Question

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

Answer 1

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.

$$ p=\frac{1}{2}\alpha$$

If we defines $$ p $$ 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.

$$ tan(p)=\frac{8.33 \quad light \quad minutes}{d}$$

Or;

$$ d=\frac{8.33 \quad light\quad minutes}{tan(p)}$$

If we convert $$ p $$ to radians, than we can use the small angle approximation, $$ tan(\theta)\approx \theta$$.

$$ 1 arc-second=4.85 \times 10^{-6} $$ radians

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

$$ d=\frac{8.33 \quad light\quad minutes}{4.85\times 10^{-6}}$$

=$$ 1.72 \times 10^6 $$ light-minutes

=$$ 3.26 $$ light years