Editing Parsec (section)

== History and derivation ==
{{See also|Stellar parallax}}
Imagining an elongated [[right triangle]] in space, where the shorter leg measures one au ([[astronomical unit]], the average [[Earth]]–[[Sun]] distance) and the [[subtended|subtended angle]] of the vertex opposite that leg measures one [[arcsecond]] ({{frac|3600}} of a degree), the parsec is defined as the length of the [[Trigonometry#Trigonometric_ratios|''adjacent'']] leg. The value of a parsec can be derived through the rules of [[trigonometry]]. The distance from Earth whereupon the radius of its solar orbit subtends one arcsecond.

One of the oldest methods used by astronomers to calculate the distance to a [[star]] is to record the difference in angle between two measurements of the position of the star in the sky. The first measurement is taken from the Earth on one side of the Sun, and the second is taken approximately half a year later, when the Earth is on the opposite side of the Sun.{{efn|name=orbit|Terrestrial observations of a star's position should be taken when the Earth is at the furthest points in its orbit from a line between the Sun and the star, in order to form a right angle at the Sun and a full au of separation as viewed from the star.}} The distance between the two positions of the Earth when the two measurements were taken is twice the distance between the Earth and the Sun. The difference in angle between the two measurements is twice the parallax angle, which is formed by lines from the Sun and Earth to the star at the distant [[Vertex (geometry)#Of an angle|vertex]]. Then the distance to the star could be calculated using trigonometry.<ref name="NASAparallax">{{Cite web |title=Deriving the Parallax Formula |url=http://imagine.gsfc.nasa.gov/YBA/HTCas-size/parallax1-derive.html |last=[[High Energy Astrophysics Science Archive Research Center]] (HEASARC) |website=NASA's Imagine the Universe! |publisher=Astrophysics Science Division (ASD) at [[NASA]]'s [[Goddard Space Flight Center]] |access-date=26 November 2011}}</ref> The first successful published direct measurements of an object at interstellar distances were undertaken by German astronomer [[Friedrich Wilhelm Bessel]] in 1838, who used this approach to calculate the 3.5-parsec distance of [[61&nbsp;Cygni]].<ref>{{Cite journal |last=Bessel |first=F.&nbsp;W. |author-link=Friedrich Wilhelm Bessel |date=1838 |title=Bestimmung der Entfernung des 61sten Sterns des Schwans |trans-title=Determination of the distance of the 61st star of Cygnus |url=https://zenodo.org/record/1424605 |url-status= |journal=[[Astronomische Nachrichten]] |volume=16 |issue=5 |pages=65–96 |bibcode=1838AN.....16...65B |doi=10.1002/asna.18390160502 |archive-url= |archive-date=}}</ref>

[[Image:ParallaxV2.svg|thumb|left|upright=1.36|Stellar parallax motion from annual parallax|alt=Diagrams illustrating the apparent change in position of a celestial object when viewed from different positions in Earth's orbit.]]
The parallax of a star is defined as half of the [[angular distance]] that a star appears to move relative to the [[celestial sphere]] as Earth orbits the Sun. Equivalently, it is the subtended angle, from that star's perspective, of the [[semimajor axis]] of the Earth's orbit. Substituting the star's parallax for the one arcsecond angle in the imaginary right triangle, the long leg of the triangle will measure the distance from the Sun to the star. A parsec can be defined as the length of the right triangle side adjacent to the vertex occupied by a star whose parallax angle is one arcsecond.

The use of the parsec as a unit of distance follows naturally from Bessel's method, because the distance in parsecs can be computed simply as the [[multiplicative inverse|reciprocal]] of the parallax angle in arcseconds (i.e.: if the parallax angle is 1&nbsp;arcsecond, the object is 1&nbsp;pc from the Sun; if the parallax angle is 0.5&nbsp;arcseconds, the object is 2&nbsp;pc away; etc.). No [[trigonometric function]]s are required in this relationship because the very small angles involved mean that the approximate solution of the [[skinny triangle]] can be applied.

Though it may have been used before, the term ''parsec'' was first mentioned in an astronomical publication in 1913. [[Astronomer Royal]] [[Frank Watson Dyson]] expressed his concern for the need of a name for that unit of distance. He proposed the name ''astron'', but mentioned that [[Carl Charlier]] had suggested ''[[siriometer]]'' and [[Herbert Hall Turner]] had proposed ''parsec''.<ref name=dyson /> It was Turner's proposal that stuck.

=== Calculating the value of a parsec ===

By the 2015 definition, {{Val|1|u=au}} of arc length subtends an angle of {{Val|1|u=arcsecond}} at the center of the circle of radius {{Val|1|u=pc}}. That is, 1 pc = 1 au/tan({{Val|1|u=arcsecond}}) ≈ 206,264.8 au by definition.<ref>{{cite journal|author=B. Luque|author2=F. J. Ballesteros| title=Title: To the Sun and beyond| date=2019|doi=10.1038/s41567-019-0685-3| journal=[[Nature Physics]]| volume=15|issue=12 | pages=1302|bibcode=2019NatPh..15.1302L |doi-access=free}}</ref> Converting from degree/minute/second units to [[radians]],

:<math>\frac{1 \text{ pc}}{1 \text{ au}} = \frac{180 \times 60 \times 60}{\pi}</math>, and
:<math>1 \text{ au} = 149\,597\,870\,700 \text{ m} </math> (exact by the 2012 definition of the au)

Therefore,
<math display="block">\pi ~ \mathrm{pc} = 180 \times 60 \times 60 ~ \mathrm{au} = 180 \times 60 \times 60 \times 149\,597\,870\,700 ~ \mathrm{m} = 96\,939\,420\,213\,600\,000 ~ \mathrm{m}</math> (exact by the 2015 definition)

Therefore,

<math display=block>1 ~ \mathrm{pc} = \frac{96\,939\,420\,213\,600\,000}{\pi} ~ \mathrm{m} = 30\,856\,775\,814\,913\,673 ~ \mathrm{m}</math> (to the nearest [[metre]]).

