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If, in free space, I throw two objects towards each other, they can either miss each other and fly apart (if the velocity is enough and there's not enough gravitational attraction between them), or they can be attracted enough to each other that they eventually crash together. In between these two cases is a (seemingly) impossibly minute set of conditions that would lead to the two objects perfectly orbiting around one another. Like tossing a bowling pin, and having it land upside-down and perfectly balanced.

And yet there are a zillion celestial objects perfectly orbit each other (like the moon orbiting the Earth). Why is that? Am I underestimating how easy/likely it is for two random objects in space thrown at each other to start orbiting (as opposed to flying apart/crashing together)? Or is this just a case of the universe being huge, and so after a gazillion tries, you'd expect to see tons of bowling pins land upside-down and perfectly balanced?

Btw, here's an expertly drawn diagram that took 1000 hours in MS Paint to show things visually. scenarios

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    $\begingroup$ Being attracted to each other in no way means they will (or even can) ‘crash together’. Angular momentum makes it hard. $\endgroup$
    – Jon Custer
    Commented Nov 22, 2023 at 20:54
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    $\begingroup$ I can't help but think this is also some strange version of survivor bias in a ways, you see all the objects that are orbiting and assume that this is just the case all the time. However many objects have certainly collided with one another all over the observed universe (for example comets hitting the large gaseous planets in our solar system and galactic mergers). Maybe not but that's how I interpreted the question. $\endgroup$
    – Triatticus
    Commented Nov 23, 2023 at 0:49
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    $\begingroup$ Related: physics.stackexchange.com/q/12140/2451 , physics.stackexchange.com/q/26083/2451 , physics.stackexchange.com/q/93830/2451 and links therein. $\endgroup$
    – Qmechanic
    Commented Nov 23, 2023 at 4:13
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    $\begingroup$ The gravitational cross-section is of order $\pi R^2$, with $R$ the sum of the objects' radii, which is obviously minuscule for astronomical distances, cf. aquarid.physics.uwo.ca/kessler/DerivationofCollisionProb.pdf. $\endgroup$
    – auxsvr
    Commented Nov 23, 2023 at 9:10
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    $\begingroup$ Re "perfectly orbit each other (like the moon orbiting the Earth)": That is far from perfect; the moon's orbit is one of the most complex known orbits (more than 1,000 parameters to describe it adequately within the measurement errors). The Sun's gravity is of comparable magnitude at the moon's distance (which makes it extremely complicated) and some of the Earth's rotational energy is also "stolen" to change the orbit. $\endgroup$ Commented Nov 23, 2023 at 15:56

7 Answers 7

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Your intuition that two isolated objects approaching each other and not already in a closed orbit will either collide or will fly apart again is substantially correct. If the objects are not already in a closed orbit and so their relative speed that is greater than their escape velocity then they cannot enter an orbit unless there is a mechanism for carrying away energy from the system. Other answers here that suggest otherwise are wrong.

However, your intuitive image of planets and their satellites flying randomly through space until they happen to reach a stable configuration is incorrect. The actual process was a bit more complicated than that.

Planets and their satellites emerged in the early solar system from a process of accumulation. The young Sun was surrounded by a cloud of gas and dust called a protoplanetary disk. Small dust particles in this disk banged together, lost energy as heat, and some eventually stuck together to form pebbles. The pebbles banged together, lost energy as heat, and some eventually stuck together to form rocks. The rocks banged together ... and so on ... eventually forming planetesimals. And some of the planetesimals banged together and eventually formed the planets - but a lot of planetesimals were left over, and now form the objects in the asteroid belt, as well as many more trans-Neptunian objects in the outer reaches of the solar system.

As it formed, each planet was surrounded by a circumplanetary disk. Some satellites formed from the material in this disk; others satellites were asteroids that collided with other objects orbiting a planet, lost energy, and were captured by the planet's gravity. We believe our own Moon had a particularly violent origin when a Mars-sized protoplanet collided with the early Earth, ejecting a massive amount of debris into orbit, which later formed the Moon (this is the "giant impact hypothesis").

Using powerful telescopes such as Hubble and JWST, we can now observe various stages of this process of planetary formation around nearby stars such as HL Tauri.

