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I understand Planetary Eccentricity and Inclination. However when it applies to a star, does the centre or ecliptic plane refer to the Sun or the Galactic Centre?

The star Sirius for example, 136.336 ± 0.040° inclination, the angle at which it is inclined to Sun, Earth or Galactic Centre?

The star Sirius has an Eccentricity of 0.59142 ± 0.00037. The only thing the star orbits is the Galactic Centre (Sagittarius A*) so I guess that is the centre?

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Sirius is a binary system, composed of a main sequence star (Sirius A) and a white dwarf (Sirius B). The two orbit each other with a period of about 50 years, and the eccentricity given on Wikipedia is the eccentricity of the stars' orbits around one another. The inclination, according to Bond et al. 2017 - the paper Wikipedia references for both numbers - is described as the inclination of the "relative visual orbit", leading me to believe that the inclination is defined based on a plane relative to the point of view of an observer on Earth.

Something it's important to realize is that stars don't follow Keplerian orbits, because of the non-point-like distribution of matter in the galaxy; the motion of a star around a galaxy is therefore somewhat more complicated than the motion of a star in a binary system.

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