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Is it possible (for either a satellite or a planet) to orbit around a black hole? Do they attract everything around themselves into the center? Or they just affect gravitational force just like stars?

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Absolutely possible. There's nothing magical about a black hole. The gravitational pull of a black hole reaches as far as gravity would for another object of the same mass.

If you replace the Sun with a black hole of the same mass, everything would continue to orbit it just as it currently does.

Anything with mass has a gravitational force itself, and a black hole will attract anything with mass. Again, it's the same as our star having an effect on the Earth, and the Earth having an effect on the Moon.

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The soure of misunderstanding was that I wasn't clear enough, how black holes work. I always imagined them as "suckholes" like whirlpools in the water. –  Zoltán Schmidt Oct 7 '13 at 17:24
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They are like whirlpools but just like a whirlpool, it only reaches so far. If you've got a bigger hole in the middle of the whirlpool (i.e. higher mass) then it will suck in stuff from further away. No different than the effect a large star of the same mass would have. The different being, that with a black hole, there is a point-of-no-return, where if anything gets pasts that point, there's no way for it to escape. –  Carl Oct 7 '13 at 21:49

They don't attract gravitational force; they have mass, so they exert a gravitational influence upon other objects.

So yes, it is possible for an object to indefinitely orbit around a black hole. Just because the mass it is orbiting is called a black hole does not mean the object is doomed to spiral in on the black hole.

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proof exists that something can orbit around a black hole. The star S2 orbits around the central 4 million solar mass black hole of our galaxy.

http://en.wikipedia.org/wiki/S2_(star)

Note that if you find an object orbiting around your target object, then you can calculate the mass of your target object.

In fact, for something orbiting around a black hole, it is very difficult to fall in. This is for the same reason why it is easier to send a probe from Earth to Mars (outward from Sun), than from Earth to Mercury (inward toward Sun), and also the same reason why it is impractical to get rid of toxic waste by throwing it into the Sun. It would require an enormous amount of energy to reach the Sun.

See basics of space flight for details.

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I've wonder why scientists say that when the milky way and andromeda collide that the massive black holes in their centers will merge. Why won't they just enter orbits around each other or continue on in the same directions they are headed now? –  Jason Goemaat Dec 12 '13 at 22:01

I think blackholes are dead star? Is not it? And i believe our nearest star is sun , and there is difference between star and blackholes(dead star) . And i strongly believe that blackholes ( dead star ) they attract everything around themselves into the center. And after that they must change the property of that object(which is at center), and then spread it's anu parts into the universe. :)

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For the purposes of comparison, here's flat, Minkowski spacetime in spherical coordinates: $$\mathrm{d}s^2 = -\mathrm{d}t^2 + \underbrace{\mathrm{d}r^2 + r^2(\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2)}_\text{Euclidean 3-space}\text{.}$$

The soure of misunderstanding was that I wasn't clear enough, how black holes work. I always imagined them as "suckholes" like whirlpools in the water.

That is not entirely incorrect. The Schwarzschild spacetime of an uncharged, nonrotating black hole in the Gullstrand-Painlevé coordinates is $$\mathrm{d}s^2 = -\mathrm{d}t^2 + \underbrace{\left(\mathrm{d}r + \sqrt{\frac{2M}{r}}\,\mathrm{d}t\right)^2}_\text{suckhole} + r^2(\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2)\text{.}$$ Where it deviates from ordinary, flat Minkowski spacetime is entirely in the middle square term. Here, the time coordinate $t$ is not the Schwarzschild time, but rather the time measured by an observer free-falling from rest at infinity. The last bit, if adjoined with the $\mathrm{d}r^2$ term one would get by multiplying out the middle part, is ordinary Euclidean $3$-space written in spherical coordinates.

If you recognize from Newtonian gravity the quantity $\sqrt{2M/r}$, or $\sqrt{2GM/r}$ in ordinary units, as the escape velocity, then the picture is very peculiar indeed: according to an observer free-falling from rest at infinity, Euclidean space is sucked into the singularity at the local escape velocity. The event horizon is the surface at which the speed at which space is "falling" at the speed of light.

This is an additional reason why sonic black holes are good analogues to their gravitational counterparts. In a sonic black hole, there can be an actual "suckhole" that drains a low-viscosity fluid at an increasing velocity, up to and faster than the speed of sound in that fluid. This forms an acoustic event horizon that is one-way to sound and is expected to have an analogue of Hawking radiation.

The corresponding structure for charged black holes is similar, and for a rotating one more complicated, although can still be described as "sucking" with a certain additional twist that rotates the free-falling observers.

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