Outer Gap Model Research Articles

The Galactic population and properties of young, highly energetic pulsars

ABSTRACT The population of young, non-recycled pulsars with spin-down energies $\dot{E} \gt 10^{35}$ erg s−1 is sampled predominantly at γ-ray and radio wavelengths. A total of 137 such pulsars are known, with partial overlap between the sources detectable in radio and γ-rays. We use a very small set of assumptions in an attempt to test whether the observed pulsar sample can be explained by a single underlying population of neutron stars. For radio emission we assume a canonical conal beam with a fixed emission height of 300 km across all spin periods and a luminosity law which depends on $\dot{E}^{0.25}$. For γ-ray emission we assume the outer-gap model and a luminosity law which depends on $\dot{E}^{0.5}$. We synthesize a population of fast-spinning pulsars with a birth rate of one per 100 yr. We find that this simple model can reproduce most characteristics of the observed population with two caveats. The first is a deficit of γ-ray pulsars at the highest $\dot{E}$ which we surmise to be an observational selection effect due to the difficulties of finding γ-ray pulsars in the presence of glitches without prior knowledge from radio frequencies. The second is a deficit of radio pulsars with interpulse emission, which may be related to radio emission physics. We discuss the implications of these findings.

Modelling pulsed gamma-ray emissions from millisecond pulsars with double peaks

ABSTRACT Pulsed γ-ray properties of 22 millisecond pulsars (MSPs) with double peaks are studied in the revised versions of the outer gap (OG) and the stripped wind (SW) models. The major differences between these two models are magnetic field structures, γ-ray production sites, and radiation mechanisms. In the models, γ-ray light curves of these MSPs are calculated through a Markov chain Monte Carlo (MCMC) method to pick best-fitting model parameters. Our results indicate that (1) both models can reproduce observed double-peak structures of the MSPs well, for most MSPs, a relatively large magnetic inclination angle (α ∼ 50°–90°) and small viewing angle (ζ ∼ 30°–90°) are obtained in the OG model, but a relatively small α ∈ (20°, 60°) and large ζ ∈ (70°, 150°) in the SW model; (2) phase-averaged spectra, cut-off energy, and γ-ray luminosity calculated in both models are consistent with observed those by Fermi-Large Area Telescope (LAT). Therefore, it may be concluded that the OG and the SW models for describing high-energy emissions of MSPs cannot be ruled out at present, and further studies are required.

Searching for High-energy, Horizon-scale Emissions from Galactic Black Hole Transients during Quiescence

Abstract We search for the gamma-ray counterparts of stellar-mass black holes using the long-term Fermi archive to investigate the electrostatic acceleration of electrons and positrons in the vicinity of the event horizon. We achieve this by applying the pulsar outer-gap model to their magnetospheres. When a black hole transient (BHT) is in a low-hard or quiescent state, the radiatively inefficient accretion flow cannot emit enough MeV photons that are required to sustain the force-free magnetosphere in the polar funnel via two-photon collisions. In this charge-starved gap region, an electric field arises along the magnetic field lines to accelerate electrons and positrons into ultra-relativistic energies. These relativistic leptons emit copious Gamma-rays via the curvature and inverse-Compton (IC) processes. It is found that these gamma-ray emissions exhibit a flaring activity when the plasma accretion rate typically stays between 0.01% and 0.005% of the Eddington value for rapidly rotating, stellar-mass black holes. By analyzing the detection limit determined from archival Fermi/Large Area Telescope data, we find that the 7-year averaged duty cycle of such flaring activities should be less than 5% and 10% for XTE J1118+480 and 1A 0620-00, respectively, and that the detection limit is comparable to the theoretical prediction for V404 Cyg. It is predicted that the gap emission can be discriminated from the jet emission if we investigate the high-energy spectral behavior or observe nearby BHTs during deep quiescence simultaneously in infrared wavelength and very-high energies.

