Pulsars. The maximum angular frequency of a spinning star can be found by equating the centripetal and gravitational acceleration M R 2 R 3 G M

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1 Pulsars Pulsating stars were discovered in 1967 via radio dipole antennae by Jocelyn Bell and Anthony Hewish Pulse period of PSR is s Most pulsars have periods between 0.25 s and 2 s The periods are exceptionally well defined Periods slow down at a rate of about Only spinning neutron stars can produce periods in this range Orbiting or pulsating stars are ruled out as are white dwarfs

2 Pulsars The maximum angular frequency of a spinning star can be found by equating the centripetal and gravitational acceleration 2 ω max R = G M R 2 This yields for the minimum period (for a 1.4 M sun neutron star) P min = 2 π R 3 G M ½ = s In addition to their rotational frequency, pulsars usually have a large relative velocity This can be explained by slightly asymmetric supernova explosions

3 Pulsars Young pulsars, like the Crab and the Vela pulsar, emit flashes also in the ranges from radio waves to gamma rays They also show sudden changes in their periods every few year, called glitches Vela pulsar glitches From P.M. McCulloch et al., Aust. J. Phys. 40, 725 (1987)

4 Synchrotron Radiation The Crab nebula from 1054 is closely related to the pulsar at its center The white glow was explained in 1953 by the Russian astronomer I. Shklovsky as synchrotron radiation Relativistic electrons spiral along magnetic field lines An accelerated charge emits radiation F Lorentz = q ( v B ) If the radiation is emitted primarily along the magnetic field lines, it is called curvature radiation Both radiations are linearly polarized and can be distinguished from blackbody radiation

5 Synchrotron Radiation The glow of the Crab nebula was identified as linearly polarized synchrotron radiation This requires a magnetic field of ~10-7 T to permeate the Crab nebula However, the expansion of the nebula should have weakened the field much more and electrons should have been depleted The energy needed to inject new electrons and power the magnetic field is supplied by the rotating neutron star Calculate the rate of energy loss I ω 2 K = = 2 d K dt = 2 π 2 I P 2-4 π 2 I Ṗ P 3

6 Synchrotron Radiation Assume the neutron star to be a uniform sphere of R = 10 km and 1.4 solar masses We obtain for the moment of inertia 5 M R 2 I = = kg m 2 2 Now using P = s and Ṗ = yields for the energy loss d K dt = W This turns out to be roughly the amount of energy need to power the expansion of the nebula

7 Pulsar Model Due to the rotation of the neutron star, the magnetic field changes rapidly at any point in space This will induce an electric field (Faraday)

8 Pulsar Model Far from the star (at the light cylinder) these varying electric and magnetic fields form an electromagnetic wave in form of magnetic dipole radiation The huge electric field at the surface of the neutron star is also responsible for removing charged particles from the star Electrons spiraling around the magnetic field lines cause the polarized synchrotron radiation responsible for the light glow of the nebula

9 Pulsar Model The radiation is in the form of magnetic dipole radiation The energy emitted per second by a rotating magnetic dipole is given by d E dt = - 32 π 5 B 2 R 6 sin 2 θ 3 µ 0 c 3 P 4 Assuming that all the rotational kinetic energy is lost via magnetic dipole radiation, we obtain for the magnetic field at the poles - 4 π 2 I Ṗ B = P 3 = 1 2 π R 3 sin θ - 32 π 5 B 2 R 6 sin 2 θ 3 µ 0 c 3 P 4 3 µ 0 c 3 I P Ṗ 2 π ½

10 Period Derivatives The distribution of period derivatives as a function of the period shows clustering The millisecond pulsars a re clearly separated from the ordinary pulsars They also coincide with pulsars in binary systems Some of the binary pulsars are actually speeding up with negative period derivatives

11 Pulsar Model Near the magnetic poles the main component is radiation along the magnetic field lines in form of gamma ray photons This curvature radiation is responsible for the sharply directed light rays The energetic gamma rays undergo pair production, which in turn leads to more gamma rays (cascading) Crab Pulsar HST & Chandra

12 Crab Pulsar Simplified sketch of the Crab pulsar Hester et al., Ap. J. 448, 240 (1985)