Gravitation Near Earth's Surface — Question 10

Question 10

A gravity gradient distinguishes a field change from a scale offset

Two calibrated gravimeters are vertically separated by $H\ll R$ near a spherical nonrotating planet. They read downward accelerations $g_{low}$ and $g_{high}$. A proposed disturbance adds the same unknown constant reading offset $b$ to both instruments.

1. Derive a first-order estimate of $g_{low}-g_{high}$.

2. Can their difference reject a common offset, and can it alone determine both $M$ and $R$?

Question 10 - Solution

A gravity gradient distinguishes a field change from a scale offset

(a) Derivation and calculation

At radii $R$ and $R+H$, $$g_{\mathrm{low}}-g_{\mathrm{high}}=\frac{GM}{R^2} \left[1-(1+H/R)^{-2}\right].$$ For $H/R\ll1$, $$\boxed{\Delta g\simeq\frac{2GMH}{R^3}=\frac{2g_0H}{R}}.$$

(b) Result and interpretation

A common additive offset cancels exactly: $$(g_{\mathrm{low}}+b)-(g_{\mathrm{high}}+b)=\Delta g.$$ However, the measured gradient fixes only $$\boxed{GM/R^3\simeq\Delta g/(2H)}.$$ The transformation $R\mapsto kR$, $M\mapsto k^3M$ leaves this combination unchanged. Additional independent information is required to separate mass and radius.

Physical check

The differential signal rejects a common offset but remains sensitive to unequal instrument offsets.