Planetary motion — OCR A-Level Physics
Test yourself on Planetary motion with OCR A-Level practice questions.
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Planetary motion explained
Kepler's three laws describe how planets move around the Sun.
Read the full explanation
The first law states that each planet moves in an ellipse with the Sun at one focus. The second law states that the line joining a planet to the Sun sweeps out equal areas in equal times, so a planet moves faster when nearer the Sun. The third law states that the square of the orbital period T is proportional to the cube of the semi-major axis a: T² ∝ a³, or T²/a³ is constant for all planets orbiting the same central body. For circular orbits a is the orbital radius r, giving T² ∝ r³. These laws are empirical but follow from Newton's law of gravitation and centripetal force.
(b) the centripetal force on a planet is provided by the gravitational force between it and the Sun
A planet moving around the Sun follows a curved path, so it must have a centripetal acceleration directed towards the Sun. The only significant force acting on the planet is the gravitational attraction between the planet and the Sun, so this gravitational force provides the required centripetal force. Equating the two for a circular orbit gives GMm/r² = mv²/r, which simplifies to v² = GM/r. Combining this with v = 2πr/T leads to T² = 4π²r³/(GM), which is Kepler's third law. This shows that the gravitational force is not an additional force but the same force that supplies the centripetal requirement.
(c) the equation T² = (4π²/GM)r³
This equation links a satellite's orbital period T to its orbital radius r around a central mass M. It comes from equating gravitational force GMm/r² to centripetal force m(4π²/T²)r. The mass m cancels, giving T² = (4π²/GM)r³. The equation shows T² is directly proportional to r³ for a given central body. For example, a satellite orbiting Earth at radius r has M equal to Earth's mass, not the satellite's mass. Rearranging gives M = 4π²r³/(GT²), which allows astronomers to find the mass of a planet or star from the orbit of a moon or satellite. The equation assumes a circular orbit and that the central mass is much larger than the orbiting mass.
(d) the relationship for Kepler’s third law T 2 ∝ r 3 applied to systems other than our solar system
Kepler's third law states that T² is proportional to r³ for any object orbiting a central mass under gravity. This applies not only to planets orbiting the Sun but also to moons orbiting planets, satellites orbiting Earth, and exoplanets orbiting distant stars. For a moon orbiting Jupiter, T² ∝ r³ with the constant of proportionality depending on Jupiter's mass. Similarly, for a satellite orbiting Earth, the constant depends on Earth's mass. The law allows comparison of orbital periods and radii for different systems, provided the central mass is the same. For example, if two moons orbit the same planet, the ratio T₁²/T₂² equals r₁³/r₂³. This relationship is used to determine masses of astronomical bodies from observed orbits.
(e) geostationary orbit; uses of geostationary satellites.
A geostationary orbit is a circular orbit around Earth's equator with a period of one sidereal day (approximately 24 hours, or 86 164 s), so the satellite appears stationary above a fixed point on the equator. It must orbit in the same direction as Earth's rotation (west to east) and at a specific altitude of approximately 3.6 × 10⁷ m above Earth's surface. Geostationary satellites are used for telecommunications, television broadcasting, and weather monitoring because their fixed position allows a ground antenna to point permanently at the satellite. For example, a satellite dish for satellite TV is aimed at a fixed point in the sky. The orbit is a special case of a geosynchronous orbit, which has a 24-hour period but may be inclined rather than equatorial.
Your focus
- State Kepler's three laws of planetary motion.
- Explain the physical meaning of each law for a planet orbiting the Sun.
- Use T² ∝ a³ to compare orbital periods and radii for planets around the same central body.
Show all 15 objectives
- Explain that the gravitational force provides the centripetal force for a planet in orbit.
- Derive the relationship v² = GM/r for a circular orbit.
- Show how equating gravitational and centripetal forces leads to Kepler's third law.
- Derive T² = (4π²/GM)r³ from Newton's law of gravitation and circular motion.
- Apply the equation to calculate orbital period, orbital radius or central mass.
- Explain the meaning of each symbol and the assumption of a circular orbit.
- Describe how Kepler's third law applies to systems other than the solar system.
- Use T² ∝ r³ to compare orbital periods and radii for objects orbiting the same central mass.
- Calculate the mass of a central body from the orbital period and radius of an orbiting object.
- Describe the properties of a geostationary orbit.
- List and explain uses of geostationary satellites.
- Calculate the orbital radius of a geostationary satellite using Kepler's third law.
Planetary motion exam tips
Marking Points
- States Kepler's first law: planets move in ellipses with the Sun at one focus.
- States Kepler's second law: the radius vector from Sun to planet sweeps equal areas in equal times.
- States Kepler's third law: T² ∝ a³ for planets orbiting the same central mass.
- Explains that the second law implies a planet's orbital speed is greatest at perihelion and least at aphelion.
- Applies T² ∝ r³ to circular orbits, where the semi-major axis equals the orbital radius.
