Sheet 2 Explanatory Notes — Gravitational Energy Here we calculate the gravitational potential difference that would be met by the decay-heat-induced expansion of Mars, which is currently divided into layers with the labels Core, Mantle and Crust, as we had not enough information to be able to identify the dimensions of separate inner and outer cores. As it is assumed to be initially cool and solid, with a homogeneous mantle, Mars is initially partitioned into Core Initial and Mantle Initial, with trial values for initial radii. Densities emerging from these trials are then fed into Sheet 3 for a calculation of initial temperature. The trial values are then adjusted to yield an initial temperature of order 300 K (27°C). There are, listed and derived above, formulae used to calculate potential energy for spherically symmetric bodies. References for input data are highlighted in yellow, with cited sources in the References table. If data is taken from another cell in the same sheet, the cell reference is shown, highlighted in cyan. If it is from another sheet, the sheet number appears before the cell reference. Along with key results, a magenta highlight indicates a trial input, such as initial surface and core radii, that would support the initially ambient temperature before thermal expansion. References Sheet 3 Explanatory Notes — Overall Heat Balance Here we calculate the heat available from accelerated decay, after No. Source 1 https://en.wikipedia.org/wiki/Gravitational_constant dUCore Initial = –GMdM/R = –GM4πρR 2dR/R = –GM4πρRdR = –G(4π)2ρ2R4dR/3 Initial Conditions Current Conditions UMantle Initial = –GρMI[2π(MCI – ρMIVCI)(RMI 2 – RCI 2) + 16π2ρ MI(RMI 5 – RCI 5)/15] UMantle = –3GMM[(MC(RM 2 – RC 2) + (4πρ M/3){RC 3(RC 2 – RM 2) + 2(RM 5 – RC 5)/5}]/[2(RM 3 – RC 3)] UCore = –3GMCo 2/(5RCo) UMantle = –G[2πρMa(MCo – ρMaVCo)(RMa 2 – RCo 2) + 16π2ρ Ma 2(RMa 5 – RCo 5)/15] UCore Initial = –G(4π) 2ρ CI 2RCI 5/15 = 3GMCI 2/(5RCI) dUMantle Initial = –GMdM/R = –G[MCI + ρMI(4πR 3/3 – VCI)]ρMI4πRdR UMantle Initial = –G[2πρMI(MCI – ρMIVCI)(RMI 2 – RCI 2) + 16π2ρ MI 2(RMI 5 – RCI 5)/15] UMantle = –3GMM[(MC(RM 2 – RC 2) + MM{RC 3(RC 2 – RM 2) + 2(RM 5 – RC 5)/5}/(RM 3 – RC 3)]/[2(RM 3 – RC 3)] UMantle = –3GMM[(MC(RM 2 – RC 2) + (MM/5)(5RC 5 – 5RC 3RM 2 + 2RM 5 – 2RC 5)/(RM 3 – RC 3)]/[2(RM 3 – RC 3)] UMantle = –(3GMM/2){(MC(RM 2 – RC 2) + (MM/5)(3RC 5 – 5RC 3RM 2 + 2RM 5)/(RM 3 – RC 3)}/(RM 3 – RC 3) UCrust = –G[2πρCr(MCo+Ma – ρCrVCo+Ma)(RCr 2 – RMa 2) + 16π2ρ Cr 2(RCr 5 – RMa 5)/15] InTerms of Mass (Simple Core-Mantle System) Expansion Energy Formulae UMantle = –GρM[2π(MC – ρMVC)(RM 2 – RC 2) + 16π2ρ M(RM 5 – RC 5)/15] UMantle = –GMM[2π(MC – ρMVC)(RM 2 – RC 2) + 16π2ρ M(RM 5 – RC 5)/15]/[4π(RM 3 – RC 3)/3] UMantle = –GMM[(MC – ρMVC)(RM 2 – RC 2) + 8πρ M(RM 5 – RC 5)/15]/[2(RM 3 – RC 3)/3] UMantle = –3GMM[(MC(RM 2 – RC 2) + ρ M(4πRC 3/3)(RC 2 – RM 2) + 8πρ M(RM 5 – RC 5)/15]/[2(RM 3 – RC 3)] STERNBERG Craters and cracks 2023 ICC 54
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