How did the Apollo astronauts get past the van Allen belt?

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The Van Allen belts are areas of highly charged particles, mainly electrons and protons, but also with smaller amount of alpha particles.
The dangers to the craft are in the form of radiation which can interfere with, disrupt and damage sensitive electrical equipment. In some space flights electrical equipment has been turned off whilst passing through the belts in order to prevent damage. Of course, space craft also have built-in screening to minimise radiation risks.
To protect astronauts from radiation dangers, spacecraft are fitted with various types of shielding, such as aluminum shielding, and the astronauts also wear protective clothing during this part of their journey. Also the courses/paths were plotted so that the craft would travel through the parts of space where the Van Allen belts are at their thinnest. Despite these precautions, there were and are still serious radiation considerations.
Using calculations of the amount of radiation, the actual time spent passing through the belts, and the shielding used by the Apollo crew, scientists reckoned that the Apollo spacecrafts would pass through the Van Allen belts so quickly that space travellers would not receive any significant life or health-threatening levels of exposure to radiation risks during their flights.
But the matter is not clear cut. There was and is a risk, even if infinitely small. For example, cancers are caused by cell mutation. And sometimes it takes many years for cancer to become evident in a person exposed to harmful radiation. In the Apollo missions the probability of risk was considered to be so small as to be 'nil' for all practical purposes.

Post-flight Developments: In more recent years NASA has reported cases of eye cataracts developing in the majority of astronauts after returning to earth, some within 4 to 5 years of returning, other cases taking 10 or more years to become apparent. Scientists have long known that there is a causal relationship between radiation exposure and cataracts, but whether the astronauts' eye disorders occurred as the direct consequence of passing through the Van Allen belts, or by other radiation in space, or by entirely non-space-flight-related factors is unknown.

Summary: The astronauts got through the Van Allen belts by speed, shielding and carefully planned flight paths, but whether they did so without harm has yet to be determined.

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