Editing Parsec (section)

=== 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}}.