with a faintest flux of ∼6 × 10⁻⁸ cm^-2 s⁻¹; there are 16 non-millisecond radio-selected pulsars to this flux limit. Of course, some gamma-ray-selected objects can indeed be detected in the radio (Camilo et al. 2009). Indeed, the detection of PSR J1741−2054 at L_1.4 GHz ≈ 0.03 mJy kpc² underlines the fact that the radio emission can be very faint. Deep searches for additional radio counterparts are underway. However, with deep radio observations of several objects, e.g., Geminga, PSR J0007+7303, PSR J1836+5925 (Kassim & Lazio 1999; Halpern et al. 2004, 2007), providing no convincing detections, it is clear that some objects are truly radio faint. The substantial number of radio faint objects suggests that gamma-ray emission has an appreciably larger extent than the radio beams, such as expected in the outer gap (OG) and slot-gap/two pole caustic (SG/TPC) models.

Population synthesis studies for normal (non-millisecond) pulsars predicted that LAT would detect from 40 to 80 radio-loud pulsars and comparable numbers of radio-quiet pulsars in the first year (Gonthier et al. 2004; Zhang et al. 2007). The ratio of radio-selected to gamma-ray-selected gamma-ray pulsars has been noted as a particularly sensitive discriminator of models, since the outer magnetosphere models predict much smaller ratios than polar cap models (Harding et al. 2007). Studies of the MSP population (Story et al. 2007) predicted that LAT would detect around 12 radio-selected and 33–40 gamma-ray-selected MSPs in the first year, in rough agreement with the number of radio-selected MSPs seen to date (searches for gamma-ray selected MSPs have not yet been conducted). Thus, in the first six months the numbers of LAT pulsar detections are consistent with the predicted range, and the large number of gamma-ray-selected pulsars discovered so early in the mission points toward the outer magnetosphere models.

We can in fact use our sample of detected gamma-ray pulsars to estimate the Galactic birthrates. For each object with an available distance estimate, we compute the maximum distance for detection from D_max = D_est(F_γ/F_min)^1/2, where D_est comes from Table 5, the photon flux F₁₀₀ from Table 4, and F_min from Figure 9. We limit D_max to 15 kpc, and compare V, the volume enclosed within the estimated source distance, to V_max, that enclosed within the maximum distance, for a Galactic disk with radius 10 kpc and thickness 1 kpc. If we assume a blind search threshold 2× higher than that for a folding search at a given sky position, the inferred values of 〈V/V_max〉 are 0.49, 0.59, and 0.55 for the radio-selected young pulsars, MSPs, and gamma-ray-selected pulsars, respectively. These are close to 0.5, the value expected for a population uniformly filling a given volume (Schmidt 1968); the MSP value is somewhat high as our sample includes three objects detected at <5σ. The value for the gamma-ray-selected pulsars is also high but is controlled by PSR J2032+4127. If we exclude this object from the sample, we get 〈V/V_max〉 = 0.5 at an effective threshold of 3× the ephemeris-folding value.

Although we do not attempt a full population synthesis here, if we assume that the pulsar characteristic ages are the true ages, our sample can give rough estimates for local volume birthrates: 8 × 10⁻⁵ kpc⁻³ yr⁻¹ (young radio-selected), 4 × 10⁻⁵ kpc⁻³ yr⁻¹ (young gamma-ray-selected, 2× threshold), and 2 × 10⁻⁸ kpc⁻³ yr⁻¹ (MSP). Note that only half of the gamma-ray-selected objects have distance estimates. If we assume that the set without distance information has comparable luminosity, the gamma-ray-selected birthrate is ∼2× larger. These estimates retain appreciable uncertainty; for example, if the effective blind search detection threshold is 3× that for folding, the inferred gamma-ray-selected birthrate increases by an additional ∼65%. If we extrapolate these local birthrates to a full disk with an effective radius of 10 kpc, we get 1/100 yr (radio-selected young pulsars), 1/100 yr (gamma-ray-selected pulsars) and 1/(6 × 10⁵ yr) (radio-selected MSPs). Normally in estimating radio pulsar birthrates one would correct for the radio beaming fraction. However, if young gamma-ray-selected pulsars are simply similar objects viewed outside of the radio beam, this would result in double-counting. In any case, one infers a total Galactic birthrate for energetic pulsars of ∼1/50 yr, with gamma-ray-selected objects representing half. This represents a large fraction of the estimated Galactic supernova rate, so clearly more careful population syntheses will be needed to see if these numbers are compatible.

The pulse-shape properties can also help us to probe the geometry and physics of the emission region. The great majority of the pulsars show two dominant, relatively sharp peaks, suggesting that we are seeing caustics from the edge of a hollow cone. When a single peak is seen, it tends to be broader, suggesting a tangential cut through an emission cone. This picture is realized in the OG and the high-altitude portion of the SG models.

