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.
Key Formula
T² = (4π² / GM) × a³
LaTeX: T^2 = \frac{4\pi^2}{GM} a^3
| Symbol | Meaning | Unit |
|---|---|---|
| T | Orbital period | seconds (s) |
| G | Gravitational constant | N·m²/kg² (6.674 × 10⁻¹¹) |
| M | Mass of the central body (Sun) | kg |
| a | Semi-major axis of the orbit | metres (m) |
Worked Example
Problem
Earth orbits the Sun with a semi-major axis of 1.496 × 10¹¹ m. Using the gravitational parameter GM = 1.327 × 10²⁰ m³/s², calculate Earth's orbital period.
Solution
Step 1: Use T² = (4π²/GM) × a³ Step 2: T² = (4π² / 1.327 × 10²⁰) × (1.496 × 10¹¹)³ Step 3: (1.496 × 10¹¹)³ = 3.347 × 10³³ m³ Step 4: T² = (39.478 / 1.327 × 10²⁰) × 3.347 × 10³³ Step 5: T² = 2.975 × 10⁻¹⁹ × 3.347 × 10³³ = 9.957 × 10¹⁴ Step 6: T = √(9.957 × 10¹⁴) ≈ 3.155 × 10⁷ s Step 7: Convert to years: 3.155 × 10⁷ / 3.156 × 10⁷ ≈ 1.0 year
Answer
T ≈ 3.155 × 10⁷ seconds ≈ 1 year
Orbital Parameters of the Eight Planets
| Planet | Semi-major Axis (AU) | Orbital Period (Years) | Eccentricity |
|---|---|---|---|
| Mercury | 0.387 | 0.241 | 0.206 |
| Venus | 0.723 | 0.615 | 0.007 |
| Earth | 1.000 | 1.000 | 0.017 |
| Mars | 1.524 | 1.881 | 0.093 |
| Jupiter | 5.203 | 11.862 | 0.049 |
| Saturn | 9.537 | 29.457 | 0.057 |
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.
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.
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.
From Latin "orbita" (track, course, rut of a wheel), derived from "orbis" (circle, ring). The term entered astronomical use in the 16th–17th centuries. Johannes Kepler was the first to show mathematically that planets follow elliptical, not circular, orbits.