The mass of the Sun can be calculated by observing how its gravity affects the motion of planets. We use Newton's Law of Universal Gravitation, which describes the gravitational force between two objects. This force keeps planets in their orbits around the Sun.
Newton's Law of Universal Gravitation states that every particle attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. The formula is F equals G times M times m divided by r squared, where G is the gravitational constant.
For a planet to maintain a stable orbit, the gravitational force must exactly balance the centripetal force required for circular motion. The centripetal force is mass times velocity squared divided by radius. Setting gravitational force equal to centripetal force gives us the fundamental equation for orbital mechanics.
Now we derive the formula for the Sun's mass. First, we cancel the planet's mass from both sides of our equilibrium equation. Then we substitute the orbital velocity as two pi r divided by the period P. Finally, we solve for M to get the mass equals four pi squared times r cubed divided by G times P squared.
Now we can calculate the actual mass of the Sun using Earth's orbital data. We know Earth's distance from the Sun is one astronomical unit, its orbital period is one year, and the gravitational constant. Substituting these values into our formula gives us the Sun's mass: approximately 1.989 times 10 to the 30th kilograms. This method can be applied to calculate the mass of any star with orbiting planets.