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    $\begingroup$ An issue: OP doesn't specify how hard the objects were thrown, so it is absolutely possible that they enter into orbit around one another. If the velocity of the initial throws had the speeds and directions of orbiting bodies, then of course they would. Now if it had specified that the two objects were gravitationally unbound to begin with, then sure. I understand 100% your interpretation of their question - it seems like they're asking something along the lines of how planets trap satellites. However, they didn't ask that, they asked about thrown objects. $\endgroup$ Commented Nov 23, 2023 at 13:42
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    $\begingroup$ @JasonPatterson in fairness, the OP didn't really ask about either — the question (based on the title) is about orbital stability in planetary systems. The "thrown objects" digression was their inaccurate analogy to the process by which orbital systems form. gandalf61 explains why it's not analogous, and therefore not really relevant. $\endgroup$
    – FeRD
    Commented Nov 25, 2023 at 23:47
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Btw, here's an expertly drawn diagram that took 1000 hours in MS Paint to show things visually.
scenarios

None of your three pictures really illustrates the actual physical scenario very well. (OK, picture B kind of comes close.) Here's a slightly nicer and, more importantly, more physically correct drawing I quickly whipped up in Inkscape:

More accurately drawn scenarios

The important thing to realize is that Newtonian gravity is an inverse-square force, i.e. the strength of the gravitational attraction between two bodies is proportional to the inverse of the square of the distance between them.

This abstract-sounding property has a remarkable consequence: the paths of two objects under their mutual gravity (viewed from a reference frame at rest relative to their mutual center of mass, and assuming that other forces and the gravitational effects of other objects are negligible) are always conic sections: circles, ellipses, parabolas, hyperbolas or (as a degenerate case) straight lines.

Also, the paths of the two bodies always share a common focus. And since the focus of a conic path does not lie on the path (except in the degenerate case of a straight line path), this results in an even more remarkable consequence: two objects interacting under gravity, in the absence of external forces, can only collide if they start out moving straight towards each other (or so close to straight that they pass within the sum of their radii from each other).

(To address some of the comments below, I should clarify here that by "straight towards each other" I mean straight in a reference frame where the objects' mutual center of mass is at rest. If you look at things from a transversely moving reference frame, straight paths can look curved and closed circular or elliptical orbits can look like corkscrew spirals, etc., but that that doesn't change the underlying physics. Also, in particular, if you're sitting on one of the objects and don't see the other one falling straight towards you, then it's not going to collide — at least not unless there's a third object nearby to perturb the system.)

In other words, your illustration of "scenario A" isn't physically possible. Unless the objects hit each other head on the first time they pass each other, they won't hit at all. They cannot spiral around each other before colliding, at least not unless some force other than gravity perturbs their orbits, or unless they're are close enough to each other and massive enough that the non-Newtonian effects of general relativity become significant (as happens e.g. in neutron star mergers).

And it's very rare for two objects in space to move exactly towards each other, so your scenario A is actually the rare case. It does occasionally happen, of course (otherwise there would be no meteor impacts, no protoplanetary collisions like suspected lunar formation event and no stellar collisions), but only very rarely.

Instead, the common cases are your scenarios B and C.

You've drawn scenario B pretty accurately: if the objects are moving too fast relative to each other, their mutual gravitational attraction won't be strong enough to pull them back towards each other after they've passed. In this case their trajectories relative to each other will be hyperbolic (or, if their relative speed is just enough, parabolic), and they'll never meet again after flying past each other once.

(That is, unless of course the gravity of some other body causes their paths to curve and brings them close together again later. This happens e.g. with the Earth and near-Earth asteroids, which are asteroids whose orbit around the Sun brings them repeatedly close to the Earth.)

If the objects aren't moving fast enough, however, we end up in scenario C: mutual orbit. Your drawing of this scenario has one major inaccuracy, though: in order for two objects to stay in a mutual orbit (without the influence of a third body or non-gravitational effects) they must already be in a mutual orbit!

In particular, (Newtonian) gravity is a conservative force, and thus dynamics under it are time-reversible. Since it's not possible for two objects orbiting each other to spontaneously gain enough speed to fly apart, it's also not possible for two objects not already in a mutual orbit to spontaneously lose enough speed to end up orbiting each other.

