Redukcja wolnopowietrzna

Redukcja wolnopowietrzna, lub redukcja Faye’a – redukcja stosowana w grawimetrii w celu zredukowania pomierzonej siły ciężkości w danym punkcie do powierzchni odniesienia (elipsoidy), bez uwzględnienia mas znajdujących się między punktem pomiaru a powierzchnią odniesienia.

Wyprowadzenie wzoru na redukcję dla Ziemi

Na podstawie prawa powszechnego ciążenia:

g_{0}={\frac {kM}{R^{2}}}

g_{h}={\frac {kM}{(R+h)^{2}}}

R

– promień Ziemi

M

– masa Ziemi

k

– stała grawitacji

\delta _{1}g=g_{0}-g_{h}=kM\left[{\frac {1}{R\cdot R}}-{\frac {1}{(R+h)^{2}}}\right]={\frac {kM}{R^{2}}}\left[1-{\frac {R\cdot R}{(R+h)\cdot (R+h)}}\right]=g_{0}\left[1-{\frac {1}{\left(1+{\frac {h}{R}}\right)^{2}}}\right]

Po rozłożeniu $\left(1+{\frac {h}{R}}\right)^{2}$ w dwumian Newtona i pominięciu wyrazów wyższych rzędów:

$\delta _{1}g={\frac {2g_{0}}{R}}h$

Przyjmując:

g_{0}

= 9,81 m/s²

R

= 6 371 000 m

otrzymujemy

\delta _{1}g=h\cdot 3,08/10^{6}

gdzie $\delta _{1}g$ jest wyrażone w metrach na sekundę kwadrat, a $h$ w metrach.

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