Chandra Phase-resolved Spectroscopy of the High Magnetic Field Pulsar B1509−58

Abstract We report on a timing and spectral analysis of the young, high magnetic field rotation-powered pulsar (RPP) B1509−58 using Chandra continuous-clocking mode observation. The pulsar’s X-ray light curve can be fit by the two Gaussian components and the pulsed fraction shows moderate energy dependence over the Chandra band. The pulsed X-ray spectrum is well described by a power law with a photon index 1.16(4), which is harder than the values measured with RXTE/PCA and NuSTAR. This result supports the log-parabolic model for the broadband X-ray spectrum. With the unprecedented angular resolution of Chandra, we clearly identified off-pulse X-ray emission from the pulsar, and its spectrum is best fit by a power law plus blackbody model. The latter component has a temperature of ∼0.14 keV with a bolometric luminosity comparable to the luminosities of other young and high magnetic field RPPs, and it lies between the temperature of magnetars and typical RPPs. In addition, we found that the nonthermal X-ray emission of PSR B1509−58 is significantly softer in the off-pulse phase than in the pulsed phase, with the photon index varying between 1.0 and 1.8 and anticorrelated with the flux. This is similar to the behavior of three other young pulsars. We interpreted it as different contributions of pair-creation processes at different altitudes from the neutron star surface according to the outer-gap model.

DIFFERENCES BETWEEN RADIO-LOUD AND RADIO-QUIET γ-RAY PULSARS AS REVEALED BY FERMI

ABSTRACT By comparing the properties of non-recycled radio-loud γ-ray pulsars and radio-quiet γ-ray pulsars, we have searched for the differences between these two populations. We found that the γ-ray spectral curvature of radio-quiet pulsars can be larger than that of radio-loud pulsars. Based on the full sample of non-recycled γ-ray pulsars, their distributions of the magnetic field strength at the light cylinder are also found to be different. We note that this might result from an observational bias. By reexamining the previously reported difference of γ-ray-to-X-ray flux ratios, we found that the significance can be hampered by their statistical uncertainties. In the context of the outer gap model, we discuss the expected properties of these two populations and compare with the possible differences that are identified in our analysis.

Constraining the geometry of PSR J0855−4644: A nearby pulsar wind nebula with double torus/jet morphology

Aims. PSR J0855−4644 is a fast-spinning, energetic pulsar discovered at radio wavelengths near the south-eastern rim of the supernova remnant RX J0852.0−4622. A follow-up XMM-Newton observation revealed the X-ray counterpart of the pulsar and a slightly asymmetric pulsar wind nebula, which suggests possible jet structures. Lying at a distance d ≤ 900 pc, PSR J0855−4644 is a pulsar with one of the highest Ė/d2 from which no GeV γ-ray pulsations have been detected. With a dedicated Chandra observation we aim to further resolve the possible jet structures of the nebula and study the pulsar geometry to understand the lack of γ-ray pulsations. Methods. We performed detailed spatial modelling to constrain the geometry of the pulsar wind nebula and in particular the pulsar line of sight (observer angle) ζPSR, which is defined as the angle between the direction of the observer and the pulsar spin axis. We also performed geometric radio and γ-ray light-curve modelling using a hollow-cone radio beam model together with two-pole caustic and outer gap models to further constrain ζPSR and the magnetic obliquity α defined as the angle between the magnetic and spin axes of the pulsar. Results. The Chandra observation reveals that the compact XMM source, thought to be the X-ray pulsar, can be further resolved into a point source surrounded by an elongated axisymmetric nebula with a longitudinal extent of 10′′. The pulsar flux represents only ~1% of the XMM compact source, and its spectrum is well described by a blackbody of temperature kT = 0.2 keV, while the surrounding nebula has a much harder spectrum (Γ = 1.1 for a power-law model). Assuming the origin of the extended emission is a double torus yields ζPSR = 32.5° ± 4.3°. The detection of thermal X-rays from the pulsar may point to a low value of | ζ−α | if this emission originates from a heated polar cap. Independent constraints from geometric light-curve modelling yield α ≲ 55° and ζ ≲ 55°, and 10° ≲ | ζ−α | ≲ 30°. A χ2 fit to the radio light curve yields a best fit at (α,ζPSR) = (22°,8°), with an alternative fit at (α,ζPSR) = (9°,25°) within 3σ. The lack of non-thermal X-ray emission from the pulsar further supports low values for α and ζ under the assumption that X-rays and γ-rays are generated in the same region of the pulsar magnetosphere. Such a geometry would explain, in the standard caustic pulsar model picture, the radio-loud and γ-ray-quiet behaviour of this high Ė/d2 pulsar.

