Which is the simplified version of Kepler’s third law when one of the planets being compared is Earth?(1 point) Responses
T^2 = r^3
T^3 = r^2
T = r^2
T^2 = r
T^2 = r^3
The simplified version of Kepler's third law when one of the planets being compared is Earth is T^2 = r^3.
The simplified version of Kepler's third law when one of the planets being compared is Earth is T^2 = r^3.
Kepler's third law states that the square of the orbital period (T) of a planet is directly proportional to the cube of its average distance (r) from the Sun. In this case, if we are comparing Earth to another planet, we can set up the equation T^2 = r^3.
To explain how this simplified version is derived, we need to understand the original version of Kepler's third law. Kepler's third law is usually expressed as T^2 = k*r^3, where T is the orbital period in years, r is the average distance from the Sun in astronomical units (AU), and k is a constant of proportionality.
To find the simplified version when comparing Earth to another planet, let's assume k is the same for both planets. Since we are comparing Earth to another planet, we can consider their orbital periods (T) and average distances (r). We want to find the relationship between these values.
Suppose we have two planets, Earth and Planet X, with orbital periods T1 and T2 and average distances r1 and r2 respectively. The relationship between these values can be written as:
(T1)^2 = k*(r1)^3 ... (Equation 1)
(T2)^2 = k*(r2)^3 ... (Equation 2)
Now, let's consider Earth as one of the planets, so T1 is the orbital period of Earth and r1 is the average distance of Earth from the Sun. We can express Equation 1 as:
(T1)^2 = k*(r1)^3
Since we are comparing Earth to another planet, let's call the orbital period of the other planet T and its average distance from the Sun as r. Thus, Equation 2 becomes:
(T)^2 = k*(r)^3 .... (Equation 3)
Now, we can compare Equation 3 with Equation 1 and make the assumption that k is the same for both planets (as stated above). Therefore, we get:
(T1)^2 = (T)^2
(r1)^3 = (r)^3
Simplifying these equations, we find:
T1 = T
r1 = r
Hence, we can rewrite Equation 1 as:
(T)^2 = k*(r)^3
Since we are comparing Earth to another planet, we can represent Earth as T1 and r1. Thus, the simplified version of Kepler's third law when one of the planets being compared is Earth is:
T^2 = r^3.