Build a small orbital model one step at a time

Imagine a tiny test object moving in a plane around one fixed central mass. Ignore other bodies, thrust, drag and relativity. Choose normalized units so the central gravitational parameter, conventionally called μ, equals 1. These assumptions define a teaching model, not a navigation solution for a real mission.

If the position is the vector r = (x, y), with distance r from the origin, gravitational acceleration is −r/r³ in these units. The minus sign points toward the origin. At position (1, 0), acceleration is (−1, 0). A circular orbit at radius 1 has speed 1 in this ideal model.

Use a simple velocity-first rule: calculate acceleration from the current position, update velocity by adding acceleration times the step duration, then update position using that new velocity. This is one discrete approximation to continuous motion. Changing the order produces a different numerical method.

Start at position (1, 0) with velocity (0, 1) and step duration 0.1. The new velocity is (−0.1, 1). Using that velocity, the new position is (0.99, 0.1). All values use the normalized units. This one-step trace checks the algorithm — open definition; it does not establish long-term accuracy.

One velocity-first step: normalized units

The ideal continuous circular solution keeps radius and speed constant. A numerical path should be checked against that known behavior over a specified interval and with different step sizes. An orbit-shaped drawing alone is not enough to validate the calculation.

If the path approaches the origin, this point-mass formula becomes singular at zero distance. The model needs a defined domain and stopping rule rather than dividing by zero or pretending a real planet has no surface. The lesson’s value is making assumptions and state updates inspectable before moving to more complete models.