X-RAY/GeV EMISSIONS FROM CRAB-LIKE PULSARS IN THE LMC

ABSTRACT We discuss X-ray and gamma-ray emissions from Crab-like pulsars, PSRs J0537-6910 and J0540-6919, in the Large Magellanic Cloud. Fermi-LAT observations have resolved the gamma-ray emissions from these two pulsars and found pulsed emissions from PSR J0540-6919. The total pulsed radiation in the X-ray/gamma-ray energy bands of PSR J0540-6919 is observed with efficiency (in 4π sr), which is about a factor of ten larger than of the Crab pulsar. Although PSR J0537-6910 has the highest spin-down power among currently known pulsars, the efficiency of the observed X-ray emissions is about two orders of magnitude smaller than that of PSR J0540-6919. This paper mainly discusses what causes the difference in the radiation efficiencies of these three energetic Crab-like pulsars. We discuss electron/positron acceleration and high-energy emission processes within the outer gap model. By solving the outer gap structure with the dipole magnetic field, we show that the radiation efficiency decreases as the inclination angle between the magnetic axis and the rotation axis increases. To explain the difference in the pulse profile and in the radiation efficiency, our model suggests that PSR J0540-6919 has an inclination angle much smaller than that of the Crab pulsar (here we assume the inclination angles of both pulsars are ). On the other hand, we speculate that the difference in the radiation efficiencies between PSRs J0537-6910 and J0549-6919 is mainly caused by the difference in the Earth viewing angle, and that we see PSR J0537-6910 with an Earth viewing angle (or ) measured from the spin axis, while we see PSR J0540-6919 with .

LEPTON ACCELERATION IN THE VICINITY OF THE EVENT HORIZON: HIGH-ENERGY AND VERY-HIGH-ENERGY EMISSIONS FROM ROTATING BLACK HOLES WITH VARIOUS MASSES

ABSTRACT We investigate the electrostatic acceleration of electrons and positrons in the vicinity of the event horizon, applying the pulsar outer-gap model to black hole (BH) magnetospheres. During a low accretion phase, the radiatively inefficient accretion flow (RIAF) cannot emit enough MeV photons that are needed to sustain the force-free magnetosphere via two-photon collisions. In such a charge-starved region (or a gap), an electric field arises along the magnetic field lines to accelerate electrons and positrons into ultra-relativistic energies. These relativistic leptons emit copious gamma rays via curvature and inverse-Compton (IC) processes. Some of such gamma rays collide with the submillimeter-IR photons emitted from the RIAF to materialize as pairs, which polarize to partially screen the original acceleration electric field. It is found that the gap gamma-ray luminosity increases with decreasing accretion rate. However, if the accretion rate decreases too much, the diminished RIAF soft photon field can no longer sustain a stationary pair production within the gap. As long as a stationary gap is formed, the magnetosphere becomes force-free outside the gap by the cascaded pairs, irrespective of the BH mass. If a nearby stellar-mass BH is in quiescence, or if a galactic intermediate-mass BH is in a very low accretion state, its curvature and IC emissions are found to be detectable with Fermi/LAT and imaging atmospheric Cherenkov telescopes (IACT). If a low-luminosity active galactic nucleus is located within about 30 Mpc, the IC emission from its supermassive BH is marginally detectable with IACT.

The effect of an offset polar cap dipolar magnetic field on the modeling of the Vela pulsar's γ-ray light curves.

We performed geometric pulsar light curve modeling using static, retarded vacuum, and offset polar cap (PC) dipole B-fields (the latter is characterized by a parameter ε), in conjunction with standard two-pole caustic (TPC) and outer gap (OG) emission geometries. The offset-PC dipole B-field mimics deviations from the static dipole (which corresponds to ε = 0). In addition to constant-emissivity geometric models, we also considered a slot gap (SG) E-field associated with the offset-PC dipole B-field and found that its inclusion leads to qualitatively different light curves. Solving the particle transport equation shows that the particle energy only becomes large enough to yield significant curvature radiation at large altitudes above the stellar surface, given this relatively low E-field. Therefore, particles do not always attain the radiation-reaction limit. Our overall optimal light curve fit is for the retarded vacuum dipole field and OG model, at an inclination angle [Formula: see text] and observer angle [Formula: see text]. For this B-field, the TPC model is statistically disfavored compared to the OG model. For the static dipole field, neither model is significantly preferred. We found that smaller values of ε are favored for the offset-PC dipole field when assuming constant emissivity, and larger ε values favored for variable emissivity, but not significantly so. When multiplying the SG E-field by a factor of 100, we found improved light curve fits, with α and ζ being closer to best fits from independent studies, as well as curvature radiation reaction at lower altitudes.

