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The standard atmosphere is defined in terms of geopotential altitude. The idea behind this concept is that a small change in geometric altitude will make a change in gravitational potential energy per unit mass. To make this same change in potential energy at sea level requires a equivalent change in geopotential altitude. Mathematically, this is expressed as g dZ = G dH where H stands for geopotential altitude and Z stands for geometric altitude, g is the acceleration of gravity and G is the value of g at sea level. The value of g varies with altitude and is shown in elementary physics texts to vary as

$$\frac{g}{G}={\left(\frac{E}{Z+E}\right)}^{2}$$ | (1) |

where E is the radius of the earth. So,

$$\mathrm{dH}=\frac{g}{G}\mathrm{dZ}={\left(\frac{E}{Z+E}\right)}^{2}\mathrm{dZ}$$ | (2) |

and integrating yields

$${\int}_{0}^{H}\mathrm{dH}={\int}_{0}^{Z}{\left(\frac{E}{Z+E}\right)}^{2}\mathrm{dZ}$$ | (3) |

$$H=\frac{EZ}{E+Z}$$ | (4) |

$$Z=\frac{EH}{E-H}$$ | (5) |

While Z and H are virtually identical at low altitudes, you can calculate that Z = 86 km corresponds to H=84.852 km. (Use 6356 km for the radius of the earth). At this altitude, g is 0.9735 times the value at sea level. If you don't like differentials, you can regard H=EZ / (E+Z) as the definition of H and then derive dH/dZ=g/G.