  We prove an analog of Schmid's $\text{\rm SL}_2$-orbit theorem for a class of
variations of mixed Hodge structure which includes logarithmic deformations,
degenerations of 1-motives and archimedean heights. In particular, as
consequence this theorem, we obtain a simple formula for the asymptotic
behavior of the archimedean height of a flat family of algebraic cycles which
depends only on the weight filtration and local monodromy.
