Kepler's Second Law
Kepler's Second Law, also called the Law of Equal Areas, states that a line segment joining a planet to the Sun sweeps out equal areas in equal intervals of time. This means a planet moves faster when it is closer to the Sun (at perihelion) and slower when it is farther away (at aphelion). This law is a consequence of the conservation of angular momentum and applies to any body moving under the influence of a central gravitational force.
Key Formula
dA/dt = L / (2m) = constant
LaTeX: \frac{dA}{dt} = \frac{L}{2m} = \text{constant}
| Symbol | Meaning | Unit |
|---|---|---|
| dA/dt | Rate of area swept by the radius vector | m²/s |
| L | Angular momentum of the planet | kg·m²/s |
| m | Mass of the planet | kg |
Worked Example
Problem
Earth travels at 30.3 km/s at perihelion (closest point, 147.1 million km from Sun) and slows down at aphelion (farthest point, 152.1 million km from Sun). Using Kepler's Second Law (conservation of angular momentum), estimate Earth's speed at aphelion.
Solution
Step 1: Kepler's Second Law implies: v_p × r_p = v_a × r_a (for near-circular orbits) Step 2: This comes from angular momentum conservation: m × v × r = constant Step 3: v_a = (v_p × r_p) / r_a Step 4: v_a = (30.3 km/s × 147.1 × 10⁶ km) / (152.1 × 10⁶ km) Step 5: v_a = (4,457,130 × 10⁶) / (152.1 × 10⁶) Step 6: v_a = 29.3 km/s
Answer
Earth's speed at aphelion ≈ 29.3 km/s (actual value: ~29.3 km/s)
Earth's Speed at Key Orbital Points
| Orbital Point | Distance from Sun (million km) | Orbital Speed (km/s) | Season (Northern Hemisphere) |
|---|---|---|---|
| Perihelion | 147.1 | 30.3 | Early January |
| Average | 149.6 | 29.8 | Equinoxes |
| Aphelion | 152.1 | 29.3 | Early July |
| Vernal Equinox | 149.1 | 29.8 | March |
| Autumnal Equinox | 150.0 | 29.8 | September |
Interactive Tools
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Related Terms
Kepler's First Law
Kepler's First Law, also called the Law of Ellipses, states that every planet orbits the Sun in an elliptical path, with the Sun located at one of the two foci of the ellipse (not at the centre). This was a revolutionary departure from the previous belief in perfectly circular orbits. The degree of elongation of the ellipse is described by its eccentricity (e), where e = 0 is a circle and e approaching 1 is a highly elongated ellipse.
Planetary Orbit
A planetary orbit is the curved path followed by a planet as it moves around the Sun (or another star) under the influence of gravitational attraction. According to Kepler's First Law, planetary orbits are ellipses with the Sun at one of the two foci. The shape of an orbit is described by its eccentricity, where 0 represents a perfect circle and values approaching 1 represent highly elongated ellipses.
The Sun
The Sun is the star at the centre of the Solar System, a nearly perfect sphere of hot plasma that generates energy through nuclear fusion of hydrogen into helium in its core. It accounts for about 99.86% of the total mass of the Solar System and provides the light and heat essential for life on Earth. The Sun is classified as a G-type main-sequence star (G2V) with a surface temperature of approximately 5,778 K and a diameter of about 1.39 million kilometres.
Named after Johannes Kepler (1571–1630), who derived this law from Tycho Brahe's precise observational data and published it in "Astronomia Nova" in 1609 alongside the First Law. Isaac Newton later showed it is a direct consequence of the inverse-square law of gravity and conservation of angular momentum.