Gravitation Near Earth's Surface — Question 2

Question 2

A constant-gravity model needs an explicit error convention

A calculation uses $g_0$ at altitude $h$ above a spherical nonrotating Earth. Define its fractional error relative to the true value as $(g_0-g(h))/g(h)$.

1. Find the largest $h/R$ for error no greater than $1\%$.

2. Compare with the result if error were instead normalized by $g_0$.

Question 2 - Solution

A constant-gravity model needs an explicit error convention

(a) Derivation and calculation

Set $x=h/R\ge0$. With true-value normalization, $$\epsilon_t=\frac{g_0-g}{g}=\frac{g_0}{g}-1=(1+x)^2-1.$$ The one-percent condition yields $$2x+x^2\le.01,\qquad \boxed{x\le\sqrt{1.01}-1=0.00498756}.$$

(b) Result and interpretation

Surface-value normalization instead gives $$\epsilon_s=\frac{g_0-g}{g_0}=1-(1+x)^{-2}\le.01.$$ Hence $$\boxed{x\le\frac1{\sqrt{.99}}-1=0.00503782},\qquad \epsilon_s=\frac{\epsilon_t}{1+\epsilon_t}.$$ The larger reference denominator permits a slightly larger altitude.

Physical check

Both conventions agree to first order, $\epsilon\simeq2h/R$, but differ at the precision requested.