According to Newton's version of Kepler's third law, how would the ratio T^2/r^3 change if the mass of the Sun were doubled?
T^2 = (2pi)^2 [ r^3/(G m) ]
T^2/r^3 = k /m where k is constant (2pi)^2/G
if you double m then T^2/r^3 is half
To determine how the ratio T^2/r^3 would change if the mass of the Sun were doubled according to Newton's version of Kepler's third law, we need to understand the relation between the variables and the law itself.
Kepler's third law states: T^2 = (4π^2/GM) * r^3
Where:
- T is the period of orbit
- G is the gravitational constant
- M is the mass of the central object (in this case, the Sun)
- r is the average distance between the orbiting object and the central object
Now, let's analyze how the ratio T^2/r^3 changes when the mass of the Sun doubles.
If we double the mass of the Sun (M), the new mass would be 2M. This means the equation becomes:
T^2 = (4π^2/G(2M)) * r^3
To simplify the equation, we can divide both sides by 2:
T^2/2 = (4π^2/G) * (r^3/(2M))
Now, let's compare this to the initial equation:
T^2 = (4π^2/GM) * r^3
As we can see, when the mass of the Sun is doubled, the ratio T^2/r^3 is halved. Specifically, it becomes T^2/2 compared to the original T^2.
To determine how the ratio T^2/r^3 would change if the mass of the Sun were doubled, we need to understand Newton's version of Kepler's third law, also known as the law of harmonies. This law states that the square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis (r) of its orbit.
Mathematically, it can be expressed as:
T^2 / r^3 = constant
Now, let's consider the scenario where the mass of the Sun is doubled.
If the mass of the Sun is doubled, the gravitational force between the Sun and the planets would also double according to Newton's law of universal gravitation. This means that the centripetal force acting on the planets would also increase, providing a stronger inward force.
Since the centripetal force is proportional to the mass of the Sun, the expression can be rewritten as:
F_c = m * v^2 / r
Where F_c is the centripetal force, m is the mass of the planet, v is the orbital velocity, and r is the distance from the planet to the Sun.
When the mass of the Sun is doubled, the gravitational force and the centripetal force both double. As a result, the orbital velocity of the planet remains unchanged.
Now, let's substitute the values into Kepler's third law equation:
(T^2 / r^3) = (4F_c / m * v^2) / r^3 = (4F_c / m * v^2) * (1 / r^3)
As we can see, the squared term in the equation remains the same because the orbital velocity remains unchanged. However, the centripetal force doubles due to the increased mass of the Sun. Therefore, the ratio T^2 / r^3 would increase by a factor of 4.
In other words, if the mass of the Sun were doubled, T^2 / r^3 would increase fourfold.