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From reading this question it would appear is if we do not currently have a great solution to radiation shielding in existing space craft -

Different sources have conflicting numbers:

This article states:

In practice, this shielding helps limit their overall exposure to about a year’s worth of radiation that we get on Earth per day

While this one states:

In just one week on the ISS, astronauts are exposed to the equivalent of one year’s exposure on the ground.

I am studies have been preformed, but exactly how much ionizing radiation / cosmic radiation are astronauts actually exposed to when aboard the International Space Station.

If I were to take an off-the-shelf Geiger counter with me, approximately what would it read (on average)? And which particles spcifically are more abundant, X-Ray, gamma?

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It's important to understand the different types of radiation exposure that astronauts can be exposed to.

  1. Galactic Cosmic Rays (GCR's) are a more-or-less isentropic throughout the galaxy and are thought to be caused primarily by supernova and black holes in distant systems. However, these are rarely a concern for astronauts in the ISS as GCR's typically don't have a high enough energy to penetrate the Earth's magnetosphere. At the poles, however, where the magnetosphere is the weakest, there is an increase in high energy GCR's at lower altitudes.
  2. The Van Allen Belts are zones of charged particles, typically, protons and electrons, that are trapped in an Earth orbit. While potentially extremely damaging for both humans and space electronics, these are not a concern for the ISS as the belts typically (depending on the solar cycle) start around 1000 km above the surface while the ISS orbits at a mean altitude of approximately 380 km.
  3. Solar Events are another major concern for space travelers. These events are caused by coronal mass ejections, which send huge masses of charged particles throughout out solar system. This video provides a cool visual of coronal mass ejections. One pretty cool effect of solar flares is the Forbush Decrease, in which GCR's are decreased in response to a coronal mass ejection.
  4. Neutrons are another potential source of radiation that can cause something called Displacement Damage. It doesn't seem to be a major concern for the ISS so I won't go into detail.

In order to determine the amount of dose astronauts and electronics are receiving, scientists use a combination of sensors and empirical models (see CREME96, AP9 as examples).

The actual amount of radiation that astronauts receive depends heavily on the solar cycle. The solar cycle is an 11 year cycle in which the sun's activity changes. The Earth's magnetic field grows and shrinks in response to the solar activity, resulting in changing levels of energetic particles in Low Earth Orbit.

It's difficult to come up with an average number or the amount of radiation that astronauts receive per year due to the changing solar cycle, the various activities they participate in, where they are located in the ISS, and other random factors, however the wikipedia page on spaceflight radiation carcinogenesis seems to provide a reasonable figure:

Astronauts are exposed to approximately 50-2,000 millisieverts (mSv) while on six-month-duration missions to the International Space Station (ISS), the Moon and beyond. The risk of cancer caused by ionizing radiation is well documented at radiation doses beginning at 50 mSv and above.

I found this cool reference while answering this question that I'll leave here: NASA Space Radiation EBook

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  • $\begingroup$ Although this is not your fault, I believe the upper limit of 2000 mSv in the Wikipedia article is an error. Although one of the citations provided for the claim is paywalled so I can't confidently say it doesn't support that upper limit, the other source indicates only that the average dose is 72 mSv, while the Wikipedia page on Sieverts indicates that the maximum career exposure for NASA astronauts is only 1000 mSv. $\endgroup$ – Kamil Drakari Dec 21 '18 at 14:27

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