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This paper is a study of the gravitational attraction between two uniform hemispherical masses aligned such that the pair is cylindrically symmetric. Three variations are considered: flat side to flat side, curved side to curved side, and flat side to curved side. Expressions for the second and third variation are derived from the first, with the use of superposition and the well-known gravitational behavior of a spherical mass as equivalent to a point mass at its center. The study covers two masses of equal diameter and of different diameters, such that one is four times that of the other. Calculations are done for separations from zero to fifty times the radius of the larger of the two, which is effectively the asymptotic limit. It is demonstrated that at any separation, the force can be expressed as if the two hemispheres were point masses separated by a certain distance. Expressions for that distance and the location of the (fictitious) point masses within each hemisphere are presented. Unlike the case of two spherical masses, the location within their respective hemisphere is not necessarily the same for each point and both are dependent upon the separation between the two hemispheres. The calculation for the first variation is done in two ways. The first is a “brute force” multi-dimensional integral with the help of
*Wolfram Mathematica*. The second is an axial expansion for the potential modified for off-axis locations by Legendre polynomials. With only a few terms in the expansion, the results of the second method are in extremely good agreement with those of the first. Finally, an interesting application to a split earth is presented.

With at least one exception, a perusal of online publications of the (Newtonian!) gravitational attraction between solid masses yields discussions of a pedagogical nature concerning spherically symmetric solids and shells. In a few cases, the force between a point mass and a disk or a hemisphere is derived for the point mass placed along the center line of the non-spherical object. The exception is a calculation of the force between two hemispheres with their flat surfaces in contact, as in the case of the earth [

In addition to the force calculations, it will be shown that the force can be expressed in the form of Newton’s law of gravitation for point masses, GM_{A}M_{B}/ (R_{eff})^{2}, where R_{eff} is an effective separation between the two point masses, each having the same mass as the associated hemisphere. Only the magnitude of the force is considered here. This situation is shown in _{A}A or α_{B}B from the flat surface. The two α’s

refer to the fraction of the radius, A or B, where the point mass is located. If a curved surface faces down, the position will be measured from that curved surface. In all cases, the position will be calculated as a function of S and shown to approach the center of mass of the respective hemisphere at infinite separation. The separation of 50 A is fully adequate to demonstrate that point. This center-of-mass limit ((3/8) A (or B), measured from the flat surface, and (5/8) A (or B), measured from the curved surface) can be proved analytically and can be viewed as a test of the accuracy of the calculation. Their position at any separation will be calculated, as shown in the bottom of _{B}B and calculating the force between it and the other hemisphere. That force between the two will be equated to the force previously calculated between the two hemispheres, thus determining α_{B}B. Since α A A + α B B + S = R e f f , the other location is determined.

First, let us be specific about the superposition of collinear forces, at least in one case. Others are similar. Assuming the form GM_{A}M_{B}/(R_{eff})^{2} and referring to

1 [ R e f f ( c f ; S 2 ) ] 2 = 2 [ R e f f ( p f ; S 2 ) ] 2 − 1 [ R e f f ( f f ; S 2 + B ) ] 2 (1)

where, in order of their appearance, “cf” refers to curved-flat; “pf” refers to point-flat (because the sphere acts as a point source), and “ff” refers to flat-flat, as also shown in the first panel. The separation for each R_{eff}, S_{2} and S_{2} + B, are also shown. The factor of 2 exists because the mass of the sphere is twice that of the hemisphere.

Next, we present the formulas for the magnitude of the force between a sphere and a hemisphere required in _{pf}, in

F p f = G M A ( 2 M B ) [ 3 A 3 ] [ ( 1 3 S 2 ) { ( 2 S 2 − A 2 ) A 2 + S 2 + 3 A S 2 + ( A ( A − S ) − 2 S 2 ) ( A + S ) } ] (2)

where we identify the multiplier of [GM_{A}(2M_{B})] with (1/R_{eff})^{2}, S = S_{2} + B, and M_{A} and M_{B} are the two hemispherical masses. In the case of panel 2B, A and B are reversed in the above formula. The equation for the force between the curved side facing the sphere and the sphere can be obtained by a similar integration, resulting in a similar formula. More simply, it can be obtained by subtracting from the force between two spheres the force between the sphere and the hemisphere (flat side facing the sphere), as shown in

