Binary star: Two equal-mass stars maintain a constant distance D apart (Fig. 6–34) of 8.0 x 10¹¹ m and revolve about a point midway between them at a rate of one revolution every 14.2 yr. Why don’t the two stars crash into one another due to the gravitational force between them?

Verified step by step guidance

1. The two stars do not crash into one another because they are in a stable orbit around their common center of mass. The gravitational force between the stars provides the necessary centripetal force to keep them in circular motion.
2. To analyze this, start by identifying the forces acting on one of the stars. The gravitational force between the two stars is given by Newton's law of gravitation: $F\; =\; G\; (m₁m₂)\; /\; r²$, where $G$ is the gravitational constant, $m₁$ and $m₂$ are the masses of the stars, and $r$ is the distance between them.
3. Next, recognize that the gravitational force acts as the centripetal force required for circular motion. The centripetal force is given by: $F\; =\; m\; v²\; /\; r$, where $m$ is the mass of one star, $v$ is its orbital speed, and $r$ is the radius of the orbit (which is half the distance $D$ between the stars).
4. Equate the gravitational force to the centripetal force to find the orbital speed: $G\; (m²)\; /\; r²\; =\; m\; v²\; /\; r$. Simplify to get: $v²\; =\; G\; m\; /\; r$. From this, you can calculate the orbital speed if needed.
5. Finally, use the relationship between orbital speed and the period of revolution to verify the stability of the orbit. The period $T$ is related to the orbital speed by: $v\; =\; 2\pi r\; /\; T$. Substitute the given period (14.2 years converted to seconds) and the radius ($r\; =\; D\; /\; 2$) to confirm that the forces balance and the orbit is stable.

Key Concepts

Gravitational Force

Gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. It states that the force is proportional to the product of the masses and inversely proportional to the square of the distance between their centers. In the case of binary stars, this force acts to pull the stars toward each other, but it is balanced by their orbital motion.

Orbital Motion

Orbital motion refers to the movement of an object in a curved path around a central point due to gravitational forces. In a binary star system, each star orbits the common center of mass, which is located midway between them. This motion creates a centripetal force that counteracts the gravitational pull, preventing the stars from crashing into each other.

Center of Mass

The center of mass is the point in a system of particles or bodies where their total mass can be considered to be concentrated. In a binary star system with equal masses, the center of mass is located exactly halfway between the two stars. Both stars orbit this point, and their gravitational attraction is balanced by their respective velocities, maintaining a stable distance without collision.