parsec

The parsec (symbol: pc) is a unit of length used to measure the large distances to astronomical objects outside the Solar System, approximately equal to or (au), i.e. . The parsec unit is obtained by the use of parallax and trigonometry, and is defined as the distance at which 1 au subtends an angle of one arcsecond ( of a degree). This corresponds to astronomical units, i.e. $1\, \mathrm = 1/\tan \left( \ \mathrm \right)\, \mathrm$. The nearest star, Proxima Centauri, is about from the Sun. Most stars visible to the naked eye are within a few hundred parsecs of the Sun, with the most distant at a few thousand.

The word ''parsec'' is a portmanteau of "parallax of one second" and was coined by the British astronomer Herbert Hall Turner in 1913 to make calculations of astronomical distances from only raw observational data easy for astronomers. Partly for this reason, it is the unit preferred in astronomy and astrophysics, though the light-year remains prominent in popular science texts and common usage. 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  au, or approximately  metres (based on the IAU 2012 exact SI definition of the astronomical unit). This corresponds to the small-angle definition of the parsec found in many astronomical references.

History and derivation

The parsec is defined as being equal to the length of the adjacent leg (opposite leg being 1 AU) of an extremely elongated imaginary right triangle in space. The two dimensions on which this triangle is based are its shorter leg, of length one astronomical unit (the average Earth-Sun distance), and the subtended angle of the vertex opposite that leg, measuring one arcsecond. Applying the rules of trigonometry to these two values, the unit length of the other leg of the triangle (the parsec) can be derived.

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. The star, the Sun and the Earth form the corners of an imaginary right triangle in space: the right angle is the corner at the Sun, and the corner at the star is the parallax angle. The length of the opposite side to the parallax angle is the distance from the Earth to the Sun (defined as one astronomical unit, au), and the length of the adjacent side gives the distance from the sun to the star. Therefore, given a measurement of the parallax angle, along with the rules of trigonometry, the distance from the Sun to the star can be found. A parsec is defined as the length of the 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, of arc length subtends an angle of at the center of the circle of radius . That is, 1 pc = 1 au/tan() ≈ 206,264.8 au by definition. Converting from degree/minute/second units to radians,

:$\frac = \frac$, and
:$1 \text = 149\,597\,870\,700 \text$ (exact by the 2012 definition of the au)

Therefore,

:$\pi ~ \mathrm = 180 \times 60 \times 60 ~ \mathrm = 180 \times 60 \times 60 \times 149\,597\,870\,700 ~ \mathrm = 96\,939\,420\,213\,600\,000 ~ \mathrm$ (exact by the 2015 definition)

Therefore,

$1 ~ \mathrm = \frac ~ \mathrm = 30\,856\,775\,814\,913\,673 ~ \mathrm$ (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. Thus the distance ES is one astronomical unit (au). The angle SDE is one arcsecond ( 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:

$\begin \mathrm &= \frac \\ &= \frac \\ & \approx \frac = \frac \, \mathrm \approx 206\,264.81 ~ \mathrm. \end$

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

Therefore, if ≈ ,
: Then ≈

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

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

$\text_\text = \frac$

where ''θ'' is the measured angle in arcseconds, Distance_earth-sun is a constant ( or ). The calculated stellar distance will be in the same measurement unit as used in Distance_earth-sun (e.g. if Distance_earth-sun = , unit for Distance_star is in astronomical units; if Distance_earth-sun = , unit for Distance_star is in light-years).

The length of the parsec used in IAU 2015 Resolution B2 (exactly astronomical units) corresponds exactly to that derived using the small-angle calculation. This differs from the classic inverse- tangent definition by about , i.e. only after the 11th significant figure. As the astronomical unit was defined by the IAU (2012) as an exact SI length in metres, so now the parsec corresponds to an exact SI length in metres. To the nearest meter, the small-angle parsec corresponds to .

Usage and measurement

The parallax method is the fundamental calibration step for distance determination in astrophysics; however, the accuracy of ground-based telescope measurements of parallax angle is limited to about , and thus to stars no more than distant. This is because the Earth's atmosphere limits the sharpness of a star's image. Space-based telescopes are not limited by this effect and can accurately measure distances to objects beyond the limit of ground-based observations. Between 1989 and 1993, the '' Hipparcos'' satellite, launched by the European Space Agency (ESA), measured parallaxes for about stars with an astrometric precision of about , and obtained accurate measurements for stellar distances of stars up to away.

ESA's ''Gaia'' satellite, which launched on 19 December 2013, is intended to measure one billion stellar distances to within , producing errors of 10% in measurements as far as the Galactic Centre, about away in the constellation of Sagittarius.

Distances in parsecs

Distances less than a parsec

Distances expressed in fractions of a parsec usually involve objects within a single star system. So, for example:
* One astronomical unit (au), the distance from the Sun to the Earth, is just under .
* The most distant space probe, '' Voyager 1'', was from Earth . ''Voyager 1'' took to cover that distance.
* The Oort cloud is estimated to be approximately in diameter

Parsecs and kiloparsecs

Distances expressed in parsecs (pc) include distances between nearby stars, such as those in the same spiral arm

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