Does anyone know an equation or something along those lines for calculating how many and the locations of atmospheric circulation cells for a planet? Also, does the planet's radius and atmospheric content affect it? This is my first question on here so I am open to constructive criticisms!
Firstly, you would need to define 'circulation cells'. If you're talking about, say, low pressure surface systems, these are considered 'synoptic scale', and tend to be on the order of 1000km+ of horizontal distance. These surface lows circle Antarctica, for instance, riding the upper atmospheric river of air known as the jet stream. As wind exits troughs within this jet stream, this creates upper air divergence (a vacuum). This pulls air from the surface, thus creating low pressure at the surface. This is the primary driving engine for middle latitude and sub-polar weather in the southern hemisphere.
Moreover, from a planetary perspective for earth, these synoptic scale systems (circulation cells) are driven from a planetary scale by cells - the Hadley Cells, Mid-latitude or Ferrell Cells, and Polar Cells. Please see below a diagram. This is what creates high pressure at the poles and in a band at roughly 30 N & S latitudes (think where you find the majority of earth's non-orographic deserts). This also drives a global equatorial band of rain called the Intertropical Convergence Zone (ITCZ). This band of rain basically bounces between the two high pressure belts around the equator seasonally.
Included is a diagram of the planetary cells. I've also included a picture of the Antarctic low pressure systems circling the continent (Antarctica is actually the largest desert on earth!). Coriolis (spin of the earth) also plays a role in determining circulation of high and low pressure systems by hemisphere.
So I suppose you would have to start by: (a) determine the thermal driving mechanisms for your planet (for earth it's the heat differential between the poles and equator); (b) determine the behavior of the upper atmospheric flows (the atmosphere behaves very much like a fluid); (c) come up with a basic weather modeling scheme from this (a VERY complex and difficult process). I hope this helps.