The temperature is 10 degrees F today and the birds are sitting on our pin oak and around the yard. They have very tiny feet that cannot possibly hold any long term heat but the feet do not freeze solid as physics would indicate. I know they have high metabolism but give me a break. The surface area is just too great to be warmed by blood isn’t it?
Empirical observation would say otherwise, no? 
The scales do help conserve heat, but birds don’t need to keep their feet the same temp as their torsos (ie, where the vital organs are). They actually can regulate the flow of blood to their feet, thus diverting more energy to keeping that torso at the proper temperature.
But wouldn’t they freeze solid?
Why? Warm blood is a-pumping down there from a nice, plump body that has probably 2 orders of magnitude more mass than the feet do.
Circulated via much faster heart rates than ours as well.
They also re-warm their feet by standing on one foot and tucking the other one up.
Brilliant question. And the answer is, yes, they should literally freeze solid within a matter of minutes if the responses in this thread the whole story.
Let’s assume that were accurate.
Just doing a back-of-the-envelope calculation, the normal blood exchange in the legs is 10cc/minute. And the legs are 40oC colder than the body core. And a Calorie is the amount of energy needed to warm 1CC of water by 1 degree.
So if the legs were being prevented from freezing by pumping in blood from a nice warm body, the bird would consume 400 calories/minute. Assuming the bird were living on pure fat, it would need to consume 40 grams of fat each minute simply to prevent itself from freezing solid. For a house sparrow, that means it would be consuming its own body weight in fat every minute. Even allowing that the insulation in the feet is 99% efficient, the bird would still be consuming its own body weight every hour and a half.
Quite clearly something very different is occurring here to the body core. Any attempt at warming the feet using blood would result in the bid dying of starvation within minutes. Even if a bird were able to obtain such an energy dense diet, the heat generated within the core through the metabolisation of the fat would cause the bird to spontaneously combust before the heat could be shunted to the limbs.
But as the man said, why don’t they just freeze solid? If the blood flow is sufficient to keep the feet from freezing then it is also, by definition, sufficient to transfer heat from the to the core. Sure, you can reduce the blood flow, but if the flow is providing sufficient heat to warm the legs by 15oC, then it must also be cooling the core by 15oC.
Looking at the previous figures, even if the bird were able to maintain its feet at just 5oC above freezing, that is still 150 calories a minute when the temperature is 15 below freezing. That equates to over 9000 calories an hour, or a sparrow that eats over 20 times its own body weight each hour. Once again, even assuming 90% insulation efficiency that is still a requirement to eat its body weight in pure lard every 4 hours to stay alive. Assuming a more natural diet of insects and seeds that’s its own body weight every hour, even if foraging for food requires no energy at all.
Which could only make the problem worse. Instead of pumping blood into the feet at 10CC/minute, you want to pump it in at 20 CC. So instead of requiring 400 Calories/minute your bird requires 800, plus the additional calories required to maintain that heart rate.
But it’s turtles all the way down. If the leg is losing heat and needs to be rewarmed, then it doesn’t matter one whit whether that is done continually in situ, or periodically against the body. If the volume of chilled blood and flesh is 10CC and it is 40oC below core temperature, it still requires 400 Calories to rewarm it, no matter how it is achieved. And if the leg is being allowed to cool lower than that for prolonged periods, then why doesn’t it freeze solid.
And remember all, a bird standing on a branch is not the worst case scenario. Many seabirds swim happily in water that is below freezing and stand on ice that is well below freezing. Consider that! With the huge thermal density of water, if the limbs were somehow being warmed from the core the energy drain would be immense. A human being moving a single hand in water at those temperatures would find it freezing solid in less than an hour. Yet bird weighing a few hundred grams can maintain the same surface area for days with no ill effects.
Clearly whatever is happening, it is much more than simply using the core to warm the limb.
So having said all that, whats the answer? Simply put, birds have a counter current heat exchanger built into their limbs. Instead of trying to use warm blood from the core to keep the legs warm, they use slightly less cold blood from further up the limb, and that blood is in turn warmed by slightly less cold blood from even further up the limb. So warm blood form the core never comes into contact with cold blood from the limb or the cold tissues of the limb. Warm blood only ever comes into contact with the slightly less warm blood.
By use of the counter current exchanger, coupled with an ability to maintain the limb at temperatures just above freezing, the animal can maintain a system where the temperature difference is never more than 0.5oC, and so never requires more than 1 calorie/2CC of blood flow.
FWIW, humans also have countercurrent heat exchangers in our limbs, which is why your fingers can be so cold that the sensory nerves shut down, yet your core temperature is able to be maintained at very little extra cost. If your body were actually trying to warm all the blood flow from your fingers from 15oC to 37oC, you would die of starvation within a few hours. While the human counter current exchange is no where near as efficient as in birds, it is still enough to handle those sorts of temperature gradients with minimal cost.
A gram of fat provides 10,000 calories, not 10 calories like you are assuming.
If the bird really needed 400 calories/minute it would only need to consume about 600 kCal a day, which while an impressive number for a tiny bird it is not something I would declare as outright impossible.