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Imagine that my ship is parked in LEO and needs to transfer to a higher orbit. It makes a Hohmann transfer and flies to the needed orbit by an elliptical trajectory. But this trajectory is crossing a few other orbits and the ship collides with another ship from one of those orbits and crashes.

So, how can we detect the possible collision and avoid it and reach the needed position on the needed orbit after this (collision avoidance can take some time, so that position in the orbit can be shifted a bit)?

For example, we need to make a rendezvous with another ship in other orbit, but we collided during flight to it.

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  • $\begingroup$ Are you developing a game, or is this question about real world application? $\endgroup$ Oct 12 at 17:45
  • $\begingroup$ For my game but it precisely simulates reality. $\endgroup$
    – Robotex
    Oct 12 at 18:52
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Exactly the same way you avoid collisions when not altering orbit.

Altering your orbit does not significantly alter your risk of collisions, other than possibly moving you to a higher or lesser densely populated part of orbital space.

Orbits are not neat stacks of perfect circles around the planet. All orbits are ellipses, with the perigee closer to the planet and the apogee further out. These two values can be very close to each other, but that is by no mean a given.

Additionally, even for other orbits at the exact same altitude, and the exact same plane of inclination, that plane can be oriented around virtually any point around Earth. Two craft that are both in a 400km, circular, 51.6° inclination orbit(like the ISS), can still have crossing orbits with a relative speed of 11km/s

The way to avoid collisions when staying on an orbit, is to predict your orbit position over time, do the same for everything else out there, and ensure there are no instances in the short-medium future where your craft and another object will be located in the same spot at the same time.

Similarly, when planning to change your orbit, you again predict your orbit position over time, do the same for everything else out there, and ensure there are no instances in the short-medium future where your craft and another object will be located in the same spot at the same time, exactly the same way.

You are in exactly the same danger of collision whether you are changing orbit, or remaining in the same orbit.

The only added danger is that anyone else would not have known that you were planning the change, so their predictions of your future position may be off, and the onus is on you to ensure safety.

When entering a particularly contested or valuable orbit, such as matching orbits with the ISS, or entering Geostationary orbit, you will of course need to coordinate with everyone that has an interest in that orbit.

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    $\begingroup$ "Exactly the same way" is a bit exaggerated. When you are in LEO, your orbit is most likely quasi-circular. You would ignore objects having a perigee higher than your orbit altitude. If you plan, say a GTO (Geo-transfer-orbit) from LEO, I would think that you have to watch out for many more objects to check that they would not be on your path (except perhaps for objects already "flying" below your LEO orbit). The computation may be the same, but the load is not. In driving a car, if you go straight on, you don't need to look 360°. If you plan a u-turn, you have to watch in more directions. $\endgroup$
    – Ng Ph
    Oct 11 at 16:36
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"Space is big. Really big. You just won't believe how vastly, hugely, mindbogglingly big it is. I mean, you may think it's a long way down the road to the chemist's, but that's just peanuts to space"

- Douglas Adams, The Hitchhiker's Guide to the Galaxy

Spacecraft are tiny compared to the vast amount of space you have around a planet. So collisions between two spacecraft are already pretty unlikely.

Nevertheless the risk is not zero and spacecraft are expensive. Which is why most of the larger known objects in Earth orbit - both functional craft and non-functional debris - are being tracked. Their orbits are known and can be extrapolated with high accuracy for the near future. So when a craft needs to performs a maneuver, it is possible to check the database of known orbits and ensure that there is enough safety distance to anything it might collide with.

