Chapter 2: Orbits and Gravity

This chapter is devoted to gaining a deeper understanding of orbits in general, but planetary orbits in particular. Gravity is the force that keeps planets and satellites in orbit, and therefore will also get attention.

2.1: The Laws of Planetary Motion

The full understanding of the motions of planets took about another hundred years after Copernicus published his heliocentric model. The work of Kepler was fundamental.

Kepler made a careful study of observations made by Tycho Brahe. For about 20 years Brahe made observations of the Sun, Moon, and planets from the island of Hven in the North Sea, off the coast of Denmark.

The Ellipse

Kepler began with the heliocentric model of the solar system, and looked for simple rules for the orbits of the planets. Kepler's first result was to realize that the orbits about the Sun weren't exactly circles, rather ellipses. An ellipse is a geometric figure closely related to circles. Recall that a circle can be defined as the planar figure where every point is a particular distance, called the radius, from a point called the center of the circle. An ellipse is a planar figure with two special points called the foci (plural of focus). The sum of the distances from the foci to any point on the ellipse is the same value.

Demonstrate how to draw an ellipse with two tacks and a piece of string.

The widest diameter of the ellipse is called its major axis. Half the distance across the widest diameter is called the semimajor axis. The roundness of an ellipse is determined by its eccentricity. The eccentricity is defined as the ratio of the distance between the foci to the length of the major axis.

If the foci are in the same point, then the ellipse becomes a circle, and the eccentricity is zero. (The distance between the foci is zero, so the numerator of the ratio is zero, while the denominator is not, so the result is zero.)

At the other extreme, the ellipse becomes stretched out into a line. In this limit the eccentricity is 1.

For an ellipse that is somewhere between a circle and this straight line, the eccentricity lies between 0 and 1.

Kepler's laws

From Brahe's data, Kepler deduced three laws about the orbits of the planets about the Sun.

I've already discussed most of what's in the first law. The second law reflects the fact that when a planet is in the part of its orbit nearer the Sun, it moves faster than when it is in the part that is further from the Sun. Were the orbits perfect circles, then the planets would always be the same distance from the Sun, and would move at a constant speed. See drawings in text or view a demonstration with a java applet.

The third law can be interpreted as follows: if the period of revolution is measured in years and the semimajor axes in AU, then the following relation holds:

(period)2 = (semimajor axis)3,
or in words, the period squared equals the semimajor axis cubed.

Take Mars as an example. Its period is 1.88 years and its semimajor axis is 1.52 AU.

1.88×1.88 = 3.53, and 1.52×1.52×1.52 = 3.51,
close enough to be deemed equal.

Newton's Great Synthesis

Newton had a number of great achievements, any one of which would have made him a famous scientist. First of all, he developed the calculus, the mathematics that deals with relations between how quantities change. The calculus is needed for the second development, his laws of motion. Newton's laws of motion are the starting point for most of Physics; they allow us to describe the motions of objects under everyday conditions.

Newton's Laws of Motion

Due to their great importance, I will state Newton's laws here:

Since this is not a Physics course, I will not go into further details here.

Universal Gravity

Newton's laws deal with forces acting (or not) on bodies. To understand what types of motion are possible, one must know the forces. The first fundamental force to be elucidated was gravity.

Newton's third, and possibly greatest insight, was to realize that the same force that pulls an apple to the ground (toward the center of the Earth) pulls the Moon toward the center of the Earth. That is, the Moon is constantly falling toward the center of the Earth, but because of its motion around the Earth (its transverse speed to be precise), by the time it falls a significant amount, it has moved away from where it was, such that the combination of the falling and the transverse speed results in an elliptic orbit about the Earth.

The force is referred to as universal gravity because Newton hypothesized that there is a an attraction among all objects. The Earth pulls on the apple, and the apple pulls back on the Earth; the Earth pulls on the Moon, and the Moon pulls back on the Earth (discuss tides later). Everything in the universe is affected by gravity, and exerts a gravitational force on all other objects in the universe.

Newton quantified the amount of force. This is important since it means that we can use the theory to make definite predictions which can be tested with observation. The force of gravity between two objects is proportional to the product of the masses, and inversely proportional to the square of their separation. The famous formula is written:

force = G M1 M2 / R2
where M1 and M2 are the masses of the two objects, R is their separation, and G is the universal constant of gravity. If we measure masses in kilograms (kg), distance in meters (m), and want the force in newtons (N), then G has the value
G = 6.673×10-11 N m2 / kg2.
This value is the same no matter what objects we are using: apples, people, planets, moons, stars, or galaxies.

