Predict orbital changes with radius ratios

Predict a ratio before reading a number

Set the selected radius to 2 Earth radii so it matches the reference. Both objects begin at the same direction and share the same elapsed time. Now predict the effect of changing the selected radius to 4R. Will its period double because the circumference doubles, or does the speed change too?

The larger circle has twice the circumference and a smaller circular speed. Its period therefore grows by more than a factor of two. The model lets you compare these effects separately rather than memorizing “higher means slower” without identifying which quantity is slower.

The physical relationships are v = √(μ/r) and T = 2π√(r³/μ). Here r is orbital radius from Earth’s center, v is speed, T is period and μ is Earth’s gravitational parameter. The representative Earth values come from the NASA Earth fact sheet. The following comparisons are deductions within the specified circular model.

Cancel the shared constants

When two orbits share the same central body, μ cancels from a ratio. If the second radius is k times the first, speed changes by 1/√k, period by k raised to the power 3/2, and gravitational acceleration by 1/k².

The exponent 3/2 means “take the cube, then its square root,” or equivalently “multiply k by √k.” For k = 2, the period factor is √8, about 2.8284. The speed factor is about 0.7071 and the gravity factor is 0.25. Each ratio answers a different question, so label them instead of writing an unexplained factor next to the picture.

At 2R the speed is about 5.5931 km/s and the period about 238.5703 minutes. At 4R the speed is about 3.9549 km/s and the period about 674.7787 minutes. Their ratios agree with the predictions. This agreement checks the calculation’s consistency; the two model outputs are not independent satellite observations.

Recover the period–radius relationship

Squaring T = 2π√(r³/μ) gives T² = (4π²/μ)r³. Thus T²/r³ stays constant for these circular orbits around the same modeled Earth. This is the circular version of the period scaling associated with Kepler’s third law. The broader two-body relationship uses semimajor axis for an ellipse; this lab does not solve an elliptical trajectory.

Create a paper table with radius multipliers 1, 2 and 4 relative to a chosen starting radius. Fill in predicted speed factors 1, 1/√2 and 1/2; period factors 1, √8 and 8; and acceleration factors 1, 1/4 and 1/16. Compare the relevant selectable cases afterward. Ratios allow useful predictions even when the absolute period is unfamiliar.

Keep radius and altitude distinct

Changing the control from 2R to 4R doubles radius, but altitude changes from R to 3R. That is a tripling of altitude. If someone says “doubling altitude gives a period factor √8,” ask them to convert both altitudes to center-based radii first.

There is a second interpretive trap: a higher circular orbit has lower speed, but this comparison does not prescribe how to move a spacecraft between circles. A maneuver changes energy and the trajectory during the transfer. NASA discusses the distinction between orbital motion and changes to an orbit in its gravity and mechanics chapter. Moving this slider selects independent circular solutions; it is not a thrust simulation.

At the same elapsed time, the larger circle also accumulates less angle because its period is longer. Compare total angles, not only wrapped orientations, when one object may have completed additional revolutions.

Transfer by solving backward

An ideal circular orbit around the same central body has a period eight times that of a reference circle. What is its radius ratio, speed ratio and gravity ratio? State the assumption needed to cancel μ.

Check the kepler-scaling transfer

k raised to 3/2 equals 8, so k = 4. Speed is one half and gravity one sixteenth of the reference values. Both comparisons must use the same central gravitational parameter and the stated negligible-orbiting-mass circular model.