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Astron. Astrophys. 359, 242-250 (2000)

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2. The population synthesis

2.1. The rotation period evolution

The evolution of the rotation period of a neutron star, in the framework of the magnetic dipole model, is given by (Pacini 1968)


where µ = [FORMULA] is the surface magnetic moment, B is the magnetic field, Rp and I are respectively the radius and the moment of inertia of the neutron star, c is the velocity of light and [FORMULA] is the angle between the rotation axis and the magnetic dipole. Integration of this equation with constant [FORMULA] gives the rotation period evolution of the so-called standard model.

In the absence of abrupt changes of the magnetic torque or of the internal structure of the star, solutions of Eq. (1) indicates a smooth increase of the rotation period with time. However, most pulsars do not slow down regularly, but display variations in their spin rates in the form of glitches (Shemar & Lyne 1996) and timing noise. An important aspect of glitches is the persistent increase of the spin-down rate following these events, which could be due to a sudden and permanent increase of the external torque (Link & Epstein 1997). In order to explain the glitches observed in the spin evolution of the Crab pulsar, Alpar & Pines (1993) suggested a reduction of the moment of inertia, induced by changes in the internal structure of the star. This would produce a spin-up of the star due to angular momentum conservation, but a decrease in the spin-down rate, contrary to observations. As the star spins down, it becomes less oblate, inducing stresses that may lead to starquakes if the yield strength of the solid crust is exceeded. Starquakes may affect the braking mechanism if they are able to change the position of the magnetic moment with respect to the rotation axis. If the structural relaxation after a starquake occurs asymmetrically about the rotation axis due, for instance, to magnetic stresses, the figure and spin axes may become misaligned. Under this condition, the star precesses and relaxes to a new equilibrium state, corresponding to a new orientation of the magnetic dipole with respect to the rotation axis. This possibility was examined in a recent work by Link et al. (1998). In their picture, a spin-down pulsar reduces its equatorial radius by shearing material across the equator and moving material along faults to higher latitudes. Such crustal motions may produce an increasing misalignment between the rotation and magnetic axes, providing a natural explanation for the observed increases in the spin-down rate following glitches in the Crab, PSR 1830-08 and PSR 0355+54.

The migration of the magnetic dipole fromm an initial angle [FORMULA] to an orthogonal position with respect to the spin axis was also considered by Allen & Horvath (1997). They showed that with such a variable torque, non-canonical braking indices may be explained. Here we adopt the scenario developed by Link et al. (1998), assuming that the magnetic dipole migrates from an arbitrary initial angle to a final orthogonal position in a timescale [FORMULA]. For our present purposes, this migration is modeled by the equation


We emphasize that this equation is not the result of any physical model, but only a phenomenological representation of the above picture, in the sense that a given pulsar born with an initial angle [FORMULA], will reach an orthogonal configuration in a timescale [FORMULA].

Combining Eqs. (1) and (2) and integrating, we obtain for the evolution of the rotation period


where P0 is the initial rotation period and [FORMULA] = [FORMULA] is the magnetic braking timescale. For [FORMULA] we recover essentially the standard model evolution. However, Eq. (3) allows one to explain the existence of objects with short periods and with a relatively small deceleration rate, which would be the consequence of a small initial misalignment between the magnetic and the spin axes, but not due to a small magnetic field strength. Note that in our picture these pulsars are young, in spite of having a large indicative age ts = [FORMULA].

Another important parameter characterizing the evolution of the rotation period is the so-called braking index N defined as


For the standard magnetic dipole model, the expected braking index is N = 3. However, the few pulsars with N measured accurately show deviations from the Standard model; N=2.518 for the Crab, N=2.837 for PSR1509-58, N=2.04 for PSR0540-69. The exceptionally low value for the Vela pulsar N = 1.40 was recently reported, suggesting that the braking torque is not due to a "pure" stationary dipole field (de Freitas Pacheco & Horvath 1997). From Eqs. (3) and (4), the predicted braking index for our model is


This equation indicates that for young pulsars with ages less than [FORMULA] the braking index N may be less than three, tending to the canonical value when [FORMULA].

The angle [FORMULA] may vary "discontinuously" when the crust suddenly cracks or almost "continuously" as Eq. (2) suggest. It is worth mentioning that pulsars like PSR1509-58 and PSR0540-69 have not yet displayed any glitch activity even though their braking indices are less than the canonical value N = 3. This could be an indication that other braking mechanisms are operative such as, for instance, anomalous electro-magnetic torques recently discussed by Casini & Montemayor (1998).

For the Crab pulsar, since the true age, the braking index N and [FORMULA] are known, we can compute using the equations above the dipole migration rate [FORMULA], which is about 9.2[FORMULA] rad/yr for a solution where the present angle between the spin and the dipole axes is about 64o. From glitch data spanning a time interval of about 20 yr, Allen & Horvath (1997) find a migration rate 1.8[FORMULA] rad/yr, which is about five times smaller than the predicted value. However, as those authors remarked, such a time interval is still very short and discrepancies should be expected when a comparison is performed between the migration rate derived from the average of few events and that obtained from a continuous variation model. For the other remaining three pulsars, the average "continuous" migration rate is [FORMULA] = 6.5[FORMULA] rad/yr (Allen & Horvath 1997). These rates are comparable to within an order of magnitude, reinforcing the idea of a common origin and giving some support to our adopted spin-down model.

