Orbital elements: six numbers and a date
An orbit around the Earth is an ellipse. Six numbers fix its size, shape and orientation: the mean motion (how many revolutions per day), the eccentricity (how stretched the ellipse is), the inclination (the tilt against the equator), the right ascension of the ascending node and the argument of perigee (how the ellipse is rotated) and the mean anomaly (where the satellite is on the ellipse). Add the epoch, the moment those values were measured, and a drag term, and you can predict where the satellite will be.
They are published as TLE (the classic two-line format) or as OMM (the same data in JSON or XML, which no longer runs out of digits for new catalog numbers). This site uses the OMM form distributed by CelesTrak from the data of the US Space Force.
SGP4: the model that turns elements into positions
The elements are not a perfect Kepler ellipse: they are mean elements, fitted to be used together with a specific model called SGP4 (SDP4 for orbits with a period over 225 minutes, such as GPS or geostationary satellites). It includes the bulge of the Earth, which makes orbits slowly rotate, and an estimate of atmospheric drag. The result is a position and velocity in an inertial frame, which is then rotated by the angle of the Earth at that instant to get latitude, longitude and altitude.
You must use SGP4 with elements made for SGP4. Feeding them to a plain Kepler formula gives errors of tens of kilometres.
Why predictions drift
The atmosphere is thin at 400 km but not empty, and its density changes with solar activity, so the real drag differs from the model. The error grows with the age of the elements: a few kilometres right after the epoch, more after days, and faster for low orbits. That is why trackers refresh the elements often and why this site hides satellites whose elements are more than 30 days old. Satellites that manoeuvre, like the ISS when it boosts, jump away from the prediction until new elements are published.
From position to what you see in the sky
Given your latitude, longitude and height, the position of the satellite becomes an azimuth (compass direction) and an elevation (angle above the horizon), plus the distance. The rate at which that distance changes gives the Doppler shift. Everything on the live radar is built from these steps. To use it, read how to read a satellite pass.
