The fact that the gravitational force between sun and moon >that between earth and moon.Why does Moon orbit around Earth but not the sun?

In a way the Moon DOES orbit around the Sun. Both the Earth and Moon circle the Sun once a year. If you plotted the Moon's orbit in space on a giant map of the solar system, but without showing the Earth, it would look a LOT like Earth's orbit, only slightly wavy, because sometimes the Moon is a bit closer to the Sun than Earth and sometimes a bit farther. Or if you plot both the Earth's and Moon's motion in space, they form a kind of twisted or braided pair of lines going around the Sun.

The moon orbits around BOTH the earth and the sun! As the earth moves around the sun, the moon goes with it.
The orbit that a body follows is determined by the vector sum of all sources of gravity, not by just the strongest source.

Actually, you are correct and the other answers are wrong. The Sun's gravity is stronger pulling on our Moon than the Earth's gravity. In other words, during a new moon, the Sun is pulling on the Moon stronger than the Earth is pulling(!).This was detailed nicely in an essay by Isaac Asimov. The fact is that our Moon's path around the Sun is always concave, never convex, meaning the Moon never ever "moves away" from the Sun during its orbit, unlike all other moons in our solar system. In this sense it is not really in orbit around Earth, but is in orbit around the Sun, just as Earth is. Isaac Asimov stated that our Earth-Moon system is really a double planet, not a planet-moon system. So why does it seem that our Moon is in orbit around Earth? Because that is the most energy convenient arrangement, nothing more.
Cheers.
Here is a quote from Wikipedia:
Asimov calculated tug-of-war values for several satellites of the planets. He showed that even the largest gas giant, Jupiter, had only a slightly better hold than the Sun on its outer captured satellites, some with tug-of-war values not much higher than one. In nearly every one of Asimov's calculations the tug-of-war value was found to be greater than one, so in those cases the Sun loses the tug-of-war with the planets. The one exception was Earth's Moon, where the Sun wins the tug-of-war with a value of 0.46, which means that Earth's hold on the Moon is less than half that of the Sun's. Asimov included this with his other arguments that Earth and the Moon should be considered a binary planet.[4]


We might look upon the Moon, then, as neither a true satellite of the Earth nor a captured one, but as a planet in its own right, moving about the Sun in careful step with the Earth. From within the Earth–Moon system, the simplest way of picturing the situation is to have the Moon revolve about the Earth; but if you were to draw a picture of the orbits of the Earth and Moon about the Sun exactly to scale, you would see that the Moon's orbit is everywhere concave toward the Sun. It is always "falling toward" the Sun. All the other satellites, without exception, "fall away" from the Sun through part of their orbits, caught as they are by the superior pull of their primary planets – but not the Moon.[4][5][Footnote 1]

— Isaac Asimov

Gravity isn't just about size (mass). It's about distance too. The effect of gravity between objects weakens the further away the objects are from one another. This is why the gravitational bond between the Earth and Moon is greater even though the Sun has more mass, because the Earth and Moon are much closer together. It's the same reason why the Moon has a greater effect over the ocean tides than the Sun does. The Moon is much closer.

Your statement would be true if the Sun and Earth were the same distance away from the Moon, in which case the Sun would easily win out.

Where you getting your facts from??
There not correct.
The Earth's Gravity has more effect on the Moon than the Sun's/\

The suns gravitational force is greater because it is bigger, but its effect on THE MOON are less because it is farther away than teh earth.

You have failed to consider the inverse square law.

Technically, the Earth and moon make a *system* that orbit the sun. Your question asks why the moon isn't ripped from Earth's gravity - and, that's because the distance to the moon - about 238,000 miles - makes the relative strength of Earth's gravity greater than the sun's at that point. Simply put - the moon is too deep in Earth's gravity well for the sun to pull it free.

If the moon was about 4 time further out - just shy of a million miles - that would be the point where the moon *might* be pulled from Earth's orbit...