(However, it's possible for quite distant objects to be in a mutual orbit, as long as their initial relative speed when far apart is sufficiently low. They'll gain speed as they fall closer to each other under gravity, but they'll also lose the same amount of speed when flying away from each other, until they finally end up at the same relative distance and velocity as they started. A familiar example of this are the orbits of long-period comets around the Sun.)

Remarkably, all of the above depends crucially on gravity being (at least approximately, ignoring relativistic corrections) an inverse-square force. If the strength of the gravitational attraction between two bodies was instead inversely proportional to, say, the cube of their distance, then much stranger orbits similar to your spirals would in fact be possible, and objects in space would be much more likely to either fly apart or spiral into each other.


Ps. If Newtonian gravity says that two non-orbiting bodies can't just spontaneously get captured into a mutual orbit, then how do objects end up orbiting each other in the first place? Well, there are two or three common situations:

  1. A lot of celestial bodies are formed already orbiting each other. This is true e.g. of most binary stars (which condense out of the protostellar nebula together), planets (which form out of the protoplanetary disk of gas and dust surrounding a young star) and most major moons (which form either in a similar fashion as the planets themselves, or from collisions between protoplanets).

    In a way this shouldn't be too surprising: two nearby objects in space will not orbit each other only if their relative velocity is high enough for them to escape each other's gravity (or low enough that they fall together and collide). And most of the time nearby things in space tend to have fairly similar (but not exactly the same) velocities.

  2. Another way for two objects to end up orbiting each other is through gravitational integrations with a third body. With three interacting objects, much more complex orbital dynamics are possible. In particular it's possible for one of the three bodies to be flung away, taking enough kinetic energy with it that the other two bodies can no longer escape each other but will remain in mutual orbit.

    This is believed to be e.g. how the irregular moons of the large gas planets in our solar system got captured into their current orbits: either they interacted (gravitationally or collisionally) with a pre-existing moon of the planet, or they started out as binary asteroids (which likely formed in situ, as described above) that ended up breaking apart when they passed too close to the planet, with one half of the pair flying away and the other getting stuck in orbit around the planet. Three-body (or more likely many-body) interactions likely also explain how the Earth's moon managed to end up in a stable non-Earth-intersecting orbit after its initial formation in a collision between the proto-Earth and another protoplanet.

  3. Finally, it's possible for objects in space to interact via non-gravitational forces (such as fluid drag, which is e.g. how the gas around newly forming stars coalesces into the protoplanetary disk mentioned above) or through gravitational interactions not captured in the simple Newtonian point mass model of gravity (such as tidal forces acting on extended bodies, or gravitational wave emission in general relativity).

    While such interactions tend to be fairly weak compared to gravity for things like asteroids, moons, planets and stars, they can still affect their orbits over sufficiently long time periods. In particular, once two bodies initially end up in a mutual orbit through one of the mechanisms described above, things like tidal interactions can help gradually dissipate energy and make the orbits more stable and closer to circular.

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  • $\begingroup$ Your wording of 'moving straight towards each other' is a bit misleading. I first thought it means an a straight lines as drawn in the picture but is only one of the possible cases. They can also move on the same parabola or hyperbola and will then hit each other. From far away this approximately looks like they move in straight lines towards each other at an angle but are timed just right to hit each other. $\endgroup$
    – quarague
    Commented Nov 24, 2023 at 17:51
  • $\begingroup$ @quarague: I was implicitly assuming that we're looking at the objects from a viewpoint at rest relative to their mutual center of mass. I should probably have been clearer about that. In the scenarios I believe you're talking about, the bodies are still moving along straight lines relative to their center of mass (and also an observer standing on either body will see the other approach along a straight line), with apparent curved motion only emerging if you look at them from a reference frame moving transverse relative to their separation. $\endgroup$ Commented Nov 24, 2023 at 18:08
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    $\begingroup$ … In general, there's not much point in looking at a system of gravitationally interacting bodies from a viewpoint that's not at rest relative to their center of mass (or co-moving with one of the bodies). That'd be like looking at the solar system from a perspective of a random interstellar asteroid flying past it and saying that the orbits of the planets around the sun aren't really ellipses but corkscrew spirals. Sure, technically that's a valid observation in that reference frame, but it doesn't give any useful insight into the orbital dynamics of the system. $\endgroup$ Commented Nov 24, 2023 at 18:16
  • $\begingroup$ I started thinking about the two objects like billiard balls moving on a table with an external observer looking at the table ignoring gravity. Then think about what happens with gravity. So fixed reference frame independent of the two objects. You are correct that in the reference frame centered at the mutual center of mass this would still look like your picture A. $\endgroup$
    – quarague
    Commented Nov 24, 2023 at 19:43
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    $\begingroup$ @Peter-ReinstateMonica: In the sense that nothing in the universe ever moves exactly on a straight line, I guess you're right. However, most pairs of objects that collide are moving approximately at a straight line relative to each other before they collide, because that's the only kind of two-body collision Newtonian gravity allows. They don't spiral in like in the OP's picture. […] $\endgroup$ Commented Nov 25, 2023 at 14:02
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they can either miss each other and fly apart (if the velocity is enough and there's not enough gravitational attraction between them), or they can be attracted enough to each other that they eventually crash together. In between these two cases is a (seemingly) impossibly minute set of conditions that would lead to the two objects perfectly orbiting around one another

