Spectral features in gamma-rays expected from millisecond pulsars

Abstract

In the advent of next generation gamma-ray missions, we present general properties of spectral features of high-energy emission above 1MeV expected for a class of millisecond, low magnetic field (∼10⁹G) pulsars. We extend polar-cap model calculations of Rudak & Dyks by including inverse Compton scattering events in an ambient field of thermal X-ray photons and by allowing for two models of particle acceleration. In the range between 1MeV and a few hundred GeV, the main spectral component is the result of curvature radiation of primary particles. The synchrotron component arising from secondary pairs becomes dominant only below 1MeV. The slope of the curvature radiation spectrum in the energy range from 100MeV to 10GeV strongly depends on the model of longitudinal acceleration, whereas below ∼100MeV all slopes converge to a unique value of 4/3 (in a νℱ convention). The thermal soft X-ray photons, which come either from the polar cap or from the surface, are Compton upscattered to a very high energy domain and form a separate spectral component peaking at ∼1TeV. We discuss the observability of millisecond pulsars by future high-energy instruments and present two rankings relevant for GLAST and MAGIC. We point to the pulsar J0437−4715 as a promising candidate for observations.

pulsars: general, gamma-rays: observations, gamma-rays: theory

Introduction

Millisecond pulsars are thought to be uninteresting as targets for gamma-ray experiments. This opinion is partly drawn from an argument that typical values of magnetic field B (or rather their dipolar components) inferred from spin period P and spin-down rate ṗ are in the range of 10⁸ to 10⁹G, in contrast to the range of 10¹¹ to 10¹³G for classical pulsars. If gamma-ray radiation is to be a manifestation of magnetospheric activity, it seems quite natural to think of millisecond pulsars in terms of B as of a scaled-down version of classical pulsars, which by themselves are weak gamma-ray sources; only seven of them have been firmly detected so far (Thompson et al. 1997). Unsuccessful observational campaigns in the past, mainly by CGRO instruments, e.g. EGRET (Fierro et al. 1995), confirmed this line of thinking. J0437−4715, with one of the highest spin-down fluxes among all pulsars, thus a very promising target, was not even included in the priority list of targets for the COMPTEL instrument of the Compton Gamma-Ray Observatory(CGRO) (Carraminana et al. 1995).

However, the ‘scaling-down’ argument becomes unjustified by the results of X-ray observations. Within a sample of some 30 pulsars detected with ROSAT and other satellites, there are 9 millisecond objects (Becker & Tr ümper 1999) with apparent X-ray properties similar to classical pulsars. In particular, the millisecond pulsars obey the same astonishing relation given by Becker & Tr ümper (1997) between the spin-down luminosity L_sd and the soft X-ray luminosity L_x that classical pulsars do. In most cases the X-ray spectra are presumed to be non-thermal in character, but whether they represent low-energy tails of putative gamma-ray emission is not known.

Rather few papers have been dedicated to the question of gamma-rays in the context of millisecond pulsars on the theoretical side. Wei, Cheng & Lu (1996) presented broad-band spectra of both pulsed and unpulsed emission expected in their version of the outer-gap model. Sturner & Dermer (1994) offered predictions for gamma-ray luminosity L in some popular models, but that had been done just by scaling down the values of B in simple analytical formulae for L derived for classical pulsars. In the framework of polar-cap scenarios, Rudak & Dyks (1998) [RD98] proposed a modification of the model by Daugherty & Harding (1982). This modification incorporates a contribution of electron-positron pairs to L, which becomes a non-monotonic function of B (see figs 7 and 8 of Dyks 1998). One of the consequences of this modification is that expected values of L for millisecond pulsars, including J0437−4715, are typically about one or two orders of magnitude lower than for classical pulsars. A quantitatively similar estimate, though for different reason, comes from a model of Dermer & Sturner (1994).

However, if the estimates above are correct then some millisecond pulsars, including J0437−4715, should be detected with next-generation experiments, thus providing a testing ground for magnetospheric models relevant for millisecond pulsars. The successor of EGRET -GLAST- with a possible launch in the year 2005, is expected to have sensitivity above 100MeV about 30 times higher than a EGRET, and it will reach high-energy limit at 300GeV, closing thus for the first time the energy gap between the very high energy (VHE) domain accessible with ground-based Cherenkov techniques and the high energy (HE) domain of satellite experiments (Kamae et al. 2000). Moreover, since we expect the millisecond pulsars to radiate the electromagnetic energy mostly in the range between 10 and 100GeV, this class of objects should be of interest for the upcoming Imaging Air Cherenkov (IAC) telescope MAGIC (Blanch et al. 1998).