2.2. The method

Initial spin periods and magnetic fields are important parameters tracing the formation history of neutron stars and their interaction with the surrounding medium. The pulsar population study by Narayan (1987) raised the so-called "injection" problem, which would connect the initial period and magnetic field in the sense that slow born neutron stars would be associated with high field strengths. Unfortunately, the ages of most pulsars are unknown and present measurements for estimating the period and its time derivative are insufficient in order to estimate the initial field and spin period. In this case, methods of population synthesis can be useful tools to estimate statistical values of these parameters, assuming ad-hoc distribution laws. Moreover, the observed pulsar population is strongly dominated by luminous objects, and since the radio emission depends on the spin period as well as on the spin down rate, present samples are biased, excluding long-period and high-field objects as well as old and faint pulsars. Comparison with actual data requires the inclusion of these selection and observational bias in the generation of synthetic catalogs, which are more treatable when Monte Carlo techniques are employed.

In order to generate a synthetic sample of pulsars, we need to model: (a) their distribution in the Galaxy, (b) the initial parameters affecting their evolution, (c) the effects of the interstellar medium in their detection and, (d) the sensitivity of a given survey.

The distribution in the Galaxy was modeled by assuming a constant birthrate until a maximum age tmax, considered as a free parameter in our numerical experiments. The ratio between the total number Np of generate pulsars and tmax gives the mean birthrate. For a given experiment, Np is the number of input pulsars required to reproduce the number of objects in the data sample, when observational and selection constraints are imposed.

Pulsars are strongly concentrated in the galactic plane and their distribution along the z-axis can be represented by an exponential law with a scale of height of about 100 pc. Some previous studies suggest a possible correlation between the height above the galactic plane and age or magnetic field (Narayan & Ostriker 1990), but here we assume that the initial z-coordinate is a statistically independent variable. In the galactic plane, the density of pulsars probably decays exponentially, in the same way as population I objects. Recent galactic mass profile models, based on infrared data, suggest a short distance scale of about 2.3 kpc (Ruphy et al. 1996); this was adopted in our numerical experiments. In order to compute heliocentric distances, the galactocentric distance of the sun was assumed to be 8.5 kpc. Some pulsars have significant proper motions, implying large initial velocities that must be taken into account when computing distances, since they affect the orbital motion in the Galaxy. An isotropic Gaussian velocity distribution with 1-D velocity dispersion of 100 km s-1 (Lorimer et al. 1993) was adopted to simulate the pulsar kick velocity, but our results are not significantly changed if the velocity dispersion is increased up to 200 km s-1. Once the initial R and z coordinates for a given pulsar were established, if its kick velocity is zero, we assumed that the pulsar follows a circular orbit using the galactic rotation curve by Schmidt (1985). For non-zero kick velocity, the new energy and angular momentum of the orbit were computed and the object was settled in a new trajectory. The heliocentric distance is computed after a time t (the age of the pulsar), taking into account the differential rotation between the sun and the pulsar. The present galactic coordinates (l, b) of the object are also computed, since these are required to establish if the pulsar should be included or not in the area of the sky covered by a given survey, as we shall see later.

In the present simulations we have assumed that the initial rotation period of pulsars can be represented by a Gaussian distribution, characterized by a mean value P0 and a dispersion [FORMULA]. We have assumed that the break-up velocity is about 12000 rad s-1, thus pulsars generated with rotation periods less than 0.5 ms were not included in the simulated sample. These parameters were allowed to vary in order to verify their effects on the results of different numerical experiments. A similar procedure was adopted for the parameter [FORMULA] (see, Eq. (3)), which is related to the magnetic field of the pulsar, for which a log-normal distribution was assumed. Simulations by Bhattacharya et al. (1992) and by Mukherjee & Kembhavi (1997) suggest that the timescale for magnetic field decay is longer then the pulsar lifetime ([FORMULA] 100-160 Myr), thus we have assumed a constant field in our numerical experiments. The initial angle [FORMULA] (or, equivalently, the value of n0) between the magnetic dipole and the spin axes is calculated supposing a random distribution. Once these parameters are assigned for a given pulsar ([FORMULA] is kept fixed during a giving experiment), the period and the time derivative can be computed from Eq. (3).

