Krdytkyu

I was thinking about how many things are affected by the Earth's rotation, and was wondering how much more difficult a retrograde re-entry would be (where the Earth's rotation direction is assumed prograde). I know about the Coriolis force, which states:

The Coriolis effect is the behavior added by the Coriolis
acceleration. The formula implies that the Coriolis acceleration is
perpendicular both to the direction of the velocity of the moving mass
and to the frame's rotation axis. So in particular:

[...]

• if the velocity is against the direction of local rotation, the acceleration is inward to the axis. (On Earth, this situation occurs
  for a body on the equator moving west, which would deflect downward as
  seen by an observer.)

Meaning that we'd essentially be seeing the Coriolis effect added to the gravity vector (right?). My guess is that the craft would basically accelerate towards Earth at a greater rate as opposed to being "ejected"? I've not really done calculations on exactly how much the Coriolis effect would be in a retrograde re-entry (because I don't really know how to).

Getting back to the point, I'd like the answer to the question to focus on:

• What are adjustments we'd need to make for a retrograde (against rotation) re-entry on Earth?


• If none, would there be a planet in our solar system that would require adjustments?


• Am I misunderstanding any key concepts here?

Things I'd like to ignore:

• In-feasibility of retrograde launches, orbits, or anything else to do with getting there.
  
  
  
  
  • The craft came back on the wrong side of the Earth from a Mars voyage or something.
  
  
  
  

I was wondering if anyone could explain why the re-entry retrograde would be more delta-v, and by how much.

The Earth rotates eastward, carrying the atmosphere with it. For prograde reentry, you're moving "with the wind" so to speak, so the relative speed between you and the atmosphere is lessened; for retrograde reentry you're going into a headwind; the relative speed is increased. Earth's rotation speed at the equator is around 460 m/s eastward, so retrograde reentry adds around 920 m/s to the entry speed.

LEO orbital speed is about 7800 m/s, so for retrograde you're coming in $frac {7800 + 460} {7800 - 460} $ or 12.5% faster than prograde. Kinetic energy is proportional to the speed squared, so you have to dump 27% more energy.

Coming from the moon, where reentry speed is about 11 km/s, your speed would be 9% higher and energy 19% greater if you entered retrograde.

Some reentry-heating considerations go up as higher powers than the square of speed, though I don't fully understand them, and you'd want to take those into consideration.

What are adjustments we'd need to make for a retrograde (against rotation) re-entry on Earth?

You'd need a slightly more robust thermal protection system, primarily, and you'd suffer a little more g-force on re-entry -- the exact amounts would depend on the exact trajectory, and there's a lot of room for tradeoff - less g-force means a longer time being heated; a high-g reentry gives a relatively short heat pulse but higher peak temperatures.

If I remember rightly, the heat shield of the Apollo command module was substantially stronger than was actually needed for lunar reentry, and off the top of my head I would guess that it could survive a retrograde reentry without modification. (I might think about using any remaining propellant in the SPS to decelerate before entry, though!)

I'd assume these numbers are lessened by the density of the atmosphere, so it really wouldn't be as big a deal landing on Mars "the wrong way"?

For Mars, the surface speed is a lot lower (~240 m/s); low Mars orbit speed is ~3360 m/s, so it's 15% faster to enter retrograde from LMO.

For Earth-Mars transit to direct entry, this WP page says Pathfinder hit at 7.3km/s, so ~7% speed difference prograde vs retrograde.

The lower density of the atmosphere doesn't affect how much total energy needs to be dissipated; it just constrains how you do so.

I'd like to add to Russell Borogove's spot-on answer. You mentioned Coriolis forces on the retrograde object. The Coriolis "force" is actually a fictitious force required by a rotating frame of reference. Here's an example showing why this force is fictitious.

Imagine your head is at the center of an inertial (non-rotating, non-accelerating) reference frame with a small object, maybe a ball, placed some distance away—for our purposes, assume it's a couple of meters—and motionless in the inertial frame. Ignoring gravitational forces, which will be really small, the ball will just sit there: no applied force, no motion (Isaac Newton nods approvingly).

