## 2. The population synthesis## 2.1. The rotation period evolutionThe evolution of the rotation period of a neutron star, in the framework of the magnetic dipole model, is given by (Pacini 1968) where 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 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 . 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 , will reach an orthogonal configuration in a timescale . Combining Eqs. (1) and (2) and integrating, we obtain for the evolution of the rotation period where P Another important parameter characterizing the evolution of the
rotation period is the so-called braking index For the standard magnetic dipole model, the expected braking index
is This equation indicates that for young pulsars with ages less than
the braking index The angle 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 For the Crab pulsar, since the true age, the braking index ## 2.2. The methodInitial 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 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 t 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 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 P The comparison between a simulated sample with actual data requires
the evaluation of the expected radio flux density
at a given frequency (400 Mhz in
general). To compute , 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 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.kpc 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 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 is in degrees and 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 where W is the observed (broadened) pulse width,
T In this equation, G where is the effective sampling time, is the broadening due to diffusion through the interstellar medium and is the channel broadening. We have used the relation given by Bhattacharya et al. (1992) for , namely, where DM is the electron dispersion measure in the line of sight of
the pulsar in pc.cm where, following Stokes et al. (1986) with B
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 and Kolmogorov-Smirnov tests. © European Southern Observatory (ESO) 2000 Online publication: June 30, 2000 |