Our aim is to present major spectral features in the energy range above ∼10MeV of pulsed gamma-ray emission which arises as a result both of ultrarelativistic primary particles (electrons) and of cascades of secondary particles above polar caps of millisecond pulsars with dipolar magnetic fields of ∼10⁹G. In particular, we allow for different structures of the accelerating potential, and we show how this affects the slope of the predicted energy spectrum in the range of 100MeV to 100GeV. Also, Compton upscattering of ambient X-ray photons off ultrarelativistic electrons is taken into account. In Section 2 we present details of modelling the processes responsible for the formation of gamma-rays in millisecond pulsars. Section 3 describes spectral properties of two distinct components, arising from curvature radiation (CR) and Compton scattering (ICS). Conclusions with a discussion of J0437−4715 are given in Section 4.

The model of high-energy radiation

We use a polar cap model, with most ingredients as postulated by Daugherty & Harding (1982) in the context of classical pulsars. High-energy emission is a superposition of curvature radiation (CR) of ultrarelativistic primary electrons, and synchrotron radiation (SR) of secondary particles (e ^± pairs) created via magnetic absorption of photons. Additional features include (1) two models of electron acceleration and (2) inverse Compton scattering (ICS) of thermal X-ray photons on electrons. The first feature is important for the history of the cooling of primary electrons and consequently it affects the slope of gamma-ray spectrum above 100MeV. The second (ICS) is not important for the overall cooling of electrons, but the resulting TeV component in the gamma-ray spectra is of potential interest. A field of ambient photons necessary for the ICS is likely to be present within the magnetosphere of some millisecond pulsars, according to some interpretations of available X-ray and extreme ultraviolet observations (e.g. Zavlin & Pavlov 1998;Edelstein, Foster & Bowyer 1995). We list the details of the model in the following subsections.

Geometry

We assume that the geometry of the magnetic field of a neutron star is well described by a static, axisymmetric dipole. In polar coordinates its field lines satisfy rsin⁻² θR₀, where the dipole constant R₀ defines a set of field lines which differ by azimuthal angle only. Polar caps are defined as (two) regions on the neutron star surface crossed by all lines for which the condition R₀ ⩾R_lc is satisfied, where R_lccP/2 π is a light-cylinder radius for a spin period P. The polar cap radius is then r_pcR_ns ×(R_nsR_lc)^1/2, where R_ns is the radius of the neutron star for which we assume the canonical value of 10⁶cm. Beam particles (primary electrons) are ejected from the outer rim of the polar cap and move along the magnetic field lines.

When modelling an extended source of soft photons, which are to be targets for electrons in Compton scatterings, we consider two cases of its geometry. In the first case, the soft photons come from the hot polar cap with a uniform temperature T_pc ≳10⁶K. In the second case the soft photons come from the entire surface of the neutron star, however now the temperature is lower: T_surf ≳10⁵K.

Electron acceleration

Primary electrons are injected along magnetic field lines into the magnetosphere from the outer rim of the polar cap. The structure of the longitudinal electric field responsible for their acceleration remains unknown. However, the electric field must be strong enough to ensure electron-positron pair creation. The presence of pairs is a necessary condition for a magnetized neutron star to be a radio pulsar. Thus the requirement of creating pairs in the magnetosphere is the primary condition that the model has to satisfy.

We use two simple models for particle acceleration. The first one (denoted model A) assumes instant acceleration to ultrarelativistic energy at the moment of electron injection on the surface of the neutron star. The values of are set to be just above the threshold value for creation of secondary pairs which in turn is a function of magnetic field strength at the polar cap B_pc and the spin period P (for details see Rudak & Dyks 1998; Dyks 1998). For a pulsar with B_pc=10⁹G and P=3ms, the initial energy is For J0437−4715 we took B_pc=7.4 ×10⁸G, P=5.75ms, and where we used the magnetic field derived from Ṗ kinematically corrected by Camilo, Thorsett & Kulkarni (1994).

For comparison we consider another model (model B) of particle acceleration, in which the longitudinal electric field ℰ is set at a constant level with an exponential cut-off:  

(2)

where the characteristic scaleheight of the electric field is assumed to be equal to the radius of the polar cap r_pc. For a pulsar with B_pc=10⁹G and P=3ms the value of ℰ at the polar cap surface was set at the level of V₀r_pc ≈10⁹Vcm⁻¹ to ensure a total yield of secondary pairs per primary electron similar to that in model (A). The initial value of the electron energy was assumed.