Your assessment of the likelihood is substantially off. For two objects interacting gravitationally there are two quantities that determine this behavior: mechanical energy and angular momentum.

For the two objects to collide the angular momentum has to be almost zero. Otherwise they will miss each other. Of the three options, this is the one with the very small set of conditions.

If the angular momentum is large enough that they don’t collide, then we turn to the energy. If the energy is positive then they fly apart. If the energy is negative then they orbit.

That is it. Orbiting just requires negative mechanical energy and a non-minuscule angular momentum. It is not a particularly restrictive set of requirements.

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    $\begingroup$ Picture C, orbit, in OP is physically impossible. The objects start in a hyperbolic trajectory and will remain so. $\endgroup$
    – Taemyr
    Commented Nov 23, 2023 at 8:49
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    $\begingroup$ "It is not a particularly restrictive set of requirements" - this is incorrect. If the objects are not already in a closed orbit then their gravitational PE plus their KE is positive, and will remain so unless there is some mechanism for removing energy from the system. Without such a mechanism, two objects that are not already in a closed orbit cannot enter a closed orbit. $\endgroup$
    – gandalf61
    Commented Nov 23, 2023 at 9:53
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    $\begingroup$ @gandalf61 the OP specified “in free space, I throw two objects towards each other”. Nothing about that precludes “throwing” into a closed orbit. I am completely correct in my assessment that it is not a particularly restrictive set of requirements. Out of all possible “throws” a large portion result in a closed orbit $\endgroup$
    – Dale
    Commented Nov 23, 2023 at 12:58
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    $\begingroup$ @Vincent kinetic energy is non-negative, but potential energy is $-GMm/r$ which is negative. If the PE is a bigger negative quantity than the KE, then the sum is negative and the orbit is an ellipse. If PE is a smaller negative quantity than the KE then the sum is positive and the orbit is a hyperbola. See en.m.wikipedia.org/wiki/Gravitational_energy $\endgroup$
    – Dale
    Commented Nov 24, 2023 at 12:25
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    $\begingroup$ @Vincent that was probably using $mgh$ where $h=0$ is an arbitrarily chosen reference, so PE can be negative or positive. In $-GMm/r$ the reference is $r=\infty$ so PE is always negative. $\endgroup$
    – Dale
    Commented Nov 24, 2023 at 19:18
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Partly it's about how galaxies form (see another excellent answer about that).

But do also consider that you're seeing the survivor bias of 13.8 billion years. If they weren't in a stable orbit, the chances of you being there to see them existing are very very very low.

Statistically, it's pretty likely that Earth does have a few meteorites captured which happened to hit our gravity just right. They're small bits of dark coloured rock though, so we'd be really unlikely to ever find them.

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chausies asked: "Why are there so many objects perfectly orbiting each other?"

Apart from the fact that most orbits are not perfectly circular but more or less elliptical: if two objects fly past each other or crash into each other they only do so for a short time, while if they orbit each other they do so for a long time, that's the main reason for your observation.

If you look for a short time you only see the moon orbiting earth, but if you look longer you see more than one asteroid flying past the earth or crashing into the atmosphere. If you look at moon's craters you'll notice it also has a lot more objects crashing into it than orbiting it. Jupiter is also a metaphorical magnet for objects to crash into it, far more than it has moons.

