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In the video The Moon Trees That Flew To The Moon On Apollo 14 after about 08:47 Scott Manley says that Apollo 14 "flew up over the top" of the Van Allen radiation belts and simultaneously shows this retro-looking graphic of them.

Is this claim itself "over the top" or did the mission (or all Apollo missions) indeed fly up over the tops belts and avoid exposing the astronauts to the radiation trapped within them? Did the spacecraft do the same on the return trip as well?

So the trajectories that they took to the Moon flew up over the top of the belts, so that they minimized the dose.

enter image description here

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Not exactly "over the top" (still through the outer portion of the belts), but yes, it appears so, according to this source.

There are simplified trajectory plots shown in the link.

Also this source

quotes a letter from Van Allen himself:

...the outbound and inbound trajectories of the Apollo spacecraft cut through the outer portions of the inner belt and because of their high speed spent only about 15 minutes in traversing the region and less than 2 hours in traversing the much less penetrating radiation in the outer radiation belt. The resulting radiation exposure for the round trip was less than 1% of a fatal dosage - a very minor risk among the far greater other risks of such flights.

From Apollo 11 Lunar Surface Journal:

Total, uncorrected dosages received by the [Apollo 11] crew during the mission was about 0.25 rad. Post-mission corrections gave true readings of 0.18 rad, most of which was received during passage through the Van Allen Belts on the way out and on the way back. In comparison, the Apollo 14 crew received an average dose of 1.14 rad, in part because their trajectory took them closer to the center of the Belts than any of the other crews. Doses at these levels do not present significant medical risks - certainly not in comparison with other risks that are inherent to a lunar mission.

Wikipedia quotes 1000rad being a fatal dose, 100 to 200 rad causes acute radiation syndrome which is not fatal.

As additional information, here's plot of Apollo-13 reentry ground track

Apollo-17 outbound and inbound groundtracks

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    $\begingroup$ Excellent and speedy answer, thank you! Those images are beautiful 1, 2, 3 They are 2D projections and I assume somehow "unwind" the trajectory's rotational component around the Earth's axis to make these plots. I'll have to think further to try to understand what they might look like in 3D. $\endgroup$ – uhoh Nov 24 '19 at 4:36
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    $\begingroup$ @uhoh You're welcome) I would like to see that real trajectory too. $\endgroup$ – Sergiy Lenzion Nov 24 '19 at 4:45
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    $\begingroup$ @uhoh Something is way off in the first image. Apollo didn't flew higher than 35° latitude. Yet it is drawn as if they left over central Europe at 50°. $\endgroup$ – asdfex Nov 24 '19 at 12:02
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    $\begingroup$ @asdfex Interesting! According to the link that's the geomagnetic equator. It could be that the continents are not drawn correctly, but the plot is roughly correct. It's a really fascinating plot, I cant stop thinking about it!! $\endgroup$ – uhoh Nov 24 '19 at 12:07
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    $\begingroup$ I'd find it interesting what exactly “1% of a fatal dosage” refers to. This actually sounds like quite a lot; it's not clear whether it's just a vague upper bound on the risk or he means a specific definition in which this is actually the ballpark risk. $\endgroup$ – leftaroundabout Nov 25 '19 at 0:52

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