Need help in understanding accelaration due to gravity

Question

I've been having some trouble in understanding acceleration due to gravity.

On earth, the acceleration due to gravity is an average of $9.80$ $m/s^2$. The mass of the earth is approximately $5.972 \times 10^{24} kg$. The acceleration due to gravity on the surface of the sun is $273.7$ $m/s^2$ and its mass is about $1.989 \times 10^{30}$.

So

$$ \frac{\textrm{Mass of Sun}}{\textrm{Accelaration at Sun surface}} = \frac{\textrm{Mass of Earth}}{\textrm{Accelaration at Earth surface}}$$

Why don't the above numbers equal each other? Is it because I am doing mass divided by acceleration?

Answer 1

The gravitational force is given by $F_g=\frac{GMm} {r^2}$ where symbols have their usual meaning. And the acceleration due to gravity at the surface of a body (uniform and spherical) is $a_g=\frac{GM} {R^2} $, where R is the radius of the body.

What you are doing is just dividing M by $a_g$ which gives you $\frac{MR^2} {GM} = \frac{R^2} {G} $. So do you see the problem? Saying that $\frac{M} {a_g} $ is same for Sun and Earth would imply that their radii are equal, which is clearly not true.