A planet's gravity extends far out into space. Despite what you have heard about "weightlessness" in space, the space station and its crew experience gravity about 80% as strong as gravity on Earth's surface. The weightlessness effect is due to both station and astronauts "falling" along the same path, so the astronauts don't seem to move inside the station. If you were in a falling elevator, you would also float like the astronauts- until the elevator hit the basement floor. The space station doesn't hit the ground because it orbits at 5 miles per second, which is so fast that the Earth's surface curves out from under the station as it falls. The station falls in a circle, along with its astronauts inside.
Isaac Newton discovered that the force of gravity between any two objects is F=G*m1*m2/r^2, where the m's are the masses of the objects, r is the distance between their centers, and G is a constant number. For your question, the important thing is that the force falls off with the square of the distance between the objects. One additional Earth radius (~4000 miles) above our heads, Earth's gravity is 1/4 as strong as it is down here. 8000 miles above our heads (3 Earth radii from Earth's center), gravity is 1/9 as strong, etc.
Most planets have moons, and from studying their orbits, it is easy to determine the strength of gravity required to keep the moons in their orbits. You get the planet's gravity at the distance of the moon's orbit, but knowing this, just multiply by (Moon's orbit radius / planet's radius)^2, and you get the gravity on the planet's surface.
For moonless planets, surface gravity can still be determined by measuring how their gravity causes slight changes in the orbits of neighboring planets. Scientists determine the strength of the gravity which made the perturbation, and then they multiply by (distance between planets / planet's radius)^2, as before, to get the surface gravity. This method was actually used to discover Neptune- they noticed a slight wobble in Uranus's orbit, and calculated that it could only have been caused by a planet in Neptune's position.
The only way to remotely determine the gravity of lone asteroids is to send a probe flying past and measure its deflection based on the timing of its radio signals.
There are a couple of ways to tell how strong the gravity is on a planet you've never been to. If you know the mass and size of an object, you can tell how strong the gravity it produces is. You can measure the size of a planet by taking a picture of it through a telescope, and you can guess how much it weighs by knowing that most planets are made up of rock and so they have about 5 grams of mass for every cubic centimeter of the planet's volume (i.e., the planet has a density of 5 g/cc or a specific density of 5). So then you have the size and the mass of the planet, and you can calculate the gravity from that. Since figuring out the mass of the planet involves some guesswork about the planet's density (which varies somewhat depending upon how which kinds of rocks and metals are in the planet), there is another, more accurate method, which is this: if the planet has moons, the amount of time it takes for the moon to complete an orbit of the planet is a direct measure of the planet's gravity. This is because it is the planet's gravity which causes the moon to orbit.
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