# what is parsec and how is it derived

Parsec is the unit for larger distances .It is the distance at which star would make parallax of one second of arc.

1 parsec=3.26 light years

### Calculating the value of a parsec

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 arc second (1/3600 of a degree) so by definition D is a point in space at a distance of one parsec from the Sun. By trigonometry, the distance SD is

$SD = frac{mathrm{ES}}{tan 1^{primeprime}}$

Using the small-angle approximation, by which the sine (and, hence, the tangent) of an extremely small angle is essentially equal to the angle itself (in radians),

$SD approx frac{mathrm{ES}}{1^{primeprime}} = frac{1 , mbox{AU}}{(tfrac{1}{60 times 60} times tfrac{pi}{180})} = frac{648,000}{pi} , mbox{AU} approx 206,264.81 mbox{ AU} .$

One AU ≈ 149597870700 metres, so 1 parsec ≈ 3.085678×1016 m3.261564 ly.

A corollary is that 1 parsec is also the distance from which a disc with a diameter of 1 AU must be viewed for it to have an angular diameter of 1 arc second (by placing the observer at D and a diameter of the disc on ES).

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