Kepler’s First Law

Before Johannes Kepler published his laws in 1609, astronomers assumed planets travelled in perfect circles — an idea inherited from ancient Greece and championed even by Copernicus. Kepler shattered this assumption. By painstakingly analysing decades of precise naked-eye observations made by Tycho Brahe, he discovered that the orbit of Mars simply could not be a circle. The shape that fitted the data perfectly was an ellipse. Kepler's First Law states: every planet moves in an elliptical orbit with the Sun located at one of the two foci of that ellipse. An ellipse is not just a 'squashed circle' — it has a precise mathematical definition. It is the set of all points for which the sum of distances to two fixed points (the foci) is constant. This means the Sun is not at the centre of a planet's orbit; it sits slightly off-centre at one focus, while the other focus is empty space. As a result, the planet is sometimes closer to the Sun (at a point called perihelion) and sometimes farther away (at aphelion), giving rise to the changing seasons and varying orbital speeds we observe. Kepler's First Law laid the geometric foundation for understanding the Solar System. It replaced circular perfection with mathematical elegance: ellipses are conic sections, shapes obtained by slicing a cone at different angles. Newton later explained why ellipses arise naturally from an inverse-square gravitational force — but Kepler found the pattern purely from observation, making it one of history's greatest examples of empirical discovery.

Key takeaways

What is an ellipse?
Eccentricity — measuring the 'squashedness'
Perihelion and Aphelion
Why not a circle? Newton's explanation

What is an ellipse?

An ellipse has two special interior points called foci (singular: focus). If you pin two drawing pins into a board, loop a piece of string around both pins, pull the string taut with a pencil, and trace a curve, you draw a perfect ellipse. The sum of distances from any point on the ellipse to both foci equals the total length of the string — this is constant. The longest diameter of the ellipse is the major axis (length 2a, where a is the semi-major axis), and the shortest is the minor axis (length 2b).

Eccentricity — measuring the 'squashedness'

Eccentricity e describes how elongated an ellipse is. It is defined as e = c/a, where c is the distance from the centre of the ellipse to each focus, and a is the semi-major axis. For a perfect circle, the two foci merge into one centre, so c = 0 and e = 0. As the ellipse becomes more elongated, e approaches 1. Earth's orbit is nearly circular with e ≈ 0.017, so the Sun sits only 1.7% of the semi-major axis away from the centre. Pluto's orbit has e ≈ 0.25 — noticeably elliptical.

Perihelion and Aphelion

Because the Sun sits at one focus rather than the centre, the planet's distance from the Sun changes throughout its orbit. The closest point is called perihelion (from Greek: peri = near, helios = Sun), at distance r_min = a(1 − e). The farthest point is aphelion, at distance r_max = a(1 + e). For Earth, perihelion occurs in early January (~147 million km) and aphelion in early July (~152 million km) — counter-intuitively, Earth is closest to the Sun during the Northern Hemisphere's winter.

Why not a circle? Newton's explanation

Kepler found the ellipse empirically, but Newton proved it theoretically in his Principia (1687). Using calculus and his law of universal gravitation — which states that the gravitational force is proportional to 1/r² — Newton showed that any bound orbit under such a force must be an ellipse (or in special cases, a parabola or hyperbola for unbound objects). The ellipse is therefore not a coincidence; it is the inevitable geometric consequence of an inverse-square attractive force. This deep connection unified terrestrial and celestial mechanics for the first time.

The Key Equation

r = \frac{a(1 - e^2)}{1 + e\cos\theta}

r — orbital radius (m (or AU)): Distance from the Sun (at the focus) to the planet at angle θ
a — semi-major axis (m (or AU)): Half the length of the longest diameter of the ellipse; sets the overall size of the orbit
e — eccentricity (dimensionless): Measure of how elongated the ellipse is; 0 for a circle, approaching 1 for a very elongated ellipse
\theta — true anomaly (degrees or radians): Angle measured at the Sun between the perihelion direction and the planet's current position

Worked Example

Problem. Mars has a semi-major axis of a = 1.524 AU and an orbital eccentricity of e = 0.093. Calculate (a) the perihelion distance and (b) the aphelion distance of Mars from the Sun, giving your answers in AU.

r_{\min} = 1.38\text{ AU},\quad r_{\max} = 1.67\text{ AU}

Answer. Perihelion distance ≈ 1.38 AU; Aphelion distance ≈ 1.67 AU

Figure

Diagram

Origins · 1 minJohannes Kepler and how this was discovered

Johannes Kepler

1571–1630 · German

Kepler formulated his three laws of planetary motion between 1609 and 1619, using the extraordinarily precise observational data of his mentor Tycho Brahe. His First Law — that planets orbit the Sun in ellipses — overthrew two millennia of belief in circular celestial motion and provided the geometric framework that Newton would later explain dynamically. Kepler's work was the essential bridge between the observational astronomy of Brahe and the theoretical mechanics of Newton. Kepler suffered from severe myopia and multiple vision problems caused by a bout of smallpox in childhood, making his life's work of precision astronomical analysis even more remarkable. He also had to spend considerable effort defending his mother against charges of witchcraft — a court case that dragged on for six years.

Kepler's Second Law — The Second Law (equal areas in equal times) directly follows from the elliptical orbit shape and conservation of angular momentum, completing the description of how fast a planet moves at each point in its elliptical path.
Kepler's Third Law — The Third Law links the orbital period to the semi-major axis (T² ∝ a³), connecting the geometry of the ellipse defined by the First Law to the timescales of planetary motion.
Newton's Law of Universal Gravitation — Newton proved mathematically that an inverse-square gravitational force is precisely the cause of elliptical orbits, providing the physical explanation for Kepler's empirical First Law.
Orbital Mechanics and Satellite Motion — Artificial satellites follow the same elliptical orbit principles as planets; understanding Kepler's First Law is essential for calculating satellite trajectories and geostationary orbits.