Kepler’s Laws

FIELDS OF STUDY: Classical Mechanics

ABSTRACT: This article defines Johannes Kepler’s three laws of orbital motion, which describe the motions of planets around a sun-centered solar system. Isaac Newton later developed the universal law of gravitation that gave a more general explanation of planetary motion. These laws can be applied to the motion of any satellite orbiting a massive central body.

Principal Terms

• Cartesian coordinates: a pair or triplet of numbers that indicate the location of a point in a plane or in a three-dimensional space and are the signed distances from the origin.
• centripetal force: the force acting on an object moving along a circular path that is directed toward the center of the path or axis of rotation.
• ellipse: an oval surrounding two focal point points where the sum of the distances from any point on the oval to the two focal points is constant.
• Lagrangian points: positions where the gravitational pull of two large masses equals the centripetal force needed for a small object to maintain a stable position in orbit with the two large masses.
• perturbation: a disturbance in the motion or orbit of a massive body due to the gravitational pull of or impact with another object.
• polar coordinates: a pair of numbers indicating a point’s length (radius) from a fixed center, called the pole or origin, and the angle between the radial and the polar axes.
• two-body problem: a mathematical description of the motion in space of two rigid, point-like objects interacting with each other.

Kepler’s Laws

German astronomer Johannes Kepler (1571–1630) used observations collected by Danish astronomer Tycho Brahe (1546–1601) to develop his three laws of orbital motion. Kepler’s first law states that planets move in elliptical orbits with the sun at one focus point. More generally, the law says that one object moves around another object in an elliptical orbit with the system’s center of mass located at one focus. An orbit can be described by its semimajor axis and eccentricity. In an ellipse, the long axis is referred to as the "major axis" and the short is referred to as the "minor axis." The semimajor axis is half the length of the major axis. Eccentricity describes the shape of an object’s orbit, with an eccentricity of zero referring to a circular orbit and an eccentricity of 1 being very flat orbit. The eccentricities of the orbits of the planets in the sun’s solar system are small (almost circular).

Kepler’s second law states that the line connecting a planet and the sun sweeps out equal areas in equal amounts of time. A planet moves slower as it travels farther from the sun and faster as it travels closer to the sun. The point where the planet is closest to the sun is called the perihelion, and the point where it is farthest, the aphelion. The aphelion and perihelion distances (R_a and R_p) can be related to the semimajor axis (a) and the eccentricity (e) of an orbit by the following equations:

R_a = a(1 + e)

R_p = a(1 − e)

Kepler’s third law states that the square of a planet’s orbital period (T) is proportional to the cube of the orbit’s semimajor axis (a), written as T² ∝ a³.

A planet’s orbital period is the time it takes for a planet to complete a full orbit around the sun. Thus, as the semimajor axis of an orbit increases, the more time it takes the planet to orbit the sun. Also, the ratio of the square of a planet’s orbital period to the cube of the orbit’s semimajor axis has the same value for every planet in the solar system.

Newton’s Version

Newton combined gravitation and circular acceleration to obtain a relation that included period, distance, and a central body’s mass. The attractive gravitational force (F_g) between two objects with masses (m₁) and (m₂) that have distance (r) between them is:

This equation is referred to as the universal law of gravitation. The constant G is the gravitational constant and is equal to 6.67384 × 10⁻¹¹ m³/kg·s². The centripetal force (F_cp) needed for a planet to orbit the sun is the product of the velocity (v) and mass of the planet (m₂), divided by its orbital radius (r).

The velocity of the planet (v) is equal to the circumference of the orbit (2πr) divided by the period (T).

Substituting this value for v in the equation for centripetal force, one arrives at:

The centripetal force is then set equal to the gravitational force, with m₁ as the mass of the sun and m₂ as the mass of the planet.

Rearranging terms and eliminating m2, one arrives at the following:

For circular motion, the semimajor axis (a) replaces the distance (r) and the expression can be written simply as T² = a³.

This relationship between period and semimajor axis was predicted by Kepler’s laws. However, Kepler’s laws apply only approximately to the solar system, since the mass of the sun is so much greater than the mass of the planets. A more precise version of the law can be written as follows:

Newton’s model better fits real observations and two-body problems, such as the orbit of a moon around a planet or a binary star system. The units for the semimajor axis of an orbit are normally expressed in kilometers (km) or astronomical units (AU), with the units for period expressed in days. An astronomical unit is about the average distance from Earth to the sun and is equal to about 149.6 × 10⁶ kilometers (about 93 million miles).

Perturbations are differences in the motion of a planet from Kepler’s laws that result from gravitational attraction by other planets or, sometimes, collisions. If another body is added to a system of two bodies, there are positions where the satellite can maintain a stable position. For example, if a satellite is added to the system of Earth and sun, the gravitational pull of the sun and Earth produces the centripetal force needed for the satellite to orbit with them. There are five Lagrangian points in the orbital plane, referred to as L1, L2, L3, L4, and L5. The points L1, L2, and L3 are positioned on a line that connects Earth and sun. L4 and L5 are located at the tip of two equilateral triangles with the sun and Earth at the other points. L1, L2, and L3 are unstable points, while L4 and L5 are very stable. A satellite positioned at L1, L2, and L3 will quickly be thrown off orbit.

There are different coordinate systems and methods for labeling points in a plane, including Cartesian coordinates and polar coordinates. Polar coordinates describe planetary motion best. A planet’s position can be described by the radial distance between the planet and sun (r) and the angle between the planet and sun (θ).

Sample Problem

Using Kepler’s laws, calculate the Earth’s farthest (aphelion) and closest (perihelion) distances from the sun if Earth’s semimajor axis is 149.6 x 10⁶ km and its eccentricity is 0.0167.

Answer:

To calculate the farthest and closest that Earth travels to the sun, use the equations below that relate aphelion and perihelion distances, semimajor axis, and eccentricity.

R_a = a(1 + e)

R_p = a(1 – e)

Plug in the values for Earth’s semimajor axis (a) and eccentricity (e). Then compute.

R_a = a(1 + e)

R_a = 149.6 × 10⁶ km (1 + 0.0167)

R_a = 152.1 × 10⁶ km

R_p = a(1 − e)

R_a = 149.6 × 10⁶ km (1 − 0.0167)

R_a = 147.1 × 10⁶ km

The farthest Earth gets from the sun is 152.1 x 10⁶ km, and the closest Earth gets to the sun is 147.1 x 106 km. The slightly elliptical nature of Earth’s orbit changes its distance from the sun at certain points along its travel.

Note that these distances do not coincide with warm- or cold-weather periods in either the Northern or the Southern Hemisphere. Seasons are determined by Earth’s tilt on its axis, not by its distance from the sun.

Impact of Kepler’s Laws

Kepler developed laws of orbital motion that made it apparent that the sun and the other planets do not orbit Earth, as was claimed by the Greek Egyptian astronomer Ptolemy (ca. 100–ca. 170 CE). Instead, Earth and other planets orbit the sun. Kepler’s laws described the motion of the planets but not why the planets moved in this manner. Isaac Newton (1642–1727) improved upon Kepler’s laws by showing that those laws are directly connected to the law of gravitation. These laws are now applied to not only planets orbiting the sun, but also to natural and artificial satellites orbiting Earth or other celestial bodies.

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