Is it possible to have a planetary system with planets having perfectly synchronized orbits?
I'm imagining a system with a star and something like 6-8 planets. The planets' orbits are (relatively) close to each other, and all share an orbital period that is exactly the same. I.e. the length of a year would be exactly the same on all planets. I've even gone so far as to imagine a system where the planets are all in a line, with a gravity elevator linking each planet to its neighbor(s). Something like this:
(star) A-----B-----C-----D-----F-----G
I understand that they would have to be offset a bit (or in slightly different orbital planes) in order to not perpetually eclipse each other. Assume that technology exists allowing a species to exactly place a planet into the desired orbit, and even make routine corrections (though I'd prefer to not have to if possible). In other words, you can almost treat each planet as a giant spaceship as long as it would not need to use any thrust 99% of the time.
In this scenario the planets would all be roughly Earth-sized and have somewhat Earth-like climates, though probably the innermost would be hotter and the outermost colder.
From what I understand, a planet orbiting at x (average) distance from the star has a specific range of velocities it must adhere to--if it is too slow it would crash into the star, and if it is too fast it would escape the system altogether. I also know that the innermost planets would travel slower and the outer ones much faster in order to make one revolution in the same period. Finally, I'm guessing the distance between each planet would vary throughout their orbits since the orbits would be elliptical (so the gravity elevators would be long and flexible). But I don't understand the math enough to do the calculations.
So specifically I'd like to know:
Is is possible for such a system to exist?
If so, are there are limitations/constraints on the length of a year in this system, or on the type of star, distance from the star, etc?
How far apart would the planets have to be in order for the gravity of each (assuming Earth-like mass) to not pull its' neighbors out of orbit?
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1 answer
Is it possible for such a system to exist?
I'm sorry, but no. At least not according to orbital mechanics as currently understood.
Kepler's third law of planetary motion is one of the old workhorses of orbital mechanics, and applies in this case. As translated and summarized by Wikipedia, it states that:
The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.
or, mathematically,
$$ P^2 \propto a^3 $$
or, stated differently, there exists some constant $k$ such that
$$ P^2 = k a^3 $$
The semi-major axis is one of the defining parameters of an ellipse. (Put simply, the semi-major axis is the longer radius of the ellipse.) Since orbits are ellipses (also), this applies.
Consequently, the least you change the distance at which the planet orbits from the star, the orbital period will change, however little. If the orbital periods are different, then the planets will drift apart over time, however slowly. (They will occasionally line up, assuming that the orbits are themselves stable and closed. A variant of this occured for some of the planets in our solar system in the 1970s-1980s, giving our solar system grand tour taken by the Voyager 1 and Voyager 2 probes.) Ergo, the system you describe cannot exist when all planets are in the same plane.
If the planets are in different planes (technically, have different inclinations relative to the solar system ecliptic, which in this case could probably conveniently be defined as the equator plane of the star), then the distance between the planets will change as they move through their orbits. You can visualize this by considering two planets, orbiting the same star with the same velocity but at different inclinations; if you trace their orbital trajectories, you will see that the distance between the two planets varies throughout their orbits. Any kind of rigid construction attaching them to one another would interfere with their movement and either cause them to crash into each other, tear the structure apart, or tear the structure from one or both of the planets involved. Either way, having the planets in different planes is not an option either.
So, sorry, no, you can't have what you want.
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