OBSERVATION AND SIMULATION OF THE VARIABLE GAMMA-RAY EMISSION FROM PSR J2021+4026

ABSTRACT Pulsars are rapidly spinning and highly magnetized neutron stars, with highly stable rotational periods and a gradual spin-down over a long timescale due to the loss of radiation. Glitches refer to events that suddenly increase the rotational speed of a pulsar. The exact causes of glitches and the resulting processes are not fully understood. It is generally believed that couplings between the normal matter and superfluid components, and starquakes, are the common causes of glitches. In this study, one famous glitching pulsar, PSR J2021+4026, is investigated. PSR J2021+4026 is the first variable gamma-ray pulsar observed by Fermi. From gamma-ray observations, it is found that the pulsar experienced a significant flux drop, an increase in the spin-down rate, a change in the pulse profile and a shift in the spectral cut-off to a lower energy, simultaneously around 2011 October 16. To explain these effects on high-energy emissions by the glitch of PSR J2021+4026, we hypothesized the glitch to be caused by the rearrangement of the surface magnetic field due to crustal plate tectonic activities on the pulsar, which was triggered by a starquake. In this glitch event, the inclination angle of the magnetic dipole axis was slightly shifted. This proposition is then tested by numerical modeling using a three-dimensional two-layer outer gap model. The simulation results indicate that a modification of the inclination angle can affect the pulse profile and the spectral properties, which can explain the observation changes after the glitch.

Theoretical Study of Gamma-ray Pulsars

We use the non-stationary three dimensional two-layer outer gap model to explain gamma-ray emissions from a pulsar magnetosphere. We found out that for some pulsars like the Geminga pulsar, it was hard to explain emissions above a level of around 1 GeV. We then developed the model into a non-stationary model. In this model we assigned a power-law distribution to one or more of the spectral parameters proposed in the previous model and calculated the weighted phase-averaged spectrum. Though this model is suitable for some pulsars, it still cannot explain the high energy emission of the Geminga pulsar. An Inverse-Compton Scattering component between the primary particles and the radio photons in the outer magnetosphere was introduced into the model, and this component produced a sufficient number of GeV photons in the spectrum of the Geminga pulsar.

Pulsed γ-ray properties of Crab pulsar in a retarded dipole with a current-induced magnetic field

Motivated by the Fermi observations of some γ-ray pulsars in which the phases of radio and γ-ray peaks are almost the same, we investigate the outer gap model in a retarded dipole with a current-induced magnetic field and apply it to explain pulsed γ-ray properties of the Crab pulsar. Our results show that the observed γ-ray energy-dependent light curves, which almost align with the radio light curve and phase averaged spectrum for the Crab pulsar, are reproduced well.

EXPERIMENTAL CONSTRAINTS ONγ-RAY PULSAR GAP MODELS AND THE PULSAR GeV TO PULSAR WIND NEBULA TeV CONNECTION

The pulsar emission mechanism in the gamma-ray energy band is poorly understood. Currently, there are several models under discussion in the pulsar community. These models can be constrained by studying the collective properties of a sample of pulsars, which became possible with the large sample of gamma-ray pulsars discovered by the Fermi Large Area Telescope (Fermi-LAT). In this paper we develop a new experimental multi-wavelength technique to determine the beaming factor $\left( f_\Omega \right)$ dependance on spin-down luminosity of a set of GeV pulsars. This technique requires three input parameters: pulsar spin-down luminosity, pulsar phase-averaged GeV flux and TeV or X-ray flux from the associated Pulsar Wind Nebula (PWN). The analysis presented in this paper uses the PWN TeV flux measurements to study the correlation between $f_\Omega$ and $\dot{E}$. The measured correlation has some features that favor the Outer Gap model over the Polar Cap, Slot Gap and One Pole Caustic models for pulsar emission in the energy range of 0.1 to 100 GeV, but one must keep in mind that these simulated models failed to explain many of the most important pulsar population characteristics. A tight correlation between the pulsar GeV emission and PWN TeV emission was also observed, which suggests the possibility of a linear relationship between the two emission mechanisms. In this paper we also discuss a possible mechanism to explain this correlation.

