Gravitational mass centres
Summary
The inverse square law for the action of gravity
F = m_{1} m_{2} G / r^{2}
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 (nonintersecting) 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 closeapproach 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 halfmass 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
rodshaped mass pointing towards the observer, and display this as a
function of its length.

Fig. 3: Endon 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
wellexplored 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
inline 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.