Please Explain the Velocity Numbers

[Donnie B.]

One way to look at this question is to assume that the CSM barely made it to the Moon-Earth crossover point before falling the rest of the way back to Earth.  It's not particularly difficult to calculate the resulting velocity at the atmospheric interface.  Can you do this calculation?  If so, then all you need to do is add the Earth-relative velocity of the CSM at crossover (hint: it's non-zero but pretty small) and you have the velocity at interface.  In fact, it can't possibly be any higher (unless they deliberately increased it by an SM engine burn).

[Mag40]

First, at the end of the Trans Earth Injection Burn, Apollo 11 had three different relative velocities, which were approximately:  (1) 2,624 meters per second relative to the Moon;

No (1) seems to be correct.

(2) 3,624 meters per second relative to the Earth;

And that is incorrect. The TEI is performed on the far side of the Moon and brings the craft out in a clockwise elliptical orbit against the Moon's orbital speed.

http://www.braeunig.us/apollo/hybrid-profile.htm

[Jason Thompson]

First, at the end of the Trans Earth Injection Burn, Apollo 11 had three different relative velocities, which were approximately:  (1) 2,624 meters per second relative to the Moon; (2) 3,624 meters per second relative to the Earth; and (3) 33,044 meters per second relative to the Sun.

Point 2 is not correct, as has been pointed out several times already.

To go into orbit around the moon, the spacecraft approached the leading edge of the Moon and swung behind it. When it was behind the Moon it was moving in the opposite direction to the moon's own orbit round the Earth. The TEI burn occurred on the far side of the Moon. This is something you are surely already aware of given how much is made of the Apollo 8 TEI burn and the anxious wait in mission Control to regain contact when they re-emerged from behind the Moon after the burn.

Given that, at the end of the TEI burn it is moving at a lower orbital speed than the Moon in relation to Earth, not a higher one, so it was never at escape velocity from Earth at lunar distance.

[PUshift]

...You do not get a perigee point unless you are in orbit.

but yes, you get a perigee indeed even if not being in orbit, because you pass a celestial body at any closest approach. General word is Periapsis. Think of probes like Cassini flying by at Jupiter on the way to Saturn. What you don´t have while still not in orbit is the opposite, maximum approach (Apoapsis)

It is been said that Apollo crews never actually left earth orbit, even if beeing closed to it. If not maybe a little bit above.
Assuming you are not yet in orbit, still above escape velocity. Now inside the atmosphere velocity decreases gradually. On the way to periapsis the hyperbolic curve turns into an ellipse (now orbit), then further down to a parabola. You land and ask what the problem is.

For Jupiter Periapsis is called Perizene/Perijove.
For the moon: Periselene/Pericynthion/Perilune
Perigee for earth
(all from 'http://en.wikipedia.org/wiki/Apsis' )

I strongly recommend reading the pages of Bob Braunig http://www.braeunig.us/space/index.htm or working with Orbiter Flight Simulator http://orbit.medphys.ucl.ac.uk/.

[pleasedebunkme]

Assuming that chart noted above might be correct, 1,529 meters per second at the distance of the Moon is still escape velocity with respect to the Earth, no? 

[Glom]

Assuming that chart noted above might be correct, 1,529 meters per second at the distance of the Moon is still escape velocity with respect to the Earth, no?

Don't think so. It's a little below.

[PUshift]

@pleasedebunkme
Orbiter Space Flight Simulator, hopefully realistic enough to show this in general. Good for people like me who can´t do the maths but have fun to try to apprehend the numbers.

1 hour before reentry, earth still to small to display it as a circle. Trajectory initialised by TEI.
'Focus PeD' is said 'aim' of 40km, in this case 6410 - 6371 = 39km
Since we have an apogee from twice the distance to the moon (more than 800 thousand km) we are still in Orbit. Just even! Closed to escape velocity.


While reentry, 1 minute before reaching perigee, altitude 50 km.
Velocity still above 10km/s


at the initial perigee of 40km, still in an elliptical orbit:


...and in 36km altitude, orbital energy is nearly utilized. Perigee had swopped position with apogee and moves virtually down into the earth´s body at the opposite side of the planet. Ellipse turned into parabola since orbit went below the surface. Still in a comfortable altitude for the final fall and parachute deploy:

[scooter]

Orbiter is a great way to play with orbits, and experiment...geostationary in the DG...yeah!

When the CSM finished it's TEI burn. it was still behind the Moon, and it's speed, while noticably increased, was certainly not enough to significantly straighten it's line of travel. The Moon curved the trajectory towards Earth, actually somewhat past it.  Earth's gravity later became dominant and started bringing the trajectory closer in. Finishing the "figure 8" that was started with TLI.
So while it did reach escape velocity from the Moon, the Moon's gravity affected (curved) the departure trajectory, sending the craft back towards Earth. Had the burn been made 90 degrees earlier or later, then there would be some "Earth escape velocity" issues. The burn on the opposite side of the Moon gave the energy and vector to get the craft into Earth's gravity domain.