For the radio-emitting pulsars, we can compare the phase lag between the radio and first gamma-ray peak δ with the separation of the two gamma-ray peaks Δ. As first pointed out in Romani & Yadigaroglu (1995), these should be correlated in outer magnetosphere models—this is indeed seen (Figure 4). The distribution can be compared with predictions of the TPC and OG models shown in Watters et al. (2009). The δ–Δ distribution and in particular the presence of Δ ∼ 0.2–0.3 values appear to favor the OG picture. However, there are a greater number having Δ ∼ 0.4–0.5, which favors TPC models. In Figure 13, we show the peak separation as a function of pulsar spin-down luminosity—the Δ distribution appears to be bimodal, with no strong dependence on pulsar $\dot{E}$ (or age). A full comparison will require detailed population models, which are being created. It may also be hoped that the precise distribution of measured values can help probe details of the emission geometry. In particular, whenever we have external constraints on the viewing angle ζ (typically from X-ray images of the PWN) or magnetic inclination α (occasionally measured from radio polarization), then the observed values of δ and Δ become a powerful probe of the precise location of the emission sheet within the magnetosphere. This can be sensitive to the field perturbations from magnetospheric currents and hence can probe the global electrodynamics of the pulsar magnetosphere.

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If one examines the energy dependence of the light curves of both the radio-selected and gamma-ray-selected pulsars, a decrease in the P1/P2 ratio with increasing energy seems to be a common feature. However, the P1/P2 ratio evolution does not occur for all pulsars, notably J0633+0632, J1028−5819, J1124−5916, J1813−1246, J1826−1256, J1836+5925, J2021+3651, and J2238+59. Most of these pulsars have two peaks with phase separation of ∼0.5 and little or no bridge emission between the peaks. Perhaps the lack of P1/P2 energy evolution is connected with an overall symmetry of the light curve.

The LAT pulsar sample also shows evidence of trends in other observables that may offer additional clues to the pulsar physics. While the detected objects have a wide range of surface magnetic fields, their inferred light cylinder magnetic fields B_LC are uniformly relatively large (≳10³ G). Indeed, the LAT-detected MSPs are those with the highest light cylinder fields with values very similar to those of the detected normal pulsars. Comparison of the spectral cutoff E_cutoff with surface magnetic field shows no significant correlation. This evidence argues against classical low-altitude polar cap models supported by γ–B cascades. However, there is a weak correlation of E_cutoff with B_LC, as shown in Figure 7. It is interesting that the values of E_cutoff have a range of only about a decade, from 1 to 10 GeV, and that all the different types of pulsars seem to follow the same correlation. This strongly implies that the gamma-ray emission originates in similar locations in the magnetosphere relative to the light cylinder. Such a correlation of E_cutoff with B_LC is actually expected in all outer magnetosphere models where the gamma-ray emission primarily comes from curvature radiation of electrons whose acceleration is balanced by radiation losses. In this case,

$$\begin{equation} E_{\rm cutoff} = 0.32 \lambda _c \left(E_{\parallel }\over e \right)^{3/4} \rho _c^{1/2} \end{equation} \tag{ 8 }$$

in m_ec², where λ_c is the electron Compton wavelength, E_∥ is the electric field that accelerates particles parallel to the magnetic field, and ρ_c is the magnetic field radius of curvature. In both SG (Muslimov & Harding 2004) and OG (Zhang et al. 2004; Hirotani 2008) models, E_∥ ∝ B_LC w², where w is the gap width. All these models give values of E_cutoff that are roughly consistent with those measured for the LAT pulsars. Although ρ_c ∼ R_LC, the gap widths are expected to decrease with increasing B_LC, so that E_cutoff is predicted to be only weakly dependent on B_LC in most outer magnetosphere models, as observed.

The detection of pulsed flux at E_max> a few GeV provides additional, physical motivation for high-altitude emission, since one expects strong (hyper-exponential) attenuation from γ–B $\longrightarrow e^+e^-$ absorption in the high magnetic fields at low altitudes of a near-surface polar gap. For a polar cap model with spin period P and surface field 10¹²B₁₂G, Equation (1) of Baring (2004) gives r ≳ (E_maxB₁₂/1.76 GeV)^2/7 P^−1/7 10⁶ cm. While the largest minimum r is derived from the 25 GeV detection of the Crab by MAGIC (Albert et al. 2008), significant minimum altitudes of 2 to 3 stellar radii are found for many LAT pulsars with large B₁₂ and high E_cutoff, assuming maximum energies set at around 2.5E_cutoff. Such altitudes are inconsistent with the r ≲ 1–2 stellar radii of polar cap models, hence implicating outer magnetosphere (e.g., TPC or OG) models where the bulk of the emission occurs at tens to hundreds of stellar radii.

In Figure 8, we see a general trend for the young pulsars to show a softer spectrum at large ${\dot{E}}$, although there is a great deal of scatter; a similar trend was noted in Thompson et al. (1999). This may be indicative of higher pair multiplicity, which would steepen the spectrum for the more energetic pulsars, either by steepening the spectrum of the curvature radiation-generating primary electrons (Romani 1996) or by inclusion of an additional soft spectral component associated with robust pair formation (Takata & Chang 2007; Harding et al. 2008). In either case, one would expect steepening from the simple monoenergetic curvature radiation spectrum Γ = 2/3 for the higher $\dot{E}$ pulsars. Interestingly, the MSPs do not extend the trend to lower ${\dot{E}}$. Of course, EGRET (and now the LAT) find strong variations of photon index with phase for the brighter pulsars. A full understanding of photon index trends will doubtless require phase-resolved modeling.