The comparison between a simulated sample with actual data requires the evaluation of the expected radio flux density [FORMULA] at a given frequency (400 Mhz in general). To compute [FORMULA], one needs to know the distance and the luminosity of the pulsar. A theory able to predict the pulsar radio luminosity is still missing, but the spin period and its time derivative are expected to be the main variables on which the radio power depends, since the magnetic field can be expressed in terms of these variables in the standard model. Different power laws have been suggested in the literature connecting P and [FORMULA] (Lyne et al. 1975; Stollman 1987; Emmering & Chevalier 1989). When these laws are compared with observational data a large dispersion is found, which can be explained by a bad understanding of the radio emission mechanism and/or by errors in the distance, estimates of which are derived by using the dispersion of radio waves through the interstellar plasma and by adopting a model for the electron density distribution inside the galactic disk. The effects of the adopted luminosity law in numerical simulations were discussed by Lorimer et al. (1993), who concluded that, at the present time there appears to be no satisfactory model to account for observed pulsar luminosities.

The radio luminosity ought to be a decreasing function of the spin period, since slow pulsars missing in different surveys are probably too weak to be detected. Models assuming a radio luminosity proportional to the cube root of the spin-down rate lead to an inverse dependence on the period (Narayan 1987), while other proposed power laws indicate even higher exponents. In the present work, we have adopted a different empirical luminosity law, imposing the following exponential fate on the radio emission


The parameters in this relation were determined by an optimised fitting procedure, using the upgraded pulsar catalog of Taylor et al. (1993). If the luminosity is given in mJy.kpc2, and the period is in seconds, the resulting numerical values of the parameters are: A=3.3[FORMULA], [FORMULA] = 0.34 and [FORMULA] = 1.02. These figures are slightly different from those given by de Freitas Pacheco & Horvath (1997) because here we have used a larger sample in the fitting procedure.

The radio emission is not isotropic, implying that pulsars are not visible from all directions, and this imposes another constraint on their detectability. Following the approach used by Emmering & Chevalier (1989), which will be adopted here, the beaming fraction of the sky covered by the pulsar emission cone is


where [FORMULA] is the aperture of the emission cone. Since short period pulsars seem to have wider emission cones, we have adopted the beaming model proposed by Biggs (1990), which gives for the cone aperture


where [FORMULA] is in degrees and P is in seconds. It is worth mentioning that the study by Rankin (1990) of a large sample of pulsars also indicates a variation of [FORMULA] inversely proportional to the square root of the period.

Our data are a culled sample of 389 pulsars extracted from the catalog of Taylor et al. (1993), where only objects that have originated from population I stars and with all required parameters measured were considered. Pulsars not included in any of the original surveys were also discarded. Since this catalog is a compilation of different catalogs covering specific sky areas and having a well defined sensitivity, the objects were distributed in sub-classes, according to the original surveys in which they were first detected.

Each simulated catalog to be compared with these data is prepared with the following prescriptions. For each pulsar generated with a given set of initial parameters, we have followed its evolution in order to compute the present P, [FORMULA] values and galactic coordinates, according to the scheme described above. Then the radio luminosity and the flux density [FORMULA] = [FORMULA] are calculated, as well as the probability for the pulsar beam to cross the earth. The galactic coordinates define the survey (or surveys) included in the general catalog that covered this region. Then, we compare the expected flux density with the minimum detectable flux density of that survey through the relation (Dewey et al. 1984; Manchester et al. 1996)


where W is the observed (broadened) pulse width, Tr and Tsky are respectively the system and the sky background temperature in the considered line of sight. A0 is a normalization constant, defined as


In this equation, Ga is the antenna gain, np is the number of polarizations used, [FORMULA] is the receiver bandwidth, tint is the integration time, S/N is the signal-to-noise ratio and [FORMULA] is a numerical factor, including other loss processes. The variation of the sky temperature with the galactic coordinates was estimated using the relation by Narayan (1987). The intrinsic pulse width We was assumed to be equal to 4% of the period (We=0.04P) and to model the pulse broadening due to instrumental effects, dispersion and scattering, we follow the approach by Dewey et al.(1984) (see also Lorimer et al. 1993), writing for the relation between the observed and the intrinsic pulse width


where [FORMULA] is the effective sampling time, [FORMULA] is the broadening due to diffusion through the interstellar medium and [FORMULA] is the channel broadening. We have used the relation given by Bhattacharya et al. (1992) for [FORMULA], namely,


where DM is the electron dispersion measure in the line of sight of the pulsar in pc.cm-3. For the channel broadening, we have used the relation


where, following Stokes et al. (1986)


with Bch being the channel bandwidth and [FORMULA] the observing frequency. Table 1 gives the main parameters characterizing the different catalogs adopted in our simulations. Note that values inside braces correspond to high latitude data.


Table 1. Survey parameters

Our procedure insures that the simulated catalog will have the objects distributed in the same proportion as in the global catalog, taking into account the specific sensibility and the sky area covered by each survey. The numerical experiments generate objects according to these prescriptions until the simulated number of detectable pulsars is equal to the number in our data sample. For a given experiment, defined by a set of initial parameters, the number of runs is comparable to the number of objects in the sample and the final result is a suitable average, in order to avoid statistical fluctuations.

For each experiment so defined, we have compared the resulting distributions with those derived from actual data. We searched for optimize the various input parameters that describe the pulsar population, controlling the fit quality through [FORMULA] and Kolmogorov-Smirnov tests.

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Online publication: June 30, 2000