I became curious about the maximum mass in a star's accretion disc while watching an episode of Star Trek involving a Dyson Sphere. I wondered if some maximum amount of stellar material would limit natural and artificial structures around any system of one or more stars. What affects these limits? Is there a logarithmic/linear/exponential correlation between stellar mass and maximum accretion mass?
The accretion disc is formed by material in orbital motion against a central body, which can be a star.
The size, mass and other characteristics are usually determined by the central object, in this case the star.
In general, the protoplanetary accretion discs are the largest ones (with the largest mass) and as the age on the central star increases, the average size decreases.
This is because as the age of the system increases, more mass is drawn into the system and less is left out in the disc itself.
The physics of black hole accretion disc is entirely different and is possible to derive a relation between the black hole mass and size of the accretion disc. The relation appears logarithmic.
Any natural or artificial structure around a star would feel the gravity of the star and consequently, it is the mass of the star that determines amount of the counteracting force (rotation, radiation pressure etc.) required to maintain stability.
For example, as the stellar mass increases, the rotation speed must increase if the orbit is maintained at the present position or the orbit must become larger. In case of stellar radiation pressure is used to counteract gravity, as the mass (and gravity) increases, the system becomes unstable (as the ratio of radiation pressure to gravity is constant).