  Barchans are dunes of high mobility which have a crescent shape and propagate
under conditions of unidirectional wind. However, sand dunes only appear above
a critical size, which scales with the saturation distance of the sand flux [P.
Hersen, S. Douady, and B. Andreotti, Phys. Rev. Lett. {\bf{89,}} 264301 (2002);
B. Andreotti, P. Claudin, and S. Douady, Eur. Phys. J. B {\bf{28,}} 321 (2002);
G. Sauermann, K. Kroy, and H. J. Herrmann, Phys. Rev. E {\bf{64,}} 31305
(2001)]. It has been suggested by P. Hersen, S. Douady, and B. Andreotti, Phys.
Rev. Lett. {\bf{89,}} 264301 (2002) that this flux fetch distance is itself
constant. Indeed, this could not explain the proto size of barchan dunes, which
often occur in coastal areas of high litoral drift, and the scale of dunes on
Mars. In the present work, we show from three dimensional calculations of sand
transport that the size and the shape of the minimal barchan dune depend on the
wind friction speed and the sand flux on the area between dunes in a field. Our
results explain the common appearance of barchans a few tens of centimeter high
which are observed along coasts. Furthermore, we find that the rate at which
grains enter saltation on Mars is one order of magnitude higher than on Earth,
and is relevant to correctly obtain the minimal dune size on Mars.
