Solar system: Kepler's laws
Model the four terrestrial planets (Mercury, Venus, Earth, Mars) orbiting the Sun with their real distances and speeds in the FizziQ Web Orbits and Gravitation simulation.
Activity Summary
The student configures a five-body system in the Orbits and Gravitation simulation: the Sun and the four terrestrial planets (Mercury, Venus, Earth, Mars) with their real masses, distances and orbital speeds. The student runs the simulation and observes the four nested coloured orbits around the Sun. The student measures each planet's orbital period and compares it to astronomical values (88, 225, 365 and 687 days). The student discovers that a more distant planet orbits more slowly and computes T²/r³ ratios to highlight a pattern common to all four orbits.
Introduction
This activity studies the orbital motion of the four terrestrial planets around the Sun using the FizziQ Web Orbits and Gravitation simulation. The measured quantities are the orbital periods of Mercury, Venus, Earth and Mars. Trajectories result from applying Newton's universal law of gravitation to a five-body system.
Learning Objectives
- Enter the real masses, distances and orbital speeds of the four terrestrial planets
- Measure each planet's orbital period and compare to known astronomical values
- Identify the qualitative relationship between distance to the Sun and the duration of one revolution
- Discover a numerical regularity common to all four orbits (T² ÷ r³ ≈ 1 in years and AU)
Scientific Concepts
Instruments and sensors
Scientific instruments
- Orbits and Gravitation simulation (FizziQ Web)
FizziQ Features
- Experiment notebook — Automatically receives each planet's x/y position data exported from the simulation, used to plot x versus time and read off each orbital period.
Required Materials
- - Computer, tablet or smartphone with FizziQ Web - FizziQ experiment notebook - Note: the protocol remains adaptable to any comparable orbital-mechanics simulation and data-analysis tool.
Experimental Protocol
Open the Orbits and Gravitation simulation in FizziQ Web (Experiment → Simulations → Orbits and gravitation).
Set the distance scale to 1,500,000 km/pixel (ruler icon) to see orbits up to Mars, and the time scale to 6 hours per frame (speed icon) to observe several revolutions.
Select body 1 and configure it as the Sun: mass 333,000 M⊕ (slider on "Sun"), speed 0 km/s, angle 0°. Leave it at the centre of the screen.
Select body 2 and configure it as Mercury: custom mass 0.055 M⊕, speed 47.4 km/s, angle -90°. Drag it to the right of the Sun until the "distances" panel reads about 58 million km.
Add a new body via the "+" tab and configure it as Venus: mass 0.815 M⊕, speed 35.0 km/s, angle -90°, distance to Sun 108 million km.
Add the Earth body: mass 1 M⊕, speed 29.8 km/s, angle -90°, distance to Sun 150 million km.
Add the Mars body: mass 0.107 M⊕, speed 24.1 km/s, angle -90°, distance to Sun 228 million km.
Assign a different colour to each planet via the palette button to distinguish them easily.
Click the body 1 (Sun) centering button in the Centering area to lock the view on the Sun throughout the simulation.
Click the red REC button to start recording. The simulation launches automatically and traces the four nested coloured orbits around the Sun.
Let the simulation run until Mars completes at least one full revolution (~687 days, about 30 seconds on screen). During this time, Mercury will have orbited about 8 times.
Click REC again to stop recording. The x_i and y_i positions of each planet are automatically exported to the experiment notebook.
In the experiment notebook, plot for each planet the x coordinate versus time. The curve is sinusoidal: the duration between two successive maxima gives the period T.
Fill in a 5-column table: Planet, Distance r (million km), T measured (days), T real (days), Deviation (%). Expected real values: Mercury 88 d, Venus 225 d, Earth 365 d, Mars 687 d.
Convert T to years (T_days ÷ 365.25) and r to astronomical units (r_M_km ÷ 150). Compute T² ÷ r³ for each planet and verify that this ratio is almost 1 for all four planets: this is Kepler's regularity.
Expected Results
The four planets appear in nearly circular nested orbits around the Sun, all rotating in the same direction. Mercury, the closest, completes one revolution in about 88 days; Venus in 225 days; Earth in 365 days; and Mars, the farthest, in about 687 days. Measured periods match astronomical values within a few percent due to the symplectic Euler numerical integration and limited simulation duration. When converting T to years and r to astronomical units (Earth = 1 year, 1 AU), the T²/r³ ratio is almost exactly 1 for all four planets: this is the regularity Kepler discovered in 1619. This result illustrates that the same law of gravitation governs all the planets in the solar system.
