Question

How was it determined that a parsec is 3.26 light years?

Answer 1

A parsec is defined to be the distance to an object that shows a parallax angle of `1 " arc-second"`. That distance happens to be `3.26 " light years"`.

Explanation:

Parallax is an effective way to measure distance to nearby stars because it relies on geometry. When astronomers use parallax, they are measuring how a star appears to move against its background. The unit parsec refers to the distance that an object would have to be from the Earth to have a parallax angle of 1 arc-second.

In the figure above, the angle, `alpha` is the angle that is measured by the Earth on opposite sides of the sun. The parallax angle, `p` is half of this angle.

`p=1/2 alpha`

If we define `p` to be `1 " arc-second"`, then our object will be 1 parsec away. Since light from the sun takes 8 minutes and 20 seconds to reach the Earth, we know that;

`1 " AU" = 8.33 " light minutes"`

We can use this information to convert our parsec into light years with the tangent formula.

`tan(p) = (8.33 " light minutes")/d`

Or;

`d = (8.33 " light minutes")/tan(p)`

If we convert `p` to radians, than we can use the small angle approximation, `tan(theta) ~~ theta`.

`1" arc-second" = 4.85 xx 10^-6 " radians"`

Plugging this in for `p` and using the small angle approximation;

`d = (8.33 " light minutes")/(4.85 xx 10^-6)`

`= 1.72 xx 10^6 " light-minutes"`

`= 3.26 " light years"`