Orbital Period Simulator

1. Real‑world phenomenon

Planets, moons, and spacecraft all trace graceful paths around more massive bodies under gravity’s relentless tug. Knowing how long an orbit takes—the orbital period—is essential for satellite communications, mission planning, and understanding celestial mechanics. Traditional calculators plug numbers into Kepler’s third law, but they leave the motion to the imagination. The simulator above preserves that numerical purpose while layering on a responsive HTML5 canvas that shows a satellite swooping around a central mass. As you adjust the radius, central mass, or initial speed, the orbit stretches into an ellipse or contracts into a circle, energy gauges pulse in real time, and a caption narrates the motion for screen‑reader users. The tool transforms Kepler’s nineteenth‑century insight into an interactive laboratory for the twenty‑first century.

2. Variables and assumptions

The system models a light satellite orbiting a massive, stationary body. The satellite’s mass is negligible compared to the central mass $M$, so the center of mass coincides with the heavy body. Initial position is on the positive $x$-axis at radius $r$, and initial velocity is perpendicular to the radius with magnitude $v$₀=fGMr, where $f$ is the “speed multiplier” input. When $f=1$, the orbit is circular; smaller values yield ellipses, and values above √2 allow escape but are clamped here to avoid numerical blow‑up. Gravity follows Newton’s inverse‑square law with constant $G=\mathrm{6.674\times 10^-11}$ m³/(kg·s²). Units remain strictly SI. The satellite motion occurs in a plane, ignoring perturbations from other bodies or relativistic effects.

3. Governing equations

$\frac{d\mathrm{v\_x}}{dt}=-\frac{GMx}{r^3}$

$\frac{d\mathrm{v\_y}}{dt}=-\frac{GMy}{r^3}$

with $\frac{dx}{dt}=\mathrm{v\_x}$ and $\frac{dy}{dt}=\mathrm{v\_y}$.

Kinetic and potential energies are $\mathrm{KE}=\frac{1}{2}{\mathrm{v\_x}}^2+{\mathrm{v\_y}}^2$ and $\mathrm{PE}=-\frac{GM}{r}$.

4. Numerical scheme

This scheme is second‑order accurate but preserves a close analogue of the system’s symplectic structure, keeping energy oscillations bounded rather than drifting. The time‑step input clamps $\mathrm{\Delta t}$ between 0.1 and 100 s; smaller steps better resolve perigee passages, while larger steps accelerate long-term evolution. The energy drift ΔE reported beside the gauges warns when $\mathrm{\Delta t}$ is too large.

5. Worked example

Imagine a satellite at 400 km altitude around Earth. Enter $M=5.972\times 10$²⁴ kg and $r=6.771\times 10$⁶ m. With $f=1$ the calculator reports an orbital period of about 5,542 s (92 min). Press Play and the dot glides in a nearly circular path; the energy gauges remain steady, demonstrating conservation. If you reduce the speed multiplier to 0.8, the satellite lacks sufficient tangential velocity, falling into an elongated ellipse with a lower apogee and a shorter period, while the potential-energy share climbs near perigee. Increasing $f$ to 1.2 stretches the ellipse outward; the kinetic-energy share spikes at perigee and the period grows. Values above about 1.4 approach escape velocity; the script prevents >1.5 to keep the satellite bound.

6. Comparison table

The table samples three cases using Earth’s mass: a circular low Earth orbit, a slower suborbital path, and a faster elongated orbit. Periods come from the simulation data.

Scenario	Speed Multiplier	Semimajor Axis (m)	Period (s)
Circular LEO	1.0	6.771×10⁶	5,542
Suborbital	0.8	5.7×10⁶	4,400
Elongated	1.2	8.2×10⁶	7,320

Reducing initial speed shortens the period and lowers the apogee, while boosting speed extends both. Such comparisons show how kinetic energy trades for gravitational potential in shaping orbital size and timing.

7. How to read the animation

The canvas centers the massive body as a filled disk. The orbiting satellite appears as a smaller dot leaving a blue trail. The axes are not drawn, but the bottom of the canvas corresponds to negative $y$. Keyboard users can focus the canvas and press the space bar to toggle play and pause. The share readouts illustrate how kinetic and potential energy trade dominance; when the potential percentage outweighs the kinetic portion the orbit is safely bound, and similar percentages hint at a near-parabolic escape. The caption narrates time, altitude, and energy so the experience remains accessible without color vision or even sight.

8. Limitations

The model neglects atmospheric drag, nonspherical gravity, and third‑body perturbations. Large time steps or extreme speed multipliers can still introduce numerical drift despite the symplectic integrator. The central mass is fixed, so mutual gravitational attraction in similar‑mass binaries is not represented. Relativistic corrections important near compact objects are ignored. Nonetheless, the simulation captures the essential features of two‑body motion under Newtonian gravity.

9. Suggested extensions

Possible enhancements include adding multiple satellites, displaying velocity vectors, or plotting phase space ($\mathrm{v\_r}$ vs. $r$). Implementing adaptive time stepping would maintain accuracy during perigee while accelerating through apogee. Modeling perturbations like J₂ oblateness or third‑body gravity would extend the tool to mission design scenarios.

10. References and related tools

Further reading can be found in Bate, Mueller, and White’s Fundamentals of Astrodynamics. For related computations, explore the Orbital Velocity Calculator, analyze fuel needs with the Rocket Equation Calculator, or estimate decay with the Satellite Orbit Decay Time Calculator.