When we view Alpha Centauri from Earth over a normal human lifespan, we see that both stars move in a long, stretched out narrow ellipse. This knowledge of this well established orbit has been obtained from many carefully made direct measures of the apparent relative placement of these two stars in observations going back more than two-an-a-half centuries. This important historical data eventually enabled us to learn about the size and shape of the apparent orbit, and allowed us calculate the necessary orbital parameters describing its observed ellipse. While this usefully tells us very much about the past and future respective motion of the components, it fails to give the real picture of the true orbit. By using some extra and rather complicated geometrical mathematics via various long-held standard orbital methods dating to the early 20th Century, we can also determine the true size and shape of most binary star orbits — including Alpha Centauri. (See Figure 3-1 > )
Like all visual binaries, we can subdivide the given orbital elements into three parts;
Firstly, the so-called dynamical elements, being the period (P), the eccentricity (e) [See Note] and the fixed time of the closest approach of the components, periastron (T). All these elements remain the same in both the apparent orbit and the true orbits.
Second is the size of the semi-major axis (a), which sets the dimensions of the whole orbit. However, due to the different orientations of the apparent orbit, this parameter does corresponded exactly to what we see. I.e. Maximum separation in the apparent orbit is about 21.7 arcsec, while in the true orbit it is 26.7 arcsec.
And thirdly, the orientation elements, being the inclination (i), and the Ascending Node position angle (Ω) and the longitude of perihelion (ω). In the apparent orbit describes the orientation of the orbit in respect to the observer. To determine the true orbit, these values become zero, resulting in the attached figure.
NOTE: Orbital shape of the apparent ellipse is measured by the ‘apparent’ eccentricity or e1. This is quite different than the true eccentricity (e) given with the normal orbital elements, whose transformation is mainly due to the orbital perspective of the orientation. I.e. The value of e=0.5179 is the true orbit, while the observed highly elliptical e1=0.9814 is in the apparent orbit.
Observationally, most orbits are described in seconds of arc, which adequately describes the separation in the telescope but does not give the dimensions in Astronomical Units (AU) or kilometres. To do so requires some knowledge of the distance of the binary star in question, by either the trigonometric parallax or the inferred dynamical parallax. In Alpha Centauri’s case we know this value very well, so it can be calculated that at the distance 1.3478 parsecs or 4.396 light years, 1.0 arcsec equals 1.3478 AU or approximately 201.6 million kilometres. Knowing this means we can calculate all the physical lengths and dimensions of the orbit, from the calculated value used to compute binary star ephemerides, often known as ‘r’. (As also shown in the Figure.)
P and A are the points of periastron and apastron, respectively, which for the α Centauri system ranges between 11.42 A.U. (1.71 billion km.) and 34.95 AU (5.23 billion km.) If you could look directly down on top of the orbit from the similar distance of 4.3 light-years between α Centauri and the Sun, both stars during the 79.92 year period (P) would vary on average between 8.5 arcsec at closest approach, 26.5 arcsec when furthest apart or 17.5 arcsec for the size of the semi-major axis (a).
|True Orbit||Apparent Orbit|
|Semi-Major Axis (a)||17.57||17.57|
|Semi-Major Axis (2a)||35.14||35.14|
|Semi-Minor Axis (b)||15.03||15.03|
(i. ) 1.75 arcsec in September 2037
(ii. ) 21.75 arcsec in May 1980 then in December 2057
(iii) Ascending Node of the apparent orbit will be on August 2050 at 18.84 arcsec.
Periastron (P.), when the stars are closest together, last occurred in early July 1955 and will do so again in June 2035 AD. Apastron (A) when farthest away, will similarly occur in June 1995 and again in late-April 2075 AD. Successive dates are spaced by fractions of either half or one whole period of the binary 79.92 years. At the widest observed positions in the apparent orbit do not exactly correspond with the maximum distance between the stars. This is mostly due to the orientation and the direction of the line between periastron and apastron (and of the orbital ascending and descending nodes), which do lie quite askew to the line-of-sight of the observer.