Question

Suppose the two stars in a binary system have masses of 1.5 and \(2.5 \mathrm{M}_{\odot} .\) What will happen to the separation of the two stars if the \(1.5 \mathrm{M}_{\odot}\) star sheds mass onto the \(2.5 \mathrm{M}_{\odot}\) star?

Short answer

The separation between the stars increases.

Step-by-step solution

Understanding the System

The binary system consists of two stars with masses 1.5 \(\mathrm{M}_{\odot}\) and 2.5 \(\mathrm{M}_{\odot}\). Initially, the gravitational force holds them in orbit around their common center of mass.

Analyzing Mass Transfer

When the \(1.5 \, \mathrm{M}_{\odot}\) star sheds mass onto the \(2.5 \, \mathrm{M}_{\odot}\) star, the total mass of the system remains constant but the distribution of mass changes.

Applying Conservation of Momentum

Since the center of mass of the system must remain in the same position, any change in mass distribution will affect the orbital separation. According to Kepler's laws and the conservation of angular momentum, the separation between the two stars will change.

Effect on Orbital Separation

As the \(1.5 \, \mathrm{M}_{\odot}\) star loses mass, it becomes less massive, and the more massive \(2.5 \, \mathrm{M}_{\odot}\) star gains that mass. As a result, the orbital separation between the two stars increases to maintain the conservation of angular momentum.

Key concepts

Mass Transfer in Binary Star Systems

In many binary star systems, mass transfer plays a significant role in their evolution. This process involves one star losing mass to its companion. When mass is transferred, it typically occurs because one of the stars expands during its life and reaches a point where its gravity cannot hold onto its outer layers, often when it becomes a red giant.
In the case of our binary system, the smaller star, originally at 1.5 \(M_{\odot}\), loses some of its mass to the larger star, which is 2.5 \(M_{\odot}\). This transfer of mass results in changes not just to the individual stars but to the binary system as a whole.
For students studying astrophysics, understanding mass transfer is crucial because it's a central mechanism in the life and death of stars. Through mass transfer, stars can change dramatically, possibly igniting new nuclear reactions, altering their lifespan, and influencing their fate.

Orbital Mechanics of Binary Systems

Orbital mechanics, the study of how objects move under the influence of gravitational forces, is essential for understanding binary star systems. Binary systems are two stars orbiting a common center of mass, also known as the barycenter. Each star's path is dictated by the gravitational pull of the other star as well as their mutual center of mass.
In our example, mass transfer alters the dynamics of these orbits. Once the mass of one star changes, the center of mass can shift.
This shift may lead to changes in the orbits. Because stars in a binary system are bound by mutual gravity, the mass transfer affects their relative separations and velocities. It's these changes that students must understand to predict movements within binary systems accurately.
The principles of orbital mechanics are governed by Kepler's laws and Newtonian mechanics, providing a mathematical description of how celestial bodies, such as stars in a binary system, interact.

Conservation of Angular Momentum

The principle of conservation of angular momentum is a key concept in physics, especially in celestial mechanics. In a binary star system, angular momentum must be conserved when mass is transferred between stars.
Angular momentum is a measure of the quantity of rotation of a body, and it heavily influences the motion in an orbiting system. For our binary stars, when the 1.5 \(M_{\odot}\) star transfers mass to the 2.5 \(M_{\odot}\) star, the way this momentum distributes can change. However, the total angular momentum of the system remains constant.
This means that as mass is transferred and the primary star becomes less massive, the system compensates by increasing the distance between the two stars. As the separation between the stars increases, the system can maintain its conserved angular momentum without losing or gaining energy.
Such changes are vital for students examining how binary star systems evolve and how they maintain stability despite internal changes.