One morning you are lying in a meadow under the blue summer sky of Connecticut. You notice a small bird (likely a song sparrow but unfortunately you have forgotten your binoculars so you just assume) perched on a branch in the shade of an old oak tree. You also notice another small bird (again likely a song sparrow) standing on top of a rock under the beautiful June sun. You wonder, why they are not both in the same type of environment. One of them must be expending more energy to keep themselves in heat balance.

Question

You know from your research on song sparrows that the body temperature of a song sparrow is usually 39ºC, their total body surface area is about 0.0036m2, the surface area of their feet is about 0.0008m2, and their emissivity is 1. Their feathers are about 0.04m thick with a thermal conductivity coefficient of 0.033 W/m ºC. Their evaporative water loss averages 0.0023g/m2sec. Average metabolic rate is 10W/m2. So you decide to measure some temperatures. The temperature of the branch the bird is sitting on is 28 ºC, the temperature of the rock the bird is standing on is 35 ºC. You figure out that the radiant energy absorbed from the tree to the bird is 0.0003W and the radiant energy absorbed from the rock to the bird is 0.0010W. Since there is no wind present, you assume there is no convection happening. In addition, the birds are in the midst of the breeding season and are not storing any extra nutrients.

Figure out which bird is expending more energy to keep themselves in heat balance.

I think I figured it out, but I am not sure if Im correct. Does anybody want to take a shot and help me out here because Im curious if I approached this question correctly.