Dynamical Age and Shell Mass of the Helix Nebula NGC 7293 from Its Expansion Field

This paper computes a self-contained budget for NGC 7293 from published observational values, with every derived number produced by the deposited computation script. The Helix Nebula is the nearest large planetary nebula and the standard example of an evolved, low-density shell. Its angular size is enormous, its surface brightness is low, and its cometary knots trace a radially expanding system whose speed has been measured both spectroscopically and by proper motions. Combining the physical radius at the accepted nearby distance with the expansion speed yields a dynamical age that can be compared with the post-asymptotic-giant-branch crossing time of its hot central star. The headline results are: R_pc = 0.3; v_exp_km_s = 13.5; dynamical_age_yr = 21729; electron_density_cm3 = 120; thermal_pressure_Pa = 3.41e-11; ionised_mass_Msun = 0.235; kinetic_energy_J = 4.27e+37; emitting_volume_fraction = 0.6; sound_speed_km_s = 11.9; mach_number = 1.13. The dynamical age is long, as expected for a nebula whose central star has already faded well past its maximum temperature. The low present density explains the faint emission measure: the shell has expanded and diluted by a large factor since ejection. The kinetic energy still stored in the expansion is small compared with the radiative energy the central star has emitted over the same interval, so the expansion is a fossil record of the ejection rather than an actively driven wind bubble. 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.