Planetary orbits

Planets in the solar system revolve around the sun in elliptical orbits at speeds that vary with distance from the sun. Laws that govern these motions were first deduced by Johannes Kepler and later quantified by Sir Isaac Newton.

Overview

Planets in the solar system move around the sun in elliptical orbits. Those whose orbits are closest to the sun move more rapidly than those that are farther away. These simple, universally accepted observations form the basis of the knowledge of planetary motions. Gravity—the force that causes apples to fall from trees and keeps humans firmly planted on Earth’s surface—plays the central role in the mechanics of planetary motions.

A simple experiment illustrates the energy relationships inherent in orbiting bodies. If a person attaches a string to a small rubber ball and the ball is swung around the person’s head in a horizontal circle, the tension along the string that holds the ball in its “orbit” is analogous to the sun’s gravity pulling on a bound planet. The English astronomer and mathematician Sir Isaac Newton (1642–1727) explained how the force of gravity affects planetary motion. Newton proved in his laws of motion that once an object is in a straight-line motion, it will continue on that course with no further input of energy (law of inertia) unless its motion is interrupted by an unbalanced force. In the case of planets, this force is provided by the gravitational attraction of the sun (or a more massive planet, in the case of a satellite). Depending upon the magnitude of the orbiting body’s “kinetic energy” (energy of motion), the body will move in either a circular orbit or, with greater kinetic energy, an elliptical orbit. Kinetic energy counters the attractive force of gravity, thus preventing a planet from falling into the sun, or the orbiting ball, as shown in the example, from striking the experimenter.

The scientist who first showed that the orbits of the planets are actually ellipses rather than circles was Johannes Kepler (1571–1630). A German mathematician, astronomer, and astrologer, Kepler worked previously as an assistant to the Danish observational astronomer Tycho Brahe. After Brahe’s death, Kepler used detailed position measurements of the planet Mars to plot an orbit that was not circular. Up to this time, planetary orbits—including that of the moon—were believed to be circular in accordance with precepts developed by the Greek philosopher Aristotle.

A circle is the locus of points all the same distance from a given center. An ellipse differs from a circle in being oval-shaped. An ellipse contains two internal, evenly spaced points called foci. It is important to understand how the foci of an elliptical orbit relate to the positions of an orbiting planet and the sun. This relationship is expressed by Kepler’s first law: each planet moves around the sun in an orbit whose shape is that of an ellipse, with the sun at one focus. The other focus is empty. Thus, the sun is not precisely in the middle of the ellipse but displaced somewhat to the side. The degree of displacement determines the orbit’s eccentricity. As a result, planets move between a minimum distance from the sun in their orbit, called perihelion, and a maximum distance from the sun, called aphelion. Planetary orbits have this repetitive pattern as a result of the central character of gravity; that is, gravity acts along the line between the gravitationally interacting bodies. The magnitude of the gravitational force follows an inverse square law with regard to its dependence upon distance between the interacting masses. If one doubles the distance between the two objects, their gravitational attraction diminishes not by a factor of two, but by four.

Kepler’s second law was actually discovered before his first law. It describes the fact that planets move more slowly when farther away from the sun (their slowest speed is at aphelion). They also move more rapidly when closer to the sun (their maximum speed is at perihelion). This observation logically would lend support to the idea that the planet’s orbit is anything but circular. The second law states that a straight line joining the planet and the sun sweeps out equal areas in space in equal intervals of time. Imagine a string attached to a planet at one end and the sun at the other. When the planet is near aphelion (farthest from the sun), it moves slowly, so that the triangular sector swept out by the string during a given time will resemble a long, slender piece of pie. In contrast, near perihelion over the same time period, the planet will move farther (because it is going faster), so that the sector swept out by the string resembles a fatter slice of pie. Kepler’s second law states that these two pie slices, or triangular sectors—although quite different in radius and opening angle—should have equal areas. This exercise is a mathematical way of stating that planets move more slowly as the sun-planet distance increases. Planetary orbits obey Kepler’s second law of motion as a consequence of conservation of angular momentum.

Kepler’s third law, formulated in 1619, was an attempt to quantify the fact that a planet moves more slowly the farther its orbit is from the sun. His task was to determine a precise mathematical relationship between a planet’s average distance from the sun and its period. Being oval, ellipses have a major axis and a minor axis of different lengths. A line passing through the two foci of the ellipse and ending at both ends of the figure defines the long axis of the ellipse and is known as the major axis. A length equal to one-half the major axis is called the semimajor axis. A line perpendicular to the major axis passing halfway between the two foci of the ellipse is the minor axis. A length equal to one-half of the minor axis is called the semiminor axis. A planet’s mean distance is half the sum of the perihelion and aphelion distances. This is equal to the average distance of a planet from the sun and also is the value of the semimajor axis. Kepler found that the cube of the mean distance for any planet is equal to the square of that planet’s period. This equation is expressed mathematically as p² = r³, where p is the planet’s period in Earth years and r is the planet’s mean distance from the sun . This is expressed in terms relative to the earth’s mean distance, 150 million kilometers, or one astronomical unit (AU). If the earth’s mean distance equals 1.0 AU, then Mars’s mean distance is 1.5 AU, Venus’s mean distance is 0.72 AU, and so on. Planetary orbits obey this third Keplerian law of motion as a result of the central character of gravity as well as its inverse-square-law nature of gravity.

