The decision to bomb Hiroshima has been covered extensively. Yet I Can’t find a lot of sources that explain why the second bomb was utilized only 3 days after. I remember reading that there was no actual specific presidential order to bomb Nagasaki, rather than an authorization to use any means necessary. Nagasaki wasn’t even one of the chosen targets initially and was only later added. Only because visibility issues over Kokura the target changed to Nagasaki.
So what I'm asking is: Why did the U.S. decide on using another atom bomb?
What was the decision-making process in that decision?
Why was Nagasaki Chosen eventually as the target?
Hi there -- u/restricteddata has written pretty extensively about this; you may want to check this section of the FAQ.
Regarding the targeting of Kokura (and Nagasaki as a secondary target) in particular, u/restricteddata addresses this in this thread.
See my comment in this thread where I go over the question of "who ordered" Nagasaki. The "decision" to use a second bomb was made by people far lower than the people who participated in the "decision" to use the bomb in general (not Truman, not Stimson).
To answer briefly your three questions:
"The US" did not decide. Groves had always thought multiple bombs would be dropped, and the order to use any bomb was open-ended. Groves et al. assumed they were authorized to drop as many bombs as they saw fit. Truman cancelled this on August 10th. I suspect Truman had no idea that two bombs would be dropped so quickly. There was no specific Presidential order to bomb Nagasaki.
The Manhattan Project had two bombs ready around August 1945. (The fact that there were two ready at the same time was really just a coincidence.) They planned to drop one around August 3 and another around August 10. Weather conditions over Japan pushed the first one back to August 6, and weather forecasts encouraged them to push the second one up to August 9. That is more or less the decision-making process: weather-related.
Nagasaki was added to the target list on July 24 because Kyoto had been decisively removed from it around then, and because they wanted (ideally) to have several backup targets for every primary target. From what I can tell, the name Nagasaki was given to Groves by Lauris Nordstad's targeting staff. It was a low-priority target.