Approximately,

:[[Image:Parsec (1).svg|400px|Diagram of parsec.]]

In the diagram above (not to scale), '''S''' represents the Sun, and '''E''' the Earth at one point in its orbit (such as to form a right angle at '''S'''{{efn|name=orbit}}). Thus the distance '''ES''' is one astronomical unit (au). The angle '''SDE''' is one arcsecond ({{sfrac|3600}} of a [[degree (angle)|degree]]) so by definition '''D''' is a point in space at a distance of one parsec from the Sun. Through trigonometry, the distance '''SD''' is calculated as follows:

<math display=block>
\begin{align}
\mathrm{SD} &= \frac{\mathrm{ES} }{\tan 1''} \\
&= \frac{\mathrm{ES}}{\tan \left (\frac{1}{60 \times 60} \times \frac{\pi}{180} \right )} \\
&  \approx \frac{1 \, \mathrm{au} }{\frac{1}{60 \times 60} \times \frac{\pi}{180}} = \frac{648\,000}{\pi} \, \mathrm{au} \approx 206\,264.81 ~ \mathrm{au}.
\end{align}
</math>

Because the astronomical unit is defined to be {{Val|149597870700|ul=m}},<ref>{{Citation |title=Resolution B2 |date=31 August 2012 |contribution=Resolution B2 on the re-definition of the astronomical unit of length |contribution-url=http://www.iau.org/static/resolutions/IAU2012_English.pdf |place=Beijing |publisher=[[International Astronomical Union]] |quote=The XXVIII General Assembly of the International Astronomical Union recommends [adopted] that the astronomical unit be redefined to be a conventional unit of length equal to exactly {{Val|149597870700|u=m}}, in agreement with the value adopted in IAU 2009 Resolution B2}}</ref> the following can be calculated:

{| style="margin-left:1em"
|-
|rowspan=5 valign=top|Therefore, 1 parsec
|≈ {{Val|206264.806247096}} astronomical units
|-
|≈ {{Val|3.085677581|e=16}} metres
|-
|≈ {{Val|30.856775815}}&nbsp;trillion [[kilometre]]s
|-
|≈ {{Val|19.173511577}}&nbsp;trillion [[mile]]s
|}

Therefore, if {{Val|1|ul=ly}} ≈ {{Convert|1|ly|m|disp=out|sigfig=3}}, 
: Then {{Val|1|u=pc}} ≈ {{Val|3.261563777|u=ly}}

A corollary states that a parsec is also the distance from which a disc that is one au in diameter must be viewed for it to have an [[angular diameter]] of one arcsecond (by placing the observer at '''D''' and a disc spanning '''ES''').

Mathematically, to calculate distance, given obtained angular measurements from instruments in arcseconds, the formula would be:

<math display="block">\text{Distance}_\text{star} = \frac {\text{Distance}_\text{earth-sun}}{\tan{\frac{\theta}{3600}}}</math>

where ''θ'' is the measured angle in arcseconds, Distance<sub>earth-sun</sub> is a constant ({{Val|1|u=au}} or {{Convert|1|au|ly|disp=out|sigfig=5}}). The calculated stellar distance will be in the same measurement unit as used in Distance<sub>earth-sun</sub> (e.g. if Distance<sub>earth-sun</sub> = {{Val|1|u=au}}, unit for Distance<sub>star</sub> is in astronomical units; if Distance<sub>earth-sun</sub> = {{Convert|1|au|ly|disp=out|sigfig=5}}, unit for Distance<sub>star</sub> is in light-years).

The length of the parsec used in [[IAU]] 2015 Resolution B2<ref>{{Citation |title=Resolution B2 |date=13 August 2015 |contribution=Resolution B2 on recommended zero points for the absolute and apparent bolometric magnitude scales |contribution-url=http://www.iau.org/static/resolutions/IAU2015_English.pdf |place=Honolulu |publisher=[[International Astronomical Union]] |quote=The XXIX General Assembly of the International Astronomical Union notes [4] that the parsec is defined as exactly (648 000/<math>\pi</math>) au per the AU definition in IAU 2012 Resolution B2}}</ref> (exactly {{sfrac|{{Val|648000}}|{{pi}}}} astronomical units) corresponds exactly to that derived using the small-angle calculation. This differs from the classic inverse-[[tangent]] definition by about {{Val|200|u=km}}, i.e.: only after the 11th [[significant figure]]. As the astronomical unit was defined by the IAU (2012) as an exact length in metres, so now the parsec corresponds to an exact length in metres. To the nearest metre, the small-angle parsec corresponds to {{Val|30856775814913673|u=m}}.