I’m don’t really know what they use to shoot polar bears, but .303 would seem to be the minimum. The professional hunters in Africa recommend the largest caliber gun you can physically handle for larger sized dangerous game. Polar bears are so aggressive I’m not going out to look for one even if I make it to the great white north.
My uncle made big bore rifles up to .700 Nitro Express. One of his clients used a .460 to bag his polar bear.
I shot a .585 when I was younger. Would probably break something if I tried it now.
Because of peculiarities in the cartridge and the rifle itself, it can't be loaded "hot" and the muzzle energy is far below the .30'06 Springfield and even somewhat below the .308 Winchester. I have heard it described as about equivalent to the .30-40 Krag - but I have never owned or fired one of those.
While the .303 has been used on dangerous game in Africa, I'm inclined to think that the persons involved just got lucky, and for every lucky fellow there are probably a hundred unsung nebbishes who wound up being grease under the hooves of a Cape or between the toes of an elephant.
Absolute minimum for any bear larger than ordinary black bear in the lower 48 (for which .30'06 or .308 are perfectly adequate in the hands of a decent shot) would seem to me to be at *least* a Remington 7mm magnum, or one of the old big, heavy cartridges like a .415 Rigby. Just about all the pro guides in Africa now carry .458 Win Mag - just in case.
My dad bought a .348 Win (one of your old big heavy brush-bushing cartridges) for my mom when they were planning to hunt Kodiak bear in Alaska. He was about to buy a .375 H&H for himself when they turned up pregnant with yours truly and the hunt had to be scrubbed.
He used to look at me occasionally, shake his head, and ask sorrowfully, "Why couldn't you have waited til I got the H&H?"
There once was a fella named Jack 'something or the other', I don't remember his name but he wrote really good books on hunting game all over the world... I think he was in love with the 270.....
Anyway, his recommendation was 1800 foot pounds at the point of impact on dangerous game... Or maybe it was 2000....
Whatever..... But those numbers can be used to determine if you have enough gun at any given range for large dangerous game....