It is doubtful whether anyone knows anything at all about the mass of
the earth, its condition or its other properties at the time when it began.
So any figures concerning its mass at that time can neither be proved true
or false.
Knowledge begins at a date that may roughly be put at two thousand
million years ago and so it is relevant to consider what the maximum
possible value of the mass of the earth can have been at that time. If the
half-life of matter is 10^{9} years, its mass two thousand million years ago was
about four times its present mass. Its radius was therefore about 60 per
cent greater than it is now. But reasons will be found in this appendix, as
well as in Appendices D and E, why this is probably a gross under-estimate.
Any emotional distaste one may experience for the notion that the
earth has today but a small fraction of its initial mass must yield to some
rather cogent facts, which are partly astronomical, partly geological and
partly biological.
This expression enables us to place an upper limit to T_{m}, It has been
shown in Appendix B that a galaxy would grow without limit if the total
number of origins within its domain always exceeded the rate of extinctions there. For the size to stabilize there must be periods of time during
which the rates are reversed and during these periods the average density
in the domain must exceed the equilibrium value N_{eq}.
What causes the density to rise above N_{eq} is that the domains of the
new clouds that are growing around the older galaxy spread and encroach
on the domain of the older galaxy. Thereby the latter loses those regions
in which the density is lowest, while it retains the central region in which
the density is high. While the mass within the domain remains almost
unaltered, the volume of the domain is reduced.
If the boundary of the domain of the new clouds were to advance from
all sides up to a mid position between the centres of the new clouds and
the older galaxy, the domain of this would be reduced to about one-eighth
of its original volume. But the new clouds are much less massive and so
the encroachment cannot go so far.
The reduction of the volume of the older galaxy cannot, of course, be
sufficient to leave only one-eighth of the previous volume. At the same
time the reduction must be great enough to raise the average mass density
from less than N_{a} to more than N_{eq} , for if it were not so, if the density in
the reduced volume still remained below that given by N_{eq}, the rate of
origins within the domain would always exceed the rate of extinctions and
the older galaxy would continue without interruption to become ever more
massive.
We thus have to conclude, firstly, that fluctuations in the domain of
any galaxy range over less than one-eighth in volume, and secondly, that
the ratio N_{eq} / N_{a} = q is less than the ratio of the maximum to the minimum
volume of the domain.
It would be a complicated matter to calculate the ratio of maximum
to minimum volume that should be expected from the growth of new
clouds around an existing galaxy, though this will have to be done some
day. Meanwhile, Table II will suffice to show how q and T_{m} are interconnected by equation (Ce). In preparing the table it has been assumed
that T_{s} is 3.66 x 10_{9} years. As p is the ratio of T_{m} to T_{s} the half-life of matter
T_{m} is obtained from equation (Cc).
Table II
q |
T_{m} years |
5 |
1.46 x 10^{9} |
4 |
1.10 x 10^{9} |
3 |
0.73 x 10^{9} |
2 |
0.37 x 10^{9} |
Table I makes it appear unlikely that T_{m} could be much less than 3 x 10^{8}
years. It is unlikely that q can be much greater than 2 and so Table II
makes it appear unlikely that T_{m} can be much greater than 4 x 10^{8} years.
If various lines of approach do not allow much room for manoeuvre,
they do not at least lead to contradictory conclusions.
It will be shown in Appendix E that a different approach also suggests
a value of the order of 4 x 10^{8} years. But if the age of radio-active substances is found to be much greater than 2.6 x 10^{9} years there may be a
reason for assigning a longer half-life to matter and, on the other hand,
evolutionists may be influenced by the conclusions reached in Appendix
E to press for a shorter estimate.
The conclusion that the half-life of matter is shorter than that of some
radio-active substances may cause surprise, perhaps be even thought to
be impossible. The reason why this conclusion need not be inconsistent
with observed facts can only be given after the relation between space
and mass has been explored and will be found at the end of Appendix H.