Geophys. J . R . astr. Soc. (1966) 10, 445-447.
LRtter to the Editors
Comments on Detecting Changes in the Earth’s Radius
In a recent paper, van Hilten (1963), using paleomagnetic data, suggests that there
has been a noteworthy increase in the Earth’s radius (upwards of 10%) in the last
few hundred million years. In a paper published at about the same time (Ward
1963), I had reached the conclusion that these data could not be considered to
indicate any increase in the Earth’s radius during that time. Van Hilten (1965)
has put forward the view that the difference between our conclusions depends on
differences in interpretation and weighting of data. In the following discussion it
will be shown that van Hilten’s method is at variance with his model, and that it
is from this fact, rather than from any difference in interpretation or weighting of
data, that the real differences between our results stem. I will also clarify the
relationship between my model and method, since this may not have been made
sufficiently clear in my paper. The symbols and terms defined in my paper will be
used throughout this discussion.
1. Van Hilten’s method
The model which both van Hilten and I explicitly use has the following two
main properties : first, that as the Earth’s radius changes, the size of the continental
block remains constant, and thus the geocentric angles between the rock units
will change, and second that, as the usual consequence of assuming a dipole field,
the geocentric angle between each rock unit and the paleomagnetic pole calculated
from it remains constant as the Earth’s radius changes. Now in equation (1) of
van Hilten’s paper we see that the geocentric angle between a rock unit and the
corresponding pole is considered to change proportionately to the ancient radiuscontrary to the requirements of the model. Again, by inspection of Table 1 of his
paper, one finds that in order to calculate the ancient radius the rock units are all
kept at the same latitude and longitude (and thus the geocentric angles between
them are kept constant) and the geocentric angle between each rock unit and the
corresponding pole is varied until these poles are most nearly coincident, then
his equation (1) is used to calculate an ancient radius from this angle from each
rock-datum. Thus it is seen that van Hilten’s calculations proceed by keeping
the geocentric angles between rock units constant, and allowing the geocentric
angles between each rock unit and the corresponding ancient pole to vary, all of
which is contrary to the requirements of the model.
In order to correct for reversals, I had chosen to work with poles in the region
of what nowadays is the N. Pacific Ocean. Van Hilten has also apparently done
this. It is essential that, for a method such as the ones under discussion to be
tenable, it should be possible to work entirely with the opposite poles-that is,
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446
M. A. Ward
those in the region of what nowadays is the S. Atlantic Ocean-and obtain the
same results. This criterion holds good for my method, but not for van Hilten’s;
in fact, if he had used these opposite poles instead, none of his intersections would
have occurred, and his result would have been that the Earth has contracted rather
than expanded during the period under discussion.
2. My method
The model assumes that the continental block remains constant in size as the
Earth’s radius changes, however the shape must change slightly, as the continent
must accommodate itself to the changing curvature of the Earth’s surface. The
simplest form of this accommodation was incorporated in the model which I used :
it was assumed that there is a central point in the continent at which the strain
due to this curvature change is zero, and that the strain is axially symmetric about
that point and entirely tangential with no radial component. Thus the distance
from the central point to each rock unit will remain constant so that the corresponding geocentric angle is given by f?:/p where 0; is the present value of that
angle. Further, the angle subtended at the central point by a pair of rock units
(4;- 4;) will remain constant. Finally the declination of magnetization of each
rock unit Di,measured with respect to the central point, will remain constant, as
of course will the geocentric angle t,bi between each rock unit and the corresponding
pole. The calculation of the poles for each from these figures is now a matter of
straightforward spherical trigonometry, and the equations for this process are given
in my paper, section 2 (ii).
3. North American data
In an idealized case where the rock units all lie on the same paleomeridian,
van Hilten’s method will yield substantially the same result as my own. On the
other hand, if the rock units lie on approximately the same paleolatitude and the
pole used for analysis is more than 90” away, van Hilten’s method and my own
will yield opposing results, in the sense that if my method indicates an increase in
radius, van Hilten’s will indicate a decrease and vice versa. This absurd situation is
exemplified by the Carboniferous data from N. America.
Had van Hilten and I used exactly the same data for the Triassic of N. America,
we would have arrived at roughly the same result. The differences between our
results in this case are entirely due to a different selection of data. The only conclusion that can be drawn from this sort of situation is that either the amount of
data available or its accuracy is insufficient for the purpose-as I said in my paper.
4. Conclusion
It is clear that the differences between van Hilten’s paper and mine are not a
matter of interpretation but rather of a basic difference between our methods of
calculation. I believe that in this discussion I have established that van Hilten’s
method is at variance with the model which we agree on, and that on the other
hand my own method is in close accord with it. Thus I can see no reason to alter
the conclusion reached in my paper : that on the basis of the paleomagnetic evidence
studied, we cannot conclude that the Earth’s radius has changed markedly in the
last few hundred million years.
Finally I would like to point out two errors in my paper. First, in section 1
the supposed change in the Earth’s radius attributed to Carey should be 25 and
Comments on detecting changes in the Earth’s radius
447
not 45%. Second, in section 2(ii) the direction cosines of the paleomagnetic pole
position should be I:, mi, nf, where
1; = li cos 4;-mi sin 4;
mi
=
li sin 4; +mi cos 4f
n!I = n.1
and l i , mi, ni are as given; this error was made in exposition and not in calculation
so that my results are correct.
Department of Mathematics,
Institute of Advanced Studies,
Australian National University,
Canberra, A.C.T., Australia.
1964 March.
References
Van Hilten, D., 1963. Nature, Lond., 200, 1277.
Van Hilten, D., 1965. Geophys. J. R. astr. SOC., 9, 279.
Ward, M. A., 1963. Geophys. J. R. astr. SOC.,8, 217.
b
M. A. WARD