Stellar parallax: The parallactic ellipse

We consider in more detail the path traced out on a geocentric celestial sphere by a star throughout the year.

p = P sin θ (1)

In figure 1, the star X has heliocentric right ascension and declination α and δ but because of stellar parallax it is displaced towards the Sun S₁ by an amount XX₁, given by equation (1), namely
XX₁ = P sin XS₁. (2)
Since the Sun moves round the ecliptic in one year and the star is always displaced by parallax towards the Sun, the star’s path X₁X₂X₃ . . . during the year will be a closed curve on the celestial sphere, enclosing the star’s heliocentric position X.

Figure 1. The projection of the parallactic ellipse on the celestial sphere.
Let S₁ be the point defined by the intersection of the great circle through the north pole of the ecliptic K and X with the ecliptic. Then S₁ is the position of the Sun when its angular distance from X is least. Let S₂ be the Sun’s position three months later. By equation (1), we have
XX₂ = P sin XS₂. (3)
Three months later, the Sun is at the point S₃ and the star’s displacement to X₃ is given by
XX₃ = P sin XS₃. (4)
But S₃ is the Sun’s position six months after it occupied the position S₁. Hence, their longitudes are 180^◦ different and
XS₃ = 180^◦ = XS₁.
By equations (2) and (4), it is, therefore, seen that XX₁ = XX₃ in size. In a similar manner, it can be seen that if S₄ is the position reached by the Sun six months after it left S₂, the parallactic displacement XX₄ of the star is equal to XX₂ in size but in an opposite direction.
Now S₂ and S₄ are the positions of the Sun that are farthest from the star’s heliocentric position X, that is S₄S₁ = S₂S₁ = 90^◦, so that the star’s parallactic shifts at such times are greatest. Any other shift, such as XX₅, will lie between XX₁ and XX₂ in size. Hence, the apparent path of the star due to stellar parallax is an ellipse whose major axis is parallel to the ecliptic and has a value P arc sec. The value of the minor axis depends on the star’s position.
For example, if the star happened to be at K, the north pole of the ecliptic, the minor axis would be of length P arc sec, since all solar positions would be 90◦ from the star’s heliocentric direction K. In other words the ellipse would be a circle. If, however, the star lay on the ecliptic, the ellipse would degenerate into an arc of the ecliptic, the minor axis vanishing.