Editing Stellar parallax (section)

=== Principle ===
{{more citations needed|section|date=June 2020}}
Throughout the year the position of a star S is noted in relation to other stars in its apparent neighborhood: 
[[File:Stellar parallax movement.png|center|frameless]]
Stars that did not seem to move in relation to each other are used as reference points to determine the path of S.

The observed path is an ellipse: the projection of Earth's orbit around the Sun through S onto the distant background of non-moving stars. The farther S is removed from Earth's orbital axis, the greater the eccentricity of the path of S. The center of the ellipse corresponds to the point where S would be seen from the Sun: 
[[File:Stellar parallax right angle of observation.png|center|frameless|500x500px]]
The plane of Earth's orbit is at an angle to a line from the Sun through S. The vertices v and v' of the elliptical projection of the path of S are projections of positions of Earth E and {{prime|E}} such that a line E-{{prime|E}} intersects the line Sun-S at a right angle; the triangle created by points E, {{prime|E}} and S is an isosceles triangle with the line Sun-S as its symmetry axis.

Any stars that did not move between observations are, for the purpose of the accuracy of the measurement, infinitely far away. This means that the distance of the movement of the Earth compared to the distance to these infinitely far away stars is, within the accuracy of the measurement, 0. Thus a line of sight from Earth's first position E to vertex v will be essentially the same as a line of sight from the Earth's second position {{prime|E}} to the same vertex v, and will therefore run parallel to it - impossible to depict convincingly in an image of limited size: 
[[File:Stellar_parallax_parallel_lines_from_observation_base_to_distant_background.png|alt=|center|frameless|500x500px]]
Since line {{prime|E}}-{{prime|v}} is a transversal in the same (approximately Euclidean) plane as parallel lines E-v and {{prime|E}}-v, it follows that the corresponding angles of intersection of these parallel lines with this transversal are congruent: the angle θ between lines of sight E-v and {{prime|E}}-{{prime|v}} is equal to the angle θ&nbsp;between {{prime|E}}-v and {{prime|E}}-{{prime|v}}, which is the angle θ between observed positions of S in relation to its apparently unmoving stellar surroundings. 
[[File:Stellar_parallax_parallel_lines.png|alt=|center|frameless|500x500px]]
The distance ''d'' from the Sun to S now follows from simple trigonometry:

   tan({{sfrac|1|2}}θ) = E-Sun / d,

so that d = E-Sun / tan({{sfrac|1|2}}θ), where E-Sun is 1 AU.
[[File:Stellar_parallax_trigonometric_calculation.png|alt=|center|frameless|500x500px]]
The more distant an object is, the smaller its parallax.

Stellar parallax measures are given in the tiny units of [[arcsecond]]s, or even in thousandths of arcseconds (milliarcseconds). The distance unit parsec is defined as the length of the [[Cathetus|leg]] of a [[right triangle]] [[Adjacent side (right triangle)|adjacent to]] the angle of one arcsecond at one [[Vertex (geometry)|vertex]], where the other leg is 1 AU long. Because stellar parallaxes and distances all involve such [[Skinny triangle|skinny right triangles]], a convenient trigonometric approximation can be used to convert parallaxes (in arcseconds) to distance (in parsecs). The approximate distance is simply the [[Reciprocal (mathematics)|reciprocal]] of the parallax: <math>d \text{ (pc)} \approx 1 / p \text{ (arcsec)}.</math> For example, [[Proxima Centauri]] (the nearest star to Earth other than the Sun), whose parallax is 0.7685, is 1 / 0.7685 parsecs = {{convert|1.301|pc|ly}} distant.<ref name="Gaia">{{Cite DR2}}</ref>