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Kepler’s Law

Kepler’s Law

Content Standards

In this lesson, students will be able to:

  • State each of Kepler’s three laws correctly in scientific language.
  • Interpret diagrams of elliptical orbits (perihelion, aphelion, semi-major axis).
  • Explain why area swept per unit time remains constant.
  • Relate the third law (T² ∝ a³) to real planetary data using the table provided.

Performance Standards

Students will be able to:

  • Draw and label an ellipse and identify its foci.
  • Explain why planets move faster near the Sun and slower farther away.
  • Apply the proportionality T² ∝ a³ to compare two planets’ orbits numerically.
  • Use reasoning to connect Kepler’s 2nd Law with conservation of angular momentum.

Alignment Standards

Reference: NCERT Book Alignment 

The lesson is aligned with the NCERT Grade 11 Physics Textbook, Chapter 7: Gravitation, Section 2 – Kepler’s Laws.

Learning Objectives

By the end of the lesson, students will be able to:

  • Describe clearly how planetary orbits are elliptical, not perfectly circular
  • Explain that equal areas are swept in equal times because angular momentum is conserved
  • State that the square of a planet’s time period is proportional to the cube of its semi-major axis
  • Demonstrate numerically that distant planets take longer to orbit the Sun

Prerequisites (Prior Knowledge)

  • Idea of planetary revolution around the Sun
  • Meaning of time period (T) of revolution
  • Concept of distance between planet and Sun changing over time
  • Basic geometry of an ellipse — focus, semi-major axis (no deep conic theory required)
  • Concept of angular momentum from earlier rotational motion discussions

Introduction

Kepler’s Laws form the foundation of our scientific understanding of how planets move around the Sun.
Using the accurate astronomical data recorded by Tycho Brahe, Johannes Kepler derived three powerful mathematical laws that describe planetary motion with remarkable precision. These laws helped bridge the gap between observation (astronomy) and theory (physics), and eventually guided Newton to formulate the Universal Law of Gravitation.

Thus, this topic is not only about memorising three statements — it is about understanding how nature follows mathematical patterns.

Timeline (40 Minutes)

TitleApproximate DurationProcedureReference Material
Engage5

The teacher demonstrates how to draw an ellipse using string and 2 fixed pins (as shown in NCERT). Students try the same on paper.

Slides

Explore10

Explore VR lab to understand Kepler’s three laws of planetary motion.

Slides + Virtual Lab

Explain10

The teacher explicitly states 3 Kepler’s Laws.

Slides

Evaluate10

Students will attempt the Self Evaluation task on LMS

Virtual Lab

Extend5

Have a discussion on how the artificial satellites sent to space follow Kepler’s laws of motion.

Slides

Kepler’s Law

Introduction

Kepler’s Laws of Planetary Motion describe how planets revolve around the Sun in space. These laws are derived from careful astronomical observations and represent a major shift from the earlier belief of perfect circular orbits. The three laws reveal that planetary motion follows definite mathematical patterns. They also form an essential basis for understanding the Universal Law of Gravitation. Kepler’s work showed that nature follows precision, symmetry, and quantitative relationships.

Theory

Kepler’s Three Laws are mathematical generalisations based entirely upon precise astronomical measurements (mainly Tycho Brahe’s data). Kepler did not have the law of gravitation at that time — however, the patterns he discovered became the foundation on which Newton formulated the inverse–square law of gravitation.

Kepler’s First Law – Law of Orbits Planets revolve in elliptical paths around the Sun, and the Sun occupies one focus of the elliptical orbit. This immediately rejects the older geocentric and circular orbit models. As the planet moves along the ellipse, its Sun–planet distance continuously varies. The orbital geometry is therefore not uniform and not circular.

Kepler’s Second Law – Law of Areas The radius vector (a line joining the centre of the Sun and the centre of the planet) sweeps equal areas in equal intervals of time. This is a highly technical statement: it implies conservation of areal velocity. When the planet is near perihelion, the gravitational pull is stronger and the orbital speed increases such that the swept area per unit time remains unchanged. This is an indirect physical statement of conservation of angular momentum.

The laws of areas can be understood as a consequence of conservation of angular momentum which holds true for any central force. A central force is a force that always acts along the line joining the central body (Sun) and the orbiting planet.

Assume the Sun is at the origin. Let the planet’s position vector be r and its momentum be p. In a small time interval Δt, the area swept by the radius vector is. Mathematically,

ΔA = 1/2​(r×vΔt) = 1/2m(r x p) = L/2m

Kepler’s Third Law – Law of Periods For any planet orbiting the same central massive body, the square of its orbital time period (T²) is directly proportional to the cube of its semi-major axis (a³). Mathematically:

T² ∝ a³

This law establishes a quantitative relationship between the size of the orbit and the time taken to complete one revolution. The ratio T²/a³ is approximately constant for all planets around the same star. This single proportionality led Newton to prove that the gravitational force must be inversely proportional to the square of distance.

Vocabulary

This is the list of vocabulary terms used throughout the lesson.

  • Ellipse – A closed curved path with two distinct foci.
  • Focus (plural: Foci) – One of the two fixed points that define an ellipse.
  • Perihelion – Point in a planet’s orbit where it is closest to the Sun.
  • Aphelion – Point in a planet’s orbit where it is farthest from the Sun.
  • Semi-major Axis – Half of the longest diameter of an elliptical orbit.
  • Areal Velocity – Rate at which area is swept by the line joining planet and Sun.
  • Orbital Period (T) – Time taken by a planet to complete one full revolution around the Sun.
  • Angular Momentum – Physical quantity that remains conserved and explains speed variation in orbit.
  • Planetary Motion – Movement of planets in fixed, predictable paths around the Sun.
  • Kepler’s Constant – Value of T²/a³ which remains nearly constant for all planets of the same solar system.

Kepler’s Law

Category

Introduction

This VR lab is designed to help you visualise and interact with Kepler’s three laws of planetary motion. Instead of only reading the laws, you will explore how orbits look, how planetary speeds vary, and how orbital time relates to orbital size. This experience will allow you to interpret each law with real-time motion, sliders, and simulations, making the laws easier to connect with physical meaning.

Key Features

  • Interactive sliders to modify eccentricity and observe changes in orbital path.
  • Play/Pause controls to track area swept and time intervals in real-time.
  • Planet database input to compare semi-major axis and orbital period for different planets.
  • Visual representation of perihelion, aphelion, and changing orbital speeds.
  • Data-driven visualisations to validate T² ∝ a³ relationship.

Step-by-Step Procedure for VR Experience

 Step 1: Introduction

  • Observe the basic concept of planetary motion around the Sun.
  • Understand that orbits are not perfect circles.

 Step 2: Law Of Orbits

  • Read the explanation of elliptical orbit geometry.
  • Use the eccentricity slider to increase or decrease eccentricity.
  • Observe how the orbit shape changes as eccentricity varies.

Step 3: Law Of Areas

  • Read the explanation of equal areas being swept in equal times.
  • Use the Play/Pause button to freeze motion at different points.
  • Measure area and time from simulation controls to verify that areal velocity remains constant.

Step 4: Law Of Periods

  • Read about T² ∝ a³ relation.
  • Type the name of a planet into the input panel.
  • Observe how the semi-major axis value predicts the time period of revolution.

Step 5: Evaluation

  • After interaction, students proceed to the quiz:

    • 2  MCQs
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