Isostasy Calculator - Airy Pratt

Isostasy is hydrostatic mass balance applied to the outer Earth: below some depth of compensation, every vertical column of rock must exert the same pressure, otherwise lateral flow in the ductile asthenosphere would redistribute mass until it did. Airy and Pratt are two ways of satisfying that balance. The Airy model holds crustal density constant and lets thickness vary, so elevated topography is supported by a root extending into denser mantle. Setting the mass of the root against the mass of the topography gives root equals elevation multiplied by crustal density divided by the density contrast; for 2.7 and 3.3 g/cm3 the contrast is 0.6 and the multiplier is 4.5, so each kilometer of elevation demands about 4.5 kilometers of root. The Pratt model instead fixes the compensation depth D and lets column density vary as reference density times D divided by D plus elevation, so higher ground is underlain by lighter rock. Seismic refraction supports Airy compensation for collisional mountain belts, where crust reaches roughly 70 km beneath the Tibetan Plateau against 30 to 40 km under stable continental interiors. Pratt compensation better describes mid-ocean ridges, where thermal expansion of hot young lithosphere lowers density and lifts the seafloor without thickening the crust. Both models are local, meaning each column is assumed to float independently, and that is the principal limitation. Real lithosphere has flexural strength, with effective elastic thickness typically between 5 and 50 km, so narrow loads of less than about 50 km wavelength are supported regionally by plate bending rather than by a local root. Equilibrium is also approached slowly: Fennoscandia is still rising at up to about 10 mm per year some 10,000 years after deglaciation.

Both models were published in 1855 in response to a single measurement problem. The Great Trigonometrical Survey of India, directed by George Everest, had measured the separation between the stations at Kaliana and Kalianpur, roughly 600 km apart, by both triangulation and astronomical observation. The two results disagreed by about 5 arc seconds, and the discrepancy was attributed to deflection of the plumb line by the mass of the Himalaya. The problem was that the observed deflection was several times smaller than the value calculated from the visible topography, implying that some hidden mass deficiency was offsetting the mountains. John Henry Pratt, then archdeacon of Calcutta and a competent mathematician, analyzed the discrepancy and proposed that crustal columns beneath high terrain are less dense, all reaching a common compensation depth. George Biddell Airy, Astronomer Royal at Greenwich, replied in the same year with the alternative that the crust is uniform in density but thickened into roots beneath mountains, using the floating iceberg analogy. Neither could be tested directly at the time. The American geologist Clarence Dutton coined the term isostasy in 1889 while working on Basin and Range topography. Felix Andries Vening Meinesz, who pioneered submarine gravity measurement in the 1920s, introduced regional flexural compensation in 1931 as a physically more realistic hybrid. Seismic evidence arrived after Andrija Mohorovicic identified the crust-mantle boundary in 1909 and later refraction work mapped thick crust beneath orogenic belts.