why not escaped moon from earth to sun?

The earth exerts MORE gravity at the moon distance than the sun does.

Gravity is proportional to the inverse square of the distance.

Show your workings and we can show you your error

I think one answer which might make sense to you is that the Moon IS orbiting the sun. Forget the earth for the moment, and just draw the path that the Moon actually takes. It appears on a piece of paper as a slightly wiggly line oscillating about a circular orbit. So you can think of the Moon's orbit as being as an orbit round the sun somewhat perturbed by the presence of the Earth. We think of it as being an orbit round the Earth because it so happens that the Earth is where we are.

Does that help?

Remember to consider the force necesary to keep the moon in

orbit around the Sun while it shares Earth's velocity.

Once you subtract that, Earth's gravity is enough to keep the

Moon in Earth orbit.

The Earth-Moon system IS in orbit around the Sun. What do you mean by "escape"?

M☉ = 1.99(10³⁰) kg; M♁ = 5.98(10²⁴) kg; M☾ = 7.35(10²²) kg

R☉-☾ = 1.50(10¹¹) meters; R♁-☾ = 3.84(10⁸) meters

G = 6.67(10⁻¹¹) N (m/kg)²; F = GM'M"/R²

F☉-☾ = 6.67(10⁻¹¹) N (m/kg)² • 1.99(10³⁰) kg • 7.35(10²²) kg /(1.50(10¹¹))² =

4.34(10²⁰) Newtons

F♁-☾ = 6.67(10⁻¹¹) N (m/kg)² • 5.98(10²⁴) kg • 7.35(10²²) kg / 3.84(10⁸))² =

1.99(10²⁰) Newtons

4.34(10²⁰)/1.99(10²⁰) = 2.18