If objects don't get caught in the earth's gravitational field they still might get caught in the sun's or the galaxy's field, since higher initial velocities have lower probabilities most objects get caught at some level. There are enough fast Neutrinos or black hole jet material with high enough velocity to escape even the galaxy's field though.

chausies asked: "Am I underestimating how easy/likely it is for two random objects in space thrown at each other to start orbiting?"

It's not like you're aiming at one specific blade of grass and hitting it the first time, it's more like shooting into a whole meadow and hitting some random blade of grass, which is not hard since there are enough of them in the way.

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And yet there are a zillion celestial objects perfectly orbit each other

This is survivorship bias. Based on our current model of how we think planets form, there were many orders of magnitude smaller bodies. What happened to those? They crashed into each other to form larger bodies and crashed into larger bodies to form our planets.

Some moons may have been formed separately and captured, but many probably formed by smaller bodies starting with dust orbiting the planet crashing into each other and coalescing into moons, and some like our own moon formed when large enough other objects crashed into planets with enough force for to eject material into orbit to form the moon.

And all that time, many more orders of magnitude of 'near misses' occurred where the objects didn't crash into each other and didn't orbit each other, but missed each other and went on to crash into other bodies.

Actually your (C) isn't really possible, two bodies that aren't gravitationally bound really won't come together into orbits if they are heading towards each other. Say you had the Earth and Moon stationary relative to each other 100 million km apart in intergalactic space and you imparted just enough lateral motion so that when they passed each other they wouldn't collide. Eventually their combined gravity would draw them closer, but at the point they passed they would have the highest relative velocity, having that gravity pulling them faster and faster the whole way. After they passed, the gravity would begin slowing them down, but only the same gravity that was pulling them closer. They would end up 100 million km apart in the opposite direction before they stopped relative to each other. When we send something to orbit Mars for example, we need a deceleration burn and/or aerobraking from the Martian atmosphere to decrease the speed.

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At first glance it may seem this way, but conservation of angular momentum sort of acts as a "restoring pseudoforce" to make orbits more likely.

If one object is traveling towards another from very far away, you can calculate the "impact parameter" $b$ which tells you how far a straight trajectory would be from the COM of the planet/object.

enter image description here

You can then calculate the angular momentum of the incoming object with respect to the center of the target planet/object, i.e.

$L = mvb $

When you calculate the equations of motion, you end up with an additional term to the potential energy called the "effective" potential due to angular momentum. It looks like this (for the case when $m << M$:

$U_{eff} = \frac{L^2}{2mr^2} - \frac{GMm}{r}$

Notice the two terms dependence on r: When the object gets far away, it's dominated by an attractive force. When it's close, it's dominated by the repulsive term.

Therefore there's a decent range of possibilities where objects will never escape forever nor crash into the surface.

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    $\begingroup$ Surely conservation of energy tells us that for an object to approach from "infinity" and go into orbit, there must be some mechanism to carry away energy from the object ? $\endgroup$
    – gandalf61
    Commented Nov 22, 2023 at 21:35
  • $\begingroup$ this is a wierd comment $\endgroup$
    – Señor O
    Commented Nov 23, 2023 at 5:27
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    $\begingroup$ @SeñorO Actually this is not weird comment - in two body system, if one object comes from 'infinity', then there are two possibilities - they either crash or object flies away to infinity; it will not stay on orbit. In three and more bodies system things are different. $\endgroup$
    – Arvo
    Commented Nov 23, 2023 at 8:17
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    $\begingroup$ @SeñorO My "weird comment" is a polite way of saying that your answer is wrong. Two objects that are not already in a closed orbit cannot enter a closed orbit unless there is some mechanism (e.g. a collision or an interaction with a third body) for removing energy from them. $\endgroup$
    – gandalf61
    Commented Nov 23, 2023 at 9:56
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    $\begingroup$ @gandalf61 you're confusing "polite" with condescending. Yes you're absolutely right I left out a crucial part of the answer and only focused on why objects don't only crash or fly past each other forever. $\endgroup$
    – Señor O
    Commented Nov 23, 2023 at 12:47

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