Knowledge (and a question) for the aspiring astronauts
Let us start with some terminology for a spacecraft orbiting around the Earth.
Periapsis : The point in its orbit where a satellite is at the least distance from the Earth.
Apoapsis : The point in its orbit where a satellite is at the greatest distance from the Earth.
Prograde : The direction in which a ship is traveling along its orbital path. Since orbits are elliptical, it is always tangent to the orbit at the point where the ship is.
Retrograde : The reverse of Prograde, backwards along the orbital path.
Radial : The radial direction is the direction inside the orbit, towards the focus of the orbit. i.e towards the center of the Earth.
Anti-radial : The Anti-Radial direction is the direction outside the orbit, away from the focus. i.e away from the center of the Earth.
Neglecting the presence of an atmosphere. What is the most efficient (least fuel consuming way) to lower one's orbit and perform a reentry.
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We will be taking this problem with the classical method of elimination.
An Anti-radial burn is strictly out of question, it can circularize your orbit at best and that too in a very inefficient way. Also it will take you further away from the planet instead of performing a reentry. Hence two of the options have been eliminated.
A Pro-grade burn will increase your kinetic energy (by increasing your velocity in the direction you are moving), hence what it will really do is put you in a larger orbit instead of a smaller one. It is a simple and true statement for orbital motion. The faster you go the higher you are.
Henceforth, we have eliminated four of our eight options.
Now here's the main part. Do you burn radially towards the center or opposite to your direction of motion (i.e Retrograde).
Most of the maneuvers modern day spacecraft perform make use of the SOI (gravitational Sphere Of Influence) of a body to cut low on fuel requirements. If you burn directly towards the center of Earth, you will be wasting fuel doing what earth's gravity could easily do if you are not going so fast. This is the basic principle of what is called a Hohmann Transfer , a standard textbook maneuver and the most fuel efficient way there is.
Hence what we do is decrease our tangential velocity. Then all we have to do is wait to fall back to the surface. But wait...
What about where to perform the burn ?
The periapsis might be obvious to some but it is not so.
The green circle represents the initial orbit. The orange ellipse represents the transfer orbit and the red represents the orbit after the circularizing burn. The thing I am trying to say is , when you perform a hohmann transfer it affects the other side of the orbit. If you perform retrograde at the periapse it will lower the apoapse and vice-versa. The periapse being already lower than the apoapse ,is what you want to lower further to consume the least fuel. Hence you burn at the apoapse to lower the periapse and the answer is.
A retrograde burn at the apoapsis.
Inspired by a problem from "An Introduction to Mechanics" by Kleppner and Kolenkow and supplemented by my knowledge of Kerbal Space Program.