He is remembered in particular for his work in fundamental astronomy, and for his lunar theory, which was in widespread use for the construction of navigational ephemerides of the Moon for much of the first half of the nineteenth century.

Later he became an assistant at the Gotha Observatory and studied under Franz Xaver von Zach.

On von Zach's recommendation, he joined the observatory of the École militaire in Paris, then directed by Jérôme Lalande.

Burckhardt's lunar tables appear to have been the first to be based on a least squares adjustment of the coefficients to selected lunar observations, of which about 4000 were used; and a committee of the Bureau des Longitudes (consisting of Laplace, Delambre, Bouvard, Arago and Poisson) concluded after an early form of sum-of-squares analysis that Burckhardt's tables improved on those of Bürg.

Burckhardt's tables were eventually replaced altogether, for issues of the Nautical Almanac for 1862 and thereafter, by new computations based on P A Hansen's more comprehensive lunar theory.