Gravitational mass centres

Summary

The inverse square law for the action of gravity F = m₁ m₂ G / r² is well established and is applied using the formula with the separation of the masses r equal to the distance between their respective centres of gravity. For spheres this is the centre of the sphere, and at least for spheres with uniform density (or only radial variations), has allowed accurate calculation of the motion of (non-intersecting) bodies through the cosmos. In contradiction of widely held teachings (e.g. [1]) it is shown in what follows that the use of the centre of gravity (c.o.g.) for objects which are not spheres is unjustified. The calculations differ considerably in close-approach situations from those derived from use of c.o.g. These situations can occur with irregular asteroids, and disc or box galaxies.

Split longitudinal mass

In order to present an example without having to use anything but the inverse square law [2] and simple arithmetic, consider an observing point O at unit distance from a finite (but physically small in size) mass m.

Now imagine an object with half the mass nearer by a distance h/2, and the other half the same distance away, but still united as a single object by a rigid (negligible mass) connection. All "splits" hereafter have this latter context.

Fig. 1: Split longitudinal mass. Click image for larger version.

The nearest part will exert twice the previous force and the furthest only 2/9. The effective geometric centre (the square root of the reciprocal) will now be 0.67 instead of unity. Fig. 2 shows the calculation for an extended range of h, including the situation where the nearest mass passes the origin. The gravitational mass centre (g.m.c.) then moves steadily away beyond this mass's c.o.g. towards a factor of root 2 times its distance (to allow for the half-mass which now dominates if the two masses are taken as one).

Fig. 2: Split mass object as function of separation. Click image for larger version.

Longitudinal rod

By summing a succession of split masses over a range, it is possible to derive (by integration) the g.m.c. of a small diameter rod-shaped mass pointing towards the observer, and display this as a function of its length.

Fig. 3: End-on rod as function of length. Click image for larger version.

It is clear in this case that, while the g.m.c. departs from the c.o.g. in a similar way to the case of the split mass, the behaviour is different once the end of the rod passes the observer. A well-explored feature in Newton's Principia, is the cancellation of attractions which are symmetrically disposed and equal. This clearly takes place here, the bottom left corner of the diagram (from h = 2) showing the cancellation area in red. The g.m.c. then moves with the remainder of the rod, which is outwards, and moves towards the c.o.g. of the "uncancelled" part of the rod.

Transverse split mass or ring

Due to symmetry, the g.m.c. of a ring is the same as a split transverse mass, so Fig. 4 covers both cases. As would be expected, nearby rings will have g.m.c.'s relatively further behind the c.o.g., due to the loss of axial force from the larger angles which have to be resolved in the axial direction.

Fig. 4: The ring. Click image for larger version.

Transverse rod

By integrating the masses of the last section with respect to the vertical height, the result for the transverse (thin) rod is produced as Fig. 5. There is much less force loss due to the presence of central masses, so the g.m.c. extension is less pronounced.

Fig. 5: The transverse thin rod. Click image for larger version.

Transverse discs

The transverse disc is particularly simple to integrate as, after multiplying the ring elements by 2 π times the radius, many of the force terms cancel out, leaving only a cosine to integrate between appropriate limits.

Fig. 6: The transverse thin disc. Click image for larger version.

Displaced and rotated objects

For simplicity the above examples have all been viewed along an axis pointing at the c.o.g. and oriented either at right angles or in-line with the view line. In the general case, the objects can present any angle, and it is worth showing the result of displacements sideways from the original axis. The mass shown in Fig. 7 has been displaced sideways by its width i.e. y = h = 1. The obvious and important point is that the line to the g.m.c. only approaches the c.o.g. when the object is at x = infinity. In general it leans towards the nearest part of the object (as black line example shown). In such cases, the acceleration in this direction would give a torque about the c.o.g., and produce rotation of the object (if the object at the origin had noticeable comparative mass).

Fig. 7: The displaced split (dumbell). Click image for larger version.

Note that the r for x = 0 is, as expected, the same as derived for the split longitudinal mass, i.e. 0.67.

Conclusions

The statement [1] that the geometric centre to be used for Newton's gravity formula is the centre of gravity is only true of separated spheres (and shells). Calculating the c.o.g. for a body involves integrating mass times distance, while the g.m.c. uses mass divided by the square of distance. Apart from spherical bodies this does not give the same answer. I believe this misconception has come about from the difficulty of interpreting the language in Newton's Principia (see [3]) together with the display of an irregular object in Principia [4]. Close examination of the latter indicates this deals with equivalence of the objects' c.o.g. to an appropriate sphere, not to its geometric mass centre.

References

Andrew Zimmerman Jones (2009). "Newton's law of gravity",
http://physics.about.com/od/classicalmechanics/a/gravity.htm.
About.com Physics.
Isaac Newton (1687). Philosophiae naturalis principia mathematica, vol. 1, p. 193.
Michael Hoskin (1996). The Cambridge illustrated history of astronomy, p. 163.
Isaac Newton (1687). Philosophiae naturalis principia mathematica, vol. 1, p. 217, proposition LXXXVIII, theorem XLV.

G.M. Clarke

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