Satellites and Kepler's Laws

Read this text, which includes visual diagrams of Kepler's Laws of Planetary Motion, which describe the motion of planets around the sun. We can also apply these laws to explain the motion of satellites around planets.

1. Kepler's First Law of Planetary Motion states that planets move around the sun in an ellipse shaped orbit with the sun at the center of the ellipse (see Figure 6.29).
2. Kepler's Second Law of Planetary Motion states that planets move so that a point on the planet sweeps an equal area in equal times (see Figure 6.30).
3. Kepler's Third Law of Planetary Motion refers to the relationship between the time it takes for two planets to revolve around the sun, and their distances from the sun: $\frac{T_1^2}{T_2^2}=\frac{r_1^3}{r_2^3}$, where $T_{1}$ and $T_{2}$ are periods of orbit while $r_{1}$ and $r_{2}$ are radii for planets one and two.

We can use Kepler's Third Law to solve problems to determine the period for planetary or satellite orbits. See a worked example of using the equation from Kepler's Third Law to determine the period of a satellite in Example 6.7. Pay attention to the derivation of Kepler's Third Law using the concept of centripetal forces.

The Case for Simplicity

The development of the universal law of gravitation by Newton played a pivotal role in the history of ideas. While it is beyond the scope of this text to cover that history in any detail, we note some important points. The definition of planet set in 2006 by the International Astronomical Union (IAU) states that in the solar system, a planet is a celestial body that:

1. is in orbit around the Sun,
2. has sufficient mass to assume hydrostatic equilibrium and
3. has cleared the neighborhood around its orbit.

A non-satellite body fulfilling only the first two of the above criteria is classified as "dwarf planet".

In 2006, Pluto was demoted to a 'dwarf planet' after scientists revised their definition of what constitutes a "true" planet.

Parent Satellite Average orbital radiusr(km) PeriodT(y) r3/T2(km3/ y2)
Earth Moon $3.84 \times 10^{5}$ 0.07481 $1.01 \times 10^{19}$
Sun Mercury $5.79 \times 10^{7}$ 0.2409 $3.34 \times 10^{24}$
Venus $1.082 \times 10^{8}$ 0.6150 $3.35 \times 10^{24}$
Earth $1.496 \times 10^{8}$ 1.000 $3.35 \times 10^{24}$
Mars $2.279 \times 10^{8}$ 1.881 $3.35 \times 10^{24}$
Jupiter $7.783 \times 10^{8}$ 11.86 $3.35 \times 10^{24}$
Saturn $1.427 \times 10^{9}$ 29.46 $3.35 \times 10^{24}$
Neptune $4.497 \times 10^{9}$ 164.8 $3.35 \times 10^{24}$
Pluto $5.90 \times 10^{9}$ 248.3 $3.33 \times 10^{24}$
Jupiter Io $4.22 \times 10^{5}$ 0.00485 (1.77 d) $3.19 \times 10^{21}$
Europa $6.71 \times 10^{5}$ 0.00972 (3.55 d) $3.20 \times 10^{21}$
Ganymede $1.07 \times 10^{6}$ 0.0196 (7.16 d) $3.19 \times 10^{21}$
Callisto $1.88 \times 10^{6}$ 0.0457 (16.19 d) $3.20 \times 10^{21}$

Table 6.2 Orbital Data and Kepler's Third Law

The universal law of gravitation is a good example of a physical principle that is very broadly applicable. That single equation for the gravitational force describes all situations in which gravity acts. It gives a cause for a vast number of effects, such as the orbits of the planets and moons in the solar system. It epitomizes the underlying unity and simplicity of physics.

Before the discoveries of Kepler, Copernicus, Galileo, Newton, and others, the solar system was thought to revolve around Earth as shown in Figure 6.31(a). This is called the Ptolemaic view, for the Greek philosopher who lived in the second century AD. This model is characterized by a list of facts for the motions of planets with no cause and effect explanation. There tended to be a different rule for each heavenly body and a general lack of simplicity.

Figure 6.31(b) represents the modern or Copernican model. In this model, a small set of rules and a single underlying force explain not only all motions in the solar system, but all other situations involving gravity. The breadth and simplicity of the laws of physics are compelling. As our knowledge of nature has grown, the basic simplicity of its laws has become ever more evident.

Figure 6.31 (a) The Ptolemaic model of the universe has Earth at the center with the Moon, the planets, the Sun, and the stars revolving about it in complex superpositions of circular paths. This geocentric model, which can be made progressively more accurate by adding more circles, is purely descriptive, containing no hints as to what are the causes of these motions. (b) The Copernican model has the Sun at the center of the solar system. It is fully explained by a small number of laws of physics, including Newton's universal law of gravitation.