|We can use Newtonian mechanics to calculate some useful formulae relating the properties of planetary orbits to the mass of the Sun. Again for simplicity, lets assume that the orbits are circular (not a bad approximation in most cases, and Newtons laws easily extend to cover the more general case of eccentric orbits). Consider a planet of mass m moving at speed v in an orbit of radius r around the Sun, of mass M. The planets acceleration (see More Precisely 2-2) is
so, by Newtons second law, the force required to keep it in orbit is
Comparing this with the gravitational force due to the Sun, it follows that
so the circular orbit speed is
Dividing this speed into the circumference of the orbit (2r), we obtain a form of Keplers third law (equivalent to the formula presented in the text):
where P = 2r/v is the orbital period.
Now we turn the problem around. Because we have measured G in the laboratory on Earth and because we know the length of a year and the size of the astronomical unit, we can use Newtonian mechanics to weigh the Sun. Rearranging the above equation to read
and substituting the known values of v = 30 km/s, r = 1 A.U. = 1.5 1011 m, and G = 6.7 10-11 N m2/kg2, we calculate the mass of the Sun to be 2.0 1030 kgan enormous mass by terrestrial standards. Similarly, knowing the distance to the Moon and the length of the (sidereal) month, we can measure the mass of Earth to be 6.0 1024 kg.
In fact, this is how basically all masses are measured in astronomy. Because we cant just go out and attach a scale to an astronomical object when we need to know its mass, we must look for its gravitational influence on something else. This principle applies to planets, stars, galaxies, and even clusters of galaxiesvery different objects but all subject to the same physical laws.