Suppose the distance between two stars doubled while the mass of each star doubled as well. What would happen to the orbital period of the two stars?
Well, if the distance between the stars doubled, they would need to hire a personal trainer to get back into orbit shape! But seriously, if the distance between the stars doubled, their gravitational pull on each other would weaken. This would result in a longer orbital period, meaning it would take more time for the stars to complete one orbit around their common center of mass. So, in short, their orbital period would increase. Just like how a Sunday morning always feels longer when you're waiting for brunch!
To determine the effect on the orbital period of the two stars when the distance between them and their masses double, we can use Kepler's third law of planetary motion.
Kepler's third law states that the square of the orbital period is directly proportional to the cube of the average distance between the two objects. Mathematically, it can be expressed as:
T^2 ∝ r^3
Where T is the orbital period and r is the average distance between the two objects.
Now, let's consider the scenario given:
1. Initially, the distance between the two stars was d, and the mass of each star was m.
2. After doubling the distance, the new distance between them becomes 2d.
3. The mass of each star also doubles, so the new mass becomes 2m.
Now, we can compare the new values with the initial values:
Initial: T^2 ∝ d^3
New: T'^2 ∝ (2d)^3
Simplifying the new equation, we have:
T'^2 ∝ 8d^3
Since the distance term (d^3) is the same in both equations, we can conclude that the new orbital period T' would be √8 times the initial orbital period T.
In other words, the new orbital period would be approximately 2.83 (which is the square root of 8) times longer than the initial orbital period.
Therefore, when the distance between two stars doubles while their masses also double, the orbital period of the stars would increase approximately 2.83 times.