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GM⊕/R⊕3≈1.541×10−6 s−2GM_\oplus/R_\oplus^3 \approx 1.541\times10^{-6}\,\mathrm{s^{-2}}

Why this formula appears here

For Earth modelled as a point mass, Φ(r)\Phi(r) = -GM/r gives a radial second derivative of magnitude 2GM/R3R^3 and, since Φ\Phi is harmonic in vacuum, transverse second derivatives of magnitude GM/R3R^3 with the opposite sign — a trace-free tensor in the exact ratio 2:{-1}:{-1} . With the standard gravitational parameter GM⊕M_\oplus = 3.986×\times10^{14}\,m3 s−2\mathrm{m^3\,s^{-2}} and mean radius R⊕R_\oplus = 6.371×\times10^{6}\,m\mathrm m , this gives GM⊕M_\oplus/R⊕3R_\oplus^3 ≈\approx 1.541×\times10^{-6}\,s−2\mathrm{s^{-2}} and a radial magnitude of 3.08×\times10^{-6}\,s−2\mathrm{s^{-2}} , matching the standard geodetic free-air gravity gradient of about 3.086×\times10^{-6}\,s−2\mathrm{s^{-2}} , or 0.3086 milligal per metre,…

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GG

Symbol G

G is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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M⊕M_\oplus

Symbol M_oplus

MoM_oplus is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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R⊕3R_\oplus^3

Symbol R_oplus^3

RoR_oplus3s^3 is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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Published contexts (1)

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GM⊕/R⊕3≈1.541×10−6 s−2GM_\oplus/R_\oplus^3 \approx 1.541\times10^{-6}\,\mathrm{s^{-2}}

Equation 131 · Evolutionary Physics

The Bend an Elevator Cannot Fake

This equation gives an approximation: it relates the quantities while allowing an approximation.

For Earth modelled as a point mass, Φ(r)\Phi(r) = -GM/r gives a radial second derivative of magnitude 2GM/R3R^3 and, since Φ\Phi is harmonic in vacuum, transverse second derivatives of magnitude GM/R3R^3 with the opposite sign — a trace-free tensor in the exact ratio 2:{-1}:{-1} . With the standard gravitational parameter GM⊕M_\oplus = 3.986×\times10^{14}\,m3 s−2\mathrm{m^3\,s^{-2}} and mean radius R⊕R_\oplus = 6.371×\times10^{6}\,m\mathrm m , this gives GM⊕M_\oplus/R⊕3R_\oplus^3 ≈\approx 1.541×\times10^{-6}\,s−2\mathrm{s^{-2}} and a radial magnitude of 3.08×\times10^{-6}\,s−2\mathrm{s^{-2}} , matching the standard geodetic free-air gravity gradient of about 3.086×\times10^{-6}\,s−2\mathrm{s^{-2}} , or 0.3086 milligal per metre,…

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