Theme D — Fields
D.1.9 Kepler’s Laws of Planetary Motion — IB Physics HL Revision Notes & Topical Questions
5 min read 10 parts Lesson 11 of 39 1 quiz
Curriculum Equation Key قوانين الرقم الهيدروجيني والاتزان الأيوني - pH & Acid-Base Equilibrium FormulasFast Exam Solution
pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴
Word Equation & Conditions: الرقم الهيدروجيني = -لوغاريتم تركيز أيون الهيدروجين | pH + pOH = 14
Location & Specifications: الاتزان الأيوني، كيمياء 3 ثانوي، التحصيلي، و AP Chemistry.
Lesson summary
D.1.9 Kepler's Laws of Planetary Motion — IB PHYSICS SL notes: State Kepler's three laws, derive T² ∝ r³ for circular orbits and use it to find masses and orbital periods.Kepler's work — Johannes Kepler analysed Tycho Brahe's careful measurements of planetary positions and summar…
- Explanation
- Position of the Sun
- Consequence
- Mass independence
- T2 against r3
- Astronomical units
Part 1 of 8Origin of the laws
Kepler's work
- Johannes Kepler analysed Tycho Brahe's careful measurements of planetary positions and summarised them in three laws
Explanation
- Newton's gravity later explained the laws
Scope
- Scope they apply to any object orbiting a much larger central mass, such as moons round planets, satellites round the Earth and exoplanets round stars
Part 2 of 8First law
1Statement
- Statement the orbit of a planet is an ellipse with the Sun at one of the two foci
2Ellipse
- Ellipse a squashed circle with two foci
3Examples
- Pluto has a highly elliptical orbit and Earth a near-circular one
4Position of the Sun
- Position of the Sun it is at a focus, not at the centre of the ellipse
Part 3 of 8Second law
1Statement
- Statement a line segment joining the Sun to a planet sweeps out equal areas in equal time intervals
2Consequence
- Consequence planets move faster nearer the Sun and slower further away
- The planet is closest at perihelion and farthest at aphelion.
- To sweep equal areas in equal time it must move faster near the Sun.
3Explanation
- Explanation angular momentum is conserved because the gravitational force is central and exerts no torque about the Sun
- It is also consistent with energy conservation, since moving closer converts potential energy into kinetic energy.
4Circular orbit
- Circular orbit the speed is constant, so the law adds nothing new
Part 4 of 8Third law
Statement
- Statement for bodies in circular orbits about the same central body, the square of the period is proportional to the cube of the orbital radius, T2 ∝ r3
Meaning
- Meaning the law links the time of an orbit to its radius, so outer planets have longer years
- T is the period in seconds and r is the mean orbital radius in metres.
Ellipses
- Ellipses r is replaced by the semi-major axis
Part 5 of 8Deriving the third law
1
For a circular orbit of radius r around a mass M, gravity provides the centripetal force, GMm/r2 = mv2/r.
2
The orbital speed is v = 2πr/T.
3
Eliminate v and the orbiting mass m.
4
Square and rearrange to get T2 = (4π2/GM) r3.
Symbols
- M is the mass of the central body, not the orbiting one.
- r is measured centre to centre, not from the surface.
Constant
- 4π2/GM depends only on the central mass, so all bodies orbiting the same body share the same T2/r3
Mass independence
- Mass independence a satellite and an astronaut orbit with identical period at the same radius
Part 6 of 8Graphs and ratios
T2 against r3
- T2 against r3 a straight line through the origin with gradient 4π2/GM, so the gradient gives M
Log plot
- Log plot log T against log r is a straight line with gradient 3/2 and a negative y-intercept
- A graph of T against r would not be straight.
