Planetary Motion and Gravity
The Scientific Revolution in Astronomy
The shift from the geocentric to the heliocentric model was not a single event — it was a 150-year process involving several scientists whose contributions built on each other in a remarkable chain. Understanding who did what, and why each step mattered, is central to this lesson.
Copernicus (1473–1543) published his heliocentric model in De revolutionibus orbium coelestium (1543, the year of his death). He placed the Sun at the centre, with Earth and the other planets orbiting it. Crucially, he explained retrograde motion as a line-of-sight effect when Earth overtakes an outer planet — a far simpler explanation than Ptolemy’s epicycles. However, Copernicus still assumed perfectly circular orbits, which meant his predictions were not significantly more accurate than Ptolemy’s.
Tycho Brahe (1546–1601) was the greatest naked-eye observer of his era. Working from the island of Hven in Denmark with instruments of his own design, he compiled two decades of planetary position data accurate to about one arcminute — roughly ten times more precise than any previous systematic record. Brahe himself favoured a hybrid model (planets orbit the Sun, but the Sun orbits a stationary Earth), but his data were the raw material that Kepler would use to demolish all circular-orbit models.
Tycho Brahe’s data were essential because only observations accurate to one arcminute or better could distinguish between circular and elliptical orbits. Without Brahe’s numbers, Kepler could not have shown that orbits are ellipses.
Kepler’s Three Laws
Kepler (1571–1630) worked as Brahe’s assistant, inheriting his data after Brahe’s death in 1601. After years of calculation, Kepler published his laws in 1609 and 1619. Each law replaced a previous assumption about planetary motion:
First Law: Each planet moves in an elliptical orbit with the Sun at one focus of the ellipse. The Sun is not at the centre of the ellipse, but at one of its two foci. The point in the orbit nearest the Sun is called perihelion; the furthest point is called aphelion. For orbits around Earth (e.g. the Moon or artificial satellites), the equivalent terms are perigee and apogee. Earth's orbit is only slightly elliptical (eccentricity 0.017), but for some comets the eccentricity approaches 1, giving a very elongated orbit.
Second Law: A line joining a planet to the Sun sweeps out equal areas in equal time intervals. This means planets move faster when they are near perihelion (close to the Sun) and slower when they are near aphelion (far from the Sun). Earth moves at about 30.3 km s⁻1 in January (near perihelion) and 29.3 km s⁻1 in July (near aphelion). This explains why Northern Hemisphere winters (when Earth is near perihelion) are slightly shorter than summers — Earth covers the perihelion arc faster.
Kepler’s Third Law — Period and Distance
Third Law: The square of a planet’s orbital period is proportional to the cube of its mean orbital radius. In equation form:
$$\frac{T^2}{r^3} = \text{constant}$$
where T is the orbital period in years and r is the mean orbital radius in astronomical units. The constant is the same for all objects orbiting the same central body (the Sun). This is the most practically useful of the three laws because it allows you to calculate the orbital radius of any planet or moon if you can measure its orbital period, and vice versa.
Worked example: Mars has an orbital period of 1.881 years. Find its mean distance from the Sun in AU.
$$\frac{T^2}{r^3} = 1 \text{ (in the appropriate units)}$$ $$r^3 = T^2 = 1.881^2 = 3.538$$ $$r = \sqrt[3]{3.538} = 1.52 \text{ AU}$$
This matches the accepted value of 1.52 AU for Mars. Always state the formula, substitute, rearrange and give the final answer with units when answering calculation questions.
Kepler’s Third Law: \(T^2 / r^3 = \text{constant}\). For objects orbiting the Sun with T in years and r in AU, the constant = 1. So \(T^2 = r^3\). Rearrange to find T or r as required.
Newton’s Explanation — Universal Gravitation
Kepler’s laws were empirical — they described the data precisely but gave no reason why planets should obey these rules. Isaac Newton (1643–1727) provided the physical explanation with his law of universal gravitation, published in the Principia Mathematica (1687). Newton showed that any two masses attract each other with a gravitational force that:
- increases with the mass of each object
- decreases with the square of the distance between their centres (the inverse square law)
Newton demonstrated mathematically that if the gravitational force follows the inverse square law, then orbiting bodies must follow elliptical paths (Kepler’s first law) and the equal-areas law (second law) follows automatically from conservation of angular momentum. He also derived the constant in Kepler’s third law in terms of the mass of the central body and the gravitational constant G.
For a stable circular orbit, the gravitational force provides exactly the centripetal force needed to keep the orbiting body on its circular path. If the orbital speed is too low, gravity pulls the object inward (it spirals in or falls); if the speed is too high, the object escapes gravity and follows a parabolic or hyperbolic path. At exactly the right speed for the orbital radius, a stable circular orbit results. This is the principle behind the orbits of planets, moons, and artificial satellites.
The geocentric-to-heliocentric transition illustrates how science progresses: Copernicus proposed a new framework that was simpler in concept but initially not more accurate. Brahe’s meticulous observations provided the data needed to test models. Kepler found laws that fitted the data only when he abandoned the assumption of circular orbits — a result that required him to discard years of earlier work. Newton then explained why those laws hold, elevating them from description to physical theory. This chain — hypothesis, observation, law, theory — is a textbook example of the scientific method operating over multiple generations.