Ten naturally-occuring long-lived -emitting radionuclides
exist on Earth:
Nd,
Sm,
Sm,
Gd,
Hf,
Os,
Pt,
Th,
U and
U. For the purpose of helium-thermochronology, all but the
heaviest three of these nuclides can often be neglected because of
their low abundance and low helium-yield. For example, only one
-particle is produced per
Sm, whereas six to eight are
formed in the Th and U decay series. Further simplification is
possible because the present-day
U/
U-ratio is constant
in the solar system (= 137.88; Steiger and Jäger, 1977).
Therefore, the ingrowth of helium with time (t) can be written as a
function of the elemental U, Th and He abundances or concentrations:
with
,
and
the decay
constants of
Th,
U and
U, respectively.
Equation 2 has no analytical solution but is easy to
solve iteratively. However, for young ages (t
1/
), a reasonably accurate linear approximation also
exists:
with P the present-day helium production rate:
The accuracy of this solution will be discussed in Section
4. Besides being easy to implement, the linear age
equation is useful for illustrative purposes and opens up some new
applications which will be discussed in Section 3.
Meesters and Dunai (2005) introduced an alternative direct solution to the (U-Th)/He age equation:
with
the weighted mean decay constant:
As shown by Meesters and Dunai (2005) and in Section 4, this solution is remarkably accurate for all practical applications.