Ratios
- (T1/T2)2 = (r1/r2)3, which avoids large numbers
Astronomical units
- Astronomical units for orbits around the Sun, with r in AU and T in years, T2 = r3
Growth of period
- Growth of period the period increases faster than the radius, as r3/2
Part 7 of 8Applications
Masses of planets
- Masses of planets the period and radius of a moon give the mass of its planet
Masses of stars
- Masses of stars a planet's period and radius give the mass of the star
Averaging
- Averaging plotting T2 against r3 for several orbits reduces random error
Exoplanets
- Exoplanets the period of a star's wobble reveals orbiting planets
Geostationary orbit
- Geostationary orbit it follows from setting T = 24 h
Double stars
- Double stars for two comparable masses the separation and total mass are used instead
Real orbits
- Real orbits they are slightly elliptical, so the mean distance or semi-major axis is taken
Part 8 of 8More from the specification
More from the specification
Centripetal gravitational forces in orbital mechanics
- Centripetal gravitational forces in orbital mechanics inverse-square attractive gravitational forces provide the required centripetal forces dictating planetary orbits.
Gravitational Field Strength
- Defined as the gravitational force per unit mass exerted on a small test mass (g = F/m), directed radially inward towards the centre of mass of a spherical body.
Newton's Gravitational Field Equation
- For a uniform spherical mass distribution, field strength at distance r from its centre obeys g = GM/r².
- (sphere of mass M)
Radial Gravitational Field Lines
- Field lines indicate the direction of gravitational attraction;
- their spacing represents field strength density, converging closer together near the surface.
Weight and Planetary Orbit Centripetal Force
- An orbiting body's weight provides the inward centripetal force required to maintain its Keplerian elliptical or circular orbit around a central star.
Gravitational field strength
- Gravitational field strength gravitational force exerted per unit mass, pointing radially towards the centre of a planetary sphere.
Radial field lines
- Radial field lines field lines around spherical masses converge towards the center, whereas near surface fields are approximately uniform.
Planetary density and mass
- Planetary density and mass mean planet density determines total gravitational mass and resulting surface weight of orbiting objects.
Uniform sphere
- Sphere with its mass distributed evenly.
Inversely proportional
- One quantity falls as the other rises in the same ratio.
Radial field
- Field directed to a center and varying with distance.
Uniform
- Same density throughout, so the weight acts at the centre.
Density
- Mass per unit volume.
Field
- Region in which an object experiences a force at a distance
Weight
- Gravitational force on a mass.
Gravitational field
- Region where mass experiences gravitational force.
Field strength
- G = Fg/m.
Field lines
- Lines showing the direction and strength of a magnetic field.
Uniform field
- Field of constant strength and direction.
Key terms
| Term | Meaning |
|---|---|
| Ellipse | A closed oval curve with two foci; planets orbit with the Sun at one focus. |
| Perihelion and aphelion | Closest and farthest points of an orbit around the Sun. |
| Semi-major axis | Half the longest diameter of an ellipse; it replaces r in the third law. |
| Orbital period T | Time for one complete orbit. |
| Astronomical unit | The mean Earth–Sun distance, about 1.50 × 10¹¹ m. |
Avoid these mistakes
1
Using the mass of the orbiting body in T^{2} = 4π^{2}r^{3}/GM — M is the central mass.
2
Using the height above the surface instead of the orbital radius.
3
Thinking planets move at constant speed in an ellipse.
4
Saying the Sun is at the centre of the ellipse — it is at a focus.
5
Forgetting to convert days or hours to seconds.
Marked practiceQuizzes for this lesson
Last updated:
Physics 0625 & 9702 SUVAT Mastery
Graph Interpretation
Constant Acceleration Equations (SUVAT) & Slopes
v = u + at | s = ut + ½at2 | v2 = u2 + 2as | s = ½(u + v)t
- Displacement-Time Graph: Gradient = Velocity (curve = accelerating).
- Velocity-Time Graph: Gradient = Acceleration; Area under curve = Distance travelled / Displacement.
- Free Fall in vacuum:
a = g ≈ 9.81 m/s2downwards (air resistance negligible).