  Achernar, the closest and brightest classical Be star, presents rotational
flattening, gravity darkening, occasional emission lines due to a gaseous disk,
and an extended polar wind. It is also a member of a close binary system with
an early A-type dwarf companion. We aim to determine the orbital parameters of
the Achernar system and to estimate the physical properties of the components.
We monitored the relative position of Achernar B using a broad range of high
angular resolution instruments of the VLT/VLTI (VISIR, NACO, SPHERE, AMBER,
PIONIER, GRAVITY, and MATISSE) over a period of 13 years (2006-2019). These
astrometric observations are complemented with a series of more than 700
optical spectra for the period from 2003 to 2016. We determine that Achernar B
orbits the Be star on a seven-year period, eccentric orbit (e = 0.7255 +/-
0.0014) which brings the two stars within 2 au at periastron. The mass of the
Be star is found to be mA = 6.0 +/- 0.6 Msun for a secondary mass of mB = 2.0
+/- 0.1 Msun. We find a good agreement of the parameters of Achernar A with the
evolutionary model of a critically rotating star of 6.4 Msun at an age of 63
million years. We also identify a resolved comoving low-mass star, which leads
us to propose that Achernar is a member of the Tucana-Horologium moving group.
Achernar A is presently in a short-lived phase of its evolution following the
turn-off, during which its geometrical flattening ratio is the most extreme.
Considering the orbital parameters, no significant interaction occurred between
the two components, demonstrating that Be stars may form through a direct,
single-star evolution path without mass transfer. Since component A will enter
the instability strip in a few hundred thousand years, Achernar appears to be a
promising progenitor of the Cepheid binary systems.
