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 ( 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. 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. Converting from degree/minute/second units to
radians, : \frac{1 \text{ pc}}{1 \text{ au}} = \frac{180 \times 60 \times 60}{\pi}, and : 1 \text{ au} = 149\,597\,870\,700 \text{ m} (exact by the 2012 definition of the au) Therefore, \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} (exact by the 2015 definition) Therefore, 1 ~ \mathrm{pc} = \frac{96\,939\,420\,213\,600\,000}{\pi} ~ \mathrm{m} = 30\,856\,775\,814\,913\,673 ~ \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 ( 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{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} 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 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: \text{Distance}_\text{star} = \frac {\text{Distance}_\text{earth-sun}}{\tan{\frac{\theta}{3600}}} where
θ is the measured angle in arcseconds, Distanceearth-sun is a constant ( or ). The calculated stellar distance will be in the same measurement unit as used in Distanceearth-sun (e.g. if Distanceearth-sun = , unit for Distancestar is in astronomical units; if Distanceearth-sun = , unit for Distancestar 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 length in metres, so now the parsec corresponds to an exact length in metres. To the nearest metre, the small-angle parsec corresponds to . == Usage and measurement ==