The energy balance of the Crab pulsar and its nebula

The measured spin period of the Crab pulsar and its derivative are Pp = 33.41 ms and Pp = 4.228 x 10"13, which correspond to the current spin frequency Qp = 2n/Pp = 188 s"1 and Qp = —2.38 x 10"9 s"2. The pulsar spin energy powers the acceleration of the nebula, the emission of electromagnetic waves, and the sweeping away the interstellar matter. The value of Q can be related to Q by

where K and n are constants determined from the pulsar timing (see, e.g., Shapiro & Teukolsky 1983; also see § 1.4.4). For the Crab pulsar, we get K = 4.68 x 10"15 (in CGS units). The braking index n is obtained through the measurable timing parameters Q, Q, and Q as n = QQ/Q2 . From the timing of the Crab pulsar in the period from 1982 to 1987 one has n = 2.509 ± 0.001 (Lyne et al., 1988). We adopt the standard assumption that n and K are fixed after the pulsar formation (e.g., in less than a few months after the neutron star birth). In what follows we will count the pulsar age from that moment. The integration of Eq. (9.30) from t = 0 gives the well known pulsar spindown law

where Qo is the initial spin frequency. Putting the current pulsar age tp = 950 years we get Q0 = 326 s"1 and the initial period P0 = 2n/Q0 = 19.3 ms. The spin-down energy loss rate is

where I is independent of Q because Q is much smaller than the mass-shedding limit (§ 6.10). The pulsar spin energy transforms into electromagnetic radiation of the nebula (with the luminosity Erad) and into the accelerated expansion of the nebula. When calculating the power of this expansion, EEexp, one should take into account that the nebula sweeps the interstellar medium:

the integration goes over the nebula volume V. The nebula mass Mneb is thought to be mostly contained in luminous filaments. The velocity vectors

Figure 9.9. Expansion of the Crab Nebula. Arrows show motion of 50 optical filaments in the next 250 years at the current expansion speed. From Nugent (1998), with the kind permission of the author.

of the filaments are nearly radial and have approximately the same magnitude (Fig. 9.9). Treating v as constant in space, we get

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