As electrons accelerate they move along the magnetic field lines and are slowed down by the curvature radiation reaction. We calculate the electron energy budget as a function of the distance travelled in the magnetopshere. Only a small fraction (less than 1per cent, see Fig. 1) of the electrons will scatter on thermal photons originating on the polar cap or on the stellar surface. Thus scattering is not an important electron deceleration process.

Figure 1.

Cumulative optical depth τ_e for non-magnetic scattering of an electron injected at the surface level (h=0) and crossing a blackbody radiation field. The left panel is for the radiation fields arising from the hot polar cap (T_cap=10⁶K and 3 ×10⁶K). Solid lines are for instant acceleration with electron energy E_init=10⁷MeV. Dashed lines are for continuous acceleration in electric field ℰ (equation 2). The right panel is for the radiation fields arising from the hot stellar surface (T_surf=2 ×10⁵K and 5 ×10⁵K). The pulsar period is P=3 ×10⁻³s and we used 10⁵ electrons in the simulation.

Curvature and synchroton processes

We follow the model described by Rudak & Dyks (1999) who calculated broad-band energy spectra of high-energy emission for two distinct pulsar groups: low magnetic field pulsars (‘millisecond pulsars’) and classical pulsars.

The dominant cooling process for primary electrons is the curvature radiation,  

3

where γ_e is the electron Lorentz factor, ρ is the curvature radius, and r₀ is the electron radius. When considering secondary electrons (pairs) one has to consider also the synchrotron radiation. The rate of SR cooling is

 

4

where ψ is the pitch angle of the pairs. The photon spectra arising from curvature and synchrotron processes are calculated following Daugherty & Harding (1982). A detailed discussion of the role of these processes in the formation of gamma-ray spectra of pulsars is given in Rudak & Dyks (1999).

Scattering

Our numerical treatment of scattering process follows Daugherty & Harding (1989). As the electrons travel up in the magnetosphere they may scatter off thermal photons originating from the stellar surface or the polar cap. For millisecond pulsars the magnetic field at the surface is typically 10⁹G or less, i.e. much weaker than the critical field B_crit=4.414 ×10¹³G. Thus the scattering process will be well described by the Klein-Nishina relativistic non-magnetic cross-section

 

5

where x=2p^νk, p^ν is the electron four-momentum, and k is the photon four-wave vector. The scattering rate (in the ‘lab frame’K) by an electron travelling with velocity β in a photon field is given by

 

6

where Ω is the solid angle subtended by the source of soft photons, σ_KN is the total cross-section, f is the photon occupation density, and n#x005E;_i is the unit vector representing the direction of the incoming soft photon. Taking the Planck function for ɛ²f this equation reads  

where ɛ˜ is the photon energy in the units of electron mass and T∼ is the photon temperature in the units of electron mass.

We draw the energy and direction of the incoming photon from the distribution of ɛ²f, and Ω, respectively (as used in equation 6). Then we transfer them to the electron instantaneous rest frame K_e and there we find the photon scattering angle θ using the Klein-Nishina differential cross-section

 

(8)

where μ is the cosine of θ. The energy ε′ of the scattered photon in K_e is obtained from the Compton formula

 

(9)

Finally, the energy of the outgoing photon is transformed back to K.

At each step, as the location of the electron changes, the solid angle Ω is corrected for the finite size of the source of soft photons (the polar cap or the stellar surface). Thus, we take into account the effect of the decreasing photon number density with height (geometrical dilution) and then we calculate the scattering rate ℛ at a given height above the neutron star surface.

Magnetic photon absorption

Despite low magnetic field values, two conditions for the process of pair creation via magnetic photon absorption (γ+B→e ^±) — the energy threshold condition, and the high optical thickness τ_γB of the magnetosphere — may well be met in the context of millisecond pulsars. For instance, in the model spectra calculated numerically by Rudak & Dyks (1999) photons with energy ∼10²GeV or higher were subject to the magnetic absorption reprocessing.

In our simulations we use the absorption coefficient as described by Erber (1966):

 

(10)

where α is a fine structure constant, λ̄_c is a Compton wavelength, B_⊥ is the component of the magnetic field perpendicular to the photon momentum, and is the Erber parameter χ. For the function T(χ) we use its approximation T(χ) ≈0.46exp(−4f/3χ), valid for χ≲0.2 (for χ ≳0.2 this approximation starts to overestimate η). The function f is the near-threshold correction introduced by (Daugherty & Harding 1983). In the case of millisecond pulsars equation (10) is a good approximation even with f=1, since magnetic photon absorption occurs well above the threshold. Electron-positron pairs created through the magnetic absorption radiate then via the synchrotron process.

