Ionised Mass and Mean Density of the Dumbbell Nebula M27

This paper computes a self-contained budget for M27 (NGC 6853) from published observational values, with every derived number produced by the deposited computation script. The Dumbbell Nebula is a large, evolved bipolar nebula whose proximity makes it a benchmark for the late stages of nebular expansion. Its electron density is low, its shell is thick, and a significant fraction of its volume emits only faintly. The ionised-mass budget therefore depends strongly on the filling factor: at low density the difference between a filled volume and a clumped one translates directly into a large difference in inferred mass, with consequences for how much envelope mass the progenitor must have ejected. The headline results are: R_pc = 0.4; local_density_cm3 = 300; filling_factor = 0.35; rms_density_cm3 = 177; ionised_mass_Msun = 0.814; recombination_time_yr = 421; emission_measure_pc_cm6 = 2.52e+04; electron_temperature_K = 1.05e+04. With the filling factor applied, the ionised mass is a substantial fraction of a solar mass, appropriate for a nebula produced by a progenitor that lost most of its envelope. The mean density is low enough that the recombination time is no longer negligible compared with evolutionary times, unlike in compact young nebulae. M27 thus sits near the transition where nebulae stop responding instantly to their central stars and begin to preserve a longer ionisation memory. The calculation is presented with its inputs, relations and sensitivities stated openly, so that any revised distance, density or stellar parameter can be substituted and the verdict recomputed directly.