Editing Stellar parallax (section)

== Parallax method {{anchor|Method}} ==

=== 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>

=== Variants ===

Stellar parallax is most often measured using '''annual parallax''', defined as the difference in position of a star as seen from Earth and Sun, i.e. the angle subtended at a star by the mean radius of Earth's orbit around the Sun. The [[parsec]] (3.26 [[light-year]]s) is defined as the distance for which the annual parallax is 1&nbsp;[[arcsecond]]. Annual parallax is normally measured by observing the position of a star at different times of the year as Earth moves through its orbit.

The angles involved in these calculations are very small and thus difficult to measure. The nearest star to the Sun (and also the star with the largest parallax), [[Proxima Centauri]], has a parallax of 0.7685&nbsp;±&nbsp;0.0002&nbsp;arcsec.<ref name="Gaia" /> This angle is approximately that [[subtended]] by an object 2 centimeters in diameter located 5.3 kilometers away.

=== Derivation ===
For a [[right triangle]],
: <math>\tan p = \frac {1\,\text{au}} {d} ,</math>
where <math>p</math> is the parallax, {{convert|1|au|km | abbr=on | sigfig=4 }} is approximately the average distance from the Sun to Earth, and <math>d</math> is the distance to the star.
Using [[small-angle approximation]]s (valid when the angle is small compared to 1 [[radian]]),
: <math>\tan x \approx x\text{ radians} = x \cdot \frac {180} {\pi} \text{ degrees} = x \cdot 180  \cdot \frac {3600} {\pi} \text{ arcseconds} ,</math>
so the parallax, measured in arcseconds, is
:<math>p'' \approx \frac {1 \text{ au}} {d} \cdot 180 \cdot \frac{3600} {\pi} .</math>
If the parallax is 1", then the distance is
:<math>d = 1 \text{ au}  \cdot 180 \cdot \frac {3600} {\pi} \approx 206,265 \text{ au} \approx 3.2616 \text{ ly} \equiv 1 \text{ parsec} .</math>
This ''defines'' the [[parsec]], a convenient unit for measuring distance using parallax. Therefore, the distance, measured in parsecs, is simply <math>d = 1 / p</math>, when the parallax is given in arcseconds.<ref>Similar derivations are in most astronomy textbooks. See, e.g., {{harvnb|Zeilik|Gregory|1998|loc=§ 11-1}}.</ref>

===Error===
Precise parallax measurements of distance have an associated error. This error in the measured parallax angle does not translate directly into an error for the distance, except for relatively small errors. The reason for this is that an error toward a smaller angle results in a greater error in distance than an error toward a larger angle.

However, an approximation of the distance error can be computed by
:<math>\delta d = \delta \left( {1 \over p} \right) =\left| {\partial \over \partial p} \left( {1 \over p} \right) \right| \delta p ={\delta p \over p^2}</math>
where ''d'' is the distance and ''p'' is the parallax. The approximation is far more accurate for parallax errors that are small relative to the parallax than for relatively large errors. For meaningful results in [[stellar astronomy]], Dutch astronomer Floor van Leeuwen recommends that the parallax error be no more than 10% of the total parallax when computing this error estimate.<ref name=van_leeuwen2007>{{cite book | first1=Floor | last1=van Leeuwen | title=Hipparcos, the new reduction of the raw data | volume=350 | series=Astrophysics and space science library | publisher=Springer | date=2007 | isbn=978-1-4020-6341-1 | page=86 | url=https://books.google.com/books?id=oU72XEf_5lEC&pg=PA86 | url-status=live | archive-url=https://web.archive.org/web/20150318002121/http://books.google.com/books?id=oU72XEf_5lEC&pg=PA86 | archive-date=2015-03-18 }}</ref>