Kepler’s Laws

Newton’s generalization of Kepler’s laws

Applications of the generalized forms of Kepler’s laws

Resources

German astronomer and mathematician Johannes Kepler (1571–1630) made it his life’3s work to create a heliocentric (sun-centered) model of the solar system that would accurately represent the observed motion in the sky of the moon and planets over many centuries. Models using many geometric curves and surfaces to define planetary orbits, including one with the orbits of the six known planets fitted inside the five perfect solids of Pythagoras, failed.

Kepler was able to construct a successful model with the Earth, the third planet out from the sun after more than a decade of this trial and error. His model is defined by the three laws named for him. He published the first two laws in 1609 and the last in 1619. They are:

1. The orbits of the planets are ellipses with the sun at one focus (F1) of the ellipse.
2. The line joining the sun and a planet sweeps out equal areas in the planet’s orbit in equal intervals of time.
3. The squares of the periods of revolution “P„ (the periods of time needed to move 360º) around the sun for the planets are proportional to the cubes of their mean distances from the sun. This law is sometimes called Kepler’s Harmonic Law. For two planets, planet A and planet B, this law can be written in the form:

A planet’s mean distance from the sun (a) equals the length of the semi-major axis of its orbit around the sun.

Kepler’s three laws of planetary motion enabled him and other astronomers to successfully match centuries-old observations of planetary positions to his heliocentric solar system model and to accurately predict future planetary positions. Heliocentric and geocentric (Earth-centered) solar system models that used combinations of off-center circles and epicycles to model planetary orbits could not do this for time intervals longer than a few years; discrepancies always arose between predicted and observed planetary positions.

Newton’s generalization of Kepler’s laws

The fact remained, however, that, in spite of Kepler’s successful modeling of the solar system with his three laws of planetary motion, he had discovered them by trial and error without any basis in physical law. More than 60 years after Kepler published his third law, English physicist and mathematician Sir Isaac Newton (1642—1727) published his Principia, in which he developed his three laws of motion and his theory and law of universal gravitation. By using these laws, Newton was able to derive each of Kepler’s laws in a more general form than Kepler had stated them, and, moreover, they were now based on physical theory. Kepler’s laws were derived by Newton from the basis of the two-body problem of celestial mechanics. They are:

1. The orbits of two bodies around their center of mass (barycenter) are conic sections (circles, ellipses, parabolas, or hyperbolae) with the center of mass at one focus of each conic section orbit. Parabolas and hyperbolas are open-ended orbits, and the period of revolution (P) is undefined for them. One may consider a circular orbit to be a special case of the ellipse where the two foci of the ellipse, F1 and F2, coincide with the ellipse’s center (C), and the ellipse becomes a circle of radius (a).
2. The line joining the bodies sweeps out equal areas in their orbits in equal intervals of time. Newton showed that this generalized law is a consequence of the conservation of angular momentum (from Newton’s third law of motion) of an isolated system of two bodies unperturbed by other forces.
3. From his law of universal gravitation, which states that two bodies of masses, M1 and M2, whose centers are separated by the distance “r” experience equal and opposite attractive gravitational forces (Fg) with the magnitude

Where G is the Newtonian gravitational factor, and from his Second Law of Motion, Newton derived the following generalized form of Kepler’s third law for two bodies moving in elliptical orbits around their center of mass where p is the ratio of the circumference of a circle to its diameter, “a” is the semi-major axis of the relative orbit of the body of smaller mass, M2, around the center of the more massive body of M1.

Some of the applications of these generalized Kepler’s laws are briefly discussed below.

Applications of the generalized forms of Kepler’s laws

First consider applications of Kepler’s third law in the solar system. Let M1 represent the sun’s mass and M2 represent the mass of a planet or another object orbiting the sun. Then, adopt the sun’s mass (M1 = 1.985 x 1030 kg) as the unit of mass, the astronomical unit (a.u.; 1 a.u. = 149,597,871 km) as the unit of length, and the sidereal year (365.25636 mean solar days) as the unit of time. Then (4π2/G) = 1, (M₁ + M₂) = 1 (where the planet masses M2 except those of the Jovian planets in the most precise calculations can be ignored), and the formidable equation above is reduced to the simple algebraic equation P2 = a3 where “P” is in sidereal years and “a” is in astronomical units for a planet, asteroid, or comet orbiting the sun. Approximately the same equation can be found from the first equation if the Earth is allowed to be Planet B, since FB = 1 sidereal year and aB is always close to 1 a.u. for Earth.

Return to the generalized form of Kepler’s third law and apply it to planetary satellites; except for Earth-moon and Pluto-Charon systems (these are considered double planets or double dwarf planets, respectively), one may neglect the satellite’s mass (M2 =0). Then, solving the equation for M1,

Measurements of a satellite’s period of revolution (P) around a planet and of its mean distance “a” from the planet’s center enable one to determine the planet’s mass (M1). This allowed accurate masses and mean densities to be found for Mars, Jupiter, Saturn, Uranus, and Neptune. The recent achievements of artificial satellites of Venus have enabled the mass and mean density of Venus to be accurately found. Also the total mass of the Pluto-Charon double dwarf planet system has been determined.

