Question

The Crab Nebula One of the most studied objects in the night sky is the Crab Nebula. It is the remains of a supernova explosion observed by the Chinese in 1054. In 1968 it was discovered that a pulsar-a rapidly rotating neutron star that emits a pulse of radio waves with each revolution-lies near the center of the Crab Nebula. The amount of time required for each rotation of this pulsar is \(33 \mathrm{~ms}\). What is the angular speed (in \(\mathrm{rad} / \mathrm{s}\) ) of the pulsar?

Short answer

The angular speed of the pulsar is approximately 190.9 rad/s.

Step-by-step solution

Identify the given information

The time period for one complete rotation of the pulsar is given as \(33\, \mathrm{ms}\).

Convert time period to seconds

Since the time period is in milliseconds, we need to convert it to seconds to use it in calculations. \[33\, \mathrm{ms} = 33 \times 10^{-3}\, \mathrm{s} = 0.033\, \mathrm{s} \]

Understand the relationship between time period and angular speed

The angular speed \(\omega\) is related to the time period \(T\) by the formula: \[ \omega = \frac{2\pi}{T} \] where \(T\) is the time taken for one complete revolution.

Calculate the angular speed

Substitute the value of \(T = 0.033\, \mathrm{s}\) into the formula:\[ \omega = \frac{2\pi}{0.033} \approx 190.9\, \mathrm{rad/s} \]

Key concepts

Crab Nebula

In the vast expanses of space, the Crab Nebula stands out as one of the most intriguing objects. It is a cloud-like remnant of a supernova explosion, observed by Chinese astronomers in the year 1054. This fascinating nebula is visible through telescopes due to its brightness and the way it illuminates the surrounding space.

Here are some interesting facts:

• The Crab Nebula is approximately 6,500 light-years away from Earth.
• It is located in the constellation of Taurus.
• Despite its distance, it can be studied to understand the life cycle of stars.

The nebula is composed of gas and dust that are the remnants of the star that exploded. It continues expanding at an incredible speed, shedding light on the dynamics of an explosion's aftermath.

Supernova Explosion

A supernova explosion is one of the most powerful events in the universe. It happens when a star exhausts its nuclear fuel and its core collapses under gravity, often leading to an enormous explosion. This process scatters the outer layers of the star into space.

The 1054 supernova that created the Crab Nebula is a classic example:

• When the explosion occurred, it was so bright that it could be seen in daylight for weeks.
• Supernovae play a crucial role in dispersing elements across the cosmos.
• They are vital for the creation of elements heavier than iron found in the universe.

The energy from the explosion can also trigger the formation of new stars and solar systems, making these events critical to the dream-like recycling of cosmic materials.

Pulsar

At the heart of the Crab Nebula lies a pulsar, a highly magnetic and rapidly spinning neutron star. Pulsars are known for their regular pulses of radiation, like radio waves, emitted as they rotate. These pulses occur because beams of radiation are emitted from the magnetic poles of the star.

Key characteristics of pulsars include:

• They have extremely high densities, with a mass greater than that of the sun.
• The rotation period can be milliseconds or up to a few seconds, making them accurate cosmic clocks.
• Pulsars are born from the remnants of a supernova, in this case, from the one that created the Crab Nebula.

The pulsar at the center of the Crab Nebula rotates every 33 milliseconds. This rapid spin is one of the reasons the nebula continues to glow so brightly, as it releases energy into the surrounding cloud.

Neutron Star

A neutron star is an incredibly dense remnant left behind following the core collapse of a massive star during a supernova. What remains is primarily made up of tightly packed neutrons, hence the name. Neutron stars are among the smallest and densest stars known, with diameters just around 20 kilometers.

Important aspects of neutron stars include:

• Their gravitational pull is incredibly strong due to their compact nature.
• They often form part of binary systems, orbiting another star.
• Neutron stars can sometimes be strong sources of X-rays, due to high-energy emissions.

The neutron star that favors the Crab Nebula is also a pulsar. This demonstrates how these remnants of stellar evolution can continue impacting their surroundings long after the initial supernova explosion.