Light-curve modelling constraints on the obliquities and aspect angles of the youngFermipulsars

In more than four years of observation the Large Area Telescope on board the Fermi satellite has identified pulsed $\gamma$-ray emission from more than 80 young pulsars, providing light curves with high statistics. Fitting the observations with geometrical models can provide estimates of the magnetic obliquity $\alpha$ and aspect angle $\zeta$, yielding estimates of the radiation beaming factor and luminosity. Using $\gamma$-ray emission geometries (Polar Cap, Slot Gap, Outer Gap, One Pole Caustic) and radio emission geometry, we fit $\gamma$-ray light curves for 76 young pulsars and we jointly fit their $\gamma$-ray plus radio light curves when possible. We find that a joint radio plus $\gamma$-ray fit strategy is important to obtain ($\alpha$, $\zeta$) estimates that can explain simultaneous radio and $\gamma$-ray emission. The intermediate-to-high altitude magnetosphere models, Slot Gap, Outer Gap, and One pole Caustic, are favoured in explaining the observations. We find no evolution of $\alpha$ on a time scale of a million years. For all emission geometries our derived $\gamma$-ray beaming factors are generally less than one and do not significantly evolve with the spin-down power. A more pronounced beaming factor vs. spin-down power correlation is observed for Slot Gap model and radio-quiet pulsars and for the Outer Gap model and radio-loud pulsars. For all models, the correlation between $\gamma$-ray luminosity and spin-down power is consistent with a square root dependence. The $\gamma$-ray luminosities obtained by using our beaming factors not exceed the spin-down power. This suggests that assuming a beaming factor of one for all objects, as done in other studies, likely overestimates the real values. The data show a relation between the pulsar spectral characteristics and the width of the accelerator gap that is consistent with the theoretical prediction for the Slot Gap model.

An assessment of the pulsar outer gap model – I. Assumptions, uncertainties, and implications on the gap size and the accelerating field

The popular outer gap model of magnetospheric emission from pulsars has been widely applied to explain the properties observed in $\gamma$-rays. However, its quantitative predictions rely on a number of approximations and assumptions that are usually overlooked. Here we examine them, reviewing the main ingredients entering in the model, evaluating their range of uncertainties. Usually, in the quantitative applications of the model, key parameters like the radius of curvature and the energies of the interacting photons are taken to be a fixed, single value. Instead, here we explore their realistic ranges, and the impact of these on the consistency of the model itself. We conclude that the popular evaluation of the trans-field size of the gap as a function of period and period derivative, is unreliable and affected by a huge dispersion. Last, the exploration of the possible values for the radius of curvature, the local magnetic field and other quantities deserve more attention for quantitative applications of the outer gap model, like the calculation of $\gamma$-ray spectra, which is the subject of an accompanying paper.

An assessment of the pulsar outer gap model – II. Implications for the predicted γ-ray spectra

One of the most important predictions of any gap model for pulsar magnetospheres is the predicted $\gamma$-ray spectra. In the outer gap model, the properties of the synchro-curvature radiation are sensitive to many parameters, whose realistic ranges have been studied in detail in an accompanying paper. There we demonstrated that the uncertainty in the radius of curvature, the magnetic field geometry, and the X-ray surface flux may affect by orders of magnitude the predicted flux and spectral peak in the $\gamma$-ray regime. Here, we present a systematic, numerical study of the impact of the different parameters on the particle dynamics along the gap and calculate the emitted synchro-curvature radiation along the trajectory. By integrating the emitted radiation along the gap and convolving it with a parametrized particle distribution, we discuss how the comparison with the wealth of {\em Fermi}-LAT data can be used to constrain the applicability of the model. The resulting spectra show very different energy peaks, fluxes and shapes, qualitatively matching the great variety of the observed {\em Fermi}-LAT pulsars. In particular, if we see a large fraction of photons emitted from the initial part of the trajectory, we show that the spectra will be flatter at the low-energy {\it Fermi}-LAT regime (100 MeV -- 1 GeV). This provides a solution for such observed flat spectra, while still maintain synchro-curvature radiation as the origin of these photons.

Why the distance of PSR J0218+4232 does not challenge pulsar emission theories

Recent very long baseline interferometry (VLBI) measurements of the astrometric parameters of the millisecond pulsar J0218+4232 by Du et al. have suggested that this pulsar is as distant as 6.3 kpc. At such a large distance, the large γ-ray flux observed from this pulsar would make it the most luminous γ-ray pulsar known. This luminosity would exceed what can be explained by the outer gap and slot-gap pulsar emission models, potentially placing important and otherwise elusive constraints on the pulsar emission mechanism. We show that the VLBI parallax measurement is dominated by the Lutz–Kelker bias. When this bias is corrected for, the most likely distance for this pulsar is | $3.15^{+0.85}_{-0.60}$ | kpc. This revised distance places the luminosity of PSR J0218+4232 into a range where it does not challenge any of the standard theories of the pulsar emission mechanism.

