Final answer:
To approximate the Sun's mass relative to Earth's, we use a modified form of Kepler's third law, which relates the orbital period and distance of the Moon from the Earth to Earth's mass. By comparing it to Earth's orbit around the Sun, we can estimate the Sun's mass. Despite the Sun's greater mass, the tides are predominantly due to the Moon's closer proximity to Earth.
Step-by-step explanation:
To evaluate the ratio of the mass of the Sun to that of Earth using only the lengths of the year and the lunar month, we can utilize a form of Kepler's third law that was modified by Newton. Kepler's law relates the orbital period of a satellite to the mass of the body it is orbiting. Based on the average orbital period of the Moon around the Earth (about 0.08 years) and the average distance from the Moon to Earth (approximately 0.0027 astronomical units or AU), we can estimate Earth's mass. Since the Earth orbits the Sun in one year and one AU is the average distance from the Earth to the Sun, similarly, we can use this data to calculate the Sun's mass.
Kepler's third law states that the square of the period (T) of an orbit is proportional to the cube of the semi-major axis (r) of the orbit. When comparing the orbits of two bodies, the formula is T12/r13 = T22/r23 = 4π2/Gravitational Constant (G)×Mass of central body (M).
Substitute the respective values for Earth and Moon, and simplify to find the ratio of the Sun's mass to Earth's mass. We know the mass of Earth, so this ratio can give us an approximate value for the mass of the Sun. However, we need to remember that the calculated value is approximate due to the assumptions we've made (ignoring the masses of the Moon and the Planets, etc.).
The importance of the Moon in causing tides is due to the difference in gravitational force – known as the tidal force – which depends not only on the object's mass but also on its distance from Earth. Although the Sun is more massive, the Moon is significantly closer to Earth, resulting in a stronger effect on the tides compared to the Sun.