Chapter 2 Summary

2.1 Tycho Brahe was the most skillful of the pretelescopic astronomical observers. His accurate observations of planetary positions provided the data used by Johannes Kepler to derive the three fundamental laws of planetary motion that bear his name: (1) planetary orbits are ellipses (a figure described by its semimajor axis and eccentricity) with the Sun at one focus; (2) in equal intervals, a planet’s orbit sweeps out equal areas; and (3) if times are expressed in years, and distances in astronomical units, the relationship between period (P) and semimajor axis (D) of an orbit is given by D³ = P² = D³.

2.2 In his Principia, Isaac Newton established the three laws that govern the motion of objects: (1) Bodies continue at rest or in uniform motion unless acted upon by an outside force; (2) an outside force causes an acceleration (and changes the momentum) of an object; and (3) for each action there is an equal and opposite reaction. Momentum is a measure of the motion of an object and depends on both its mass and its velocity. Angular momentum is a measure of motion of a spinning or revolving object. The density of an object is its mass divided by its volume.

2.3 Gravity, the attraction of all mass for all other mass, is the force that keeps the planets in orbit. Newton’s law of gravity relates gravitational force to mass and distance (F = GM₁M₂/R²). Newton was able to show the equivalence of gravitational force (weight) on Earth to the gravitational force between objects in space. When Kepler’s laws are re-examined in the light of gravitational theory, it becomes clear that the masses of both Sun and planet are important for the third law, which becomes D³ = (M₁ + M₂) x P² = D³. Mutual gravitational effects permit us to calculate the masses of astronomical objects, from comets to galaxies.

2.4 The lowest point in a satellite orbit around the Earth is its perigee, and the highest point is its apogee (corresponding to perihelion and aphelion for an orbit about the Sun). The planets all follow orbits about the Sun that are nearly circular and in the same plane. Most asteroids are found between Mars and Jupiter in the asteroid belt, whereas comets generally follow orbits of high eccentricity.

2.5 The orbit of an artificial satellite depends on the circumstances of its launch. The circular satellite velocity at the Earth’s surface is 8 km/s, and the escape velocity is 11 km/s. There are many possible interplanetary trajectories, including those that use gravity-assisted flybys of one object to redirect the spacecraft toward its next target.

2.6 Gravitational problems that involve more than two interacting bodies are much more difficult to deal with than two-body problems. They require large computers for accurate solutions. If one object dominates gravitationally, it is possible to calculate the effects of a second object in terms of small perturbations. This approach was used by Adams and Leverrier to predict the position of Neptune from its perturbations of the orbit of Uranus and thus discover a new planet mathematically.