There are a few things you may find worth observing. You will learn more about them when you study orbital energy. First, what is the change

in the speed of the rocket after firing its rear (Forward thrust) engine? What happens to its speed after a few moments? If youqualitatively

consider the total energy of the ship, can you explain you what is going on? You may want to consider it akin to what happens if you throw a

ball straight up into the sky.

If you cannot dock but are able to leave and return to the initial circular orbit, you can consider your mission achieved.

12.13 - Kepler’s first law

The law of orbits: Planets move in elliptical

orbits around the Sun.

The reason Newton’s comparison of the Moon’s motion to the motion of an apple was

so surprising was that many in his era believed the orbits of the planets and stars were

“divine circles:” arcs across the cosmic sky that defied scientific explanation. Newton

used the fact that the force of gravity decreases with the square of the distance to

explain the geometry of orbits.

Scientists had been proposing theories about the nature of orbits for centuries before

Newton stated his law of gravitation. Numerous theories held that all bodies circle the

Earth, but subsequent observations began to point to the truth: the Earth and other

planets orbit the Sun. The conclusion was controversial; in 1633 the Catholic Church

forced Galileo to repudiate his writings that implied this conclusion.

Even earlier, in 1609, the astronomer and astrologer Johannes Kepler began to

propose what are now three basic laws of astronomy. He developed these laws through

careful mathematical analysis, relying on the detailed observations of his mentor, Tycho

Brahe, a talented and committed observational astronomer.

Kepler and Brahe formed one of the most productive teams in the history of astronomy.

Brahe had constructed a state of the art observatory on an island off the coast of

Denmark. “State of the art” is always a relative term í the telescope had not yet been

invented, and Brahe might well have traded his large observatory for a good pair of

current day binoculars. However, Brahe’s records of years of observations allowed

Kepler, with his keen mathematical insight, to derive the fundamental laws of planetary

motion. He accomplished this decades before Newton published his law of gravitation.

Kepler, using Brahe’s observations of Mars, demonstrated what is now known as

Kepler’s first law. This law states that all the planets move in elliptical orbits, with the

Sun at one focus of the ellipse.

The planets in the solar system all move in elliptical motion. The distinctly elliptical orbit

of Pluto is shown to the right, with the Sun located at one focus of the ellipse. Had Kepler been able to observe Pluto, the elliptical nature of

orbits would have been more obvious.

The other planets in the solar system, some of which he could see, have orbits that are very close to circular. (If any of them moved in a

perfectly circular orbit, they would still be moving in an ellipse, since a circle is an ellipse with both foci at its center.) Some of the orbits of these

other planets are shown in Concept 2.

Kepler’s first law

Planets move in elliptical orbits with the

Sun at one focus

Solar system

Most planetary orbits are nearly circular

12.14 - More on ellipses and orbits

The ellipse shape is fundamental to orbits and can be described by two quantities: the

semimajor axis a and the eccentricity e. Understanding these properties of an ellipse

proves useful in the study of elliptical orbits.

The semimajor axis, represented by a, is one-half the width of the ellipse at its widest,

as shown in Concept 1. You can calculate the semimajor axis by averaging the

maximum and minimum orbital radii, as shown in Equation 1.

The eccentricity is a measure of the elongation of an ellipse, or how much it deviates

from being circular. (The word eccentric comes from “ex-centric,” or off-center.)

Mathematically, it is the ratio of the distance d between the ellipse’s center and one

focus to the length a of its semimajor axis. You can see both these lengths in Equation

2. Since a circle’s foci are at its center, d for a circle equals zero, which means its
   eccentricity equals zero.
   Pluto has the most eccentric orbit in our solar system, with an eccentricity of 0.25, as
   calculated on the right. By comparison, the eccentricity of the Earth’s orbit is 0.0167.
   Most of the planets in our solar system have nearly circular orbits.
   Comets have extremely eccentric orbits. This means their distance from the Sun at the
   aphelion, the point when they are farthest away, is much larger than their distance at the perihelion, the point when they are closest to the Sun.

Elliptical orbits

Semimajor axis: one half width of orbit

at widest

Eccentricity: elongation of orbit

(^232) Copyright 2000-2007 Kinetic Books Co. Chapter 12