Question

Most stars maintain an equilibrium size by balancing two forces - an inward gravitational force and an outward force resulting from the star's nuclear reactions. When the star's fuel is spent, there is no counterbalance to the gravitational force. Whatever material is remaining collapses in on itself. Stars about the same size as the Sun become white dwarfs, which glow from leftover heat. Stars that have about three times the mass of the Sun compact into neutron stars. And a star with a mass greater than three times the Sun's mass collapses into a single point, called a black hole. In most cases, protons and electrons are fused together to form neutrons-this is the reason for the name neutron star. Neutron stars rotate very fast because of the conservation of angular momentum. Imagine a star of mass \(5.00 \cdot 10^{30} \mathrm{~kg}\) and radius \(9.50 \cdot 10^{8} \mathrm{~m}\) that rotates once in 30.0 days. Suppose this star undergoes gravitational collapse to form a neutron star of radius \(10.0 \mathrm{~km} .\) Determine its rotation period.

Short answer

Answer: The rotation period of the neutron star after gravitational collapse is approximately \(_\Box_\) days.

Step-by-step solution

Recall the equation of angular momentum

The equation for angular momentum (L) is given by: $$L = Iω$$ where I is the moment of inertia and ω is the angular velocity.

Calculate the initial angular momentum

Before the collapse, the star has mass \(M_1 = 5.00 \cdot 10^{30} \mathrm{~kg}\), radius \(R_1 = 9.50 \cdot 10^{8} \mathrm{~m}\), and rotation period of \(P_1 = 30.0 \mathrm{~days}\). The moment of inertia for a sphere is given by: $$I_1 = \dfrac{2}{5}M_1R_1^2$$ To find the initial angular velocity, we can use the formula: $$ω_1 = \dfrac{2π}{P_1}$$ Note that we need to convert the period from days to seconds. The initial angular momentum, \(L_1\), is given by: $$L_1 = I_1ω_1$$

Calculate the final moment of inertia and angular momentum

After the collapse, the star has a new radius, \(R_2 = 10.0 \mathrm{~km}\). We assume that the mass remains the same during the collapse, so \(M_2 = M_1\). The final moment of inertia, \(I_2\), is given by: $$I_2 = \dfrac{2}{5}M_2R_2^2$$ Using the conservation of angular momentum, we know that the final angular momentum, \(L_2\), is equal to the initial angular momentum, \(L_1\) i.e. $$L_2 = L_1$$

Calculate the final angular velocity and rotation period

Now, we can find the final angular velocity, \(ω_2\), using the final angular momentum and moment of inertia: $$ω_2 = \dfrac{L_2}{I_2}$$ To find the final rotation period, \(P_2\), we use the equation: $$P_2 = \dfrac{2π}{ω_2}$$ Finally, we can convert the rotation period from seconds back to days.

Key concepts

Gravitational Collapse

When stars exhaust their nuclear fuel, they can no longer produce the outward pressure needed to counterbalance the inward pull of gravity. This imbalance leads to a natural process known as gravitational collapse. During this event, a star’s matter contracts towards its core. The outcome of the collapse depends on the mass of the original star.

• Stars similar in mass to our Sun transform into white dwarfs.
• Stars with masses greater than about three times that of the Sun can end their journey as neutron stars or, if massive enough, become black holes.

The immense pressure during collapse forces protons and electrons to combine, forming neutrons, thus making a neutron star extremely dense and compact.

Neutron Stars

Neutron stars are fascinating cosmic remnants that arise from the gravitational collapse of massive stars. Once the outer layers are expelled, the core—primarily composed of neutrons—remains. Neutron stars have some unique properties:

• They are incredibly dense; a sugar-cube-sized amount of neutron-star material would weigh about a billion tons on Earth.
• Despite having just 10-20 kilometers in radius, they contain more mass than our entire Sun.
• Neutron stars exhibit rapid rotation rates due to angular momentum conservation.
• Because of their extreme density, they also possess a very high surface gravity.

These stars' magnetic fields are billions of times stronger than that of Earth, contributing to phenomena like pulsars—neutron stars that emit beams of radiation.

Angular Momentum

Angular momentum is a fundamental concept in physics, crucial for understanding stellar behaviors post-collapse. It is a conserved quantity, meaning it remains constant in an isolated system. For a collapsing star, this means that the star's rotation speeds up as it shrinks.
In a simplified way, angular momentum can be imagined as a spinning ice skater pulling in their arms to spin faster. The formula for angular momentum is:\( L = I \omega \), where:

• \( L \) is the angular momentum,
• \( I \) is the moment of inertia,
• \( \omega \) is the angular velocity.

Stars conserve their angular momentum during collapse, so as their radius decreases drastically, their rotation speeds increase proportionally. This conservation helps maintain a balance even in massive cosmological events.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation. It plays an essential role when considering the dynamics of stellar collapse. The moment of inertia depends on the mass distribution of the object. For spherical objects like stars, it is calculated as:\[ I = \frac{2}{5}MR^2 \]This formula shows that if a star changes its radius while maintaining mass, its moment of inertia will change significantly, resulting in changes to its rotation speed.
When a massive star collapses into a neutron star, its radius shrinks but its mass remains almost the same. As the moment of inertia decreases, the angular velocity must increase to keep the angular momentum constant, leading to the neutron star spinning extremely fast. Understanding the balance between these quantities is vital for describing the mechanics of stellar evolution.