Please Explain the Velocity Numbers

[ka9q]

I have the relevant formula on a NASA bumper sticker on our refrigerator:

What part of
GxMxm/R = mxVesc²/2
Don't you understand? It's only rocket science!

It was apparently designed by an English major who didn't really understand it, so it should have read

GM/R = Vesc²/2

which can be rearranged to read

Vesc = sqrt(2GM/R)

GM for the earth is 398,600.4418 km³/s². For the moon it's 4902.8 km³/s². So at a distance of 400,000 km from the earth, Vesc = 1.412 km/s. For the moon at 1850 km (approx radius of Apollo from moon's center), Vesc is 2.3 km/s.

[Abaddon]

Maybe this might help. Pioneer 10, pioneer 11, Voyager 1 and Voyager 2 all did not enter orbits around Jupiter, but they all had a perijove.
http://www.google.ie/url?sa=i&source=images&cd=&cad=rja&docid=g7fY5KsD4SgQeM&tbnid=zbXxbRisruGLlM:&ved=0CAgQjRwwAA&url=http%3A%2F%2Fwww.daviddarling.info%2Fencyclopedia%2FP%2FPioneer_anomaly.html&ei=B3yeUfG_BMWf7Ab2pYDICg&psig=AFQjCNELAqC0gSrEqEYGznvd3JNyZlxQTA&ust=1369427335158588

[ka9q]

Not only did they all have a perijove, all four also had perikrones. And Voyager 2 even had a periuranion and a periposeidion too.

And in just a few years New Horizons will reach perihadion, after having had a perijove some years ago.

[PUshift]

Not only did they all have a perijove, all four also had perikrones. And Voyager 2 even had a periuranion and a periposeidion too.
And in just a few years New Horizons will reach perihadion, after having had a perijove some years ago.

...and when did we reach Perigalacticon again?
Orbital period 225–250 Myr. - but could be tomorrow
Seriously, do we know that?

[pleasedebunkme]

Thank you, ka9q.  So, after the Trans Earth Injection Burn, Apollo 11 was indeed traveling escape velocity with respect to both the Moon and the Earth.  How did it get home? 

[Chew]

Thank you, ka9q.  So, after the Trans Earth Injection Burn, Apollo 11 was indeed traveling escape velocity with respect to both the Moon and the Earth.  How did it get home? 

Did you read my post???

By the time the spacecraft was 16,000 km from the Moon it had slowed to below Earth's escape velocity.

[Echnaton]

Thank you, ka9q.  So, after the Trans Earth Injection Burn, Apollo 11 was indeed traveling escape velocity with respect to both the Moon and the Earth.  How did it get home? 

The failure to show your work indicates you can't actually do the math.  So, please stop the presumption that your ignorance somehow needs to be disproved and tell us why it shouldn't made it home?

[Glom]

You seem to labouring under the misconception that once a satellite has escape speed it must shoot away from the attractor. Not so. If the direction is straight at the attractor, it will hit it.

In astrodynamics, direction is as important as speed.

[Jason Thompson]

Thank you, ka9q.  So, after the Trans Earth Injection Burn, Apollo 11 was indeed traveling escape velocity with respect to both the Moon and the Earth.  How did it get home? 

Which part of the many responses that pointed out your error in adding the spacecraft velocity to lunar velocity rather than subtracting it, since it was travelling in the opposite direction to the Moon's orbital path at the end of TEI, and the Moon's gravity did the slowing, did you not understand?

After TEI it is the Moon's gravity that swings the spacecraft round the edge of the Moon and on a path back to Earth that intersects its atmosphere. It doesn't matter if you have escape velocity at any given altitude if that velocity vector puts you on a course that intersects the thing you have escape velocity from!

[Chew]

You seem to labouring under the misconception that once a satellite has escape speed it must shoot away from the attractor. Not so. If the direction is straight at the attractor, it will hit it.

In astrodynamics, direction is as important as speed.

I think he is laboring under the misconception that once escape velocity has been reached the spacecraft will never slow down again; it becomes immune to gravity.

