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Chapter 10: Problem 18

A pulsar is a rapidly rotating neutron star that emits a radio beam the way a lighthouse emits a light beam. We receive a radio pulse for each rotation of the star. The period \(T\) of rotation is found by measuring the time between pulses. The pulsar in the Crab nebula has a period of rotation of \(T=0.033 \mathrm{~s}\) that is increasing at the rate of \(1.26 \times 10^{-5} \mathrm{~s} / \mathrm{y} .\) (a) What is the pulsar's angular acceleration \(\alpha ?\) (b) If \(\alpha\) is constant, how many years from now will the pulsar stop rotating? (c) The pulsar originated in a supernova explosion seen in the year 1054 . Assuming constant \(\alpha\), find the initial \(T\).

Short Answer

Expert verified
(a) \( -0.000723 \ \mathrm{rad/s^2} \), (b) 288,270 years, (c) 0.025 seconds.

Step by step solution

01

Calculate Angular Acceleration 伪

The angular velocity \( \omega \) of the pulsar can be represented as \( \omega = \frac{2\pi}{T} \). The angular acceleration \( \alpha \) is the rate of change of angular velocity with time, which can be determined from the change in period \( T \): \( \alpha = \frac{d\omega}{dt} = -\frac{2\pi \cdot \text{rate of change of } T}{T^2} \). Given that \( \frac{dT}{dt} = 1.26 \times 10^{-5} \ \mathrm{s/year} \), plug in the values:\[ \alpha = -\frac{2 \pi \times 1.26 \times 10^{-5}}{(0.033)^2} \approx -0.000723 \ \mathrm{rad/s^2} \]
02

Find Time Until Pulsar Stops Rotating

The pulsar will stop rotating when its angular velocity \( \omega = 0 \). Initially, \( \omega_0 = \frac{2\pi}{T} \). Using the equation \( \omega = \omega_0 + \alpha \cdot t \), and setting \( \omega = 0 \), we find:\[ 0 = \frac{2\pi}{0.033} + (-0.000723) \cdot t \] Solving for \( t \), we get:\[ t = \frac{2\pi/0.033}{0.000723} \approx 288,270 \ \mathrm{years} \]
03

Calculate Initial Rotation Period T0

Given that the supernova was observed in 1054, and assuming no change in \( \alpha \), the time elapsed since then is \( t = 2023 - 1054 = 969 \ \mathrm{years} \). With \( \omega = \omega_0 + \alpha t \), solve for initial period \( T_0 \):\[ \frac{2\pi}{T} = \frac{2\pi}{T_0} + \alpha \cdot 969 \] Solving for \( T_0 \)\[ T_0 = \frac{2\pi}{\frac{2\pi}{0.033} + (0.000723 \times 969)} \approx 0.025 \ \mathrm{seconds} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pulsar
A pulsar is a fascinating celestial object that astronomers study to understand more about the universe. It is essentially a highly magnetized, rotating neutron star that emits a beam of electromagnetic radiation. This radiation is often visible as radio waves.

Pulsars are typically formed from the remnants of massive stars that have gone supernova. As they rotate, the beam of radiation that they emit can sweep across the Earth, which makes the pulsar appear like it is pulsing, hence the name.
  • Essentially, they are like cosmic lighthouses, giving off light at regular intervals.
  • The rotation period of a pulsar indicates how fast they spin, which can vary from milliseconds to seconds.
  • Pulsars are used as precise cosmic clocks due to their predictable timing.
Studying pulsars can provide significant insights into the behavior of neutron stars and the conditions of the interstellar medium.
Neutron Star
A neutron star forms when a massive star exhausts its nuclear fuel and undergoes a catastrophic collapse, a process known as a supernova. This results in an incredibly dense object, typically only about 20 kilometers in diameter but with a mass up to twice that of the sun.

These stars are composed nearly entirely of neutrons, which are subatomic particles with no charge. This gives them their extreme density and makes them fascinating objects in astrophysics.
  • They have a solid crust of atomic nuclei and a fluid core, primarily of neutrons.
  • Their magnetic fields are tremendously powerful, often billions of times stronger than Earth's.
  • This magnetic strength contributes to phenomena like pulsars.
The study of neutron stars helps scientists understand the extremes of matter and the nature of gravitational forces.
Crab Nebula
The Crab Nebula is a remnant of a supernova explosion that was observed on Earth in 1054 by Chinese and Arab astronomers. This explosion left behind the pulsar that we study today, which lies in the heart of the nebula. Located about 6,500 light-years away from Earth, the Crab Nebula is a beautiful and dynamic astronomical object.

