Addendum 7: Stellar Death, Neutron Stars/Pulsars (Chapter 18)
First define some constants and dimensional units needed below
1. Rotational period vs. radius for a spinning star. As a star contracts to a white dwarf or neturon star, it conserves its spin angular momentum L:
where I is the moment of inertia. For a uniform density sphere:
So the spin angular momentum is:
If the star doesn't lose mass, this means that the ratio R2/P is a constant as it contracts.
Example 1: Estimate the spin period of the Sun after it becomes a white dwarf. The present spin period is P ~ 25 days; assumew the white dwarf radius is Rwd ~ 1 Re.
Example 2: A star with an initial spin spin similar ot the Sun collapses to a pulsar (neutron star, radius ~ 10km). What is its expected spin period?
Note: the fastest observed spin period of a pulsar is ~1 ms.
4. Spin-down rate of a pulsar. As a pulsar emits radiation, the energy is extracted from its rotation. By observing the luminosity of a pulsar's nebula (powered by the directed 'searchlight' beams of the pulsar), we can estimated a rate at whichteh pulsar is slowing down.
The rotational energy of a uniform spinning sphere is:
The luminosity of the nebula L is of energy emitted per unit time, which must be extracted from the pulsar's rotation:
Solving for the rate of period change we obtain:
Example: The Crab nebula has a luminosity L = 2 x 1031 W. What is the expected period change of the Crab pulsar (observed period 33 ms, assumed radius 10 km, mass 1.5 solar masses) in sec/sec?
Note: the observed rate is 4.2x10-13 s/s; not all the energy released from spin-down is converted to luminous energy