Orbit Likelihood: The Odds of Objects Falling into Stable Orbits Explained

[Alltimegreat1]

How likely or unlikely is it for a passing object to fall into a stable orbit of another object? For example, if an Earth-sized rogue planet came near the sun, would the odds be something like 10 million to 1 that it would fall into a stable orbit?

 

[mfb]

It depends on how the object is falling in. In particular, it depends on its initial velocity (far away from the Sun) and its impact parameter (without gravity, how close would it come?). Let's call the rogue planet X. If you just consider the sun/X-system, there is no way sun can capture the object. It will come close and leave again. To get captured, you need an interaction with something else in the solar system. Where "something else" is most likely Jupiter (300 times the mass of Earth) or Saturn (100 times). As long as the planet is light compared to those two, its mass does not matter.

Some mathematics ahead:

Let's consider a specific model, just with Jupiter, and with some approximations made on the way: Far away from the sun X moves with a velocity of 5 km/s (quite slow), it comes in orthogonal to the ecliptic, and its impact parameter is somewhere between 0 and 30 AU (orbit of Neptune). At the same distance to sun as Jupiter (ignoring that Jupiter doesn't have a perfectly circular orbit), it will have a speed of 19.05 km/s relative to Sun. If the gravitational interaction with Jupiter slows it down to 18.38 km/s, it gets captured. It has to change its speed by at least 0.67 km/s. Jupiter will actually increase the speed with a probability of more than 50%, and the probability that the direction of velocity change is right depends on the speed change, but I don't want to start doing 3D integrals, so let's just take the probability of a 1 km/s speed change and divide it by 2.

Relative to Jupiter, X moves at a speed of about 23 km/s. A speed change of 1 km/s is a deflection by 1/23, this needs an eccentricity of $e=\frac {1} {\cos(\pi/2-1/46)} \approx 46$. The semi-major axis is $a=\frac{-Gm}{v_\infty^2} = -240,000~ km$. The distance of closest approach, which is nearly the same as the initial impact parameter, is then $d=-a(e-1)=10.8 \cdot 10^6~ km$. Our planet X has to come closer than 10 million kilometers to Jupiter. In the disk given by the 30 AU from above, that gives a probability of 3 parts in a million that the rogue planet doesn't leave the solar system again immediately. This is not yet a stable orbit!

Even if Jupiter slows it down significantly - it will get an orbit that crosses the Jupiter orbit. It will encounter Jupiter again quickly, getting another chance to escape (or hit Jupiter). It will continue to do so unless interactions with other planets change its orbit sufficiently to avoid Jupiter's orbit. Those interactions are dangerous on their own, and can kick out our planet as well. To stay for geological timescales, the planet needs an orbit that is far away from all existing planets, and it has to get there via very weak interactions of the other planets. I don't know how likely this part of the process is, but it is certainly not frequent. Overall, I guess the probability is significantly below 10 million to 1.

Edit: forgot the factor 2. Fixed.

 

[tony873004]

The object can also come in as a binary (or more). The Sun splits the pairing, one escapes with even more energy than before, while the other one loses some energy stays behind. This could leave it in a non-planet crossing orbit. Some hypothesize that that's how Neptune captured Triton, and how the Sun captured Sedna.