For demonstrative purposes it is useful to present the condition τ_γB ⩾1 in an analytical way. It is straightforward to show that a photon created with a momentum parallel to the local magnetic field line at a height h above the neutron star surface will undergo magnetic absorption if its energy satisfies approximately the following inequality:

 

(11)

This formula is valid for dipolar magnetic field, and the field line is attached to the outer rim of the polar cap, and it is in very good agreement with numerical calculations of magnetic photon absorption.

Results

Electrons are either accelerated instantly (model A) or accelerated by the electric field given by equation (2) (model B), and they are decelerated mainly by the curvature radiation. In model B the deceleration rate is insignificant near the polar cap because it increases with the electron energy and is small. On the other hand, the acceleration rate does not depend on the electron energy and it ceases at the height h∼r_pc (equation 2). There exists a region in between where both competing processes balance. An example calculation of the electron energy is shown in Figs 2 and 3. The top left panel of each figure presents electron energy within model A, with The bottom left panel shows the electron energy within model B; the electron reaches a maximum energy of a few TeV at the height h≃4 ×10⁴cm (i.e. hr_pc≃1/6).

Figure 2.

The thick solid lines in the left panels show the dependence of the mean energy of unscattered primary electrons as a function of the height. The top left panel corresponds to the case of instantaneous acceleration (model A), while the bottom left panel corresponds to case of accleration described by equation (2) (model B). The dots show the energies of the Compton upscattered photons and height where the scatterings took place (in these simulations we propagated 10⁵ primary electrons in the magnetosphere). The thin solid line shows the optical depth to magnetic absorption equal to unity, according to equation (11). The right panels show the energy spectra formed for model A (top right panel) and model B (bottom right panel). The soft photons originate on the surface of the neutron star with temperature T_surf=5 ×10⁵K. The CR and the ICS spectral components are the result of a single primary electron injected at the outer rim of the polar cap of the standard millisecond pulsar. We normalize the spectra with - maximum energy of the electron, mediated between acceleration and cooling rates. For the top left panel, is equal to the initial energy Pulsar parameters are B_pc=10⁹G, and P=3ms and for model A.

Figure 3.

Same as Fig. 2 but for the case of thermal emission from the polar cap with the temperature 3 ×10⁶K.

The optical depth τ_e for non-magnetic scattering of an electron is small (Fig. 1) and thus scattering is not a relevant deceleration process of electrons. The optical depth for scattering scales as ∝T², where T is the temperature of the radiation field. There are several factors that influence the scattering rate as a function of the height above the surface of the neutron star h. The photon density decreases with height, but the electron spends much more time in the upper part of the magnetosphere, thus increasing the chance of scattering there. We find that the latter effect is dominant and most of the scatterings take place in the upper part of the magnetosphere between 10⁵ and 10⁷cm, where the electron energy is close to the value of a few TeV. This is illustrated in Figs 2 and 3 (left panels), where the electron energy is plotted as a function of h. We show the energies of the ICS photons and the locations of the scattering events. The intrinsic spectrum of the ICS photons has a width of about one decade in energy centred around a few TeV, corresponding to the range of electron energies in the region where most of the scattering occurs. In Fig. 4 we illustrate the effects of reprocessing in the pulsar magnetosphere (taking model A as an example) and present both the intrinsic spectrum of ICS photons (dotted line) and the spectrum after reprocessing because of magnetic absorption (solid line). Magnetic absorption removes the high-energy part of the intrinsic ICS spectral peak, decreasing the strength of the peak by a factor of more than 10, and redistributing the ICS photon energy to the lower part of the spectrum. The ICS photons subject to magnetic absorption reprocessing are located in Figs 2 and 3 (left panels) above the thin line corresponding to τ_γB=1 (equation 11). In the case when the soft photon source is the hot polar cap most of the ICS photons will be reprocessed; see Fig. 3. However, in the case when the photon source is a warm surface of a neutron star most of the scattering events take place at a distance of a few neutron star radii from the surface and most of the ICS photons will escape freely; see Fig. 2. The escaping ICS photons have typical energies equal to the electron energy at the height of few neutron star radii above the stellar surface and thus the spectrum of the ICS photons exhibits a strong peak at the energy of ≈1TeV. The height of the ICS peak depends on the number of the soft thermal photons in the magnetosphere, and so it is directly related to the temperature and the radiating area (polar cap or entire stellar surface).

