Problem 71 - University Physics with Modern Physics

Question

Binary Star-Equal Masses. Two identical stars with mass \(M\) orbit around their center of mass. Each orbit is circular and has radius \(R,\) so that the two stars are always on opposite sides of the circle. (a) Find the gravitational force of one star on the other. (b) Find the orbital speed of each star and the period of the orbit. (c) How much energy would be required to separate the two stars to infinity?

Step-by-Step Solution

Answer

(a) \( F = \frac{G M^2}{4R^2} \), (b) \( v = \sqrt{\frac{G M}{2R}} \), \( T = 2\pi \sqrt{\frac{2R^3}{G M}} \), (c) \( E = \frac{G M^2}{2R} \).

Step 1: Understanding the Force Between Stars

The gravitational force between two stars can be calculated using Newton's law of universal gravitation. The formula is: \[ F = \frac{G M^2}{(2R)^2} = \frac{G M^2}{4R^2} \] where \(G\) is the gravitational constant, and \(2R\) is the distance between the centers of the two stars.

Step 2: Calculating Orbital Speed

To find the orbital speed of each star, we use the centripetal force equation where the gravitational force provides the necessary centripetal force: \[ \frac{G M^2}{4R^2} = \frac{M v^2}{R} \] Solving for \(v\), the orbital speed, we get: \[ v = \sqrt{\frac{G M}{2R}} \].

Step 3: Determining the Period of Orbit

The period \(T\) can be found from the relationship between the circumference of the orbit and the orbital speed:\[ T = \frac{2\pi R}{v} \]Substituting \(v = \sqrt{\frac{G M}{2R}} \), we find: \[ T = \frac{2\pi R}{\sqrt{\frac{G M}{2R}}} = 2\pi \sqrt{\frac{2R^3}{G M}} \].

Step 4: Calculating Energy Required to Separate Stars

The energy required is equal to the gravitational potential energy. For two point masses, this is given by: \[ U = -\frac{G M^2}{2R} \]Thus, the energy needed to separate them is \(-U\), which becomes: \[ E = \frac{G M^2}{2R} \].

Key Concepts

Gravitational Force

In a binary star system, two stars exert a gravitational force on each other. This force is determined by Newton's law of universal gravitation. The gravitational force is the attractive force that pulls the two stars towards each other. It is governed by the equation:

• \( F = \frac{G M^2}{(2R)^2} = \frac{G M^2}{4R^2} \)

Here, \( G \) represents the gravitational constant, \( M \) is the mass of each star, and \( 2R \) is the distance separating the stars. The force arises from their mutual pull due to gravity.
This force helps maintain their stable orbit around the common center of mass. It ensures that despite their motion, they remain at a constant distance apart. Understanding this force is essential in studying celestial mechanics and dynamics of any two-body system.

Orbital Speed

The orbital speed of a star in a binary system is crucial for understanding its trajectory. The speed with which each star orbits the center of mass ensures that it remains in a stable circular orbit. This requires balancing the gravitational force and centripetal force.
The equation for orbital speed \( v \) in this system is derived from equating the gravitational force and the centripetal force:

• \( \frac{G M^2}{4R^2} = \frac{M v^2}{R} \)
• Solving for \( v \) gives: \( v = \sqrt{\frac{G M}{2R}} \)

The formula shows that the speed depends on the star's mass \( M \) and the radius \( R \) of the orbit. Understanding this concept helps in determining the motion of stars and planets in various gravitational systems.

Orbital Period

The orbital period is the time taken by a star to complete one full orbit around the center of mass. In our binary star system, this period provides insight into the dynamics and timing of the stars' movements.
To calculate the orbital period \( T \), we must consider the circumference of the orbit and the orbital speed:

• \( T = \frac{2\pi R}{v} \)
• Substituting \( v = \sqrt{\frac{G M}{2R}} \) results in: \( T = 2\pi \sqrt{\frac{2R^3}{G M}} \)

The period depends on the radius \( R \) and the mass \( M \), reflecting the time for gravitational interaction to occur over the pathway. Measuring the orbital period is fundamental in astronomy for predicting star positions and system behavior over time.

Gravitational Potential Energy

Gravitational potential energy in a binary star system relates to the work needed to separate the stars to infinity. Initially, the stars are bound by their gravitational attraction, with potential energy given by the formula:

• \( U = -\frac{G M^2}{2R} \)

This equation quantifies the energy due to their positions in the gravitational field. To separate the stars completely, an amount of energy equal to the magnitude of this potential energy needs to be provided.
The energy required to achieve this separation is:

• \( E = \frac{G M^2}{2R} \)

This highlights the strength of the gravitational bond between the two masses. Understanding gravitational potential energy helps explain phenomena like star system stability and dynamics when external forces are applied.