Orbital Motion and Mass

Using Newton's laws of motion and universal gravity, we can derive Kepler's third law, the relationship between the period of a planet's orbit and its distance from the Sun. Not only do we get Kepler's third law, we get a modification which accounts for the planet's mass:

D3 = (M1 + M2) × P2.
In this formula, M1 is the Sun's mass, and M2 is the planet's mass, both expressed in units of the Sun's mass (that is, M1 = 1). Since the panets' masses are significantly smaller than the Sun's, the sum of the masses is only slightly greater than 1. For Earth the sum is 1.000003, and for Jupiter, the most massive planet, it is 1.001.

Orbits in the Solar System

The path of any object through space is referred to as its orbit. The orbits are ellipses, as noted by Kepler. The orbits are characterized by thir size (semimajor axis), shape (eccentricity), and period of revolution.

Additionally, there are two points along an orbit about the sun that are given special names:

For a satellite orbiting the Earth we use the corresponding terms perigee and apogee. "Helion" comes from the Greek word Helios for Sun, and "gee" comes from the Greek word Geos for Earth.

Orbits of the Planets

Table 2.2 summarizes the data for the orbits of the planets, plus the largest asteroid, Ceres. You can use the values for the semimajor axis and period to check Kepler's third law, D3 = P2. Note that the eccentricity is rather small for most of the planets. For example, the Earths eccentricity is 0.02. This means that the orbits are very nearly circular (a circle has eccentricity of 0). The exceptions are the planets Mercury and Pluto which have eccentricities of 0.21 and 0.25 respectively.

The planets orbit in a common plane, near the plane of the Earth's orbit (called the ecliptic). The exception that makes the rule is Pluto who's orbit is inclined about 17° to the others.

Orbits of Asteroids and Comets

There are many smaller objects in the solar system, including two classes of objects called asteroids and comets. (see Figure 2.10) The asteroids orbit primarily in the region between the orbits of Mars and Jupiter, a region called the asteroid belt.

Comets generally have orbits or larger size and high eccentricity (0.8 or higher).

2.5 Motions of Satellites and Spacecraft

Space Flight and Satellite Orbits

The law of gravity and Kepler's and Newton's laws apply to man-made spacecraft and satellites as well. Since Sputnik in 1957, we've placed thousands of satellites in orbit around the Earth. (Figure 2.12) We've also sent spacecraft to the Moon, and every planet except Pluto.

Once the rocket's engines have finished firing, and a satellite is deployed, its orbit obeys the same laws as the orbits of natural objects, such as the Moon's orbit about Earth. Satellites in high orbit will remain there indefinitely. Satellites in low orbit experience some drag (friction) from the outermost part of the Earth's atmosphere. This drag slows the satellite, eventually causing it to fall out of orbit and re-enter the atmosphere. Most satellites burn up on re-entry from the heat caused by friction with the atmosphere. Notable exceptions are some exceedingly large objects, such as skylab, which can survive the heat. Nowadays space agencies plan for this, and try to bring the satellites out of orbits on purpose so they can control where it impacts the Earth.

Interplanetary Spacecraft

We have sent spacecraft from the Earth to every planet, except Pluto, and to several comets and asteroids. To escape Earth's gravity, the spacecraft must achieve escape velocity. The Earth's escape velocity is about 11km/s (about 25,000mph). An object leaving Earth with this velocity is capable of escaping the pull of Earth's gravity and coasting out of the solar system. (Figure 2.13)

Small thruster rockets are used to adjust the path of the spacecraft on the way to its destination. Close encounters with planets also alter the path; such encounters can even be used to slingshot the spacecraft to a higher speed! Elsewhere, the craft follows a Keplerian orbit.

To put a spacecraft in orbit around a planet, it must be slowed with a rocket as it nears its destination. Additional rocket thrusts are required if we want to bring the vehicle down for a landing on the surface. If the craft is then to return to Earth, still more rocket power is needed to boost it off the planet on its return trip. Through clever design, a manned trip to the Moon was possible. A manned trip to Mars is being discussed and will require equal ingenuity.

2.6 Gravity With More Than Two Bodies

To keep things simple, we have only considered the gravitational attraction between two bodies at any time: the Sun and a planet, or the Earth and Moon. But all objects are constantly exerting a gravitational force on all other objects. This greatly complicates any calculations, and generally requires computer calculations.

The Interactions of Many Bodies

As we've seen, the orbits of the planets are well described by Newton's and Kepler's laws. But this description is not exactly right, and accurate measurements can detect the small deviations. Scientists were able to treat the effects of other planets on the orbits as small perturbations to the Keplerian orbits, and thereby account for the small deviations.

The Discovery of Neptune

Eventually, deviations of the orbit of Uranus were seen that couldn't be accounted for by the known planets (primarily Jupiter and Saturn). A mathematical analysis led to the hypothesis that an as yet unknown planet lay beyond the orbit of Uranus. Both an Englishman, Adams, and a Frenchman, Leverrier, came to this conclusion. The estimation of the expected position was accurate enough that the planet was observed by Galle after only one night! The discovery was a triumph for Newton's laws and theory of gravity.

© Robert Harr 2004