Neutron stars are what are thought to cause pulsars. A pulsar is an astronomical radio source
that emits pulses of radiation, as seen from earth. The regularity of the pulses initially caused
the discoverers Jocelyn Bell Burnell and Antony Hewish to speculate that the signals were from
another civilization. The pulses were so regularly spaced that it was thought that pulsars could
be used as a time standard. However, it was soon seen that the rate of pulse emission slows, but
then abruptly increases in events known as glitches.
The current model (highly simplified) for a pulsar is a neutron star, one in which the electrons
have been pushed down into the nucleuses by extreme gravitational forces to combine with the
protons to form neutrons. Material falling onto the surface of the neutron star gets superheated
and shoots off in the form of a jet. If the jet doesn’t coincide with the rotational axis of the star,
the jet will precess around the axis. If the earth happens to lie on the cone of precession,
radiation will arrive at the earth and the pulsar can be detected. The pulsing effect is due to the
fact that the jet only points at the earth once per precession. An analogy is that of a lighthouse;
although the beam is always on, one can see the beam only when it is pointing directly at the
observer.
The slow decrease in pulse rate is due to the fact that the pulsar is emitting energy, some of
which comes from the kinetic energy of rotation. As the star slows, the crust of the star is placed
under stress. When the strain becomes too much, the surface breaks and resettles at a smaller
distance from the center; this is called a star�quake. Since the mass of the star is now closer to
the axis of rotation, the moment of inertia decreases, and, much like the skater pulling in her
arms, the angular speed increases.
A neutron star with angular speed �‰i = 70.4 rad/sec experiences a glitch such that its angular
speed increase by a factor Î��‰/�‰i = 2.01ï‚´10â€�6. If the initial radius were 11km, by how much was
the star’s radius decreased?
Assume that the star is a uniform sphere if you like, although this assumption is not really
necessary
The Correct Answer and Explanation is:
Answer:
We are given:
- Initial angular speed: ωi=70.4 rad/s\omega_i = 70.4 \, \text{rad/s}
- Relative increase in angular speed: Δωωi=2.01×10−6\frac{\Delta \omega}{\omega_i} = 2.01 \times 10^{-6}
- Initial radius: Ri=11 km=11,000 mR_i = 11 \, \text{km} = 11,000 \, \text{m}
- Model: Uniform solid sphere
For a rotating body, conservation of angular momentum applies when no external torques act: Iiωi=IfωfI_i \omega_i = I_f \omega_f
For a uniform sphere, the moment of inertia: I=25MR2I = \frac{2}{5} M R^2
So: 25MRi2ωi=25MRf2ωf⇒Ri2ωi=Rf2ωf\frac{2}{5} M R_i^2 \omega_i = \frac{2}{5} M R_f^2 \omega_f \Rightarrow R_i^2 \omega_i = R_f^2 \omega_f
We define the change in angular speed: ωf=ωi+Δω=ωi(1+Δωωi)=ωi(1+2.01×10−6)\omega_f = \omega_i + \Delta \omega = \omega_i (1 + \frac{\Delta \omega}{\omega_i}) = \omega_i (1 + 2.01 \times 10^{-6})
Substitute into conservation equation: Ri2ωi=Rf2ωi(1+2.01×10−6)⇒Rf2Ri2=11+2.01×10−6R_i^2 \omega_i = R_f^2 \omega_i (1 + 2.01 \times 10^{-6}) \Rightarrow \frac{R_f^2}{R_i^2} = \frac{1}{1 + 2.01 \times 10^{-6}}
Take square root: RfRi=(1+2.01×10−6)−1/2≈1−12(2.01×10−6)=1−1.005×10−6\frac{R_f}{R_i} = \left(1 + 2.01 \times 10^{-6}\right)^{-1/2} \approx 1 – \frac{1}{2} (2.01 \times 10^{-6}) = 1 – 1.005 \times 10^{-6}
Now calculate radius decrease: ΔR=Ri−Rf=Ri(1−RfRi)=11,000×1.005×10−6=0.011055 m≈1.11 cm\Delta R = R_i – R_f = R_i \left(1 – \frac{R_f}{R_i}\right) = 11,000 \times 1.005 \times 10^{-6} = 0.011055 \, \text{m} \approx \boxed{1.11 \, \text{cm}}
Explanation
Neutron stars are dense remnants of massive stars that have undergone a supernova explosion. Their extreme density means that matter is compressed to the point that electrons and protons merge to form neutrons, creating a city-sized object with a mass greater than the sun. Pulsars are a type of neutron star that emit beams of electromagnetic radiation. As the star rotates, if the beam of radiation points toward Earth during each rotation, it appears as a “pulse,” similar to the effect of a rotating lighthouse beam.
The pulses are generally very regular, but small irregularities known as “glitches” sometimes occur. These glitches are believed to result from sudden structural changes in the neutron star’s crust, which is under constant stress due to the immense gravitational and rotational forces. When the crust “cracks” and settles into a more compact configuration (called a starquake), the distribution of mass shifts closer to the rotation axis. As a result, the moment of inertia decreases.
Conservation of angular momentum dictates that if moment of inertia decreases, angular speed must increase. This is akin to a figure skater pulling in their arms to spin faster.
In this problem, we are given that the neutron star’s rotation rate increased slightly after a glitch. Using the conservation of angular momentum and modeling the neutron star as a solid sphere, we find that the star’s radius must have decreased slightly to account for the increased spin. The calculated decrease is approximately 1.11 cm, a small change in radius but significant enough to alter the rotational dynamics of such a massive and compact object. This underlines how sensitive the rotation of neutron stars is to changes in their internal structure.
