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I'm only in high school, so this will probably have fatal flaws.

So basically in space, there is bound to be stray radiation, whether from the stars, or cosmic background, floating around right. And if I built a spaceship and flew it through space, it would absorb some of that radiation. So the way I imagine it, in space these electromagnetic waves are flying through the vacuum in a similar way to rain falling on the ground. And the question I posed myself was hypothetically if I wanted the spaceship to get hot in a certain amount of time how fast should I drive it, as fast as possible or not?

The analogy I tried to use to understand this is the problem of do you get more wet walking or running in the rain on earth, except that the rain doesn't just fall downwards, it falls in all directions, and at the speed of light.

So here I am, pondering this question, and my original theory was "Well the faster I drive, the more radiation I'm bound to hit, therefore the hotter I get". But then I remembered that the speed of light is constant no matter which reference frame you are in? Does that mean that whatever speed I drive at (50 m/s, 10000 m/s) that the radiation from all sides will always be approaching me at the speed of light, and therefore the amount of radiation my spaceship absorbs does not depend whatsoever on the speed I am travelling at? Or is that wrong?

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Not only does your speed affect the amount of radiation that you receive, but this actually happens to the Earth and has been measured experimentally.

You say:

So basically in space, there is bound to be stray radiation, whether from the stars, or cosmic background, floating around right.

and the most obvious example of this is the cosmic microwave background that pervades the whole universe. If you're at rest with respect to the average distribution of matter in the universe then the CMB looks isotropic - that is, it looks the same in all directions. However if you're moving then the CMB is blue shifted in your direction of motion and red shifted in the opposite direction. That means the CMB gets hotter and more intense in your direction of motion and cooler and less intense in the opposite direction.

Because the Solar System is moving wrt the average matter distribution we see this effect in measurements of the CMB made by the the Planck mission. The CMB is very slightly hotter in our direction of motion. This effect is called the dipole anisotropy.

If you're interested in pursuing this further see the question How is the Plane of the Solar System oriented to the Sun's motion through space?

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    $\begingroup$ but if the CMB is blue shifted in one direction and redshifted in the other, doesn't that mean if I wanted to gather heat, that both the effects would cancel off, as it is implying one side of my spaceship would get hotter than the other side would, but on average it would be the same as just staying still? $\endgroup$
    – QCD_IS_GOOD
    Commented Sep 29, 2014 at 7:56
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    $\begingroup$ @JoshuaLin: If you're stationary the CMB temperature is about 2.7K. So the biggest redshift possible is only -2.7K i.e. down to absolute zero. On the other hand the blue shift can be arbitrarily big if you go fast enough. So it would most definitely not average out to be the same as staying still. Well, it would, but only for velocities well below the speed of light (like the Earth's). $\endgroup$
    – John Rennie
    Commented Sep 29, 2014 at 8:48
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Firstly, the immediate answer is no, the intensity of radiation is higher when you fly into it, like the rain situation (but the maths is different).

There are three interrelated effects that you need to know about regarding your question, which are frequently omitted in popular or introductory SR: aberration of light, the full angle-dependent doppler effect, and the headlight effect (the one that pertains most closely to your question). I'm not going to quote Wikipedia at you because to be honest it's not the best place to start in this case, but the ancient link below should give you an idea of what is involved, and some useful search terms to follow up with. There are some low-rez but informative animations there too.

This is the very old link

There is also a more recent site with a downloadable program by the same author here.

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There is another effect to take into account as you accelerate your ship. It is the Unruh Effect

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