Scientific Questions
- What would happen if the Sun were removed from the simulation? How would the planets' trajectories evolve?
- Why is the T²/r³ ratio nearly identical for Mercury, Venus, Earth and Mars?
- Why aren't the orbits perfectly circular even with a fixed initial speed?
- How could Kepler's regularity be used to predict the period of a planet whose distance to the Sun is the only known value?
Scientific Background
The **solar system** is composed of the Sun and eight planets. The four closest to the Sun — **Mercury, Venus, Earth and Mars** — are called **terrestrial planets** because they have rocky surfaces. The four farther out (Jupiter, Saturn, Uranus, Neptune) are gas giants, much more massive.
All planets orbit the Sun thanks to **universal gravitation** described by Isaac Newton. The Sun, with its enormous mass (**333,000 times that of Earth**), attracts each planet and keeps it in a nearly circular orbit.
The **closer** a planet is to the Sun, the stronger the gravitational attraction it experiences, and the faster it must move to avoid falling into the Sun. Mercury races at **47 km/s**, while Mars, farther away, moves at only **24 km/s**.
The **farther** a planet is, the longer the path to travel and the slower it moves: its **orbital period** is therefore much longer. The period goes from 88 days for Mercury to 687 days for Mars, almost 8 times longer.
The **astronomical unit (AU)** equals 150 million km, the mean Earth-Sun distance. Mercury is at 0.39 AU, Venus at 0.72 AU, Earth at 1 AU and Mars at 1.52 AU.
In 1619, **Johannes Kepler** discovered an astonishing regularity: for all planets, the ratio **T² / r³ is constant**. If T is expressed in years and r in astronomical units, this ratio equals **1** for the four terrestrial planets as for the others.
The simulation reproduces this regularity because it applies the same law of gravitation to all planets: this result illustrates that the same Newtonian physics governs all bodies in the solar system.
Extensions
- Increase or decrease the Sun's mass (×2 or ÷2 for example) and observe how the planets' periods evolve
- Give planets initial speeds slightly different from the theoretical circular speeds to observe elliptical orbits
- Replace the Sun with a red dwarf (less massive) and find the orbital speed that keeps Earth in stable orbit
- Compute the T²/r³ ratio for the four planets and compare with the outer planet values (Jupiter, Saturn, Uranus, Neptune) found in the documentation
Frequently Asked Questions
Why can't I include all eight planets of the solar system?
The simulation accepts at most 5 simultaneous bodies. So we can model the Sun and 4 planets only, which is enough to study the fundamental principles and Kepler's regularity.
How can I place a planet precisely at the right distance from the Sun?
Drag the body with the mouse and watch the "distances" panel in the upper right, which displays in real time the distance between all bodies. The scale indicator at the bottom left also helps to estimate.
Why are all initial angles set to -90°?
So that all planets orbit in the same direction around the Sun and their initial velocity is perpendicular to the Sun-planet direction — the condition for a circular orbit.
Why do my orbits cross or become unstable?
The initial speeds must match the real circular orbital speeds (47.4 / 35.0 / 29.8 / 24.1 km/s). An error of a few km/s makes the orbit elliptical; too large an error can make the planet escape or fall into the Sun.
How do I express T² ÷ r³ in years² per AU³?
Convert T to years (T_days ÷ 365.25) and r to astronomical units (r_km ÷ 150,000,000). With these units, the ratio equals 1 for Earth and stays very close to 1 for all other planets in the solar system.
Detailed Description
The student configures a five-body system in the Orbits and Gravitation simulation: the Sun and the four terrestrial planets (Mercury, Venus, Earth, Mars) with their real masses, distances and orbital speeds. The student runs the simulation and observes the four nested coloured orbits around the Sun. The student measures each planet's orbital period and compares it to astronomical values (88, 225, 365 and 687 days). The student discovers that a more distant planet orbits more slowly and computes T²/r³ ratios to highlight a pattern common to all four orbits. FizziQ Web's Orbits and Gravitation simulation lets the student build and run this five-body system directly on screen, then automatically exports each planet's x and y positions to the FizziQ experiment notebook for period measurement and graphing.
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