The question is about the planar version of the "circular restricted three-body problem", which is a specific, highly simplified celestial mechanics problem involving point masses. So by definition, there are only two degrees of freedom.

@PeterErwin I dunno. The plane can have any orientation and the masses can get kicked out by a comet hitting one of the masses. They are free to do so. The degrees of freedom are not realized but the masdes still have them. If the motion was physically restricted by two parallel planes (true planes) then the degrees would be truly restricted. But without them the masses are free to leave their orbit (if the occasion presents itself). A particle traveling on a line in space has three degrees of freedom too (so not one).

You're making it too complicated. The "coplanar circular restricted three-body problem", which is what the OP is asking about, is a particular mathematical simplification of celestial mechanics: three point masses, one of which has only infinitesimal mass, the other two in circular orbits. No comets or other complexity allowed; it's not the real universe.

@PeterErwin In that case yes. But I thought he was talking about the real stuff.

@PeterErwin Which one has infinitesimal mass? The asteroid? I wouldnt call that infinitesimal. The orbits can have any desired orientation. It rotational invariant due to the rotationally symmetric force. Hence the point masses have three degrees of freedom each.

A real asteroid is not infinitesimal. The third body in the restricted three-body problem is. (So the latter is a simplified analogy for the former, though it's a pretty good one, since the mass of an asteroid is negligible compare to that of Jupiter and the Sun.)

@PeterErwin If a particle is connected with a rod to a center then indeed there are just two degrees of freedom. Even if the particle moves in a plane.

@DescheleSchilder we don't get to re-invent the question in order to write the answer we want to write. The first sentence of the question is absolutely clear: "How many degrees of freedom does a mechanical system consisting of three bodies, the Sun, Jupiter and an asteroid, have in the restricted circular coplanar problem of the three bodies?" This does not answer that question. It is a 2D math problem, it is not embedded in three dimensions. The math problem has been well-defined for centuries, these are point masses in a plane with $m_i /r_{i}^2$ accelerations of 3 from $i=1, 2$

@uhoh Then whats the Op confused about? The three bodies can still have different initial velocities. Each in three different directions giving rise to different planes. Are you saying that a particle traveling through space has one degree of freedom only? Whats the origin of the coplanar motion?

@DescheleSchilder he's confused about what he asks: the number of degrees of freedoms, thus the independent variables of their problem. Given the exact words they asked about, Peter's answer is absolutely correct and given the wording of the question there's not much room for interpretation to generalize it to real bodies with real masses and real extent and spin and movement out of the orbital plane of the two main bodies.

@planetmaker The confusion cimes exactly from treating it mathematically. There is restricted motion but there is nothing that prevents the bodies from moving out of the plane. It's a rhree body problem with no constraints. If the initial velocities are given a certain configuration will arise (of which planar motion is one). In the Lagrange equations there is nothing to prevent another motion (which wiuld be the case if the bodies were bound to their coplanar motion by two parallel rigid plates above and below the plane of motion, which obviously is not the case).

@planetmaker Im not saying its an incorrect answer but its only correct in an imaginary world and not in the reaal world of astronomy. Like this the OP might think that Saturn or whatever planet can never move out of the planetary system... The physical degrees of freedom are just 9. You cant argue with that.

@DescheleSchilder the OP clearly does not think that. "restricted circular coplanar three body problem" is a well-defined term from theoretical mechanics and not open to free interpretation. Peter's answer basically explains the wording. And then draws the appropriate conclusion for the dimensionality of the problem.

@DescheleSchilder all of stellar astrophysics was first done in 1 dimension. 1, 2 Astronomers use lower dimensional approximations all the time! Math is an astronomical tool and learning about how to apply it is certainly on-topic here. Why are there Trojan asteroids at Lagrange points? Those are 2D concepts.

@planetmaker Exactly. It is not open to free interpretation. And that is exactly what is done in the first answer. Its freely interpreted.

No, the words are explained. And then a conclusion is drawn in the last two paragraphs / sentences.

@uhoh Astronomy is about the real universe. Not about a mathematical one.

@planetmaker So the planets are not able to move out of the solar system?

@planetmaker So if you look at the situation physically there are still three degrees of freedom for evefy planet. There is nothing to take their three degrees of freedom away. Not even coplanar motion. The bodies just dont use the other degrees.

@DescheleSchilder your view on what Astronomy is or isn't doesn't match the facts.

@uhoh Isnt it about the real universe?

@DescheleSchilder Astronomy is not a story about something. It is a field of scientific research that uses the same mathematical tools that other hard sciences use. The main tools are telescopes and mathematics. You always have to abstract a problem somewhat in order to apply math to it.

@uhoh Yes ineed. The math is just a tool. Different from telescopes though but that aside. But math cant take degrees of freedom away in the real physical world. From the stand of theoretical physics (Lagrangean mechanics) the three point masses still have three degrees of freedom. Thid is how I look to the problem.

You look at a different problem than the OP asks about. The OP asks about a well-defined problem with a limit wrt mass ratios and about a limitations of the relative movement options.

@planetmaker But there is no limitation wrt relative movement. The relative movements can have any three directions. For every mass. Every mass can be moving in three different relative dirrctions even if they are restricted to the plane. The whole plane can have thrre orientations, so. I think its a question of how you look at it. I think we are both right. :)

@DescheleSchilder No, you are not both right.

@DavidHammen Then Im right only...Ask Jupiter. Answer only this question: is Jupiter able to move out of the solar system?

@DescheleSchilder you should learn about the choice of appropriate coordinate systems. Just because the plane of motion can be arbitrarily oriented doesn't mean you have all three degrees of freedom.

@planetmaker Of course the three dgrees are restricted. But that doesn't mean they still have them. If a comet comes by this means they arent able to move outside of the plane. Its physics pure and simple. You should learn about that.

@planetmaker In every choice of coordites you will find the degrees are not reduced. The equations describing three body motion are invariant under rotation and translation so no degree of the masses is reduced. It are simply 3 masses freely interacting with each other. The interaction is not inducing a reduction in the degrees of freedom. The masses simply move in a plane. But you can easily let the plane move in the z direction with every speed you like

@DescheleSchilder Quoting Will Rogers, "If you find yourself in a hole, stop digging."

@DavidHammen Exactly! Then stop digging! Im already outside the hole... However deep you dig the answer is clearly not that there are not two degrees. Can planets moving in a plane escape the plane? Obviously yes. That they dont actually do is another point.

@DescheleSchilder our arguments are just as much "not about the real universe". You are ignoring all the degrees of freedom of all the individual particles making up each object, as well as all their internal degrees of freedom. And you seem to be assuming Newtonian gravity, which is not the real universe. What about the degrees of freedom in the metric field, and the effects of passing gravitational waves? Why do you think it's OK to ignore those?

(Also, you keep saying there's "a" degree of freedom for rotation. This is wrong. Even for the simplification of rigid-body motion, there are three degrees of rotational freedom. (Of course, this is irrelevant to the three-body problem.)