# How do scientist measure the mass of the planets?

Jun 12, 2016

The mass of a planet can be determined using Kepler's 3rd law and by gravitational effects.

#### Explanation:

The most accurate way of measuring the mass of a planet is to send a spacecraft to it and measure the acceleration due to gravity as the spacecraft passes by it.

Alternatively, if the planet has a moon then its mass can be calculated from the moon's orbit.

First of all we need to know how far the planet is from the Earth. In the case of Venus we do this by bouncing radar signals off the planet and measuring the time it takes for the radar to return. Given the distance of Venus from Earth at its closest point, we can calculate the distance from the Earth to the Sun. Now if we measure the orbital period of any other planet we can calculate the distance using Kepler's third law.

To calculate the mass of the planet we need the distance of the planet form Earth $R$. We then need to measure the orbital period $T$ of the moon and the largest angular separation $\theta$ of the planet and the moon as the moon orbits the planet.

We can now calculate the radius of the moon's orbit $r = R \theta$. We now use Newton's form of Kepler's third law:

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

Where $G$ is the gravitational constant, $M$ is the mass of the planet and $m$ is the mass of the moon.
Rearranging the equation gives:

$M + m = \frac{4 {\pi}^{2} {r}^{3}}{G {T}^{2}}$

We now have calculated the combined mass of the planet and the moon. If the moon is small compared to the planet then we can ignore the moon's mass and set $m = 0$. This is true of most moons in the solar system.

If the moon is relatively large such as the Earth's Moon and Pluto and Charon, then we need to find the centre of mass which the planet and the moon are orbiting around. The distance $d$ from the centre of the planet to the centre of mass of the planet and moon can be used to calculate the ratio of that planet and moon masses and hence the planet''s mass.

$M d = m \left(r - d\right)$

This gives the planet's mass as:

$M = \frac{4 {\pi}^{2} {r}^{2} \left(r - d\right)}{G {T}^{2}}$