Mass From Shadow: Inverting the Two Measured Diameters and Pricing the Inversion

Running the shadow formula backwards converts a measured diameter and a distance into a mass. For M87*, the measured 42 microarcsecond shadow at 16.8 megaparsecs inverts to 6.88 billion solar masses, plus or minus 0.59, which is 1.058 times the stellar-dynamical value. For Sagittarius A*, the inferred 48.7 microarcsecond shadow at 8277 parsecs inverts to 3.929 million solar masses, plus or minus 0.565, which is 0.914 times the orbital value. The fractional price of the inversion is 8.6 per cent for M87* and 14.4 per cent for Sagittarius A*, and in both cases the diameter term, not the distance term, dominates the price. Shadow masses are therefore competitive with dynamical masses for M87* and markedly inferior to orbital masses for Sagittarius A*, a ranking that reverses the intuitive one and follows directly from which diameter, ring or shadow, is inserted. Computed from published values; simulated, not measured.