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If we had an Earth around a hotter sun in a habitable, say an F, A, or O star, would we get sunburn easier?
Yes, we very probably would, for just the same reasons. The hotter the star, the larger the fraction of its light that comes through UV, so even if we kept the total amount of sunlight at Earth the same, more of it will be causing sunburn. This could perhaps be offset by a stronger ozone layer, depending on how the planet's atmospheric chemistry works out.
Another problem with hotter stars is that even though they're more massive, they burn through their fuel much
more quickly. An A class star may stay on the main sequence for about a billion years. O class stars -- only millions. They don't live long enough to harbor habitable planets. On the other end of the scale, the cooler, lower mass M and K stars stay on the main sequence for tens of billions of years. But they may also give off flares which could be very bad for the planets in their habitable zones, being so close to them.
Lets say for a moment that the people of this planet are pretty much the same as this Earth. Despite the situation being unrealistic, it's all just there and thriving for no real reason other than it simply existing. Let's say that the Earth in this situation or rather this alternate planet has a stronger Ozone layer (by measures which I can't predict). If they are around an A or O class star, how long do you think until they found out their world is not something that should exist, and within a few million years, will be completely uninhabitable to life as they know it? Would temperature changes be noticeable in their lifetime or noticeable variations in their O or A class star be noticeable? Around an A or O class star the orbital period of the planet would take less of an importance in stellar calendars, I could see those calendars taking more importance into it's moon (s) orbits. Around an O class star, a planet would likely have an orbit in hundreds of thousands of years so such values would be rather unimportant to cultures existing around them I suppose.
They would be able to notice their sun's rotation upon developing such technology, likely finding out it has a rapid rotation rate to be under a day or less as these stars would have less time to have their rotations slow down. Other planets would appear not as bright in the sky, likely that of distant stars as they would likely be spaced out over larger distances. These planets would likely be quite volatile surfaces, hostile planets due to the amount of heat they emit from recently forming. We would likely see different orbital patterns of moons due to these planets being relatively young, accompanied by their unrealistic neighbor of our hypothetical Earth. This Earth would likely support more than one moon due to a larger roche radius. You could likely go as far until the planet appears as a mere speck and still be in it's gravitational influence because it's so far away from it's O or A companion.
Could also have asteroid belts and other things, impressive things, all orbiting around a small Earth like world. Due to distances involved, planetary colonization would be extremely difficult, though when it happens expect it to be quite advanced and able to support people on dozens of years or longer as it would take to go between planets. You'd have more massive gas giant planets, planets that would swallow up even our Jupiter or brown dwarfs orbiting this O class star with little affect to our Earth due to the massive blue sun holding them in place, not allowing barycenter like systems. A small earth next door to giants that would put our jupiter to shame.
Could be possible this planet has a property that render's it's plant life to resemble ours for some reason. By which property, I wouldn't know. Rather it could be a certain chemical existing in the atmosphere or something to do with the atmosphere / plant color. On lower class stars, such as K and M stars, we would see the star basically be massive in the sky, I don't think that would be a very pleasant thing to see. Kinda scary I guess, we would be tidal locked unless we are in a resonance which would result in massive Earthquakes due to the proximity to the star and we'd be unable to hold onto any natural moons due to the close proximity. Lower class star worlds don't really appeal all that much to me. Orbital periods would likely be in dozens or less, being rather unimportant due to how fast their orbits are.
We wouldn't be celebrating christmas every dozen days or so, likely the actual calender would depend on an outer planet. A planet that has a longer orbit that would be the year to the cultures on the planet. A very interesting world, no moons. Possibly other planets could be close in, rather other cycles depending on those planets orbital periods. With apparent brightnesses quickly fading out as they go further than the sun. You could have large fuzzy disks of planets in the sky, that are quite large but dim. Or you could have planets on the interior of the habitable world's orbit, appearing quite bright, always accompanying it's apparent massive companion of the 'sun' in the sky. It would be extremely easy to visit other planets, making colonization quick and easier.
You wouldn't have massive jupiter planets around these stars unless they are further out as they would offset the balance of the inner solar systems because they would create a binary with the smaller, cooler star. Even our Jupiter would probably do that, expect smaller worlds. Larger worlds further out would not be seen until telescopes are invented, leading these people to realize their system is far larger than they imagined. Cooler stars could have planets close in that would be easily seen, then planets further out at massive distances. Would allow for many many planets to exist, and not even be detected until they have technology like ours. Imagine knowing there are like 5 large planets out there, yet you don't know what they look like.
The question is, which of these situations would you prefer to live in? Blue or red. Blue or red pill, if you will.