## Estimate for the Mass of the Sun From Observational Data

We can estimate the mass of the Sun from observational data - Period of orbit and radius of orbit - using the equation representing Newton's Law of Gravity to estimate the mass of the Sun, or in fact the mass of any massive body being orbited by a much smaller body.Suppose we wanted to estimate the mass of the Sun. Equate the force of gravity to the centripetal force.

{jatex options:inline}\frac{GM_{Sun} m_{Earth}}{r^2}= \frac{m_{Earth} v^2}{r}{/jatex}

Rearranging gives

\[M_{Sun} = \frac{v^2r}{G}\]

The Earth orbits the Sun once a year or 365 days. In a year there are

\[365 \times \ 24 \times 60^2 = 31536000 s\]

The Earth orbits the Sun at a radius of

\[150 \times 10^9 m\]

The speed of the Earth around the Sun is

\[v= \frac{2 \pi r}{T} = \frac{2 \pi \times 150 \times 10^9}{31536000}=29.885 \times 10^3 m/s 29.995 m/s\]

The mass of the Sun is then

\[M=\frac{29885^2 \times 150 \times 10^9}{6.67 \times 10^{-11}} = 2.01 \times 10^30 kg\]

This is an estimate. The main errors arise because we have made the following approximation

Both the Sun and the Earth are spheres.

The orbit of the Earth is a circle.

The mass of the Earth is small compared with the Sun so can be ignored.