Which form of Kepler’s third law can you use to relate the period T and radius r of a planet in our solar system as long as the unit year is used for the period and astronomical unit is used for the radius?(1 point) Responses T=r2 cap t is equal to r squared T2=r3 cap t squared is equal to r cubed T3=r2 cap t cubed is equal to r squared T2=r
T2=r3 cap t squared is equal to r cubed
The correct form of Kepler's third law that can be used to relate the period T and radius r of a planet in our solar system, when the unit year is used for the period and astronomical unit is used for the radius, would be:
T^2 = r^3
The form of Kepler's third law that you can use to relate the period T and radius r of a planet in our solar system, as long as the unit year is used for the period and astronomical unit (AU) is used for the radius, is T^2 = r^3.
To understand how this formula is derived, let's break it down.
Kepler's third law states that the square of the period of revolution of a planet around the Sun is directly proportional to the cube of the semi-major axis (radius) of its orbit. Mathematically, it can be expressed as T^2 = k * r^3, where T is the period, r is the radius, and k is a constant.
Now, to specifically relate the period T and radius r of a planet in our solar system when the unit year is used for the period and astronomical unit is used for the radius, we have to match the appropriate units. Since the period is measured in years and the radius is measured in astronomical units, we need to choose a value for the constant k that makes the equation work when these units are used.
Considering that for all planets in our solar system, the ratio of T^2/r^3 is constant, we can use the ratio of the square of Earth's period (1 year)^2 to the cube of Earth's radius (1 AU)^3 as our constant. Therefore, k = (1 year)^2 / (1 AU)^3.
Substituting this value of k into the equation, we get T^2 = r^3.
So, the correct form of Kepler's third law equation to relate the period T and radius r of a planet in our solar system, using unit year for the period and astronomical unit for the radius, is T^2 = r^3.