VOL. 75, NO. 5, DECEMBER2002 335 The Gravityof Hades ANDREWJ. SIMOSON King College Bristol, TN 37620
[email protected] Imaginebeing beamedinto a chamberdeep withinthe Earth,such as depictedin the JulesVerne cavern of FIGURE1. Is the accelerationdue to gravitystronger or weaker in the chamberthan at the surfaceof the Earth?If the readeris hesitantin answering immediately,here's how LeonardEuler answered the questionin 1760 [3, volume I, LetterL, p. 182]: We are certain ... that gravity ... acts with the greatest force at the surface of the earth, and is diminished in proportion as it removesfrom thence, whether by penetrating towards the centre or rising above the surface of the globe. Figure1 Frontispiecefor Verne's1871 publication of Journeyto the Centerof the Earth, 100 miles below Iceland To answer this question, we consider various models of the Earth's structure that have been proposed over the years. We show that ? for homogeneouslydense planets-Euler's implicitmodel-gravity weakenswith descentfrom the surface; ? for a planetpossessing a homogeneousmantle that is less than 2/3 as dense as its core, a local minimumfor gravityintensity exists withina sufficientlythick mantle; ? for our own Earth,having a solid innercore, a less dense liquidouter core, and an extensiveeven less densemantle, gravity intensifies with descentfrom -9.8 m/s2 at the surfaceto an extremeof -10.8 m/s2 wherethe mantlemeets the core. We also give a conditioninvolving only surfacedensity and mean density that deter- mineswhether gravity increases or decreaseswith depthfrom the surfaceof a planet; and we concludewith an analysisof a body falling througha classicallyenvisioned hole through the Earth. 336 MATHEMATICSMAGAZINE Some preliminary classical mechanics To find the gravitational acceleration induced by the Earth on a particularpoint P, we follow Newton and first find the gravitational acceleration induced by each point of the Earth on P, and then take the aggregate of these accelerations as the total acceleration induced by the Earth.