Evaluate circular-orbit evidence and limits

Separate a solved model from a measured trajectory

A smooth orbit can look convincing without providing independent evidence. This lab evaluates the circular solution analytically: radius stays fixed, speed is selected for that radius, and angle advances according to the computed period. The path is circular because that solution is built into the calculation, not because observations were supplied and showed a circular orbit.

The physical relationships express gravity and circular motion. Their implementation can still contain mistakes, so numerical consistency checks are useful. They answer whether the program follows its equations. Testing how accurately those equations describe a particular spacecraft requires measurements and a defined prediction task.

NASA’s gravity and mechanics chapter discusses orbital free fall and real effects such as atmospheric drag. This lab deliberately uses a spherical Earth, negligible spacecraft mass and an isolated, drag-free circular orbit. It includes no thrust, third-body perturbation, Earth flattening or elliptical solver.

Audit a worked output

At radius 2R, record radius 12742 km, altitude 6371 km, speed about 5.5931 km/s, period about 238.5703 minutes and acceleration about 2.4551 m/s². Use the reported speed to calculate circumference divided by speed. Remember that distance in kilometers divided by km/s gives seconds, which must be divided by 60 for minutes.

Next compute v²/r using the same kilometer units, then multiply by 1000 to express acceleration in m/s². It should agree with the gravitational acceleration, within rounding. Finally, calculate 360 × 30/238.5703 to predict the angle at 30 minutes. These three checks link distance, speed, acceleration and time instead of testing only one displayed number.

Keep a ledger headed expected relationship, independent arithmetic and displayed result. Here “independent arithmetic” means your separate calculation, not independent physical observations. Rounding a displayed speed can produce a small mismatch even when the underlying calculation is consistent; record your precision before declaring a failure.

Look for conclusions the display cannot support

The reference radius is fixed at 2R, and both modeled objects begin on the positive horizontal axis. The comparison is constructed, not a pair of actual satellites launched together. A radius adjustment selects another circular solution and immediately supplies its corresponding speed. It does not compute the energy, engine timing or intermediate path of a transfer.

The plane is inertial for this idealized calculation, with no rotation of Earth or mapping to geographic coordinates. The diagram therefore cannot predict which city lies below the object, when an observer will see it, or when it crosses a particular longitude. A completed revolution is not a ground-track forecast.

The spherical radius and rounded μ are based on the NASA Earth fact sheet. That page also distinguishes equatorial and polar radius. Choosing a mean-radius sphere makes the exercise tractable; it does not establish that Earth’s real shape has no dynamical consequences.

Design evidence for a bounded real question

Suppose the question is whether an observed object’s motion over a short interval is adequately described by a circular approximation. You would need timestamped position estimates in a stated reference frame, their uncertainties, and a way to estimate orbital parameters without using every observation for both fitting and evaluation.

Write a plan that fits on one page. Reserve later observations before adjusting parameters, predict positions at their timestamps, and compare position residuals against the stated uncertainties. Persistent structured residuals could motivate a less restrictive model, but one residual pattern would not by itself identify drag or a particular external force. Check time standards, units and frame definitions before assigning a physical cause.

No such observations are loaded here. Your plan is proposed evidence collection, not a completed validation exercise. Keep that distinction in the conclusion accompanying your model ledger.

Transfer by repairing a mission claim

A report says: “Changing the radius from 2R to 4R showed that a spacecraft can reach the higher orbit by instantly reducing its speed to 3.95 km/s.” Rewrite the claim and identify the missing calculation.

Check the orbit-evidence transfer

The model compares separate ideal circular orbits: the 4R circle requires about 3.95 km/s. It does not show a transfer from 2R. Evaluating that maneuver requires a trajectory and thrust or impulse calculation, with appropriate initial conditions and forces, rather than an instantaneous radius selection.