# Calculating the Ratio of Planetary Orbital Periods

• lizzyb
In summary, the conversation discusses two planets, A and B, with B having twice the mass of A, orbiting the Sun in elliptical orbits. The ratio of the orbital period of planet B to that of planet A is 2, according to Kepler's Third law. The equation used to determine this ratio involves the mass and semi-major axis of the planets, as well as the mass of the Sun.
lizzyb

## Homework Statement

Two planets A and B, where B has twice the mass of A, orbit the Sun in elliptical orbits. The semi-major axis of the elliptical orbit of planet B is two times larger than the semi-major axis of the elliptical orbit of planet A.

What is the ratio of the orbital period of planet B to that of planet A?

## Homework Equations

$$T^2 = (\frac{4 \pi^2}{G M}) r^3$$

## The Attempt at a Solution

$$M_B = 2 M_A$$
$$a_B = 2 a_A$$
$$\frac{T_B}{T_A} = \frac{\sqrt{\frac{4 \pi^2}{G 2 M_A} 8 a_A^3}}{\sqrt{\frac{4 \pi^2}{G M_A} a_A^3}} = \sqrt{\frac{8}{2}} = \sqrt{ 4 } = 2$$
but that was wrong. ?

lizzyb said:

## Homework Equations

$$T^2 = (\frac{4 \pi^2}{G M}) r^3$$

## The Attempt at a Solution

$$M_B = 2 M_A$$
$$a_B = 2 a_A$$
$$\frac{T_B}{T_A} = \frac{\sqrt{\frac{4 \pi^2}{G 2 M_A} 8 a_A^3}}{\sqrt{\frac{4 \pi^2}{G M_A} a_A^3}} = \sqrt{\frac{8}{2}} = \sqrt{ 4 } = 2$$
but that was wrong. ?
Planetary Mass is not a factor. Kepler's Third law states that $T^2/a^3$ is the same for all planets. The M in your equation is the mass of the sun.

AM

thank you.

## 1. What is the ratio of planetary orbital periods?

The ratio of planetary orbital periods, also known as Kepler's third law, states that the ratio of the square of a planet's orbital period to the cube of its semi-major axis is the same for all planets in the solar system. This ratio is approximately 1:11.86 for all planets orbiting the sun.

## 2. How is the ratio of planetary orbital periods calculated?

The ratio of planetary orbital periods can be calculated by taking the square of a planet's orbital period and dividing it by the cube of its semi-major axis. This calculation is based on Kepler's third law, which relates the orbital period of a planet to its distance from the sun.

## 3. Why is the ratio of planetary orbital periods important?

The ratio of planetary orbital periods is important because it helps us understand the relationship between a planet's orbital period and its distance from the sun. This ratio allows us to make predictions about the orbital periods of other planets based on their distance from the sun.

## 4. Are there any exceptions to the ratio of planetary orbital periods?

While the ratio of planetary orbital periods is a general rule that applies to most planets in the solar system, there are some exceptions. For example, when considering the moons of a planet, the ratio will be different due to the influence of the planet's gravity.

## 5. How does the ratio of planetary orbital periods vary among different solar systems?

The ratio of planetary orbital periods can vary among different solar systems, as it is dependent on the mass and distance of the planets from their respective stars. Therefore, the ratio may be different for planets orbiting a different star compared to those in our own solar system.

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