Hello Mr. Dog,
The compressor provides two functions: it supplies the necessary oxygen for the aerobic bacteria; and the bubbles stir the tank providing an even distribution of warmth (provided by the bacterial metabolic function) & 'food' (such as it is, provided by you).
Without the compressor, the tank will stratify and the bacteria colony will basically shut down from oxygen starvation. But the real damage happens when the tiny bits of 'undigested bacterial food' get washed down the line into your leaching field and (eventually) irreparably plug it all up
.
J