The author examined the gravitational behavior of a thin spherical cap of axial symmetry, which is one of the most fundamental elements of gravity correction. Suppose a spherical capwith a uniform density p, whose geometrical shape is defined by angular distance (or truncation angle) ψ and geocentric distances γ
1 and γ
2. Then the gravity g
+ or g
- due to the sphericalcap, at a radial distance γ
+ or γ
- (γ
-<γ
1≤γ≤
2<
+) on the symmetry axis, respectively, can be written as
g
±=2πGpr
± [Y
±(t
2, Ψ)-Y
±(t
1, Ψ)].
where, G: Newtonian gravitational constant, t
1=γ1
1/γ
±, t
2=γ
2/γ
± and
Y
± (t, Ψ)=[(
±)(t
2-tcosΨ+3cos
>3Ψ-2)(
t+1-2tcosΨ
1/2+3)(cos
3Ψ-cosΨ) ⋅ln | t-cosΨ+(t
2+1-2tcosΨ)
1/2 | ±t
3]/3.
Double signs attached in the formulae should be taken in the same order (same hereinafter).A dimensionless parameter t=r/r
±, the radius of curvature normalized by r
±, takes t
±=r/r
±<1 for g
+, and t
-=r/r
->1 for g
-. In the above formula, setting t=t+Δt, Δt:
2=t
2-t
1<<t, applying the Taylor's theorem and neglecting higher terms than or equal to (Δt)
2, the gravity due to such a thin spherical cap with a thickness Δh:=r
±Δt(or a spherical membrane)can be approximated by
g
′±-2πGp⋅H
±(t, Ψ)Δh,
View full abstract