No, rate isn't distance, but they're related! Let's take some random numbers. One broadcast center produces something like 5 culture/sec. Let's say to make it easy that each jump halves the culture rate.
If you have 1 building, you have 5/s to the next planet. Then 2.5/s, then 1.25/s, and .625/s and you probably stop noticing it moving after that. So it's probably good for 3 jumps or so, though even 1.25/s is pretty slow crawling.
Now you have 4 buildings, so you have 20/s to the next planet. Then 10/s, 5/s, 2.5/s, 1.25/s, .625/s and so on. So now it spreads decently fast for a few more jumps, before it starts crawling.
Note that half is just what I used for easy math, I'm not sure on the actual reduction.