Thousands of years ago, some ancient Greek dude discovered (or maybe invented) the Conic Sections. These are the possible shapes a quadratic equation’s graph can take, and by extension, the possible orbits that can exist in nature. They are called Conic Sections because their shapes can be demonstrated by slicing a cone at different angles. Think of a cone-shaped piece of cheese, lying on its base on a table, and a sharp knife slicing through the cone at different angles, relative to the table, leaving curved “orbital” cut lines in the surface of the cheese.
For example, if the cut is made parallel to the table top, separating the cheese into the flat end and the pointy end, the shape of the cut line will be a perfect circle. If the cut is at a slight angle to the table top, the line will be an ellipse. The circle can be considered a special case of the ellipse. A cut perpendicular to the table top forms a hyperbola, and one at an acute angle forms a parabola. These curves can all be expressed as mathematical formulae, and have all sorts of interesting properties and share key parameters, such as axes, eccentricities, asymptotes and foci.
But they are not just mathematical abstractions. The conics also play a role in orbital mechanics. The possible orbits that two masses describe as they interact gravitationally can all be described as conic sections. Two masses will orbit a common center of gravity in elliptical orbits. If the orbits are carefully balanced and adjusted they will be perfect circles. Although perfectly circular orbits are rarely, if ever, found in nature, they are not dynamically impossible. A parabola is very similar to an ellipse, except it doesn’t close in on itself. It is open at one end. The two ends get closer and closer together, but increasingly slower at long distances from the focus. However, at the end nearest the focus a parabola is so close to a very skinny ellipse that it is often difficult to tell them apart just by measurement. A parabola can be thought of as an ellipse with its semi-major axis at infinity. Comets that have parabolic orbits look like they come from interstellar space, but they are probably just elliptical orbits that have been perturbed by an encounter with some third body, Hyperbolas, on the other hand, are always totally open, and the two open ends get further and further apart the farther you travel from the focus. An object traveling in a hyperbolic orbit is certainly not a bound member of the system, it comes from another place. A comet or planetoid traveling in a hyperbolic orbit has either been accelerated by some energetic episode (like a gravitational flyby, or a rocket thrust maneuver, or it comes from somewhere else. Astronomers have always carefully checked the orbital elements of newly-discovered comets in the hope of finding an extrasolar visitor. Ouamuamua was the first one so identified, but it was probably not the first–or the last.
A hyperbolic orbit implies the object has achieved escape velocity, or the speed at which the gravitational attraction of the orbited body is not sufficient to prevent the object from escaping its gravitational grasp. Consider two masses floating a great distance from one another in an empty universe, far away from the gravitational influence of any other masses.
Gravity will draw the two masses together, and as they approach they will speed up, losing potential energy and gaining kinetic energy, until they eventually collide, at escape velocity. Let us assume that one of the masses is much greater than the other, it makes the math easier! If the smaller mass experiences some external nudge that allows it to avoid a head-on collision, it will whip pass the larger body in a hyperbolic orbit, slingshotting off in the opposite direction. It is impossible for the smaller body to be “captured” by the larger unless some third body interacts with the two and absorbs the excess energy. The high prevalence of orbiting bodies found in nature is explained by the fact that they formed simultaneously as orbiting pairs in some primordial nebula. They were not “captured”. Captures are possible, but not very likely. Everything about the encounter has to be perfect. For example, an approaching spacecraft can orbit a planet it encounters, but it has to fire its thrusters at just the right moment and direction to either speed up or slow down and enter into orbit. It is unlikely to happen by accident.
v= sqrt (2GM/r)
v = escape velocity
G = universal gravitational constant
M = mass of the body to be escaped from
r = distance from the center of the mass
An object traveling at greater than escape velocity WILL escape (unless it collides first). An object travelling at less than escape velocity must eventually collide with the parent body (unless it is in orbit around it).
In general, then, two objects drifting aimlessly through space will exert a minute gravitational attraction to one another. As they fall towards each other, their velocity will increase. However, we must add (or subtract) to that velocity the original velocity the two had relative to one another. When they eventually meet, their velocity will reflect the speed gained by falling down the gravity well (which, by the way, is always the escape velocity!) plus whatever velocity relative to one another they had when they were just drifting. Therefore, any object falling towards the sun from interstellar space will be traveling at least at escape velocity when it gets here, plus whatever proper motion it had to begin with.