ALPHA CENTAURI : Part 3

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 **e ^{1}**. This is quite different than
the true eccentricity (

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).

Dimensions(arcsec) |
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 |

Maximum Separation | 26.67 | 21.75 |

Minimum Separation | 08.47 | 01.75 |

Apastron | 26.67 | 16.94 |

Periastron | 08.47 | 05.38 |

(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.