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Magazine Chapter 13: Problem 57
In a certain binary-star system, each star has the same mass as our Sun, and they revolve about their center of mass. The distance between them is the same as the distance between Earth and the Sun. What is their period of revolution in years?
Short Answer
Expert verified
The period of revolution is approximately 1.41 years.
Step by step solution
01
Understanding the Problem
We are given a binary-star system where each star has the same mass as the Sun, and the separation between them is equal to the distance between the Earth and the Sun. We need to find the period of revolution of these stars around their center of mass in years.
02
Applying Kepler's Third Law
From Kepler's Third Law for circular orbits, we know that the square of the period of revolution (\(T^2\)) is proportional to the cube of the semi-major axis (\(a^3\)) divided by the total mass of the system. For the Sun-Earth system, \[ T^2 = \frac{4\pi^2a^3}{G(M_1 + M_2)} \]where \(T\) = period of Earth's orbit (1 year), \(a\) = average distance from Earth to Sun (also considered as 1 AU), \(G\) = gravitational constant, \(M_1 = M_2 = M_{sun}\) = mass of each star (Sun's mass).
03
Expression for the Binary-Star System
In the binary-star system, both stars have mass \(M_{sun}\) and revolve around their center of mass. Because they have equal mass, the center of mass is at the midpoint of the distance between them, \(a\). Therefore, the semi-major axis \(a\) in terms of the binary system becomes half of the total distance, which is \(a = 1/2\) AU.
04
Calculated Expression for the Period
Substituting the values for the binary-star system into Kepler's Third Law:\[ T^2 = \frac{4\pi^2(0.5)^3}{G(2M_{sun})} \]Since the Sun-Earth system has a period \(T=1\) year and distance \(a =1\) AU, we simplify:\[ T_{binary}^2 = 2T_{Earth}^2 \]
05
Solving for T
The simplified form tells us \(T_{binary} = \sqrt{2} \times 1\) year. Calculating \(\sqrt{2}\) gives approximately 1.41.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binary Star System
A binary star system is where two stars orbit a common center of mass. This setup is common in our galaxy, with many stars found in pairs or larger groups. In a binary system, each star influences the other through gravity, causing them to orbit around a shared point. This center of mass serves as the balance point and is positioned based on the masses of the two stars. If both stars have equal mass, as in our example, the center of mass will be halfway between them. This arrangement is crucial for calculating their orbital characteristics, such as the period of revolution.
Orbital Period Calculation
Orbital period calculation involves determining how long an object takes to complete one full orbit around another body. In our scenario, we calculate the orbit of two stars in a binary system about their common center of mass. According to Kepler's Third Law, the period depends on the semi-major axis and the total mass of the system. For a binary star system where both stars have equal mass, the calculation simplifies, as the semi-major axis becomes half the distance between the stars. Using these knowns, we can use Kepler鈥檚 Law to find the period. This is critically important for understanding how such systems behave dynamically.
Gravitational Constant
The gravitational constant, denoted by the letter "G," is a fundamental constant in physics. It plays a crucial role in the equations of Kepler's Third Law, relating the masses involved and the distance between them to the gravitational forces. "G" is approximately equal to 6.674 脳 10^{-11} m^3 kg^{-1} s^{-2}. This value helps determine the strength of the gravitational force between two massive bodies. Understanding "G" is fundamental for astrophysics, as it enables us to calculate the forces and, subsequently, the motion within a binary star system. The gravitational pull dictates how stars within these systems orbit their center of mass.
Center of Mass
The center of mass in a binary star system is the point where the masses of both stars balance each other out. It's the weighted average location of the total mass. For two stars of identical mass, the center of mass sits precisely at the midpoint between them. This point is important because it's the focal point around which the stars orbit. Even though neither star is located at the center of mass, they are gravitationally bound to it. Determining this location is vital for understanding the dynamics of the orbit and influences how we calculate their orbital periods.
Semi-Major Axis
The semi-major axis is a key term in understanding orbital dynamics. For an elliptical orbit, it is half of the longest diameter of the ellipse. However, in the case of a circular orbit, which we consider for the binary star system, it equals the distance from the center to the perimeter. When assessing a binary system, the semi-major axis becomes half the distance between the two stars due to their equal mass. This simplification allows for easier calculations when applying Kepler's Third Law to find the orbital period. Grasping the concept of the semi-major axis is integral to understanding the structure and behavior of orbits in astronomy.