- States that the gravitational force between planet and Sun acts towards the Sun and provides the centripetal force.
- Writes the gravitational force as GMm/r² and the centripetal force as mv²/r or mω²r.
- Equates the two expressions to derive v² = GM/r for a circular orbit.
- Combines v = 2πr/T with v² = GM/r to obtain T² = 4π²r³/(GM), linking to Kepler's third law.
- Explains that no separate centripetal force exists; gravity alone fulfils that role.
- Equate gravitational force GMm/r² to centripetal force m(4π²/T²)r.
- Cancel the orbiting mass m to obtain T² = (4π²/GM)r³.
- Identify M as the mass of the central body, not the orbiting body.
- Use M = 4π²r³/(GT²) to determine the central mass from orbital data.
- Recognise that T² ∝ r³ for a fixed central mass.
- State that T² ∝ r³ applies to any system where a smaller body orbits a larger central mass.
- Give examples such as moons orbiting planets, artificial satellites orbiting Earth, or exoplanets orbiting stars.
- Explain that the constant of proportionality depends on the mass of the central body.
- Use the ratio T₁²/T₂² = r₁³/r₂³ for objects orbiting the same central mass.
- Apply the law to determine the mass of a central body from orbital data of an orbiting object.
- Define a geostationary orbit as one with a period of one sidereal day (approx 24 hours) above the equator, moving west to east.
- State that the satellite appears stationary relative to a fixed point on Earth's surface.
- Give uses such as telecommunications, television broadcasting, and weather monitoring.
- Explain that the fixed position allows continuous coverage and fixed ground antennas.
- Distinguish geostationary from geosynchronous orbits (which may not be equatorial).
Examiner Tips
- 💡Learn all three laws with their precise wording, since multiple-choice questions often test one law at a time.
- 💡Use the constant form T²/a³ when comparing two planets orbiting the same body.
- 💡Remember that the second law is about equal areas in equal times, not equal distances or equal speeds.
- 💡Link the third law to Newton's law of gravitation to show it is not an independent assumption.
- 💡Start from the force equation GMm/r² = mv²/r and cancel m before substituting numbers.
- 💡Check that M is the mass of the central body (the Sun) and m is the orbiting body (the planet).
- 💡Use v = 2πr/T when the period is given rather than the orbital speed.
- 💡State clearly that gravity provides the centripetal force, not that it is balanced by another force.
- 💡Write down the equation exactly as given before substituting numbers.
- 💡Check that r is the orbital radius from the centre of the central body, not the altitude above its surface.
- 💡Use standard SI units: metres for r, seconds for T, kilograms for M.
- 💡When comparing two orbits around the same central body, use the ratio form to avoid calculating the constant.
- 💡Ensure r is measured from the centre of the central body, not from its surface.
- 💡Remember that T must be in seconds and r in metres for calculations.
- 💡Remember the period is one sidereal day (approximately 24 hours or 86 164 s) and the orbit is equatorial.
- 💡Use the equation T² = (4π²/GM)r³ to calculate the orbital radius if needed.
- 💡Link uses to the advantage of a fixed position relative to Earth, such as not needing to track the satellite with a moving dish.
Common Mistakes
- Describing planetary orbits as perfect circles; Kepler's first law requires ellipses with the Sun at one focus.
- Stating the third law as T ∝ a³ or T² ∝ a²; the correct relationship is T² ∝ a³.
- Claiming the Sun is at the centre of the ellipse; it is at one focus, not the geometric centre.
- Applying T² ∝ a³ to planets orbiting different stars without accounting for the different central masses.
- Treating centripetal force as an extra force in addition to gravity; it is the resultant of the gravitational force.
- Using the planet's mass incorrectly in v² = GM/r; the planet's mass cancels, leaving the Sun's mass M.
- Forgetting that the gravitational force acts towards the Sun, which is the direction of the centripetal acceleration.
- Mixing up the masses in GMm/r², using the planet's mass for M instead of the Sun's mass.
- Using the satellite's mass for M instead of the central body's mass; correct by using the mass of the body being orbited.
- Forgetting to square the period when substituting into the equation; correct by ensuring T is in seconds and squared.
- Mixing units, such as using km for r and hours for T; correct by converting r to metres and T to seconds before calculation.
- Assuming Kepler's third law only applies to planets in the solar system; correct by recognising it applies to any gravitational orbit.
- Using the mass of the orbiting body instead of the central body when calculating the constant; correct by using the central mass.
- Forgetting that the law assumes circular orbits and that the central mass is much greater than the orbiting mass; correct by stating these assumptions.
- Confusing geostationary with geosynchronous; correct by noting geostationary must be equatorial and circular.
- Thinking any satellite with a 24-hour period is geostationary; correct by requiring it to be above the equator and moving west to east.
- Stating GPS uses geostationary satellites; correct by noting GPS uses medium Earth orbits, whereas geostationary is used for communications and TV.
- Assuming geostationary satellites can be used for polar observations; correct by noting they only view one region of Earth near the equator.