Figure Eight Orbits?

In your first assumption, the two stars would have to be EXACTLY the same size, mass, density and distance from the object they are orbiting. If this happened, then you would be able to view one from the other as "stationary".

The figure 8 pattern, however, is impossible. Imagine a satellite orbiting around one planet. The second planet would have to have a greater gravitational pull to force the satellite out of its original orbit and start orbiting the second planet. Then to continue the figure 8, the first planet would have to increase its gravitational pull (which is impossible) to force the satellite back. This increase/decrease would have to continue at exactly the same time as the satellite's orbit.

I love physics and these are two great questions.

Any two objects close enough together can orbit around their center of mass. For two equal size objects it is a point midway between them. For a large and small object the small object just causes the large one to wobble a bit.

Star systems with two stars orbiting each other is really a common thing, some are close to the same size and some are not. Look up binary stars.

While your stars would maintain about the same distance from each other, one would still appear to orbit around the other from their point of view. From our point of view they would take turns eclipsing the other.

The figure eight pattern you talk about is theoretically possible only if the planet found its way to and stayed in an exact orbit and there is only one possible orbit. In real life there is no thing as an exact orbit because other things in the star system would alter its orbit over time. Also the odds of something ending up in an exact orbit is zero.

Think of it this way. Cut a length of sting to be exactly a foot long is impossible. Even if you are super precise you will get a string that is something like 1.00000001 or 0.99999999 of a foot long no matter how hard you try.

It is possible. However, there are a few details, and the figure eight would not be regular. First, there is one point between two masses that has exactly the same potential. At this point, a body will have equal chance of falling toward either body. This is not a stable point, so a object will not sit there, and once off of this point, without outside forcing, will not attain rest at that position again. This should illustrate for you that the two stars do not need to be of the same mass.

Now, the goal then is to get a object into this system with orbital energy higher than that required to attain the 'saddle' point, yet lower than the escape velocity of the smallest star. Then, shazzam, figure eight orbits (albeit chaotic). However, this scenario is not very likely, as the long term stability of the path is not very nice. Each successive orbit could have significant path deviations, resulting in pretty irregular planetary properties. I would guess this would act as a serious destabilizing process on a planetoid. You might be more likely to find comets or asteroids in such orbits, but again, put two bodies into this, and the chance of collision over time is pretty high.

Yes, it is possible for two stars to revolve around a common central point. In fact, it is a fairly frequent occurence in binary star systems. The main thing you need is stars with about the same mass.

To make the stars stationary with respect to each other, you need to go to a non-intertial reference frame. If you take into account centripetal and Coriolis forces, you can do this.

As I recall (and I haven't done the calculation recently), a figure 8 orbit is possible, but it is unstable. Any deviation would send the planet away from that orbit exponentially. Too bad, eh?

BTW: be careful about saying a probability of 0 makes something impossible. Think about this: what is the probability of hitting the number 1/2 if you draw a number at random from the interval [0,1]?

No you could not have a figure eight orbit with two stars. When one works through the proof for the shapes of orbits, you arrive at only one possibility: they can be one of the three conic sections: parabolo, hyperbole, and ellipse (circle being the special case).

And, yes the two stars do revolve around a common point, which is their center of mass. If the two stars are identical, the center of mass would be at the half-way point between them. As one star gets larger than the other, the center of mass obviously moves towards the larger star. For example, it is incorrect to say the moon revolves around the center of the Earth. The Earth -moon system revolves around the center of mass, which happens to be inside the Earth. As a result, the Earth has a slight wobble, but it does look like the moon simply goes around the Earth.

The only way to get a figure eight orbit is if you had a system of more than two stars or planets. But, you would have to set them up just so in order for that to work. It wouldn't work in nature because that system would be very unstable, and eventually everything would settle into the elliptical orbits.

The short answer is no.

This is an specific example of the three-body problem.

Even if we assumed perfect mathematical descriptions of the bodies and their positions with no outside influences, while it may be possible for a body to navigate a (near) figure eight a few times, the orbit will degrade.

In fact, the only semi-stable configuration of three such bodies would place the planet at one of 5 Lagrangian points, only 2 of which are completely stable. In this configuration, the planet remains stationary with respect to the stars; i.e., gravitational pull of the two large masses precisely cancels the centripetal acceleration required to rotate with them.

The mathematics which shows that the 3 unstable lagrangian points are in fact unstable can be modified to prove that a figure 8 orbit of a planet would be equally unstable.

Unfortunately, the margin is too small to include the proof here. (Whoops! Sounds a little too much like Fermat)

I don't think a planet could orbit a star far enough from the star to also orbit its binary neighbor. If the sun had an equally-massive binary partner, I think it would be more than twice as far away as Pluto at Pluto's aphelion.

They wouldn't revolve around a fixed point of nothing, one of the stars would have to revolve around the other. As to the figure 8 I don't think it is possible.

I'm not very sure... but i think figure 8 like-orbit is not possible...

Yes, this is possible. But there are no known examples of it in nature.