The answer is 0.547.
I'm writing programs to simulate random sequential adsorption and find the fraction of space occupied beyond which one can't add any more objects, and I'm looking for the actual values to compare my results to. The one I can't find is the maximum surface occupied by discs dropped randomly onto a pad without touching each other (Wikipedia only has oblong 2d shapes, and the papers don't seem to put values in the abstracts). What is its theoretical value?
(I've seen the terms "jamming coverage", "saturated coverage" and "maximum coverage", and I'm not quite sure what they all mean but I think that's what I'm looking for.)
Details of results:
0.747 of length occupied by randomly placed segments on a line. (This is the Renyi car-parking constant so all is good.)
0.449 of surface occupied by randomly placed discs on a pad. (???)
0.267 of volume occupied by randomly placed spheres in a box. (Wikipedia has it at 0.384 so I've got some head-scratching to do.)
As far as use for others goes, the Wikipedia article could be updated with the value for discs.
.547
I am sorry for asking the question without properly searching for results.