[Zakalwe]

You seem to labouring under the misconception that once a satellite has escape speed it must shoot away from the attractor. Not so. If the direction is straight at the attractor, it will hit it.

In astrodynamics, direction is as important as speed.

I think he is laboring under the misconception that once escape velocity has been reached the spacecraft will never slow down again; it becomes immune to gravity.

and immune to aerobraking.

[pleasedebunkme]

Chew, yes I read your posts.  Thank you for taking the time.  I would like to know whether your calculation to come up with the 16,000 km distance from the Moon assumed the Moon was stationary, or did you account for the fact the Moon would be moving away at the rate of approximately 1,000 meters per second? 

And, of course I understand that the Moon's gravity was affecting Apollo 11.  But, in your calculations, wherein you came up with your 16,000 km figure, did you assume that the Moon was just making Apollo 11 slow down, or did you assume that the Moon would be pulling Apollo 11 along on the Moon's orbital trajectory?  And, if you did assume that the Moon would be pulling Apollo 11 along on the Moon's orbital trajectory, did you include some amount of added velocity for Apollo 11 to account for the fact that Apollo 11 was being pulled by the Moon?           

[ka9q]

Thank you, ka9q.  So, after the Trans Earth Injection Burn, Apollo 11 was indeed traveling escape velocity with respect to both the Moon and the Earth.  How did it get home?

No, that's not what I said (NTNWIS).

Because Apollo was in a retrograde (east-to-west) lunar orbit, when it performed trans-earth-injection behind the moon it was pointed against the moon's orbital motion. So while it achieved lunar escape velocity (otherwise it couldn't have gotten home), it was not on an earth escape trajectory. It was back in a highly elliptical earth orbit with apogee at the moon's orbit and perigee well within the earth's atmosphere. So when it got back, atmospheric drag "captured" it by taking it below even orbital velocity.

[ka9q]

But, in your calculations, wherein you came up with your 16,000 km figure, did you assume that the Moon was just making Apollo 11 slow down, or did you assume that the Moon would be pulling Apollo 11 along on the Moon's orbital trajectory?

For purposes of illustration it is possible to approximate a lunar mission as two patched conics, i.e., an ordinary 2-body orbit around the earth connected to one around the moon. But that's only a rough approximation, and what actually happens is much messier. During critical periods the spacecraft is significantly affected by both the earth and the moon and that means solving the three-body problem, made famous by the fact that several brilliant mathematicians looked for but failed to find any general analytic solutions, unlike the 2-body problem that does have such solutions.

That meant a solution was not practical until the modern digital computer could carry out a numerical integration of the basic equations of motion: add up all the forces; divide by mass to get acceleration; integrate acceleration to get velocity; integrate velocity to get position; wash, rinse & repeat from the new position.

Without rooms of IBM mainframes calculating trajectories, Apollo simply could not have happened.

I could do those same problems now on my laptop, and so can you. Get an orbital simulator program like ORBITER and play with it. Give it an Apollo trajectory. See what actually happened after each major burn: TLI, LOI, TEI. Tweak some of the parameters and see what they do. You'll learn far more about what actually happens than by waving your hands and winging it as you are currently trying to do. But you may have to let go of some of your intuitive notions.

[Chew]

Chew, yes I read your posts.  Thank you for taking the time.  I would like to know whether your calculation to come up with the 16,000 km distance from the Moon assumed the Moon was stationary, or did you account for the fact the Moon would be moving away at the rate of approximately 1,000 meters per second? 

Physics doesn't work that way. Get the notion of the Moon moving relative to the Earth somehow affecting the velocity of a spacecraft leaving lunar orbit out of your head. The spacecraft had two velocities, one relative to the Moon, the other relative to the Earth. The velocity relative to the Earth is entirely irrelevant while the spacecraft is in the Moon's sphere of influence. For a spacecraft leaving lunar orbit aiming for the Earth, it is the Earth that is moving at 1000 m/s. That must be compensated for by determining when and where to fire the rockets so the spacecraft will intercept the Earth.