As was stated earlier, this situation is the only one requiring a nontrivial calculation. It was performed in two ways. The first involves a direct integration of the force between all of the points in one hemisphere and all of the points in the

other. Because integration over two volumes was required, it started out as a six-fold integration. However, azimuthal symmetry reduced it by one integral and another integral could be done analytically, thus reducing the problem to a still-unpleasant four-fold integration. The second way involved expansions of the potential created by one hemisphere along its axis for positions z ≤ A and z ≥ A. The off-axis values were then expressed in terms of the original expansion multiplied by Legendre polynomials, as in electrostatics.

The direct integration method was performed with respect to

The basic integral expression for the point-to-point (pp) force, F_{pp}, is:

F p p = G ρ A ρ B ∬ d V A d V B cos φ R 2 (3)

where V_{A} and V_{B} are the volume elements in their respective hemisphere; R is the separation between any two points in different hemispheres; φ is the angle between R and the vertical; and ρ_{A}_{(B)} is the mass density of hemisphere A(B), assumed to be different for the sake of generality. By symmetry, the force is in the vertical direction. This equation can be written in an explicit form:

F p p = G ρ A ρ B ∫ 0 B d z B ∫ 0 ( B 2 − z B 2 ) 0.5 R B d R B ∫ 0 2π d φ B ∫ 0 2π d φ A ∫ 0 A d z A ∫ 0 ( A 2 − z A 2 ) 0.5 R A d R A ( z A + z B + s ) [ ( z A + z B + s ) 2 + r A 2 + r A 2 − 2 R A R B cos φ A ] 3 2 (4)

Writing the equation in the form of the product of the two point masses divided by the square of an effective separation between them scaled by the radius A(1/R_{eff}_{,A})^{2}, we obtain the following complicated expression:

( 1 R e f f , A ) 2 = 9 π ∫ 0 1 d z b ∫ 0 ( 1 − z b 2 ) 0.5 R b d R b ∫ 0 π d φ a ∫ 0 1 d z a ( z a + b z b + s a ) [ ] (5)

where

[ ] = [ ( z a + b z b + s a ) 2 + ( b R b ) 2 ] 1 / 2 ( b R r b sin φ a ) 2 + ( z a + b z b + s a ) 2 + ( b R b cos φ a ) ( 1 − z a 2 ) 1 / 2 − ( ( z a + b z b + s a ) 2 + ( b R b ) 2 ) ( ( b R b sin φ a ) 2 + ( z a + b z b + s a ) 2 ) X 1 2

And where

X = ( 1 − z 2 ) − ( 2 b R b cos φ a ) ( 1 − z a 2 ) 1 / 2 + ( ( z a + b z b + s a ) 2 + ( b R b ) 2 )

R_{eff}_{,A} = R_{eff}/A. All the other linear parameters in the expression are similarly scaled and use similar notation (e.g., s_{a} ≡ S/A; b = B/A; z_{a} = Z_{A}/A; r_{a} = R_{A}/A; z_{b} = Z_{B}/B; r_{b} = R_{B}/B). The integral over the azimuthal angle, φ_{B}, forthe upper hemisphere was simply replaced by 2π, and the integral over r_{A} was doable analytically. Thus, as stated above, the 6-fold integral was reduced to a 4-fold integral. Since the 4-fold integral did not appear to be analytically evaluable, it was numerically evaluated by using the Wolfram Mathematica command “NIntegrate” as applied to multiple integration. The results of the direct integration are shown in _{A}, followed by a transition to very small values of (1/R_{eff}_{,A})^{2} at large values of

S_{A}. The values for b = 1/4 are consistently larger than those for b = 1. At S = 0, for example, the two are 1.31 and 0.75, respectively. The force for b = 1 is considerably larger, however, because the product of the masses is 64 times as large. Given the value of 0.75 for (1/R_{eff}_{,A})^{2} with b = 1, an expression for the magnitude of the force between the two hemispheres in contact with one another is (1/3) [G(πρA^{2})^{2}], where ρ is the mass density of both hemispheres. This is the same result obtained in [