Now consider the use of Kepler’s laws in stellar and galactic astronomy. The equation for Kepler’s third law has allowed masses to be determined for double stars for which “P” and “a” have been determined. These are two of the orbital elements of a visual double star; they are determined from the double star’s true orbit. Kepler’s second law is used to select the true orbit from among the possible orbits that result from solutions for the true orbit using the double star’s apparent orbit in the sky. The line joining the two stars must sweep out equal areas in the true and apparent orbits in equal time intervals (the time rate of the line’s sweeping out area in the orbits must be constant). If the orbits of each star around their center of mass can be determined, then the masses of the individual stars can be determined from the sizes of these orbits. Such double stars give the only accurate information about the masses of stars other than the sun, which is very important for the understanding of star structure and evolution.

In combination with data on the motions of the sun, other stars, and interstellar gases, the equation for Kepler’s third law gives estimates of the total mass in the Milky Way galaxy situated closer to its center than the stars and gas studied. If total mass (M1 + Ms) is constant, the equation predicts that the orbital speeds of bodies decrease with increasing distance from the central mass; this is observed for planets in the solar system and planetary satellites. The orbital speeds of stars and gas further from the center of the Milky Way than the sun are about the same as the sun’s orbital speed. They do not decrease with distance from the center, which indicates that much of the Milky Way’s mass is situated further from the center than the sun. This information has led to a large upward revision of the Milky Way’s total estimated mass. Similar estimates of the mass distributions and total masses of other galaxies can be made. The results allow estimates of the masses of clusters of galaxies; from this, estimates are made of the total mass and mean density of detectable matter in the observable part of the universe, which is important for cosmological studies.

When two bodies approach on a parabolic or hyperbolic orbit, if they do not collide at their closest distance (pericenter), they will then recede from each other indefinitely. For parabolic orbit, the relative velocity of the two bodies at an infinite distance apart (infinity) will be zero, and for a hyperbolic orbit their relative velocity will be positive at infinity (they will recede from each other forever).

The parabolic orbit is important in that a body of mass M₂ that is insignificant compared to the primary mass, M₁ (M₂=0) that moves along a parabolic orbit has just enough velocity to reach infinity; there it would have zero velocity relative to M₁. This velocity of a body on a parabolic orbit is sometimes called the parabolic velocity; more often it is called the escape velocity. A body with less than escape velocity will move in an elliptical orbit around M1; in the solar system a spacecraft has to reach velocity to orbit the sun in interplanetary space. Some escape velocities from the surfaces of solar system bodies (ignoring atmospheric drag) are 2.4 km/sec for the moon, 5 km/sec for Mars, 11.2 km/sec for the Earth, 60 km/sec for the cloud layer of Jupiter. The escape velocity from Earth’s orbit into interstellar space is 42 km/sec. The escape velocity from the sun’s photosphere is 617 km/sec, and the escape velocity from the photosphere of a white dwarf star with the same mass as the sun and a photospheric radius equal to Earth’s radius is 6,450 km/sec.

The last escape velocity is 0.0215 the vacuum velocity of light, 299,792.5 km/second, which is one of the most important physical constants and, according to the theory of relativity, is an upper limit to velocities in the Milky Way’s part of the universe. This leads to the concept of a black hole, which may be defined as a volume of space where the escape velocity exceeds the vacuum velocity of light. A black hole is bounded by its Schwarzchild radius, inside which the extremely strong force of gravity prevents everything, including light, from escaping

KEY TERMS

Conic section —A conic section is a figure that results from the intersection of a right circular cone with a plane. The conic sections are the circle, ellipse, parabola, and hyperbola.

Double star —A gravitationally bound system of two stars that revolve around their center of mass in elliptical orbits.

Jovian planets —Jupiter, Saturn, Uranus, and Neptune. They are the largest and most massive planets.

Mass —The total amount of matter (sum of atoms) in a material body.

Mean density —The mass of a body divided by its volume.

Volume —The amount of space that a material body occupies.

White dwarf —A star that has used up all of its thermonuclear energy sources and has collapsed grav-itationally to the equilibrium against further collapse that is maintained by a degenerate electron gas.

to the universe outside. Light and material bodies can fall into a black hole, but nothing can escape from it, and theory indicates that all we can learn about a black hole inside its Schwarzschild radius is its mass, net electrical charge, and its angular momentum. The Schwarzschild radii for the masses of the sun and the Earth are 2.95 km and 0.89, respectively. Black holes and observational searches for them have recently become very important in astrophysics and cosmology.

Hyperbolic orbits have become more important since 1959, when space technology had developed enough so that spacecraft could be flown past the moon. Spacecraft follow hyperbolic orbits during fly-bys of the moon, the planets, and of their satellites.

See also Satellite.

Resources

BOOKS

Arny, Thomas. Explorations: An Introduction to Astronomy. Boston, MA: McGraw-Hill, 2006.

Aveni, Anthony F. Uncommon Sense: Understanding Nature’s Truths Across Time and Culture. Boulder, CO: University Press of Colorado, 2006.

Chaisson, Eric. Astronomy: A Beginner’s Guide to the Universe. Upper Saddle River, NJ: Pearson/Prentice Hall, 2004.

Ferguson, Kitty. The Nobleman and his Housedog: Tycho Brahe and Johannes Kepler, The Strange Partnership that Revolutionized Science. London, UK: Review, 2002.

Frederick R. West