Weighing the Sun with five photographs

Abstract

Using only five photographs of the Sun at different dates, we show that the mass of the Sun can be calculated using a telescope, a camera, and Kepler’s third law. With these photographs we are able to calculate the distance between the Sun and the Earth at different dates over a period of time of about three months. These distances allow us to obtain the correct elliptical orbit of Earth, proving Kepler’s first law. Analysis of the data extracted from the photographs is performed using an analytical optimization approach that allows us to find the parameters of the elliptical orbit. Also, it is shown that the five data points fit an ellipse using a geometrical scheme. The obtained parameters are in very good agreement with the ones for Earth’s orbit, allowing us to foresee the future positions of Earth along its trajectory. The parameters for the orbit are used to calculate the Sun’s mass by applying Kepler’s third law and Newton’s law for gravitation. This method gives a result which is in excellent agreement with the correct value for the Sun’s mass. Thus, over a time span of about three months, any student is capable to calculate the mass of the Sun with only five photographs, a telescope and a camera.