The Sun is keeping you close. After all, you are orbiting it just like the Earth. You don't fly off into space because the Earth and you experience the same acceleration due to the Sun's gravitational force, so you orbit together; this is sometimes called the equivalence principle.
If, however, you were floating near Earth but closer to the Sun, you would experience stronger gravity. You would be in a smaller orbit which would make you drift away from the Earth. You wouldn't fall into the Sun, though.
Edit: I forgot to say something about the outer planets, something which the other answers touch on but I think get wrong. First, we should speak of acceleration rather than force, because like I said earlier all objects at a given distance from the Sun experience different forces but the same acceleration.
You ask "how come the Sun is strong enough to keep the distant planets in orbit but I don't fall into it?". The important point is that you don't need such a huge acceleration to keep the planets in orbit, because they are far away and move very slowly.
But, the smallness of the acceleration isn't the reason you don't feel it. The reason is that you're in free fall around the Sun; even if you were zipping around kilometers from the Sun's surface, you would not feel the huge gravitational force, because it affects everything around you in exactly the same way (disregarding tidal effects).
Updated April 24, 2017
By Drew Lichtenstein
The more massive a planet or star is, the stronger the gravitational force it exerts. It is this force that allows a planet or star to hold other objects in their orbit. This is summed up in Isaac Newton's Universal Law of Gravitation, which is an equation for calculating the force of gravity.
Newton's Universal Law of Gravitation is a formula for understanding the relationship of gravity between two objects. The equation is "F = G(M1)(M2)/R," where "F" is the force of gravity, "G" is the gravitational constant, the "M"s are the masses of the objects being considered, and "R" is the radius of the distance between the two objects. Thus, the more massive either object is, and the closer they are together, the stronger the force of gravity.
Gravity is what keeps planets in orbit around the sun. The sun is extremely massive, thus it holds very distant objects, like the outer planets and comets, in its orbit. This can also be seen on a smaller scale, with planets keeping satellites in their orbits; the more massive a planet is, the more distant its satellites. For example, Saturn, one of the gas giants, has the most known moons. Stars themselves orbit around the center of the galaxy.
Newton's three laws of motion are also applicable for understanding the effects of gravity on the cosmic law, particularly the first and third law. The first law states that an object at rest or in motion will remain in that state until something acts on it; this explains why planets and moons stay in their orbits. The third law is that for every action, there is an opposite and equal reaction. While this is negligible when considering something like a planet affecting a star, this explains tides on Earth, which are caused by the moon's gravity.
Newton understood how gravity worked, but not why. It wasn't until Albert Einstein's General Theory of Relativity, published in 1915, that a theory was postulated to explain the cause of gravity. Einstein showed that gravity was not a quality inherent to objects, but instead it was caused by curves in the space-time dimensions, which is what all objects rest on. Thus, even light and other massless phenomena are affected by gravity.
According to Sir Isaac Newton’s Law of Universal Gravitation, all objects that have mass are attracted to each other. Mass is the measure of an object’s matter (what it’s made up of). The greater an object’s mass, the greater its gravitational force. The earth has a strong attracting force for objects with smaller mass (including the moon), and the sun has an attracting force on the earth and other planets in our solar system.
Albert Einstein’s General Theory of Relativity explains gravity in another way. Instead of being a force, Einstein theorized that gravity is the result of bending in space. Huge objects like the sun create a sort of well in space that causes planets to move in curved rather than straight paths. Although there is evidence to support this theory, it has not been tested enough to become a scientific law.
Weight
is determined by the force of gravity pulling on an object. The stronger the pull of gravity on an object, the greater its weight. In physics, weight is measured in newtons (N), the common unit for measuring force. To calculate your weight in newtons, measure your mass on a scale (in pounds) and multiply it by 4.5.
Unlike mass, which remains constant, weight depends on the force of gravity that is exerted on it. What do you think would happen to your weight if you were in a place where gravitational force was less? Would the weight be less or greater? On the moon, where gravity is very low, you would weigh less than on earth. On the planet Jupiter, where gravity is stronger, you would weigh much more than you do here. On the sun, gravity is so strong that you would weigh about 27 times as much as on earth! The mass of your body, however, would remain constant in all of these situations.
In the illustration, weight differences based on stronger or weaker gravity are shown. If your weight was 150 pounds (lbs) on the earth, then your weight on the moon would be about 25 pounds and your weight on the sun would be about 4060 pounds—a little over two tons!
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If we want to change the distance to the sun, in which direction do we have to change the gravity?
In any direction! The distance between the Earth and the Sun is a careful balance between the mass of the Earth, the mass of the Sun, the speed with which the Earth orbits the Sun, and the strength of gravity. If any of these things change in any direction, the orbit of the Earth would change. If the orbit of the Earth changes, then the distance between the Earth and the Sun changes.
If we kept the strength of gravity the same, and increased the mass of the Sun, the Sun would exert a stronger gravitational force on everything that orbits it, and in the absence of any other changes in the solar system, would disturb the orbits of all of the planets, pulling them closer for at least part of their orbits. Our Earth has an almost perfectly circular orbit around the sun, but that’s not required to be the case – just look at the long, looping orbits of comets around our sun. If you tug on an object once (as you would, if you suddenly increased the mass of the sun), you’ll likely pull a circular orbit into a more oval, looping orbit.
If the Earth’s mass suddenly increased, we’d have a few problems here on earth, as our bodies are calibrated to work best when we’re dealing with exactly the amount of gravitational force that we have. But on an orbital sense, the gravitational force between two objects depends on the mass of both the larger and the smaller object. So if we increase the mass of the smaller object (the Earth), that will increase the gravitational force between the Sun and the Earth, probably still pulling the Earth off of its circular orbit, but also pulling more strongly on the Sun, making the Sun wobble very slightly more as the Earth orbits around it.
It’s not hard to imagine ways of changing the distance between two objects before you get around to playing with the strength of gravity. However, changing the strength of gravity does the exact same kind of things to an orbit as changing the masses of the objects you’re interested in. Let’s say we increase the strength of gravity; that’s equivalent to making everything in the solar system more massive at once. That change would, in turn, mean that the strength of the gravitational pull between all objects gets stronger. The exact response of the orbits will depend on how different the masses are, but we can safely say that the Sun would be pulled into a more wobbly rotation around its own axis, and that the Earth would wind up closer to the Sun for at least part of the year.
If, on the other hand, we let gravity get fainter, the opposite happens. The pull between Earth and Sun grows weaker, and the planets would drift farther from the Sun, spending much more time at much greater distances from the Sun. If you’d like to play around with this more directly you can try out this solar system simulator (there are a lot of other setups you can tinker with).
So really, any change to the strength of gravity, or any change to the masses of the objects involved, would change the distance between the Earth and the Sun. However, changing the strength of gravity would have relatively catastrophic consequences for a whole series of physical processes that we currently rely on, so I wouldn’t recommend it as a scenario.
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