How do you get the rest of a rocket (which contains the astronauts) back to earth?

Answer

Of course there are formulas for that, which take into account the effect of gravity, the effects of the density of the atmosphere, the shape of the spacecraft and where exactly the center of gravity of the spacecraft is located…
Spacecraft, be it a shuttle or a Soyuz, are in relatively low orbit around the Earth. When returning, the spacecraft is directed forward with its engines and decelerated, at a precisely calculated moment, with a precisely calculated force, orientation and duration. This puts the spacecraft in a more elliptical orbit, causing it to lose altitude. For the rest, they actually let the atmosphere do the rest. The normal orbital speed of typically just under 8 km/s is slowed down by atmospheric friction, hence the need for a heat shield. Single-use heat shields are ablative, they are simply allowed to evaporate for about half their thickness, dissipating the heat.
With a Soyuz, the center of gravity is not exactly in the center of the spacecraft, so that adjustments can be made to a limited extent. To give you an idea, a Soyuz that wants to land in Russia is approaching from the south of Africa, where it slows down for the first time. He then flies almost another quarter of the Earth to land. The accuracy with which one lands is in the order of a few kilometers if everything goes normally. With a Shuttle, the accuracy is greater because a shuttle is controlled as a glider, which gives more flexibility.