Question
which expression best represents keplers third law with a planets period t and radius r in our solar system
Answers
Answered by
GPT-4o mini
Kepler's Third 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 around the Sun. In its mathematical form, the law can be expressed as:
\[
T^2 \propto r^3
\]
In a more precise form, for planets orbiting the Sun, it can be stated as:
\[
\frac{T^2}{r^3} = \text{constant}
\]
For planets in our solar system, this constant can often be expressed in terms of years and astronomical units (AU), with 1 year corresponding to the orbital period of Earth and 1 AU being the average distance from the Earth to the Sun. Specifically, for Earth:
\[
T^2 = r^3
\]
where \( T \) is in years and \( r \) is in astronomical units. This simplified expression indicates that if you know the radius of a planet's orbit in AU, you can find its orbital period in years by calculating \( T = \sqrt{r^3} \).
\[
T^2 \propto r^3
\]
In a more precise form, for planets orbiting the Sun, it can be stated as:
\[
\frac{T^2}{r^3} = \text{constant}
\]
For planets in our solar system, this constant can often be expressed in terms of years and astronomical units (AU), with 1 year corresponding to the orbital period of Earth and 1 AU being the average distance from the Earth to the Sun. Specifically, for Earth:
\[
T^2 = r^3
\]
where \( T \) is in years and \( r \) is in astronomical units. This simplified expression indicates that if you know the radius of a planet's orbit in AU, you can find its orbital period in years by calculating \( T = \sqrt{r^3} \).