Parsec

The word parsec is a shortened form of a distance corresponding to a parallax of one arcsecond, coined by the British astronomer Herbert Hall Turner in 1913. The unit was introduced to simplify the calculation of astronomical distances from raw observational data. Partly for this reason, it is the unit preferred in astronomy and astrophysics, though in popular science texts and common usage the light-year remains prominent. Although parsecs are used for the shorter distances within the Milky Way, multiples of parsecs are required for the larger scales in the universe, including kiloparsecs (kpc) for the more distant objects within and around the Milky Way, megaparsecs (Mpc) for mid-distance galaxies, and gigaparsecs (Gpc) for many quasars and the most distant galaxies.

In August 2015, the International Astronomical Union (IAU) passed Resolution B2 which, as part of the definition of a standardized absolute and apparent bolometric magnitude scale, mentioned an existing explicit definition of the parsec as exactly ⁠648000/π⁠ au, or approximately 30856775814913673 metres, given the IAU 2012 exact definition of the astronomical unit in metres. This corresponds to the small-angle definition of the parsec found in many astronomical references.

History and derivation

Imagining an elongated right triangle in space, where the shorter leg measures one au (astronomical unit, the average Earth–Sun distance) and the subtended angle of the vertex opposite that leg measures one arcsecond (1⁄3600 of a degree), the parsec is defined as the length of the adjacent leg. The value of a parsec can be derived through the rules of trigonometry. It is 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. 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. Then the distance to the star could be calculated using trigonometry. 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 Cygni.

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 reciprocal of the parallax angle in arcseconds (i.e.: if the parallax angle is 1 arcsecond, the object is 1 pc from the Sun; if the parallax angle is 0.5 arcseconds, the object is 2 pc away; etc.). No trigonometric functions 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. It was Turner's proposal that stuck.

Calculating the value of a parsec

By the 2015 definition, 1 au of arc length subtends an angle of 1″ at the center of the circle of radius 1 pc. That is, ${\displaystyle 1\text{ pc}=\frac{1\text{ au}}{\text{tan}(1\text{ arcsecond})}\approx 206,264.8\text{ au}}$ by definition

Converting from degree/minute/second units to radians,

${\displaystyle \frac{1\text{ pc}}{1\text{ au}}=\frac{180\times 60\times 60}{\pi }}$, and

${\displaystyle 1\text{ au}=149597870700\text{ m}}$ (exact by the 2012 definition of the au)

Therefore, $${\displaystyle \pi \mathrm{p}\mathrm{c}=180\times 60\times 60\mathrm{a}\mathrm{u}=180\times 60\times 60\times 149597870700\mathrm{m}=96939420213600000\mathrm{m}}$$ (exact by the 2015 definition)

Therefore, $${\displaystyle 1\mathrm{p}\mathrm{c}=\frac{96939420213600000}{\pi }\mathrm{m}=30856775814913673\mathrm{m}}$$ (to the nearest metre).

Approximately,

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). Thus the distance ES is one astronomical unit (au). The angle SDE is one arcsecond (⁠1/3600⁠ of a 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: $${\displaystyle \begin{array}{cc}\mathrm{S}\mathrm{D}& =\frac{\mathrm{E}\mathrm{S}}{\tan 1^{″}}\\ & =\frac{\mathrm{E}\mathrm{S}}{\tan \left(\frac{1}{60\times 60}\times \frac{\pi }{180}\right)}\\ & \approx \frac{1\mathrm{a}\mathrm{u}}{\frac{1}{60\times 60}\times \frac{\pi }{180}}=\frac{648000}{\pi }\mathrm{a}\mathrm{u}\approx 206264.81\mathrm{a}\mathrm{u}.\end{array}}$$

Because the astronomical unit is defined to be 149597870700 m, the following can be calculated:

Therefore, if 1 ly ≈ 9.46×10¹⁵ m,

Then 1 pc ≈ 3.261563777 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: $${\displaystyle {\text{Distance}}_{\text{star}}=\frac{{\text{Distance}}_{\text{earth-sun}}}{\tan \frac{\theta }{3600}}}$$ where θ is the measured angle in arcseconds, Distance_earth-sun is a constant (1 au or 1.5813×10⁻⁵ ly). The calculated stellar distance will be in the same measurement unit as used in Distance_earth-sun (e.g. if Distance_earth-sun = 1 au, unit for Distance_star is in astronomical units; if Distance_earth-sun = 1.5813×10⁻⁵ ly, unit for Distance_star is in light-years).

The length of the parsec used in IAU 2015 Resolution B2 (exactly ⁠648000/π⁠ astronomical units) corresponds exactly to that derived using the small-angle calculation. This differs from the classic inverse-tangent definition by about 200 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 30856775814913673 m.