Problem 23 - Astronomy Journey to the Cosmic Frontier

Question

What happens to the rotation rate of a pulsar as time passes?

Step-by-Step Solution

Answer

A pulsar's rotation rate decreases over time due to energy loss from radiation.

Step 1: Understand Pulsar Rotation

Pulsars are rapidly rotating neutron stars emitting beams of radiation. As they rotate, these beams sweep across the Earth, seen as pulsating sources of light.

Step 2: Use Conservation of Angular Momentum

The principle of conservation of angular momentum states that if no external torque acts on a system, its angular momentum remains constant. However, pulsars gradually lose energy over time due to radiation.

Step 3: Consider Energy Loss

As a pulsar emits radiation, it loses energy, which translates to a decrease in rotational kinetic energy. This affects its angular velocity and thus its rotation rate.

Step 4: Determine Rotation Rate Change

Since the pulsar loses energy and its angular momentum remains constant, it must reduce its angular velocity (rotation rate) to compensate. Therefore, its rotation rate slows.

Key Concepts

Neutron Stars

Neutron stars are the remnants of massive stars that have undergone supernova explosions. Imagine a star that was initially many times larger than our sun collapsing under its own gravity. When this happens, the core compresses to an incredibly dense state, forming a neutron star.

• Neutron stars are only about 20 kilometers in diameter, roughly the size of a city.
• Despite their small size, they can have a mass about 1.4 times that of the Sun!
• Their density is so extreme that a sugar-cube-sized amount of neutron star material weighs as much as Mount Everest.

In the pulsar state, these neutron stars rotate rapidly, often completing several revolutions per second. This rapid spinning is a key feature that helps distinguish pulsars from other celestial objects. The beams of radiation emanating from pulsars make them appear as though they are "pulsing," which is how they got their name.

Conservation of Angular Momentum

The conservation of angular momentum is a fundamental principle in physics that affects rotating bodies, like pulsars. According to this principle, a system's angular momentum will remain constant unless acted upon by an external torque.

• Angular momentum is the rotational equivalent of linear momentum and depends on the object's mass, rotation speed, and radius.
• In the case of a pulsar, as long as there is no significant external force acting on it, its total angular momentum remains the same over time.
• The equation for angular momentum is given by: \[ L = I \cdot \omega \]where \( L \) is the angular momentum, \( I \) is the moment of inertia, and \( \omega \) is the angular velocity.

For a pulsar losing energy without losing mass, the principle of conservation leads to adjustments in its rotation rate to maintain constant angular momentum.

Angular Velocity

Angular velocity is a measure of how quickly an object rotates or revolves relative to another point. In the context of pulsars, it plays an essential role since it directly affects how often their beams sweep past our observation point on Earth.

• It is expressed in radians per second and influences the pulsar's rotation rate.
• As the rotational kinetic energy of a pulsar decreases due to energy loss, its angular velocity also decreases.
• The decrease in angular velocity leads to a slower rotation rate, causing the pulsar's pulses to appear less frequent over time.

The angular velocity relationship can be observed in the equation: \[ \omega = \frac{v}{r} \]where \( \omega \) is the angular velocity, \( v \) is linear speed, and \( r \) is the radius of rotation. This underscores how tightly intertwined angular velocity is with rotational energy and how it impacts pulsar observations.

Rotational Kinetic Energy

Rotational kinetic energy is the energy an object possesses due to its rotation. For pulsars, this energy is critical because it powers their emissions.

• The formula for rotational kinetic energy is: \[ KE_{rot} = \frac{1}{2} I \omega^2 \]where \( KE_{rot} \) is the rotational kinetic energy, \( I \) is the moment of inertia, and \( \omega \) is the angular velocity.
• As pulsars emit radiation, they lose rotational kinetic energy, resulting in a decrease in their angular velocity.
• This energy loss is gradual and causes the pulsar's rotation rate to slow down over time.

Understanding rotational kinetic energy helps explain why the pulsation rate of a pulsar gradually decreases, as this energy is a key factor in maintaining its rapid spin.