I'm creating an alien species and want to have them a bit too close to their star but of course, don't want to kill them. They're not very advanced technology wise but they have powers caused by radiation from a war.
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In general, when we are talking about planets in the habitable zone, we speak of planets that could have liquid water on the surface (since that is considered a prerequisite for non-exotic life). Too close, and the water will boil away; too far, and the water will freeze. A planet on the inner edge of the habitable zone might have boiling surface water at the equator and hence lots of cloud and hot rain, something that a species should have no problem adapting to. If the planet is in a very elliptic orbit, the water might only be boiling during the closest approach, but in larger regions, making for a dangerous summer season.
Also worth considering: A planet very close to its sun is likely to be tidally locked, always facing the same side to the sun, meaning that the sunward side is blisteringly hot, while the opposite side might be frozen. Only a roughly ring-shaped twillight zone will be habitable, and the weather could be very extreme.
The planet might also be orbiting an intrinsic variable star, with massive flares at regular intervals. Life during a flare would be quite dangerous, and it is possible that any species living on such a planet would enter some kind of hibernation state, possibly underground, during such a flare.
On a different note: What kind of 'powers' caused by radiation do you imagine your species has? Outside of superhero comics, the only power you could feasibly get from radiation exposure is improved resistance to radiation. Radiation also causes mutation, and while most mutations by far are detrimental, there may be a few advantageous ones. Your species may thus exhibit great variation in (non-exotic) abilities, but the downside would be a lot more disabled people - unless we are talking so far after the radiation event that any detrimental mutations will have died out (possibly through eugenics) .