Pi is for planets, and spacecraft, for orbital dynamics and craters. It's 3.14, and it's all about circles.
How does Cassini peek at the poles of Saturn without burning ridiculous amounts of fuel? By using the gravity of Titan to get a boost and redirect! This technique is the pi transfer, where Cassini swings past Titan once, then again pi-radians (half an orbit) later.
Moving beyond pi-transfers, orbital dynamics in general is a delicate dance of quick burns at key points to get the most change for the least fuel. Cassini has been delicately dancing around Saturn and its moons for years, with endless careful calculations involving the irrational pi pointing out exactly where to burn some of that precious, limited fuel to get a new angle or flyby in Cassini's continuing explorations.
Expanding that out, pi is used to calculate just about everything fuel-related. How much fuel is in the tanks? Do some cylinder-volume calculations with pi. Need to figure out how much sloshing will take place during particular manoeuvres? More uses for pi. And how that sloshing will impact spacecraft performance? Pi.
Moving on to the gas giant next door, Jupiter's moon Europa teased scientists for decades with its mysteriously cracked surface. It wasn't until a group of researchers in Arizona started applying just how tidal forces, mass distribution, and precession would impact the water-and-ice moon that things started lining up. And yes, every part of that math relies on calculating circles with pi.
I have an enduring fondness for the impact-calculator in determining just how nasty it would be if a near-miss actually hit us. That calculator works by a combination of yield scaling, pi-scaling, Gault's semi-empirical relationship, and a handful of rules about gravity and angle of impact to determine just how big a crater would be. The impact-calculator is the prettiest implementation of this math, but the technique has been floating around in academic circles for a while.
Yes, doing this same style of calculations in reverse plays into how all that crowd-sourced citizen-science crater-identification data is translated into determining what our early solar system looked like with impactors whizzing around everywhere. (Or, at least, whizzing around until they smashed into a nice, dust-puffy moon...)
Once a star reveals the secret of its planet, the first bit of calculating we do is to compare the planet's radius (as determined by the change in brightness) and mass (as determined by orbital wobbles). Once we've got those two core numbers, pi comes into play to convert that radius to a volume. From there, it's simple division to extract density, and suddenly we know if we're looking at a gas giant or a rocky planet. Oooooh, exoplanetary classification for the win!
Density isn't the only clue pi gives us into alien worlds. Pi lets us calculate the surface area, daily rotation, and temperature gradients. That can be plugged into atmospheric models, and suddenly we're coming up with weather reports for world's we've never seen!
Exometeorology: because local weather forecasting was too easy.
The list can keep going and going, just like pi. But to avoid an infinite post, I'll cut it off at an uneven 7. (Why 7? Because if you write your dates as day-month, Pi Day is on July 22nd: 22/7.) Go forth, and spend your day appreciating circles and the irrational majesty of pi.
Cassini's orbital dynamics infographic by the New York Times. Crack propagation animation by Hoppa et al. Exoplanet brightness curve diagram credit CNES. Exoplanet atmospheric system illustration by Noah Stacy. Photographs credit NASA. Need more space-pi? JPL put out an infographic for PI Day, and this series of pi-centric math problems.