The snow in bear tracks is not melted, so it seems they let their feet get cold when they are walking (probably when swimming, too). I suspect the flow of blood is regulated to keep the feet from getting frozen, but too cold to melt snow. My experience is with Alaska brown bears, but they are very close relatives of polar bears.
I'm the poster who brought up bear tracks. Brown bears don't have hair under their feet. They do have pads, like dogs, and to a lesser extent, so do we. I have occasionally walked barefoot in the snow for a ways, and know others who have, too. Your feet get cold, but as long as the rest of your body is warm, they don't freeze.
their legs don't freeze because, looking at my biology book, polars avoid frost bite by means of a countercurrent heat exchange mechanism. In the legs of a polar bear, the main artery and vein run parallel and very close together. All along the length of the leg heat is exchanged between the artery and vein. The foot tissues are supplied with nutirents, and the tissue is held at a temperature that prevents freezing aswell as minimising the loss of heat by conduction. The blood leaving the legs is almost at the same temperature as the arterial blood.
I think a point is being missed here. With progressively better insulation, you can keep tissue even 1mm from snow at a higher temperature. However, part of the bear must actually be touching the snow, and must eventually become the same temperature, either melting the snow or freezing the paw. Therefore either the contact patch must be freezing, the snow must eventually melt, or something is wrong with our understanding of temperature (or, as always, a combination)
"However, part of the bear must actually be touching the snow, and must eventually become the same temperature, either melting the snow or freezing the paw."
That misses the point of insulation. The external surface may have a similar temperature, but the insulation maintains a high gradient, the temperature on the other side of the pad/fur (the side) is higher. Insulation means that very little energy is escaping from the foot to the snow, so the snow does not need to melt).
The other mechanisms (blood heat exchange etc.) may play a part as well with regard to the animals , but we can leave that to a biologist to confirm.
Ok - glad it seems polar bears are explained... what about swans and ducks paddling along through ice-caked lakes of water all day. Water conducts heat 25x faster than air and also a lot more than snow (how much?) Their feet and legs are very thin with large surface areas. How on earth do they stop their paddles icing up?
I don't know about swans and ducks but was told recently that some birds in the southern ocean have two circulation systems with one dedicated to keeping the bits closest to the water warm when they are on the water. The energy needed to do this means that an albatross actaully uses less energy when soaring just above the waves than when it is floating on the water.
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