The $Fermi$-LAT has revealed that rotation powered millisecond pulsars (MSPs) are a major contributor to the Galactic $\gamma$-ray source population. We discuss the $\gamma$-ray emission process within the context of the outer gap accelerator model, and use a Monte-Calro method to simulate the Galactic population of the $\gamma$-ray emitting MSPs. We find that the outer gap accelerator controlled by the magnetic pair-creation process is preferable in explaining the possible correlation between the $\gamma$-ray luminosity and the spin down power. Our Monte-Calro simulation implies that most of the $\gamma$-ray emitting MSPs are radio quiet in the present sensitivity of the radio survey, indicating that most of the $\gamma$-ray MSPs have been unidentified. We argue that the Galactic $Fermi$ unidentified sources located at high latitudes should be dominated by MSPs, whereas the sources in the galactic plane are dominated by radio-quiet canonical pulsars.
The F ermi Large Area Telescope (F ermi-LAT) has discovered over 80 γ-ray pulsars, and revealed that the γ-ray pulsars are a major class of Galactic γ-ray sources. Of these, F ermi-LAT first detected pulsed γ-ray emission from 11 millisecond pulsars (Abdo et al. 2010a(Abdo et al. , 2009a,b;,b;Saz Parkinson et al. 2010; Guillemot et al. 2011). Furthermore, the detection of radio millisecond pulsars (MSPs) associated with over 20 unidentified F ermi point sources (e.g. Ransom et al. 2011; Keith et al. 2011) has been reported, suggesting that the millisecond pulsar, as well as the canonical pulsar is one of the major Galactic γ-ray source. The γ-ray emission from pulsars have been discussed in the context of a polar cap accelerator (Ruderman & Sutherland 1975), a slot gap (Harding & Muslimov 2011) and an outer gap accelerator (Cheng, Ho & Ruderman 1986; Takata, Wang & Cheng 2010b).
The high quality data measured by the F ermi enable us to perform a detail study for population of the γ-ray pulsars. For example, Takata et al. (2010b) and Takata, Wang and Cheng (2011d) have studied the possible relation between the high-energy emission properties and pulsar spin down power in the context of the outer gap accelerator model. They proposed that the outer gap accelerator model controlled by the magnetic pair-creation process can explain the observed population statistics better than that controlled by the photon-photon pair-creation process (e.g. Zhang & Cheng 2003). Story, Gonthier & Harding (2007) studied the population of γ-ray millisecond pulsars within the context of the slot gap accelerator model, and predicted the F ermi observations. They predicted that the F ermi will detect 12 radio-loud and 33-40 radio-quiet γ-ray millisecond pulsars. With the Monte-Calro simulation of the outer gap, Takata, Wang & Cheng (2011a,b) have explained the observed distributions of the characteristics of the γ-ray pulsars detected by the F ermi with the six-month long observation. The population studies (e.g. Kaaret & Philip 1996;Takata et al. 2011b,c) have also pointed out that unidentified MSPs located at high-Galactic latitudes will associate with the F ermi unidentified sources (Abdo et al. 2010b).
In this proceeding, we will review our recent Monte-Carlo studies for the Galactic population of the γray emitting MSPs (Takata et al. 2010b;Takata et al. 2011a,b,c) and the possible association with the F ermi unidentified sources.
For the outer gap model, the luminosity of the γ-ray emissions is typically
where L sd is the spin down power of the pulsar and the gap fraction f is defined as the ratio of the gap thickness at the light cylinder to the light cylinder radius R lc = P c/2π, where P is pulsar rotation period. Zhang & Cheng (2003) have argued a self-consistent outer gap model controlled by the photon-photon pair-creation process between the curvature photons and the X-rays from the stellar surface. They estimated the gap fraction for the MSPs as f p = 7.0 × 10 -2 (P/1 ms) 26/21 (B s /10 8 G) -4/7 δr 2/7 5 ,
(2) where B s is the stellar magnetic field of the global dipole field and δr 5 is the distance (in units of 10 5 cm) from the stellar surface to the position, where the local multiple magnetic field, which dominates the global dipole field, is comparable to the dipole field, and it will be δr 5 ∼ 1 -10 cm. Takata et al. (2010b) argued that the incoming particles emit photons with an energy m e c 2 /α f ∼ 70MeV by curvature radiation near the stellar surface and eConf C110509 that these photons can become pairs via the magnetic pair creation process. For a simple dipole field structure, all pairs move inward and cannot affect the outer gap accelerator. However if the local field lines near the surface are bent sideward due to the strong multipole field, the pairs created in these local magnetic field lines can have an angle greater than 90 • , which results in an outgoing flow of pairs. In this model, the fractional gap thickness in this circumstance is
where K ∼ B -2 m,12 s 7 characterizes the local parameters. Here, B m,12 and s 7 are the strength of the local magnetic field in units of 10 12 G and the local curvature radius in units of 10 7 cm, respectively, near the stellar surface.
The γ-ray luminosity L γ (1) can be cast in terms of the spin down power L sd = 2(2π) 4 µ/(3c 3 P 4 ) as
for the outer gap controlled by the photon-photon pair-creation process, and
by the magnetic pair-creation process. Here, L sd,34 = (L sd /10 34 erg s -1 ), K 1 = K/10 and B 8 = B s /10 8 G .
In Figures 1 the model predictions given by equations (4) and ( 5) are plotted with the solid line or dashed line, respectively. The filled circles represent the MSPs detected by the F ermi-LAT. Notwithstanding the large observational errors, the data points at large L γ in Figures 1 may suggest that the magnetic pair-creation model for the gap closing process is preferred over the photon-photon pair-creation model for the L γ -L sd re
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