In precocial birds, feathers begin growing back as soon as the eggs hatch. In altricial birds (including bald eagles), patches remain functional through early brooding (my guess would be 15-20 days, or about the time some of Mom's lethargy starts to fade). Then they gradually disappear, restoring the area to non-breeding function and feather cover about the time the young are fledged.
Why do eagles sunbathe? We believe it helps kill and/or prevent parasites in less-exposed areas like wingpits, and it also looks quite comfortable. Thanks for the great capture!
- Brood patches: https://web.stanford.edu/group/stanfordbirds/text/essays/Brood_Patches.html
- Hormonal control of brood patch development: http://joe.endocrinology-journals.org/content/26/1/11