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    $\begingroup$ @NgPh It is quite true. This is the correct answer. In fact, space is so big that we generally don't worry about collisions when calculating transfer orbits. That said, control centers in the US do keep in touch with NORAD because they track everything orbiting the Earth, from one inch bolts to Santa. $\endgroup$ Oct 12 at 10:56
  • $\begingroup$ @David Hammen, I wish that you are right. And if it is "quite", "generally" not a worry, it could remain so. This is because "Newspace" is so disruptive. For example: space.com/… $\endgroup$
    – Ng Ph
    Oct 12 at 11:31
  • $\begingroup$ "Spacecraft are tiny compared to the vast amount of space you have around a planet. So collisions between two spacecraft are already pretty unlikely." In my game it is not true. In future with a billions spacecrafts is not true too. $\endgroup$
    – Robotex
    Oct 12 at 12:03
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    $\begingroup$ One cannot just ignore the danger of collisions. We actively monitor orbital craft and debris, and when a potential collision(close pass actually) is detected, the owners of both craft are notified, so that evasive maneuvers can be performed. Even with this, there have been several on-orbit collisions. see en.wikipedia.org/wiki/Satellite_collision It is not a trivial problem, it's just that the OP's question of how much altering orbit increases the danger.. it doesn't increase it much if at all. But constant vigilance is required. $\endgroup$ Oct 13 at 22:25
  • $\begingroup$ @Robotex one would assume in that future you would have a solution that looks very close to the solution for airplanes. There are designated lanes, spacing patterns, traffic control stations, etc. Of course it would all be automated with AI, which is what we are close to doing today anyway. $\endgroup$
    – eps
    Oct 19 at 19:27
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Express the position of your craft in your orbit as an equation with respect to the parameter T (time).

Express the position of something else in its orbit also as an equation with respect to T.

Now you can derive an equation for the distance D between objects on these respective orbits as an equation in T. At the point T where the differential D' of this function is zero and the second differential D'' is positive, T is the time of your closest approach and D is the distance of your closest approach.

Repeat this process for every orbiting object that you might collide with. If you get D less than 50km for any of them and T within the length of time you intend to be on that orbit, that's probably too close.

If something is too close, adjust your proposed orbit in a way consistent with applying thrust one second later (or other seemingly trivial adjustment) and try again.

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  • $\begingroup$ What does it mean? "where the differential D' of this function is zero and the second differential D'' is positive" $\endgroup$
    – Robotex
    Oct 12 at 7:59
  • $\begingroup$ Ok, imagine, that I detected the possible collision. How to avoid it? Should I decrease speed, increase, change it direction? $\endgroup$
    – Robotex
    Oct 12 at 8:00
  • $\begingroup$ This seems to be an algorithm for avoiding collision with ONE object. The real problem is more challenging. First, your 50-Km criterion is dependent on how your D(t) approximate reality. Second, you always have some goal function (minimum Delta-v, minimum time to reach target, latest time to reach target ...). Third, the complexity of the computation is a function of the number of potential objects on your path. $\endgroup$
    – Ng Ph
    Oct 12 at 8:24
  • $\begingroup$ I can check objects in some radius around the spacecraft. I don't think that there will be more than 100 objects at same time. $\endgroup$
    – Robotex
    Oct 12 at 12:05
  • $\begingroup$ @Robotex - the derivative of a function is the slope of its line when you graph it. "D' = 0' means that the distance function is not getting bigger or smaller at that point - it's 'level'. D'' is positive' means that Distance is getting smaller before that point and larger after it. Half of the places where D' is zero are closest approaches; the other half are greatest distances. Looking to see whether the Derivative OF the Derivative is positive filters out the furthest-distances. $\endgroup$
    – Edward.
    Oct 13 at 21:56
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@Ng Ph: Yes, this is an algorithm for avoiding one object. It has to be applied iteratively to check every object that might intersect.

@Robotex: You are being very optimistic. There are literally hundreds of thousands of objects orbiting Earth today, and at the relative speeds they're moving, they'll orbit the world several times before you're done with your movement. If you think you can just check a hundred of them, it will be a miracle if you can figure out which hundred. You'll need to check at least tens of thousands.

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  • $\begingroup$ What if, following the decision to avoid one object your new trajectory gets you closer to 10 other objects? What if, by avoiding getting closer to 10 objects, your trajectory gets you on quasi-collision with one object? What would be the algorithm/decision criteria for these scenarios? $\endgroup$
    – Ng Ph
    Oct 14 at 12:25
  • $\begingroup$ Then you pick a different proposed trajectory and check again. Checking thousands of possible trajectories to find a "safe" one is something that doesn't use up sweat or fuel and your navigation computer should handle it in less than a second. If it checks a million proposals and seriously can't find anything within your delta-vee and safety margin, then you either scrub the orbit change or try again next orbit or compromise your safety margin and solve for a smaller radius. $\endgroup$
    – Edward.
    Oct 16 at 20:23

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