Since the gravitational force between the two halves of a solid sphere, such as the earth, is of some interest to inquiring minds [^{1/3}, the ratio of the two forces, spheres to hemispheres, can easily be deduced as (2^{2/3}/3) ≈ 0.53. Thus, the force between the two hemispheres is almost twice that of the two spheres. The second application is concerned with the rotation rate of the earth required to separate its two halves under only the combined action of the fictitious centrifugal force and gravity. The calculation is done with reference to _{c}, is obtained by integrating the component of the force perpendicular to the interface plane over the hemisphere.

F c = 2 ω 2 ρ ∫ 0 A d z ∫ 0 ( A 2 − z 2 ) 1 / 2 r 2 d r ∫ − π 2 π 2 cos φ d φ = π 4 ω 2 ρ A 4 (6)

Equating F_{c} to [G(πρA^{2})^{2}]/3, solving for ω, and expressing ω as 2πN_{r}, where N_{r} is the number of rotations per second, we obtain N_{r} = [Gρ/(3π)]^{1/2}. This results in ≈ 20 × 10^{−5} rotations/sec. required to just separate the two halves of the sphere. The earth’s rotational speed ≈ 1.16 × 10^{−5} rotations/sec., which ≈ (1/17) of the required speed. Of course, the separation speed would be much greater if the cohesion within the earth, which serves a kind of geological corpus callosum, were taken into account. Furthermore, tidal friction is causing the earth’s rotational speed to slowly diminish. Thus, we have nothing to worry about.

In this section, we present a graphical array of results, in addition to the one already presented in _{eff}_{,A})^{2} vs. S_{A} for the various configurations in _{eff}_{,A})^{2} for S_{A} ≤ 3, b = 1, and for its three distinguishable configurations. The dotted line is added to show the linear behavior in the immediate vicinity of S_{A} = 0. The ordering in size is consistent with the amount of mass in one hemisphere that is close to the other hemisphere. That order is flat-to-flat, flat-to-curved, curved-to-curved. _{eff}_{,A})^{2} is larger than in

figurations 2A and 2B at S_{A} = 0.25, which is maintained down to S_{A} = 0. _{A} varies as 1 / S A 2 . _{A}, depending on whether

the flat or curved side faces the other hemisphere. That condition is denoted by “flat” or “curved” in ^{2} and only approaches that behavior at very large vales of S (as shown in

Without the use of high-speed computers and sophisticated computational software, such as Wolfram Mathematica or, perhaps, MATLAB, it would be extremely difficult to perform the four-fold integration that makes up Equation (5). This is particularly true because of the coupling among the variables of integration. However, the use of a power expansion of the on-axis gravitational potential of hemisphere A (or B), followed by a term-by-term multiplication by Legendre polynomials to produce off-axis values of the potential simplifies the calculation. This is commonly done when dealing with electrostatics problems containing azimuthal symmetry, and it is discussed in many textbooks [

This method of solution for configuration 1 is presented in conjunction with

Once the potential, Φ(r, θ), is determined, its negative z-derivative (the z- component of its gravitational field) is calculated and integrated over hemisphere B to determine the total force between the two hemispheres. In the case of ^{l}, whereas for R ≥ A, it can be written as a power series in R^{−(}^{l}^{+1)}. Thus, the potential at angle θ is of the form:

Φ ( R , θ ) = ∑ l = 0 ∞ C l R l P l ( cos θ ) ; R ≤ A (7a)

Φ ( R , θ ) = ∑ l = 0 ∞ D l R − ( l + 1 ) P l ( cos θ ) ; R ≥ A (7b)

C_{l} and D_{l} are parameters to be determined and P l ( cos θ ) is the Legendre Polynomial of order l, all of which are unity at θ = 0. In order to proceed, we must first determine the on-axis potential. After a simple integration it can be shown to be:

Φ ( R , 0 ) = − G ( 2 π ρ a ) [ ( A + R ) 3 3 R − ( A 2 + R 2 ) 3 2 3 R − A R − A 2 2 ] (8)

Ignoring constants in the above expression, which contribute nothing to the force, results in the following expansions for R ≤ A and R ≥ A, respectively.

