Neutron star population dynamics

We study the Ðeld millisecond pulsar (MSP) population to infer its intrinsic distribution in spin period and luminosity and to determine its spatial distribution within the Galaxy. Our likelihood analysis on data from extant surveys (22 pulsars with periods less than 20 ms) accounts for the following important selection e†ects : (1) the survey sensitivity as a function of direction, spin period, and sky coverage ; (2) interstellar scintillation, which modulates the pulsed Ñux and causes a net increase in search volume of D30%; and (3) errors in the pulsar distance scale. Adopting power-law models (with cuto†s) for the intrinsic distributions, the analysis yields a minimum-period cuto† ms (99% conÐdence), a period distribution proportional to P&[ 0.65 P~2.0B0.33, and a pseudoluminosity distribution proportional to (where is the product of L p ~2.0B0.2 L p the Ñux density and the square of the distance, for mJy kpc2). L p o 1.1 We Ðnd that the column density of MSPs (uncorrected for beaming e†ects) is kpc~2 in the D50~20 `30 vicinity of the solar system. For a Gaussian model, the z scale height is kpc, corresponding to ~0.12 `0.16 the local number density kpc~3. (For an exponential model, the scale height becomes ~11 `17 ~0.13 `0.19 kpc, and the number density kpc~3.) Estimates of the total number of MSPs in the disk of the ~16 `25 Galaxy and for the associated birthrate are given. The contribution of a di†use halo-like component (tracing the Galactic spheroid, the halo, or the globular cluster density proÐle) to the local number density of MSPs is limited to of the midplane value. [1% We consider a kinematic model for the MSP spatial distribution in which objects in the disk are kicked once at birth and then orbit in a smooth Galactic potential, becoming dynamically well-mixed. The analysis yields a column density kpc~2 (comparable to the above), a birth z kick velocity ~17 `27 km s~1, and a three-dimensional velocity dispersion of D84 km s~1. MSP velocities are smaller ~11 `17 than those of young, long-period pulsars by about a factor of 5. The kinematic properties of the MSP population are discussed, including expected transverse motions, the occurrence of asymmetric drift, the shape of the velocity ellipsoid, and the z scale height at birth. If MSPs are long-lived, then a signiÐcant contribution to observed MSP z velocities is the result of di†usive processes that increase the scale height of old stellar populations ; our best estimate of the one-dimensional velocity kick that is unique to MSP evolution is D40 km s~1 if such di†usion is taken into account. The scale heights of millisecond pulsars and low-mass X-ray binaries are consistent, suggesting a common origin and that the primary channel for forming both classes of objects imparts only low velocities. Binaries involving a common envelope phase and a neutron starÈforming supernova explosion can yield such objects, even with explosion asymmetries like those needed to provide the velocity distribution of isolated, nonspun-up radio pulsars. Future searches for MSPs may be optimized using the model results. As an example, we give the expected number of detectable MSPs per beam area and the volumes of the Galaxy sampled per beam area for a hypothetical Green Bank Telescope all sky survey. Estimates for the volume that must be surveyed to Ðnd a pulsar faster than 1.5 ms are given. We also brieÑy discuss how selection e†ects associated with fast binaries inÑuence our results. Subject headings : pulsars : general È stars : kinematics È stars : neutron È stars : rotation È stars : statistics