But now put yourself in rotation around your long axis (head-to-feet axis), with a rotating frame of reference fixed to you. Another of the rotating frame's axes has a direction from the origin (which is inside your head), and is aligned along your normal line of sight, so it emerges from your head directly between your eyes and at time zero continues straight at the ball. In the inertial frame you are rotating but the ball is still just sitting there, motionless, as you rotate. But in your rotating frame, the ball appears to be orbiting around your head with a uniform circular motion; some force (the $m(Omegatimes(Omegatimes R))$ force) appears to be making its path deviate from a straight line and instead describe a circle around your head, i.e. the origin of the rotating frame. It appears centrally directed, like the force due to gravity, except given that you're rotating at a constant rate, its magnitude increases proportionally with distance, not with the inverse square of distance as gravity would.

But if you allow yourself an out-of-body experience, and your astral projection looks at your body and the ball from a vantage point at rest in the inertial frame, you immediately know, "There's no such centrally-directed force. It's just an artifact of my rotation." And that is a perfect qualitative description of Coriolis forces: they are artifacts of the rotating frame of reference.

If you look at the prograde and retrograde entries from an inertial reference frame their trajectories are identical, in terms of speed and flight path angle vs altitude. As Russell Borogove says, the difference is that Earth's atmosphere is moving almost parallel to the prograde one, almost anti-parallel to the retrograde one. Notably, the curvature of the two trajectories is identical at the same altitude. They feel the same gravitational acceleration, and the curvature of their trajectories is identical, at the same altitude. There is no other significant force acting on the two objects. In the inertial frame!

But once you look at this situation in a frame of reference centered on and rotating with Earth, that simple situation becomes more complicated. In this frame, the velocity vector of the object entering on an eastward (prograde) trajectory has a magnitude smaller than its inertial-frame speed. The apparent centrifugal force (if treating this rotating reference frame as if it were inertial) is too small to account for the trajectory's curvature given the gravitational force and the velocity. The $-2(Omegatimes V_{pro})$ artificial force (outward) produces the proper curvature in the rotating frame.

The object entering westward has a velocity magnitude larger than its inertial-frame speed. The apparent centrifugal force is too large to account for the trajectory's observed (in the rotating frame) curvature. The $-2(Omegatimes V_{retro})$ artificial force (inward now, since the direction of $V_{retro}$ is close to opposite that of $V_{pro}$), acting somewhat like increased gravitational acceleration, produces the proper curvature in the rotating frame. Actually, it's not contributing to gravitational acceleration—it's correcting for too much centrifugal acceleration in the rotating frame! And in the prograde case, it's correcting for not enough centrifugal acceleration.

Once you return to viewing the situation from the inertial reference frame, those Coriolis forces go away. They are artifacts of the rotating reference frame.

As Russell says, the two cases have different atmosphere-relative entry speeds. They also have different atmosphere-relative entry flight path angles (EFPAs)! Relative to the atmosphere, they have identical radial (downward) speeds. But the atmosphere-relative tangential speed for the retrograde case is higher than for the prograde case, so its apparent EFPA is shallower than for the prograde case. EFPA is an important parameter for atmospheric entries: it has a large effect on peak heating rates, peak deceleration rates, and other quantities important to the design engineers.

Russell mentioned that some entry conditions are not proportional to velocity squared. One example of that is the radiative component of heating. Heat shields ("Thermal Protection Systems", TPS, in the lingo) are heated primarily by two mechanisms: convection and radiation. Air coming through the shock wave in front of the entering vehicle is compressed and heated tremendously, and when that heated air moves into contact with the TPS it heats the TPS. The air heated at the shock wave also radiates electromagnetic energy, and when that energy impinges on the TPS it heats the TPS. Typically, in a given atmosphere (composition is important!) the higher the velocity, the larger the fraction of total heating is due to radiation. I can't point to any specific references, but I participated in a workshop at NASA Ames Research Center discussing entry conditions at the gas giant planets, and one of the presenters commented that for prograde entries at Saturn the radiative heating rate goes as velocity to the 15th power!