Lunar period
Measure the Moon's orbital period around Earth and compare it to the real value of 27.3 days with the FizziQ Web Orbits and Gravitation simulation.
Learning objectives:
The student configures the Earth-Moon system in the Orbits and Gravitation simulation, runs the model and measures the time the Moon takes to complete one full revolution around Earth. The student observes the orbital trajectory, identifies extreme positions over time and compares the measured period to the real value of 27.3 days. The student discovers how Earth's mass attracts the Moon and keeps it in orbit. The activity illustrates gravitational attraction between two celestial bodies.
Level:
Author:
Middle school
FizziQ
Duration (minutes) :
30-40
What students will do :
'- Configure the Earth-Moon system in the FizziQ Web Orbits and Gravitation simulation
- Measure the Moon's orbital period in days
- Compare the experimental value to the real period of 27.3 days
- Identify the parameters that influence a satellite's trajectory (central mass, radius, speed)
- Qualitatively describe the role of gravitation in orbital motion
Scientific concepts:
'- Universal gravitation
- Orbital motion
- Orbital period
- Circular trajectory
- Natural satellite
- Gravitational attraction
- Sidereal and synodic period
Sensors:
'- FizziQ Web Orbits and Gravitation simulation
What is required:
'- Computer, tablet or smartphone with FizziQ Web
- FizziQ experiment notebook
Experimental procedure:
Open the Orbits and Gravitation simulation in FizziQ Web (Experiment → Simulations → Orbits and gravitation).
Verify the default Earth-Moon configuration: body 1 (Earth, mass 1 M⊕, speed 0 km/s) and body 2 (Moon, mass 0.012 M⊕, speed 1.022 km/s, angle -90°).
Verify the scales: distance 2,000 km/pixel and time 30 minutes per frame (settings suited to the Earth-Moon system).
Click the body 1 centering button in the Centering area to lock the view on Earth during the simulation.
Click the red REC button to start position recording. The simulation launches automatically and traces the Moon's trajectory.
Observe the Moon's coloured trajectory around Earth. Read the chronometer in the upper-left corner, which displays elapsed time in days.
Identify the moment when the Moon returns to its initial position: note this time T in days. This is the orbital period.
Let the simulation run a second revolution to verify the measured duration is consistent.
Click REC again to stop recording. The data t, x_2, y_2 are automatically exported to the experiment notebook.
In the experiment notebook, plot the graph of x_2 versus time. The curve is sinusoidal: the duration between two maxima gives the period T.
Compare the measured value to the real period of 27.3 days. Compute the percentage deviation: deviation (%) = 100 × (T_measured − 27.3) / 27.3.
Repeat the simulation by doubling Earth's mass (2 M⊕) without changing the distance. Observe that the Moon orbits faster: the period decreases.
Expected results:
The Moon's trajectory appears as a near-perfect circle centred on Earth, with a radius close to 384,400 km. The measured period is around 27.3 days, in agreement with the real value. A small deviation (of order 1 to 5%) may appear depending on simulation duration and time step, due to the numerical integration scheme. The x_2(t) curve is sinusoidal: the time between two successive maxima corresponds to the period. When Earth's mass is doubled, the Moon orbits about 1.4 times faster and the period decreases. If the initial speed is greatly increased, the Moon escapes because gravity can no longer hold it.
Scientific questions:
'- Why doesn't the Moon fall to Earth even though it is gravitationally attracted?
- What would happen if Earth were twice as massive? How would the period change?
- What would happen if the Moon's initial speed were decreased? And greatly increased?
- Why does the Moon's sidereal period (27.3 days) differ from the cycle of phases (29.5 days)?
- What role does the orbit's radius play in a revolution's duration around the same central body?
Scientific explanations:
Universal gravitation is a force of attraction that acts between all bodies with mass. Described by Isaac Newton in 1687, it is stronger when masses are larger and bodies closer together.
For the Moon and Earth, this force prevents the Moon from flying off in a straight line and keeps it in orbit. Without gravity, the Moon would continue in a straight line and drift away from Earth permanently. The Moon "falls" continuously towards Earth, but its lateral speed of about 1 km/s carries it ever further: it never touches Earth.
The orbital period is the time the Moon takes to complete one full revolution around Earth. Its real value is 27.3 days: this is the sidereal period, measured against the fixed stars.
This period differs slightly from the cycle of the Moon's phases (synodic period of 29.5 days), because Earth also moves around the Sun while the Moon orbits Earth. So it takes a bit more time to return to the same observed phase from Earth.
Two main parameters influence a satellite's period:
- The central body's mass: the larger it is, the stronger the attraction and the shorter the period.
- The orbital radius: the larger it is, the longer the period. A planet far from the Sun orbits more slowly than one nearby.
The mean Earth-Moon distance is 384,400 km and the Moon's mean orbital speed is 1.022 km/s. These two values are linked to Earth's mass through the law of gravitation.
The simulation computes the trajectory by applying Newton's law over each small time interval: this is numerical integration. The trajectory obtained is not a perfect circle because the Moon's real speed varies slightly along its orbit: the real orbit is an ellipse of low eccentricity (about 0.055).
Extension activities:
'- Why doesn't the Moon fall to Earth even though it is gravitationally attracted?
- What would happen if Earth were twice as massive? How would the period change?
- What would happen if the Moon's initial speed were decreased? And greatly increased?
- Why does the Moon's sidereal period (27.3 days) differ from the cycle of phases (29.5 days)?
- What role does the orbit's radius play in a revolution's duration around the same central body?
Frequently asked questions:
Q: Why doesn't the Moon fall to Earth?
A: The Moon moves laterally at about 1 km/s. Gravity continuously deflects it towards Earth, but its velocity carries it ever further: it "falls perpetually" around Earth without ever reaching it.
Q: Why isn't my orbit a perfect circle?
A: The initial speed (1.022 km/s) does not exactly match the ideal circular speed at this distance. The trajectory becomes a slightly elongated ellipse, like the Moon's real orbit.
Q: How can I measure the period precisely?
A: Plot the Moon's x_2 (or y_2) coordinate versus time. The curve is sinusoidal: the distance between two maxima gives the period. You can also measure the total duration of several revolutions and divide for greater precision.
Q: Why does the simulation run so fast?
A: The time scale (dt = 30 minutes per frame) greatly accelerates the simulation. A real lunar revolution of 27 days is observable on screen in less than a minute.
Q: Why is the Moon's mass set to 0.012 M⊕ and not 1 M⊕?
A: The Moon is about 81 times less massive than Earth, or 1.2% of Earth's mass. A mass that is too high would move Earth and distort the orbit observation.