ULunar Mantle = –G[2πρLM(MIC+OC – ρLMVIC+OC)(RLM 2 – ROC 2) + 16π2ρ LM 2(RLM 5 – ROC 5)/15] ULunar Crust= –G[2πρLC(MIC+OC+LM – ρLCVIC+OC+LM)(RLC 2 – RLM 2) + 16π2ρ LC 2(RLC 5 – RLM 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)] 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)] UInner Core = –3GMIC 2/(5RIC) UOuter Core = –G[2πρOC(MIC – ρOCVIC)(ROC 2 – RIC 2) + 16π2ρ OC 2(ROC 5 – RIC 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) STERNBERG Craters and cracks 2023 ICC 35
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