It serves as an excellent example of the aftermath of stellar death and how materials are recycled to form new stars and planets.
  • The nebula's colorful tendrils are indicative of its rich chemical composition.
  • It contains synchrotron radiation, a process that accelerates electrons in the magnetic fields of the pulsar.
  • Being home to the Crab Pulsar, it is highly observed for both historical and scientific purposes.
The Crab Nebula provides a rich site for examining the consequences of supernovae and their contribution to the cosmos.
Angular Velocity
Angular velocity is a measure of how fast an object rotates or revolves relative to another point. For pulsars, this concept is crucial as it describes how rapidly the neutron star spins about its axis.

In the context of our exercise, angular velocity (\(\omega\)) is derived from the pulsar's rotation period (\(T\)) using the formula \(\omega = \frac{2\pi}{T} \).
  • It is expressed in radians per second (rad/s), providing a measure of rotational speed.
  • A change in angular velocity indicates the pulsar is either speeding up or slowing down, as seen by the angular acceleration (\(\alpha\)).
  • Pulsars gradually slow down over time due to loss of energy, a phenomenon observed from their increasing period \(T\).
Understanding angular velocity helps astronomers calculate how long it will take before a pulsar, like the one in the Crab Nebula, potentially stops spinning, and it assists in measuring the past behavior of rotating objects across the universe.

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Most popular questions from this chapter

(a) What is the angular speed \(\omega\) about the polar axis of a point on Earth's surface at latitude \(40^{\circ} \mathrm{N} ?\) (Earth rotates about that axis.) (b) What is the linear speed \(v\) of the point? What are (c) \(\omega\) and \((\mathrm{d}) v\) for a point at the equator?

A thin spherical shell has a radius of \(1.90 \mathrm{~m}\). An applied torque of \(960 \mathrm{~N} \cdot \mathrm{m}\) gives the shell an angular acceleration of \(6.20 \mathrm{rad} / \mathrm{s}^{2}\) about an axis through the center of the shell. What are (a) the rotational inertia of the shell about that axis and (b) the mass of the shell?

A gyroscope flywheel of radius \(2.83 \mathrm{~cm}\) is accelerated from rest at \(14.2 \mathrm{rad} / \mathrm{s}^{2}\) until its angular speed is 2760 rev/min. (a) What is the tangential acceleration of a point on the rim of the flywheel during this spin-up process? (b) What is the radial acceleration of this point when the flywheel is spinning at full speed? (c) Through what distance does a point on the rim move during the spin-up?

Starting from rest, a disk rotates about its central axis with constant angular acceleration. In \(5.0 \mathrm{~s}\), it rotates \(25 \mathrm{rad}\). During that time, what are the magnitudes of (a) the angular acceleration and (b) the average angular velocity? (c) What is the instantaneous angular velocity of the disk at the end of the \(5.0 \mathrm{~s}\) ? (d) With the angular acceleration unchanged, through what additional angle will the disk turn during the next \(5.0 \mathrm{~s}\) ?

At \(7: 14\) A.M. on June 30,1908 , a huge explosion occurred above remote central Siberia, at latitude \(61^{\circ} \mathrm{N}\) and longitude \(102^{\circ} \mathrm{E} ;\) the fireball thus created was the brightest flash seen by anyone before nuclear weapons. The Tunguska Event, which according to one chance witness "covered an enormous part of the sky," was probably the explosion of a stony asteroid about 140 \(\mathrm{m}\) wide. (a) Considering only Earth's rotation, determine how much later the asteroid would have had to arrive to put the explosion above Helsinki at longitude \(25^{\circ} \mathrm{E}\). This would have obliterated the city. (b) If the asteroid had, instead, been a metallic asteroid, it could have reached Earth's surface. How much later would such an asteroid have had to arrive to put the impact in the Atlantic Ocean at longitude \(20^{\circ} \mathrm{W} ?\) (The resulting tsunamis would have wiped out coastal civilization on both sides of the Atlantic.)
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