  The visual A component of the Gliese 586AB system is a double-lined
spectroscopic binary consisting of two cool stars with the exceptional orbital
eccentricity of 0.976. Such an extremely eccentric system may be important for
our understanding of low-mass binary formation. We present a total of 598
high-resolution echelle spectra from our robotic facility STELLA from 2006-2012
which we used to compute orbital elements of unprecedented accuracy. The orbit
constrains the eccentricity to 0.97608+/-0.00004 and the orbital period to
889.8195+/-0.0003d. The masses of the two components are 0.87+/-0.05 Msun and
0.58+/-0.03 Msun if the inclination is 5+/-1.5degr as determined from
adaptive-optics images, that is good to only 6% due to the error of the
inclination although the minimum masses reached a precision of 0.3%. The flux
ratio Aa:Ab in the optical is betwee n 30:1 in Johnson-B and 11:1 in I. Radial
velocities of the visual B-component (K0-1V) appear constant to within 130 m/s
over six years. Sinusoidal modulations of Teff of Aa with an amplitude of apprx
55 K are seen with the orbital period. Component Aa appears warmest at
periastron and coolest at apastron, indicating atmospheric changes induced by
the high orbital eccentricity. No light variations larger than approximately 4
mmag are detected for A, while a photometric period of 8.5+/-0.2 d with an
amplitude of 7 mmag is discovered for the active star B, which we interpret to
be its rotation period. We estimate an orbital period of approx 50,000 yr for
the AB system. The most likely age of the AB system is >=2 Gyr, while the
activity of the B component, if it were a single star, would imply 0.5 Gyr.
Both Aa and B are matched with single-star evolutionary tracks of their
respective mass.