Figure 4.

The dotted line shows the intrinsic spectrum of the inverse Compton photons, i.e. with transfer effects neglected. The solid line shows the spectrum of magnetically reprocessed photons as seen by an external observer. Here we assumed: thermal emission from a polar cap with the temperature T_cap=3 ×10⁶K, surface magnetic field B=10⁹G, pulsar period P=3 ×10⁻³s, and instantaneous electron acceleration.

Another factor that influences the scattering rate is the soft photon flux anisotropy. Such anisotropy results from opacity angular dependence in strong magnetic field, and cannot be neglected in the case of classical pulsars with B ≈10¹²G. Detailed models of magnetized atmospheres show that in the case of 10⁹-G fields, the soft photon flux is almost constant up to about the angle of 60° with respect to the magnetic field axis (Pavlov et al. 1994). This result justifies our simplified treatment of the soft photon field as being isotropic.

The dominant spectral component in the energy range from 10MeV out to its cut-off at ≳100GeV is the result of curvature radiation (see the energy spectra in Figs 2 and 3, right panels). This cut-off energy is significantly higher than that expected in some classical pulsars as well as that measured in some of them ( ≈10GeV, Thompson et al. 1997). This is because of smaller curvature radii implying higher values of the characteristic photon energy, and also much weaker magnetic absorption (γB→e ^±). The slope α of radiation energy spectrum per logarithmic energy bandwidth (E²dN/dE∝E^α) below ε_break ≈100MeV is insensitive to the model of acceleration, and is close to 4/3 (see Rudak & Dyks 1999 for discussion of ε_break). Above ∼100MeV, the spectrum assumes a slope which depends on the acceleration model. For model A the slope between 100MeV and 10GeV is 1/3, in agreement with simple analytical estimates for a monoenergetic source function of beam particles. The slope for model B was found numerically to be equal to 0.84 (Bulik & Rudak 1999). The electron acceleration in model B makes the electron spectrum harder in comparison to the case of model A, which in consequence leads to a harder CR spectrum. Note that the total energy contained in gamma rays is larger in model B than in model A (compare right-hand panels of either Fig. 2 or Fig. 3). In model A the initial energy of electrons is large enough to enable pair creation, while in model B they need to overcome the CR losses to attain The work done by the electric field on the electron in model B is about 20 times larger than

Discussion

We calculated energy spectra of pulsed HE and VHE emission from millisecond pulsars for two models of beam particles acceleration. The spectra are a superposition of curvature (CR), synchrotron (SR), and Compton upscattering (ICS) components. The curvature component dominates the region between 1MeV and 100GeV. The slope of the spectrum in the range between 100MeV and 10GeV is sensitive to the details of the electron acceleration process. The synchrotron component becomes important only below ∼1MeV (Rudak & Dyks 1999). The ICS component has a narrow peak around 1TeV and its strength is related to the number density of the soft photons in the magnetosphere. Two sources of soft photons were considered: the hot polar cap with a uniform temperature T_pc ≳10⁶K, and the entire surface of the neutron star, with a lower temperature, T_surf ≳10⁵K.

Several new gamma-ray telescopes will be operational in the near future (Catanese & Weekes 1999) and thus it is interesting to discuss the observability of the HE radiation from millisecond pulsars. For our ranking we have selected the millisecond pulsars from the pulsar catalogue of Taylor et al. (1995) using the following criteria: P<60ms and For each pulsar the expected integral fluxes above 100MeV and above 100GeV was then calculated. Model A had been chosen for these calculations as a more conservative one. Moreover, two cases were considered for each pulsar: in addition to the case with the energy of injected primary electrons being chosen according to Section 2.2 (see also equation 8 of Rudak & Dyks 1998), we also allowed the electron energy to be two times higher. This was done to present how sensitive the results are to our choice of the electron's energy. We used the results of Rudak & Dyks (1998) to callibrate the fluxes of primary electrons (note that the spectra presented in Figs 2 and 3, are per single electron).

Ten best candidates in each band are presented in Figs 5 and 6 respectively. The first two candidates for GLAST observations (Fig. 5), B1534+12 and B1821−24, are objects with relatively high magnetic fields, which place them in a transition region between the domains of classical and millisecond pulsars. Because of the magnetic field strength and consequently high magnetic absorption in these two objects, the high-energy cut-off lies below 100GeV, and therefore they disappear from the ranking of Fig. 6.

