Maharashtra State Board: Class 10, 11
Formula: Kepler's Third Law
Maharashtra State Board: Class 10, 11
Law: Kepler's Third Law
Kepler's Third Law (Law of Periods)
- The square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of its orbit.
- This means a planet farther from the Sun takes a longer time to complete an orbit.
Maharashtra State Board: Class 11
Introduction
- Kepler’s Third Law explains the relationship between the time a planet takes to revolve around the Sun and its average distance from the Sun.
- This law is also called the "Law of Periods."
- The law was formulated from careful observations of planetary positions.
- Kepler found that planets farther from the Sun take more time to complete their orbit.
- His law helps us understand how the solar system works.
Maharashtra State Board: Class 11
Characteristics
- Applies to all planets revolving around the Sun.
- The ratio T2/r3 is the same for all planets.
- Helps estimate the orbital period if the distance is known.
- Based on observational data—not theoretical derivation.
- Applies to elliptical orbits (not just circular).
Maharashtra State Board: Class 11
Understanding Kepler's Third Law
- Kepler studied the solar system using regular measurements of planet positions.
- He found that for every planet, if you square the time it takes to go around the Sun (T) and divide it by the cube of its average distance from the Sun (r), you get the same value.
- Mathematically, T2 ∝ r3 or T²/r³ = constant.
- This means the further a planet is from the Sun, the longer it takes to complete its orbit.
- The data for different planets (see Table) shows that T2/r3 has almost the same value for all planets.
Table (Kepler's Third Law Verified with Planets)
| Planet |
Semi-major Axis (10¹⁰ m) |
Period (years) |
T2/r3 (10⁻³⁴ y²·m⁻³) |
| Mercury |
5.79 |
0.24 |
2.95 |
| Venus |
10.8 |
0.615 |
3.00 |
| Earth |
15.0 |
1.00 |
2.96 |
| Mars |
22.8 |
1.88 |
2.98 |
| Jupiter |
77.8 |
11.9 |
2.97 |
| Saturn |
143 |
29.5 |
2.98 |
| Uranus |
287 |
84.0 |
2.98 |
| Neptune |
450 |
165 |
2.99 |
| Pluto |
590 |
248 |
2.99 |
Maharashtra State Board: Class 11
Significance
- Helps compare the orbits of different planets in the solar system.
- Shows that solar system planets follow a regular pattern.
- Used to calculate orbital periods or distances if one of them is known.
- Confirms the predictability of planetary motion based on observations.
Maharashtra State Board: Class 11
Example
Question: What would be the Earth’s period (year duration) if its distance from the Sun became:
- Thrice the present distance
- Twice the present distance
Solution Steps:
- Let the present period T1 = 365 days, and the present distance r1.
- Use Kepler's law:
\[\frac{T_2}{T_1}=\left(\frac{r_2}{r_1}\right)^{3/2}\]
- Case 1: r2 = 3r1
\[\frac {T_2}{T_1}\] = (3)3/2 = \[\sqrt {27}\]
T2 = 365 × \[\sqrt {27}\] = 1897 days
- Case 2: r2 = 2r1
\[\frac {T_2}{T_1}\] = (2)3/2 = \[\sqrt {8}\]
T2 = 365 × \[\sqrt {8}\] = 1032 days
Maharashtra State Board: Class 11
Real-Life Examples
- Astronomers calculate the orbital period of new planets or moons using Kepler’s Third Law.
- Used in space agency missions to plan satellite orbits around planets.
- Helps predict the position of planets in the solar system for astronomy events.
