  Based on the perturbation theory up to the second order, analytical
asymptotic expressions for the variation of the population of hyperfine
component $\mu=0 $ particles in the ground state of spin-1 condensates against
a magnetic field $B$ has been derived. The ranges of $B$ in which the
asymptotic expressions are applicable have been clarified via a comparison of
the numerical results from the analytical expressions and from a
diagonalization of the Hamiltonian in a complete spin-space. It was found that,
For $^{87}$Rb, the two analytical expressions, one for a weak and the other one
for a strong field, together cover the whole range of $B$ from 0 to infinite.
For Na, the analytical expressions are valid only if $B$ is very weak or
sufficiently strong.