CONSTRAINTS ON THE EMISSION GEOMETRIES AND SPIN EVOLUTION OF GAMMA-RAY MILLISECOND PULSARS

Millisecond pulsars (MSPs) are a growing class of gamma-ray emitters. Pulsed gamma-ray signals have been detected from more than 40 MSPs with the Fermi Large Area Telescope (LAT). The wider radio beams and more compact magnetospheres of MSPs enable studies of emission geometries over a broader range of phase space than non-recycled radio-loud gamma-ray pulsars. We have modeled the gamma-ray light curves of 40 LAT-detected MSPs using geometric emission models assuming a vacuum retarded-dipole magnetic field. We modeled the radio profiles using a single-altitude hollow-cone beam, with a core component when indicated by polarimetry; however, for MSPs with gamma-ray and radio light curve peaks occurring at nearly the same rotational phase we assume that the radio emission is co-located with the gamma rays and caustic in nature. The best-fit parameters and confidence intervals are determined using a maximum likelihood technique. We divide the light curves into three model classes, with gamma-ray peaks trailing (Class I), aligned (Class II) or leading (Class III) the radio peaks. Outer gap and slot gap (two-pole caustic) models best fit roughly equal numbers of Class I and II, while Class III are exclusively fit with pair-starved polar cap models. Distinguishing between the model classes based on typical derived parameters is difficult. We explore the evolution of magnetic inclination angle with period and spin-down power, finding possible correlations. While the presence of significant off-peak emission can often be used as a discriminator between outer gap and slot gap models, a hybrid model may be needed.

VERY LONG BASELINE INTERFEROMETRY MEASURED PROPER MOTION AND PARALLAX OF THE γ-RAY MILLISECOND PULSAR PSR J0218+4232

PSR J0218$+$4232 is a millisecond pulsar (MSP) with a flux density $\sim$ 0.9 mJy at 1.4 GHz. It is very bright in the high-energy X-ray and $\gamma$-ray domains. We conducted an astrometric program using the European VLBI Network (EVN) at 1.6 GHz to measure its proper motion and parallax. A model-independent distance would also help constrain its $\gamma$-ray luminosity. We achieved a detection of signal-to-noise ratio S/N > 37 for the weak pulsar in all five epochs. Using an extragalactic radio source lying 20 arcmin away from the pulsar, we estimate the pulsar's proper motion to be $\mu_{\alpha}\cos\delta=5.35\pm0.05$ mas yr$^{-1}$ and $\mu_{\delta}=-3.74\pm 0.12$ mas yr$^{-1}$, and a parallax of $\pi=0.16\pm0.09$ mas. The very long baseline interferometry (VLBI) proper motion has significantly improved upon the estimates from long-term pulsar timing observations. The VLBI parallax provides the first model-independent distance constraints: $d=6.3^{+8.0}_{-2.3}$ kpc, with a corresponding $3\sigma$ lower-limit of $d=2.3$ kpc. This is the first pulsar trigonometric parallax measurement based solely on EVN observations. Using the derived distance, we believe that PSR J0218$+$4232 is the most energetic $\gamma$-ray MSP known to date. The luminosity based on even our 3$\sigma$ lower-limit distance is high enough to pose challenges to the conventional outer gap and slot gap models.

Phase-averaged gamma-ray spectra from rotation-powered millisecond pulsars

{\it Fermi}-LAT has detected pulsed gamma-ray emissions with high confidences from more than 40 millisecond pulsars (MSPs). Here we study the phase-averaged gamma-ray properties of MSPs by using revised version of a self-consistent outer gap model. In this model, a strong multipole magnetic field near the stellar surface for a MSP is assumed and such a field will be close to the surface magnetic fields ($\sim 10^{11}- 10^{12}$ G) of young ulsars; the outer gap of a MSP is controlled by photon-photon pair production process, where the effects of magnetic inclination angle ($\alpha$) and magnetic geometry have been taken into account, therefore the fractional size of the outer gap is a function of not only pulsar's period and magnetic field strength but also magnetic inclination angle and radial distance to the neutron star, the inner boundary of the outer gap can be estimated by the pair production process of the gamma-ray photons which are produced by the back-flowing particles through the null charge surface; inside the outer gap, a Gaussian distribution of the parallel electric field along the trans-field thickness is assumed, and the gamma-ray emission is represented by the emission from the average radial distance along the central field lines of the outer gap. Using this model, the phase-averaged gamma-ray spectra are calculated and compared with the observed spectra of 37 MSPs given by the second {\it Fermi}-LAT catalog of gamma-ray pulsars, our results show that the {\it Fermi}-LAT results can be well explained by this model. The thermal X-ray emission properties from MSPs are also investigated.