Kepler's Laws for Planets and Satellites Learning Objectives

Department of Physics United States Naval Academy Lecture 33: Kepler’s Laws for Planets and Satellites

Learning Objectives • State Kepler’s three laws and identify which of the laws is equivalent to the law of conservation of angular momentum. For an orbiting natural or artiﬁcial satellite, apply Kepler’s relationship between the orbital period and radius and the mass of the astronomical body being orbited.

Kepler’s Laws Studies of the motion of the Moon, the planets, and other celestial bodies provided valuable information for Newton in his work on gravitation. Of the many important astronomers prior to the time of Newton, one of the most famous is Johannes Kepler (1571-1630) of the famed Kepler’s laws of planetary motion.

These laws are not ”laws of nature” in the sense of Newton’s laws of motion or New- ton’s law of gravitation. Rather, Kepler’s laws are mathematical rules that describe motion in the solar system. They state that

(a) The law of orbits. All planets move in elliptical orbits with the Sun at one focus.

• Any object bound to another by an inverse square law will move in an elliptical path

• Second focus is empty

(b) The law of areas. A line joining any planet to the Sun sweeps out equal areas in equal time intervals. (This statement is equivalent to conservation of angular momentum.) Figure 1: A planet of mass m moving (c) The law of periods. The square of the orbital period T of any planet is pro- in an elliptical orbit around the Sun. The portional to the cube of the semi-major axis a of its orbit. That is, Sun, of mass M, is at one focus F of the ellipse. The other focus is F0, which is T 2 = ka3 located in empty space. The semi-major where k is a constant unique to the properties of the astronomical object at axis a of the ellipse, the perihelion (near- the focal point of the elliptical orbit. est the Sun) distance Rp, and the aphe- lion (farthest from the Sun) distance Ra For circular orbits with radius r, the universal gravitation law show that are also shown. 4π2 T 2 = r3 GM

where M is the mass of the attracting body - the Sun in the case of the solar system. For elliptical planetary orbits, the semi- major axis a is substituted for r.

2 • For motion around the Sun, the constant, which is independent of the mass of the object is: 4π /GMs = 2.97 × 10−19s2/m3

Geosynchronous Orbits: Earth-orbiting satellites used for transmitting telephone or television signals travel in geosynchronous orbits. These orbits have a period of 1 day, so these satellites move in synchrony with the Earth’s rotation and are thus always at the same position in the sky relative to a person on the ground. Sending signals to and from these satellites is thus greatly simpliﬁed because an antenna can be aligned only once and then needs no further adjustment.

From a telecommunications point of view, it is advantageous for satellites to remain at the same location relative to a location on the Earth. This can occur only if the satellite’s orbital period is the same as the Earth’s period of rotation, which is 24 h. (a) At what distance from the center of the Earth can this geosynchronous orbit be found? (b) What is the orbital speed of such a satellite?

Estimate the mass of the Sun noting that the period of the Earth’s orbit around the Sun is 1 year and its distance from the Sun is 1.496 × 1011 m.

The Sun, which is 2.2×1020 m from the center of the Milky Way galaxy, revolves around that center once every 2.5×108 years. Assuming each star in the Galaxy has a mass equal to the Sun’s mass of 2.0 × 1030 kg, the stars are distributed uniformly in a sphere about the galactic center, and the Sun is at the edge of that sphere, estimate the number of stars in the Galaxy.

A 20 kg satellite has a circular orbit with a period of 2.4 h and a radius of 8.0 × 106 m around a planet of unknown mass. If the magnitude of the gravitational acceleration on the surface of the planet is 8.0 m/s2, what is the radius of the planet?