# How do orbital spacecraft calculate reentry?

For a given spacecraft, in a given LEO orbit, assuming you have all the required data, how do you choose the needed burnout speed and timing (or position in orbit), in order to reach a certain point on earth with a ballistic trajectory?

• "How do orbital spacecraft calculate reentry?" Very carefully ;-)
– uhoh
Mar 4, 2019 at 23:53
• @uhoh: Well, Voskhod 2 crew did it less carefully but rather urgently, with a marker over the capsule window.
– SF.
Mar 5, 2019 at 10:23
• What are the equations you would need calculated using sensor data to correct in real time :)? Yes- Im actually asking. A list of some used on actual reentry vehicles to maintain trajectory would make a cool answer. Mar 8, 2019 at 0:03

In reality one re-entry burn would never suffice to reach a specific point. The most you can aim for is within a certain area (e.g. splash down in the pacific) but the uncertainties in your actual impact position will be very large. Either you have to regularly perform corrective manoeuvres or you make sure your spacecraft just burns up and never reaches the ground.

Now assuming you have a VERY accurate model of the atmosphere (including wind) and precise knowledge of the attitude of your satellite, the process of determining the required burn duration and direction is purely numerical optimization. The equations of motion of a satellite re-entering are very non-linear and there is no analytical solution.

In reality it will be a combination of numerical optimization to calculate the re-entry burn and then a sophisticated control system to keep the satellite as close to the reference trajectory as possible.

Edit: In case of spaceplanes this is not valid as they can just glide towards the spot they want to go to and have much more control over their trajectory.

• Shuttle did one re-entry burn and managed to land precisely on a runway. Not ballistic though. Mar 5, 2019 at 13:16
• @OrganicMarble, hmm yeah that's correct! I'll add to my answer that it's not valid for spaceplanes. Mar 5, 2019 at 13:22
• The last 2 Mercury orbital flights, after working out various issues in earlier flights, both got within 5 miles of their target points with no active steering after retrofire. Earlier flights were off the mark due to human error (Carpenter’s MA-7 flight) and incorrect calculations (Glenn’s MA-6). Mar 5, 2019 at 21:55

Not really a scientific answer, but it should help with the question:

An uncontrolled re-entry is mostly unpredictable. If a satellite on a low earth orbit experiences drag from atmospheric friction, predicting re-entry is almost impossible. The drag depends on tons of factors, such as sun activity and weather, and changes constantly. Since the satellite orbits around the whole earth once every ~90 minutes, if you are wrong by 45 minutes you're already on the other side of the planet (plus some shift from the earth's rotation). The closer you get the better you can calculate, often based on live measurements. That's why it's impossible to know where a decaying satellite will hit even the day before it crashes.

Controlled is a bit different. Assuming you don't have a strongly elliptic orbit, if you burn retrograde (i.e. against the orbital flight direction - basically "braking"), you create a perigee (i.e. lowest point in orbit) on the exact other side of the object you are orbiting. With a strong enough retrograde burn, you can lower that point deep into the atmosphere or into the planet. Time it right and you can create a point of re-entry by your choosing - it's not very precise, but you can at least reliably hit a specific area of an ocean of your choosing or, say, siberia.

Space Shuttles do that and add control over atmospheric flight to the mix, which gives them enough control to land on a runway.

The way spacecraft do this is via one reentry burn using the best estimates of the atmosphere, and doing some slight corrections to the path while reentering. The best example of this is the Space Shuttle, which could precisely land due to the aerodynamic surfaces, however, every spacecraft has some ability to steer itself inside the atmosphere. It is very rare to have a purely "Ballistic reentry", however, they were sometimes done, particularly in the early days of the space program.

Basically orienting the capsule will allow it to slightly change its direction. Soyuz can land with a 28 km accuracy. See How does the landing accuracy of Dragon (under parachutes) compare to Soyuz? for some comparisons to Dragon, and some raw values.

• There was a Soyuz ballistic re-entry in 2008. I guess that's early days. Mar 5, 2019 at 13:37
• There was one in 2018 too, for the abort. I said it is rare, but was more common in the early days of the space program. Mar 5, 2019 at 13:41
• 2018 technically wasn't a reentry since they didn't bust the Karman line. Mar 5, 2019 at 13:54
• It didn't? Huh, I thought it had. Well, close enough. Mar 5, 2019 at 21:59