The gravity that sun has on the moon is about 2 times more than the gravity from earth so why doesn't moon run away from earth to sun?
Please explain by equation!

The moon is in orbit about both the earth and the sun. Its motion is in equilibrium with both the pull of gravity due to the sun and the pull due to the earth. Note that the moon's mean distance from the sun is the same as the mean distance of the earth to the sun and hence its orbital period about the sun is the same as the earth's (1 year). So it doesn't matter that the sun's gravity is greater on the moon than is the earth's - just as the fact that the moon's gravitational attraction on the earth is significantly less than the sun's.

thanks, but can you explain it by equation?? Or give me a link??

The ratio of the gravitational attraction between sun and moon versus earth and moon is:


\frac {M_s R_e^2} {M_e R_s^2}
=\frac {1.99x10^{30} \times 385000^2}{5.9x10^{24} \times (1.5x10^8)^2} = 2.2

where:
M_s = mass of the sun
M_e = mass of the earth
R_s = distance from sun to moon
R_e = distance from earth to moon


Hence the ratio of \omega^2R for the two orbits should also be 2.2:

\omega_s = 1/365.25 days, \omega_e = 1/27.3 days, and the ratio is:


\frac {(1/365.25)^2 \times 150000000}{(1/27.3)^2 \times 385000} = 2.2

So it's all consistent.