# Intro - The Need for Reduction You're out of the orbital energies section - finally! Now it's time for us to embark on the two-body problems. Let's imagine - a far-flung planet with the name ["KELT-9b"](https://exoplanets.nasa.gov/exoplanet-catalog/3508/kelt-9-b/), orbiting a fiery blue B9V star, KELT-9. In fact, it's so close that its pull is forcing the centre of mass of the planetary system away from the central star. This means that both objects are orbiting the centre of mass - much like the Pluto-Charon system itself. Here's a handy diagram! ![[Pasted image 20240105000413.png]] Now suppose we wanted to model how both the star and the planet moved relative to each other. If the star weren't orbiting around the centre of mass, it wouldn't be hard - we'd be able to approximate it as such, but it's not. Unfortunately. Here's where our reduced mass comes in. With the combined masses, there won't be two bodies to have to deal with anymore. Instead, we have the >[!Success]- Reduced Mass - The Generic Proof >Let's get this straight - both planet and star is going to move. I don't think you'll find something 'stationary' in our universe, besides! Moving forward, this means that we can write an equation of motion for the two objects: >$F_{1 \to 2} = F_{2 \to 1}$ >$m_{1} a_{1} = -m_{2}a_{2}$ > >![[Pasted image 20240105015944.png]] > >We're assuming that each of the objects have this mutual, unyielding attractive force between each other. > >Reduced mass is a conversion between a two-body system to a one-body system. To do this, we're going to have to find a lot of relatives - starting from the relative acceleration. >$a_{rel} = a_{1} - a_{2} = \left( 1+\frac{m_{1}}{m_{2}} \right)a_{1} = \frac{m_{2} + m_{1}}{m_{1}m_{2}}a_{1} = \frac{F_{1 \to 2}}{\mu}$ >Where $\mu$ is the relative mass, or our reduced mass. Now rearrange to isolate $\mu$! >$\frac{m_{2} + m_{1}}{m_{1}m_{2}} = \frac{1}{\mu}$ >$\mu = \frac{m_{1}m_{2}}{m_{2} + m_{1}}$ >[!Abstract]- Reduced Mass - The Nuanced Approach # Specific Angular Momentum # Case Study 1: The Radial Velocity Sometimes, we can use this effect to our advantage - # Links https://faculty.fiu.edu/~vanhamme/ast3213/orbits.pdf # What's Next?