Figure 5.

Millisecond pulsars ranked according to their gamma ray fluxes above 100MeV expected within model A. Each vertical bar spans the range in the flux when the energy of injected primary electrons varies by a factor of two. The horizontal dashed line represents the GLAST sensitivity.

Figure 6.

Similar to Fig. 5 but the ranking is for gamma-ray fluxes above 100GeV, and the horizontal dashed line represents the MAGIC sensitivity.

The third best candidate for GLAST observations is J0437−4715, a pulsar located very close to the Earth at the distance of 140pc, with a spin period of P=5.75ms, and the magnetic field at the polar cap B_pc=7.4 ×10⁸G inferred from kinematically corrected Ṗ (Camilo et al. 1994). J0437−4715 is also our primary candidate for observations in the range above 100GeV with the future Cherenkov telescopes. The integral photon spectra of J0437−4715 obtained in the course of calculations for the purposes of the ranking are presented in Fig. 7 along with the sensitivities of GLAST, CANGAROO, and MAGIC in its ‘large zenith angle’ mode. The expected photon flux in the EGRET energy range (i.e. between 0.1 and 10GeV) is between ≈2.0 ×10⁻⁹ and ≈2 ×10⁻⁸cm⁻²s⁻¹ (for and respectively) and falls below the upper limit of 2.1 ×10⁻⁷cm⁻²s⁻¹ placed with EGRET for this object (Fierro et al. 1995). Fig. 7 shows that the flux expected from J0437−4715 in the GLAST energy range (0.1 to 200GeV) is above its sensitivity limit of 3 ×10⁻⁹cm⁻²s⁻¹ (Kamae et al. 2000).

Figure 7.

Integral spectral fluxes of photons calculated for J0437−4715 at a distance of 140pc. Model A is used with two following values of : (1) 1.54 ×10⁷MeV (after Rudak & Dyks 1998, see also Section 2.2); (2) a value a factor of 2 higher, i.e. 3.08 ×10⁷MeV. These two values give the spectra (solid lines) which limit the shaded region from the bottom and the top, respectively. The soft photons used for ICS come from the entire stellar surface with the temperature 4 ×10⁵K (Edelstein et al. 1995). Sensitivities of several high-energy detectors are also indicated.

In the list of potential targets for atmospheric Cherenkov telescopes, presented in Fig. 6, J0437−4715 clearly stands out as the strongest source, mainly because of its proximity. The next source B1257+12 is also rather strong and should be detected by the telescopes in the northern hemisphere like MAGIC and VERITAS, and also it could be seen by CANGAROO-III. The remaining objects have the expected fluxes between a few times 10⁻¹³ and a few times 10⁻¹²cm⁻²s⁻¹, which places them in a class of possible but uncertain targets.

The 100GeV part of J0437−4715 spectrum can be probed by Cherenkov telescopes in the southern hemisphere like CANGAROO-III. This object should be also visible just above the horizon by the MAGIC telescope in the Canary Islands. The expected photon flux above 100GeV is ≈10⁻¹⁰cm⁻²s⁻¹ which is almost two orders of magnitude above the sensitivity of MAGIC in its ‘large zenith angle’ mode (Blanch et al. 1998). MAGIC may also probe the ICS component of the spectrum (above 0.5TeV) provided the energy of beam particles (i.e. primary electrons) is higher than assumed in our model.

Our ranking list of Fig. 5 also includes J0218+4232, a pulsar with a likely detection by EGRET at about 3σ level (Verbunt et al. 1996; Kuiper et al. 2000). Our theoretical gamma-ray flux for this object is a factor of up to 10 below the expected GLAST sensitivity, mainly because of the large distance. Thus within the model presented above we remain skeptical of this report about its positive detection, knowing that the gamma-ray source is coincident with an AGN.

Millisecond pulsars are potentially very interesting sources of high-energy gamma-rays. The gamma-ray radiation, if detected, will provide useful information about physical conditions in a pulsar magnetosphere: the primary electron energy, the nature of the electric field accelerating the primary electrons, and indirectly soft photon density. We appeal to include the millisecond pulsars into target lists for the upcoming gamma-ray observatories, both space- and ground-based.

Acknowledgments

This reasearch has been supported by KBN grant 2P03D02117. We thank Gottfried Kannbach for useful discussions on gamma-ray detectors. Insightful comments and suggestions made by an anonymous referee allowed us to improve considerably the presentation of the paper.