Φ ( R , 0 ) = − G ( 2 π 3 ρ a A 3 ) ( 1 A ) [ − ( 3 2 ) ( R A ) + ( R A ) 2 − ( 3 8 ) ( R A ) 3 + ( 1 16 ) ( R A ) 5 − ( 3 128 ) ( R A ) 7 + ⋯ ] (9a)

and

Φ ( R , 0 ) = − G ( 2 π 3 ρ a A 3 ) ( 1 A ) [ ( A R ) + ( 3 8 ) ( A R ) 2 − ( 1 16 ) ( A R ) 4 + ( 3 128 ) ( A R ) 6 − ( 3 256 ) ( A R ) 8 + ⋯ ] (9b)

From these two equations, the constants C_{l} and D_{l} can be directly determined. To obtain Φ(R, θ), we multiply each term in the above two expansions by the appropriate Legendre polynomial, express the expansion in terms of the hemispheric coordinates z, r, φ and, to obtain the force, integrate its negative z-deriv- ative over z from s to ( s + ( B 2 – r 2 ) 1 / 2 ) and over r from 0 to B. The azimuthal integration over φ is simply replaced by 2π. In addition, we note that cos θ = z / ( z 2 + r 2 ) 1 / 2 . When all of this and a few other manipulations are completed, we identify (1/R_{eff,A})^{2} for R ≤ A as:

1 ( R e f f , A ) 2 = ( 3 b 3 ) ∫ 0 b r a d r a ∫ S a S a + ( b 2 − r a 2 ) 1 / 2 d z a ∂ ∂ z a [ 3 2 ( z a 2 + r a 2 ) 1 2 − ( z a 2 + r a 2 ) P 2 ( z a ( z a 2 + r a 2 ) 1 2 ) + ⋯ ] ; R ≤ A (10a)

We note the use of scaled coordinates and the fact that the z_{a}-integral is particularly simple because it is the integral of a derivative. The integral over r_{a} was done numerically. For R ≥ A, the results are similar:

1 ( R e f f , A ) 2 = ( 3 b 3 ) ∫ 0 b r a d r a ∫ S a S a + ( b 2 − r a 2 ) 1 / 2 d z a ∂ ∂ z a [ − ( z a 2 + r a 2 ) − 1 2 − ( 3 8 ) ( z a 2 + r a 2 ) − 1 P 1 ( z a ( z a 2 + r a 2 ) 1 2 ) + ⋯ ] ; R ≥ A (10b)

For simplicity, only the first two terms of each expansion are shown in Equations (10a) and (10b).

This paper consists of a study of the gravitational attraction between two inline hemispheres over a range of separations from zero to 50 times the radius of the larger of the two hemispheres. Two relative radii were chosen, 1:1 and 0.25:1 as sufficiently representative, along with three relative orientations: flat side facing flat side, curved side facing curved side, and flat side facing curved side. Initially, the first was calculated through a “brute force”, four-fold numerical integration with the help of Wolfram Mathematica. The second and third orientations were derived from the first by a superposition of the first with a simpler, easily calculated, intermediate configuration. The results were expressed in terms of the canonical form of two point masses separated by an effective distance. That “effective distance” was calculated, as was the location of the “point masses” within the hemispheres. An application of the 1:1, flat-to-flat configuration was made to a split, rotating earth. Finally, the flat-to-flat configuration was analyzed using a power series expansion for the on-axis potential and Legendre polynomials. With the use of very few terms in the expansion, the results agreed extremely well with the four-fold integration.

Based on the methods of calculation presented here, future work could involve other pairs of masses that form a cylindrically symmetric configuration. Elongating or compressing one or both hemispheres along their individual axes immediately comes to mind.

The author is grateful to Joseph S. Accetta, CEO of JSA Photonics, and William F. Filter, retired Sandia National Laboratories physicist, for reviewing the manuscript and making valuable suggestions for its improvement.

Weiss, J.D. (2017) The Gravitational Attraction between Hemispherical Masses. Applied Mathematics, 8, 820-834. https://doi.org/10.4236/am.2017.86064