[Chew]

However, I do not believe it is possible for the Moon’s gravity to slow down Apollo 11 enough to allow Apollo 11 to regain orbital velocity relative to the Earth.

You can use the Vis-viva equation to compute how much it was slowed down at various distances from the Moon. All you need is the Standard gravitational parameter for the Moon, semi-major axis of the orbit, and distance from the center of the Moon.

First we need to rearrange the equation to find the semi-major axis (a). Using a distance (r) of 1838 km from the center of the Moon (Moon radius of 1738 km + 100 km altitude) and a velocity of 2.624 km/s, we get a semi-major axis of -3162.2 km (a negative semi-major axis means a hyperbolic trajectory, which we know makes sense in this case because the velocity is in excess of escape velocity). Then we can solve for the velocity for any distance, keeping in mind the limits when the Earth's gravity eventually exceeds the Moon's.

Now that we know the semi-major axis we can plug in some distances and compute some velocities:
2000 km 2.54 km/s
4000 km 2.00 km/s
8000 km 1.67 km/s
16,000 km 1.47 km/s
32,000 km 1.36 km/s

For this last distance, if the spacecraft were directly between the Earth and the Moon the escape velocity from the Earth would be 1.33 km/s. But we know the spacecraft would be farther than the algebraic difference in the distances because the path back to Earth was a long ellipse. It would be 32,000 km from the Moon but it would be about 370,000 km from the Earth.

But it's not really an issue for the reasons others have stated, e.g. aerobraking and vacuum perigee.

[Jason Thompson]

General word is Periapsis.

Thank you, that's the word I couldn't think of earlier!

[Jason Thompson]

But it's not really an issue for the reasons others have stated, e.g. aerobraking and vacuum perigee.

Indeed. For an example of an object that definitely did have Earth escape velocity and yet failed to escape Earth, just look at what happened in Chelyabinsk in February. It doesn't matter if your spacecraft does have escape velocity if its perigee is in the atmosphere. That will slow it down nicely. The trick is getting it right so that it slows you down quickly enough that you don't skip out and slowly enough that you don't burn up.

[ka9q]

It is been said that Apollo crews never actually left earth orbit, even if beeing closed to it.

Not only was it said, it was also true. At no time did an Apollo spacecraft on a lunar mission achieve earth escape velocity. The eccentricity after TLI was 0.97, which is still a closed elliptical orbit with an apogee somewhat beyond the moon's orbit.

Note I said "Apollo spacecraft". Another piece of the Apollo 8, 10, 11 and 12 missions did achieve earth escape, namely the S-IVBs. They did this by slowing down relative to Apollo after separation. The S-IVB fell so far behind after three days that the moon crossed its path in the meantime. So it swung past the trailing limb of the moon, whose gravity dragged it along in the direction of its orbit. This gave the S-IVB earth escape velocity, flinging it into an independent orbit around the sun.

Apollo got to the moon's orbit ahead of the moon (and the S-IVB) so the moon's gravity acted in the opposite direction, pulling Apollo into what would have been a free return trajectory to earth if lunar orbit injection had not occurred. (This actually happened on Apollo 13.)

Now had Apollo somehow achieved a prograde (eastward) lunar orbit instead of the retrograde (westward) orbit actually achieved and then performed a prograde trans-earth injection burn on the far side, the moon's orbital velocity would indeed have been added to its own. Apollo would have broken out of both lunar and earth orbit and sailed off into an independent solar orbit like the S-IVB before it.

[Echnaton]

I believe “pericynthion” is an orbital term.  Therefore, “pericynthion” does not exist unless you are first in orbit, around something.

Calculations for spacecraft in flight are orbital calculations and all the terms apply.   Why?  Because orbital mechanics is the field of applying math to address the question of bodies in such circumstances. The popular use of the term "orbit" that you are mistakenly using suggests that an orbit is only a craft relative to a larger body that the craft is circling.  That is, the notion that Apollo left earth orbit and entered lunar orbit is a gross popular simplification of a far more complex science and engineering discussion.   As long as you favor a populist approach and refuse to address the more exacting ideas of orbital mechanics, you will never understand why your position is wrong. 

[PUshift]

General word is Periapsis.

Thank you, that's the word I couldn't think of earlier!

Thank you for your reply. May I get a t-shirt with "I brought up anything useful on the ApolloHoax.net-Forum before somebody else beat me to it" ?

[Chew]

Dang. I screwed up the escape velocity in my previous post. At 32,000 km from the Moon the escape velocity for the Earth is 1.50 km/s, not 1.33 km/s. So the spacecraft velocity is well below what is needed to go in a closed orbit around the Earth.