Question

Two neutron stars are separated by a center-to-center distance of \(93.4 \mathrm{~km}\). They each have a mass of \(1.56 \times 10^{30} \mathrm{~kg}\) and a radius of \(12.6 \mathrm{~km}\). They are initially at rest with respect to one another. (a) How fast are they moving when their separation has decreased to one-half of its initial value? \((b)\) How fast are they moving just before they collide? Ignore relativistic effects.

Short answer

To find the velocity of the neutron stars when their separation decreases to half of its initial value, apply the conservation of energy. Use the formula for gravitational potential energy for the initial and final states. The velocities of the neutron stars before they collide can also be determined by similar means.

Step-by-step solution

Understand the Problem

Two neutron stars initially at rest are moving towards each other due to the gravitational force between them. We need to find their velocities at two points: when their separation is half of the initial distance and when they are about to collide.

Apply Conservation of Energy

We can use the law of conservation of energy to analyze this problem. The kinetic and potential energies of the neutron stars are changing, but their total energy remains constant. So we can set up the equation: E_initial = E_final, where E_initial = K_initial + U_initial and E_final = K_final + U_final. Initially, the stars are at rest so the kinetic energy K_initial is 0. The potential energy is given by: \(U_initial = -G \cdot m1 \cdot m2 / r_{initial}\) where G is the gravitational constant, m1 and m2 are the masses of the neutron stars, and \(r_{initial}\) is the initial distance between the stars.

Compute the Initial Potential Energy

Let's compute the initial potential energy using the values given. Remember to use the correct units when substituting into the formula.

Compute the Final Energy when Separation is Halved

When the separation is halved, we require the final state, having velocity v and separation half of the original distance. Hence, the final potential energy can be calculated as: \(U_final = -G \cdot m1 \cdot m2 / (r_{initial}/2)\). Now, we can compute velocities using the conservation of energy principle.

Compute the Final Energy when Stars are about to Collide

In this case, the separation between the stars is equal to the sum of their radii: r_final = r1 + r2. The final potential energy in this situation is thus \(U_final = -G \cdot m1 \cdot m2 / (r1+r2)\). By applying the conservation of energy principle, we can compute the velocity of the stars just before they collide.

Key concepts

Conservation of Energy

The law of conservation of energy states that within a closed system, energy cannot be created nor destroyed, only transformed from one form to another. This fundamental principle of classical mechanics is utilized to analyze and predict the motion of objects, including the spectacular dynamics of neutron star collisions.

In the problem involving the neutron stars, conservation of energy proves crucial in calculating their velocities as they approach each other. Although their gravitational potential energy decreases (becomes more negative), their kinetic energy increases to conserve the total energy of the system. This trade-off allows us to determine their velocities without having to trace the detailed path of their movement.

Gravitational Potential Energy

Gravitational potential energy (U) is the energy an object possesses because of its position in a gravitational field. It depends on the masses of the interacting bodies and the distance between them. The formula for the gravitational potential energy between two masses, such as neutron stars, is given by:
\[ U = -G \cdot m_1 \cdot m_2 / r \]
where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses, and \( r \) is the distance between their centers.

This negative energy reflects that work must be done against the gravitational force to separate the masses. In our neutron star scenario, as the stars spiral inward, the gravitational potential energy becomes more negative, thus releasing energy that is mostly transformed into kinetic energy and radiation.

Neutron Star Collision

A neutron star collision is an awe-inspiring cosmic event where two neutron stars spiral towards each other due to gravitational attraction. Over time, their rotation speed increases dramatically as they release gravitational waves and their separation decreases.

Understanding these events requires applying our knowledge of gravitational potential energy and the conservation of energy. As the neutron stars draw closer, their potential energy decreases as velocity increases, which is consistent with the conservation laws. These collisions are not only fascinating for astronomers but they also have significant implications for physics, such as the creation of heavy elements and the study of extreme states of matter.

Classical Mechanics

Classical mechanics is the branch of physics that deals with the motion of bodies under the influence of forces, without considering quantum effects. It is governed by Newton's laws of motion and is characterized by the predictable behavior of large objects at speeds much slower than the speed of light.

In the context of the neutron stars problem, we apply Newtonian gravity and conservation laws from classical mechanics to calculate the stars' motion. While neutron stars are dense and exhibit strong gravitational fields, the motion in this problem can still be described accurately without considering general relativity, provided that the velocities are significantly less than the speed of light, so relativistic effects can be neglected.