# Converting parsecs to light years in the dumbest way possible

I have to convert 132 parsecs to light years and I'm struggling to get the expected result. Here is my method:

$$132 \text{ pc} = 132'' = 2,2' = 0,03(6)°$$

I have an equation for stellar parallax which looks like this: $$d = \Big( \dfrac{1 \text{ AU}}{\tan(0,03(6)°)}\Big)$$

This should give me the distance to the object in Astronomical Units right? Well It doesn't and I have absolutely no idea why, I'd be very glad if someone could help me out

• A distance of 1 pc has a parallax angle of 1" so angle made by something 132 pc away is 132 times smaller: 1/132=0.00757" Mar 29 at 17:45
• Yepp this seems to be the issue... thank you very much Mar 29 at 17:55
• A parsec is a distance, and so is a lightyear. There’s no need to involve trigonometry.
– pela
Mar 29 at 22:10
• Ikr? I just love making life hell for myself Mar 30 at 7:26
• @stanisławsłowiński it's always okay to post an answer to your own question. But as comments point out, the distance is 132 times farther by definition, but that doesn't mean that the parallax would be exactly 132 times smaller. It will be close, but that would not be the right way to do it.
– uhoh
Mar 30 at 9:16

## You asked for a dumb, tedious way of doing it. So here it is, in all its glory

One parsec is defined as $$\dfrac{1}{\tan(1'')} \text{ AU}=\dfrac{648000}{\pi} \text{ AU}$$

As $$1 \text{ AU} = 149597870700 \text{ m}$$, one parsec is equal to $$\dfrac{648000}{\pi} \cdot 149\,597\,870\,700 = \dfrac{96\,939\,420\,213\,600\,000}{\pi} \text{ m}$$

One light year can be calculated by multiplying $$c$$ by the number of seconds in a year: $$299792458 \cdot 86400 \cdot 365.25 = 9\,460\,730\,472\,580\,800 \text{ m}$$

Dividing the former by the latter, we get $$\dfrac{1 \text{ pc}}{1 \text{ ly}} = 3.261564$$

From Google I get: A parsec, or “parallax second,” is defined as 3.26 light-years. So using the definition of parsec you can easily determine the distance in light years.

• Yeah I know, someone smart once said that if you can do something in only one way, you dont understand it well enough (or something along these lines). So here I am converting parsecs to angles :)) Mar 30 at 7:26
• Although it's true that 1 pc = 3.26 lyr, this is not the definition of a parsec. The definition is the distance at which an object, 1 AU across, spans an angle of 1".
– pela
Apr 29 at 8:42