# Understanding the Parsec Sometimes, astronomers feel like doing things differently. They look at the stars, take out their 8-inch Dobsonian telescope and immediately think: "Where will this star be in the future?" Although that's not a terrible question in itself, the pursuit for predictability has led astronomers to come up with a new method to describe stellar distances - the parsec. The parsec is a direct offshoot of the parallax unit, which denotes the **angular movement of stars** as a function of a right triangle with base 1AU. In fact, its name is an abbreviation for **PARallax arcSECond**. Below is a diagram that should (hopefully) help out a bit: ![[Pasted image 20230709172921.png]] As the diagram shows, the Earth's perception of a faraway star will experience angular displacement in the night sky. We can therefore use this ratio in the differences of perception between the two solstices (so half a year, half an Earth orbit) to find the distance! To calculate the distance in parsecs, all we need now is to do some trigonometric identity magic. We can therefore get: $d = \frac{1}{\theta}$ >[!Abstract]- Deriving the Equation >We can let the distance to the star be $d$ and the horizontal axis of the triangle be $1AU$. This creates the relationship: >$\tan \theta = \frac{1}{d}$ >The small angle approximation of $\tan \theta$ is just $\theta$. This means we can rewrite our equation as: >$\theta = \frac{1}{d}$ >Reorganising to isolate the $d$ gives us: >$d = \frac{1}{\theta}$ This means we can now find the value for 1 parsec - plugging in one arcsecond into our equation gives us a distance of around **3.26 light years**. However, it's impossible for to plug in a value for theta in degrees and call it a day. No! There're more units we need to know - the arcminutes and arcseconds. Arc-units are typically written with dashes on top of them - an arcminute is denoted by one apostrophe, $1'$, while an arcsecond is denoted with two dashes, $1''$. An arcminute is 1/60th of a degree, and an arcsecond is denoted as 1/60th of an arcminute. This means that there are 1,296,000 arcseconds in one full circle! To find a value for the distance in parsecs, we must use a value for the parallax angle that is in arcseconds. That means if we get a value in arcminutes or degrees, we're going to have to convert it into arcseconds. Remember this! >[!Danger]- More on Parallax >The equation with the mean fluxes actually gives us the apparent magnitude of a star. As flux is just a non-corrected form (meaning it doesn't take the inverse square law into account). This allows us to create the following relationship for the absolute magnitude: >$M_{1} - M_{2} = -2.5\log_{10}\left( \frac{L_{1}}{L_{2}} \right)$ >For cases without a reference star, we can use the **zero-point luminosity**, the luminosity of a star with an absolute **bolometric** magnitude of 0, meaning it's net electromagnetic flux (in all wavelengths, not just visible) would equal to a brightness of zero on the magnitude scale. This creates the equation: >$M = -2.5\log_{10}\left( \frac{L}{3.0128 \times 10^{28}} \right)$ >Where $3.0128 \times 10^{28}$ is the zero-point luminosity of the star. ## Example Problems >[!Example] # What's Next