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We present an idealized, semi-analytic model of the evolution of a magnetized molecular cloud due to ambipolar diffusion. This model allows us to follow the quasi-static evolution of a core up to its collapse, and the subsequent evolution of the remaining envelope. Our main simplifying assumptions are: \begin{enumerate} \item Thermal stresses are negligible compared to magnetic stresses. \item The cloud is spherically symmetric, and the only non-vanishing component of the Lorentz force is given by the scalar term $\propto dB^2/dr$, where $B=B(r)$, and $r$ is the usual spherical radius. \item The evolution is quasi-static and the ion velocity is negligible compared to the neutral velocity. \end{enumerate}
For Lagrangian inital conditions of the form $\rho(t=0,M) \propto 1/(A+M)^2$, where $A$ is a free parameter, we are able to find exact, analytic\/ solutions of the MHD equations for $\rho(t,M)$, $v(t,M)$, and $r(t,M)$, and numerical solutions for $B(t,M)$ and $v_i(t,M)$ that involve only simple quadratures. The non-dimensional solutions depend on two parameters, the initial degree of concentration, which depends on $A$, and the initial ratio of the cloud's mass $M_{\rm cloud}$ to the magnetic critical mass $M_\Phi$. These solutions are valid up to the time when the core of mass $M_{\rm core} = A\,M_{\rm cloud}$ undergoes dynamical collapse, and inmediately thereafter.
By a judicious choice of $A$ and $M_{\rm cloud}/M_\Phi$ we reproduce within factors of order unity the numerical results of Fiedler \& Mouschovias (1992; ApJ 415, 680) for the physical quantities in the midplane of a collapsing, magnetized, axisymmetric cloud. In addition, we can follow the evolution of the remnant, magnetized envelope after the collapse of the core.
