Filling Factor and Ionised Mass of the Ring Nebula NGC 6720 from Density Diagnostics

This paper computes a self-contained budget for NGC 6720 from published observational values, with every derived number produced by the deposited computation script. The Ring Nebula presents a bright main ring surrounding a lower-density central cavity, with a faint halo beyond. Its appearance is a projection of a barrel-shaped shell, and that geometry matters for the mass budget: the emitting volume is a shell rather than a filled sphere. Forbidden-line ratios give the local density inside the emitting gas, while the total emission measure gives a lower root-mean-square density, and their comparison measures how much of the shell volume actually emits. The headline results are: R_pc = 0.19; local_density_cm3 = 1.4e+03; filling_factor = 0.45; rms_density_cm3 = 939; ionised_mass_Msun = 0.523; recombination_time_yr = 90.3; emission_measure_pc_cm6 = 3.35e+05; electron_temperature_K = 1.05e+04. The emitting gas fills only part of the shell volume, consistent with the clumpy and filamentary texture seen in high-resolution images. The ionised mass derived with the filling factor applied is substantially below the value a filled-shell assumption would give, which brings the nebular mass into comfortable agreement with the envelope mass a single asymptotic-giant-branch star can supply. Ignoring the filling factor would manufacture a mass problem that the data do not actually contain. 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.