Stars - Annual Parallax

 

 

    The Stellar parallax have been always a very important factor in astronomy. As George Lovi wrote: "Parallax is the only true geometrical link between us and our nearer neighbours in that vast interstellar void. It has enabled astronomers to create and calibrate procedures to take us much farther out!"

 

 

    Bellow is a scheme of the stellar parallax:

 

Fig. 01: Stellar Parallax.

 

    Let X be the position of the celestial body, C the barycentre of the solar system and E the centre of the Earth. Let also r and R be the position barycentre vectors of the object and of the Earth respectively and r' the geocentric position vector of the object. The angle between the vectors r and r' it's the Annual Parallax p.

 

    Obviously we have that r = r' + R. In the case of X be a body inside the solar system it's essential to use the previous formula as we present, however if the body is a star than we can make some adjustments. The parallax will vary during the year as also the Earth describes his orbit around the Sun.

 

    Let E bet the elongation of the Star from the Sun, i.e., E=CÊX, then

 

sinp = (R/r)sinE

 

    The stellar parallax it's general defined by p, where

 

sinp = 1/r

 

    with r in astronomical units. This value corresponds to the annual parallax when R=1 and E=90º.

 

    However to those who needs to calculate the position of the stars with very accuracy the stellar parallax is not really important, happily the stellar parallax never exceeds the 0''.8 and can be put away in the most of the cases. According to R. Burnham, only 13 stars with magnitude larger than 9.0 are less than 13 light-years (4 parsec) from us and have a stellar parallax larger than 0''.25, they are:

 

a Centauri;

Lalande 21185 (in Ursa Majoris);

Sirius;

e Eridani;

61 Cygni;

Procyon;

e Indi;

S2398 (in Draco);

Groombridge 34 (in Andromeda);

t Ceti;

Lacaille 9352 (in Piscis Austrinus);

Cordoba 29191 (in Microscopium);

Kapteyn (in Pictor).

 

    Due to the stellar parallax be always less than 0''.8, the distance to the Earth can be calculated with sufficient accuracy by:

 

d = 1/p

 

    This distance it's in general very so we have to use the right units. The most common unit is the parsec that corresponds to a parallax of an arcsecond. If p are in radians, then by the previous formula we obtain the distance in astronomical units, but, if p is in arcseconds then the distance is obtain in parsecs.