Newton later reformulated Kepler’s three laws using more sophisticated mathematics than was available to Kepler. Newton’s modification of the first law states that each planet has an elliptical orbit with the center of mass between it and the sun at one focus. The “center of mass” is a point between the two bodies (the sun and the orbiting planet) where their masses are essentially balanced. Mathematically, it is the point at which the product of mass times length is equal for the two bodies: M₁L₁ = M₂L₂, where M = mass,L = length from the center of mass, and subscripts 1 and 2 referring to bodies 1 and 2. The sun is such an extremely massive body that its center of mass with any planet lies near the sun’s own center. Therefore, the sun does lie essentially at a focus of the planetary ellipse, as Kepler stated. Its movement around the center of mass (deep within its interior) is detectable only as a slight wobble. For bodies that are more comparable to one another in terms of mass, such as pairs of stars, these objects actually revolve around a common point that lies between them. Pluto and its similarly sized satellite Charon provide a good example of that effect. Because the masses of these bodies are similar, they revolve around a common point known as the barycenter.

Newton revised Kepler’s second law as follows: Angular momentum in a two-body system is constant when no net external torque is present. This law originally described the fact that planets move more rapidly when they are closer to the sun compared to when they are farther away. Newton found that all bodies that rotate or move around some center have “angular momentum.” This quantity is expressed as a body’s mass times its speed times its distance from the center of mass (mvr, where m = mass, v = linear speed, and r = distance from the center of mass). Because angular momentum is constant for any two-body system in the absence of a net external torque, if r becomes greater, v must become smaller to compensate (mass always remains constant). On the other hand, near the center of mass (the sun, for planets), the distance r is diminished and speed v must increase to compensate. Conservation of angular momentum comes into play when a spinning skater pulls her outstretched arms close to her body, initiating a more rapid spin rate. Physicists and astronomers usually talk about planetary speeds of revolution or more properly angular velocity, which is the linear speed per unit distance from the focus. In such a discussion, angular momentum then involves the product of the moment of inertia times angular velocity. There is no net torque acting on the planet revolving about the sun, so this angular momentum expression is conserved or remains constant. That means that the distance from the sun squared times angular velocity is an invariant throughout the planet’s orbital motion.

Newton’s revision of Kepler’s third law is especially important. Newton discovered that the sum of the masses of the two bodies times the square of the period is proportional to the cube of the mean distance, which is expressed mathematically as (M₁ + M₂)P₂ = a₃. The masses must be expressed as a fraction of the sun’s mass for the calculation to be valid. The immediate consequence of this equation is that astronomers could now use this equation to calculate the masses of distant bodies given information on the mean distance and period of the orbiting bodies. In most instances, the mass of the smaller body (planet or satellite of a planet) may be neglected because that mass is so insignificant compared to the sun’s mass (1.99 10³⁰ kilograms, or 332,943 times Earth’s mass). Rearranging the equation gives M₁ = a₃/p₂. This equation can now be used to calculate the mass M₁ of any central body that has a satellite of mass M₁.

Applications

The impact of Kepler’s and Newton’s laws of planetary motion and gravity on the scientific world on the scientific world was profound not only during their own time but to this very day. The results of their work continue to be used by astronomers to solve problems. For example, the flight path of the Apollo astronauts to the moon and back was calculated using all three of Kepler’s laws. The energy required to propel the Saturn 5 rocket on its way and later to orbit the moon was calculated using Newton’s laws of gravity. The same can be said for all interplanetary spacecraft, such as Voyagers 1 and 2, which visited and photographed the outer planets, Jupiter, Saturn, Uranus, and Neptune. The two Voyager probes were assisted in their journeys by using the gravitational attraction of these massive planets to accelerate them toward their next target. Calculating gravity assists involves kinetic energy and gravitational relationships developed by Newton.

One of the most useful of Kepler’s laws for planetary astronomers is the third law as modified by Newton. This law allows the calculation of the mass of a massive body using data about the mean distance and period of one or more of its satellites. It has been used to calculate the masses of all planets that have satellites (which excludes Mercury and Venus). One of the most difficult mass determinations was that for the dwarf planet Pluto and its satellite Charon. These bodies are so far away from Earth that Charon was discovered only in 1977. Its orbital characteristics were determined, with great difficulty, sometime later. The similar masses of Charon and Pluto cause them to orbit a center of mass (a barycenter) that lies nearly halfway between them, but the location of that Barycenter is somewhat closer to Pluto than it is to Charon. The third law was used to calculate both the mass of Pluto, using data from Charon’s orbit, and the mass of Charon, using data ffom Pluto. These calculations show that both bodies have very low masses and are most likely composed of methane ice.

