Gravity and the Mass of the Sun

 As a young school boy I found it hard to believe that scientists were able to compute the mass of our sun.  After all, a galactic-sized measuring device does not exist.  But where there’s a will, there’s a way, and by the 18th Century scientists had it all figured out, thanks to the work of others before them.  Newton’s two formulas concerning gravity were key to later scientific discoveries, and we’ll be working with them again today to derive a third formula, bringing us a step closer to determining our sun’s mass.

Newton’s Second Law of Motion allows us to compute the force of gravity, F_g, acting upon the Earth, which has a mass of m. It is,

F_g = m × g                     (1)                                                                        

Newton’s Universal Law of Gravitation allows us to solve for g, the sun’s acceleration of gravity value,

g = (G × M) ÷  r²              (2)

where, M is the mass of the sun, r is the distance between the sun and Earth, and G is the universal gravitational constant.

You will note that g is a common factor between the two equations, and we’ll use that fact to combine them.  We’ll do so by substituting the right side of equation (2) for the g in equation (1) to get,

F_g = m × [(G × M) ÷  r²]

then, using algebra to rearrange terms, we’ll set up the combined equation to solve for M, the sun’s mass:

M = (F_g × r²) ÷ (m × G)           (3)

At this point in the process we know some values for factors in equation (3), but not others.  Thanks to Henry Cavendish’s work we know the value of m, the Earth’s mass, and G, the universal gravitational constant.  What we don’t yet know is Earth’s distance to the sun, r, and the gravitational attractive force, F_g, that exists between them.