Another important consequence of the laws of planetary motion involves the survival of life on the Earth. One theory suggests that periodic mass-extinction events—such as the demise of the dinosaurs—may have been caused by gigantic impacts of asteroids (rocky planetoids with diameters of less than one thousand kilometers) or comets (asteroid-sized ice balls) with the Earth. In the solar system, most asteroids are concentrated in a belt between Mars and Jupiter, while most comets originate in the outer regions of the solar system and beyond. Occasionally, collisions or gravitational perturbations from the massive gas giant planets, such as Jupiter or Saturn, cause asteroids and comets to assume orbits that carry them near the Earth. These bodies all have sufficient kinetic energy to resist the Earth’s gravitational attraction, so that objects that graze the Earth’s orbit continue by without going into orbit around the Earth. This fact explains why the Earth and other relatively low-mass planets have few or no satellites (while the gas giants—Jupiter, Saturn, Uranus, and Neptune—have many). Therefore, the bodies that do strike the Earth, causing extinctions and making huge craters if they strike land areas, must make a direct hit of a moving Earth. The chances of that occurring on a frequent basis fortunately are rather low, but not zero. Given the billions of years of the history of the earth, it is probable that an occasional body will crash into the Earth with catastrophic consequences. The high kinetic energy of these bodies is converted into heat and shock waves upon impact, causing considerable destruction. Newton’s laws of gravity and motion play a pivotal role, mostly in determining the trajectories of these dangerous visitors to the inner solar system. By the same token, Newton’s laws reveal ways that gravity could be ingeniously used to push possible impacting bodies away from a trajectory that otherwise would have them intersect with the Earth, thereby averting a possibly cataclysmic collision.

Context

The history of science closely parallels the development of astronomy in that the study of heavenly bodies and their relationship to the Earth dominated philosophical and religious thinking for millennia. One of the first scientists to study religious thinking and astronomical phenomena seriously was the Greek philosopher Aristotle (384–322 BCE). Unlike most of his contemporaries, Aristotle used some observations to prove his speculations. His major contribution to planetary motion studies was his belief that the natural state of matter is to seek the center of the Earth, which is why objects always fall when released above the Earth. Although erroneous, this and related ideas laid the groundwork for later studies by Galileo and Newton on the effects of gravity. Aristotle also believed, as did many others, that the Earth was at the center of the universe. That the sun and planets revolved around the Earth in perfectly circular orbits was advocated first by his great mentor, Plato. Later, Aristarchus (ca. 270 BCE), a Greek astronomer, adopted the idea that the sun is at the center of the known universe. That idea was forgotten until revived nearly two thousand years later by Nicolaus Copernicus, whose “heliocentric” model, published shortly after his death in a volume titled De revolutionibus orbium coelestium (1543; On the Revolutions of the Heavenly Spheres, 1952; better known as De revolutionibus), describes a system in which the planets orbit the sun in perfect circles. Although not completely accurate, the heliocentric model eventually supplanted the Earth-centered model of Aristotle and other philosophers.

In the middle of the second century CE, Ptolemy (ca. 100–178 CE), wrote a text called Mathēmatikē syntaxis (ca. 150 CE; Almagest, 1948) in which he summarized all that was known about astronomy up to that time. This book influenced astronomical thinking for the next millennium. It included a model of the solar system that was quite accurate in predicting planetary positions. Using an idea first developed by Apollonius of Perga (ca. 240–170 BCE), Ptolemy declared that planets move in perfect circles aroundthe Earth. Nevertheless, planets moved simultaneously in smaller circles called “epicycles.” These were necessary to explain why the outer planets occasionally seemed to reverse their normally eastern motion relative to the stars and move west. One of the great triumphs of Kepler’s laws is that they provide an explanation for retrograde motion (the Earth moves faster and overtakes the outer planets). Kepler, and before him, Copernicus, laid the foundations for a scientific understanding of planetary motions that broke the hold on thinking imposed by the Almagest.

Sir Isaac Newton used ideas developed by Galileo and Kepler to quantify the knowledge of planetary motion and to explain these motions in terms of gravitational forces and kinetic energies. He published his ideas in Philosophiae Naturalis Principia Mathematica (1687; The Mathematical Principles of Natural Philosophy, 1729; best known as the Principia). Although it is now known that Newton’s laws do not work well on atomic and subatomic scales (treated in the discipline of quantum mechanics) or in cases where bodies are moving relative to one another at very high speeds close to that of light (which later were addressed by Albert Einstein’s relativity theory), Newton’s laws work perfectly under everyday conditions on Earth and in the solar system.

In 2009, the National Aeronautics and Space Administration (NASA) honored Johannes Kepler in naming a powerful space telescope in his honor. The Kepler mission imaged selected sections of the galaxy with the intent to locate exoplanets outside the Earth's solar system. Before the telescope was decommissioned in 2018, the Kepler telescope was credited with having identified more than 2,600 previously unidentified exoplanets.

Newton’s and Kepler’s time-honored laws continue to be used by astronomers and other space scientists to make predictions about planetary motions and interactions. Relativity is needed, however, to explain the advance of Mercury’s perihelion as it orbits so close to the sun. Spacecraft deep in the gravitational well of the sun likewise requires calculations involving relativity to maintain them on their proper courses.

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