Astronomy

Kepler's Three Laws of Planetary Motion

Johannes Kepler formulated three laws to approximate the behavior of planets in their orbits. To understand Kepler's First Law of Planetary Motion ( Law of Ellipses), one must first be familiar with the properties and components of an ellipse. An ellipse is the path of a point that moves so that the sum of its distances from two fixed points (the foci) is constant. An ellipse has two axes of symmetry. The longer one is called the major axis, and the shorter one is called the minor axis. The two axes intersect at the center of the ellipse. Kepler's First Law states that the orbit of a planet is an ellipse with the Sun at one focus (see Figure 1). The size of an ellipse is given by the length of the semi‐major axis (half of the major axis), which is also equal to the average distance of the planet to the Sun as it travels about the Sun in its orbit. The shape of an ellipse is measured by the eccentricity, or a measure of how much an ellipse deviates from the shape of a circle (e = CF/a = (1 – b ²/a ²) ^1/2). Therefore, a circle would have an eccentricity of 0, while a line would have an eccentricity of 1. The closest approach of a planet to the Sun is known as the perihelion, a distance equal to a(1 – e). The greatest distance between a planet and the Sun is the aphelion equal to a(1 + e) (see Figure 2).

Figure 1

Figure 2

Kepler's Second Law of Planetary Motion ( Law of Areas) states that a line connecting the planet with the Sun sweeps over an area at a constant rate. In other words, if the time for an object to move from position A to position B is the same as the time to move from C to D, the areas swept out are also equal. This law is actually an alternative statement of the physical principle of the conservation of angular momentum: In the absence of an outside force, angular momentum = mass × orbital radius × the tangential velocity (that is, the velocity perpendicular to the radius) does not change. In consequence, when a planet moves closer to the Sun, its orbital velocity must increase, and vice versa.

Kepler's Third Law of Planetary Motion ( Harmonic Law) details an explicit mathematical relationship between a planet's orbital period and the size of its orbit, a correlation noted by Copernicus. Specifically, the square of a planet's period (P) of revolution about the Sun is proportional to the cube of its average distance (a) from the Sun. For example, P ² = constant a ³. If P is expressed in years and the semi‐major axis a in astronomical units, the constant of proportionality is 1 yr ²/AU ³, and the proportionality becomes the equation P ² = a ³.

Although Kepler's Laws were deduced explicitly from study of planets, their description of orbital properties also applies to satellites moving about planets and to situations in which two stars, or even two galaxies, move about each other. The Third Law, in the form as proposed by Kepler, however, applies only to planets whose masses are negligible in comparison to that of the Sun.

Newton's Three Laws of Motion and a Law of Gravitation

Newton's First Law of Motion ( Law of Inertia) states that an object continues moving at the same rate unless acted upon by an outside (external) force. If no external force interferes, a moving object keeps moving at a constant velocity (that is, both speed and direction remain the same). Similarly, an object at rest stays at rest. This tendency of matter to remain at rest if at rest, or, if moving, to keep moving in the same direction at the same speed is called inertia. Mass is what gives an object inertia. Mass is a measure of the quantity of material in an object, not its weight, which is a measure of the gravitational force exerted on an object. Newton's First Law is a statement of the modern principle of Conservation of Momentum, where momentum (p) is an object's mass (m) times its velocity (v). Momentum stays constant if the outside force is zero.

As in any mathematical expression of a physical law, each term has a precise definition and meaning. Both velocity and momentum are vector quantities; that is, each has both a size and a direction. Thus, p = mv involves both the magnitudes of the quantities involved and their directions; momentum and velocity are expressed as boldface (or sometimes with an arrow above the symbol) to remind the user that a direction is involved. This physical law is not something one would intuitively derive from real‐world observations, for virtually no real circumstances exist without outside forces—usually friction—acting on objects. Friction can't be seen; therefore, one tends to forget about it; but it is a real force.

Newton's Second Law of Motion ( Law of Force) states that if a force acts upon an object, the object accelerates in the direction of the force, its momentum changing at a rate equal to that force. Force F is the agent that causes a change in a body's momentum. Force = rate of change of momentum = rate of change of (mass × velocity) = mass × rate of change of velocity = mass × acceleration; or, more familiarly, F = m a where again the boldface indicates the vector nature of both force and the acceleration. Newton's first law is a direct consequence of the second: If no force acts, there is no acceleration. No acceleration (a change in velocity divided by the time over which the change occurred) means no change in the velocity.

The combination of the operation of these two laws is sufficient to explain orbital motion. If left alone, an object continues its motion in a straight line. Application of a gravitational force on the object produces an acceleration in the direction of the force, which also produces a velocity component in the direction of the force. The combination of the two motions produces a velocity in a new direction, along which the object continues to move as long as no other force is introduced. As a result, an object like the Moon, being acted upon by Earth's gravitational force, literally “falls” around a larger object like Earth.

Newton's Third Law of Motion ( Law of Reaction) states that forces always occur in mutually acting pairs. In other words, forces are reciprocal; for every force, there is an equal and opposite force.

Newton's Third Law has a significant consequence for Kepler's Third Law of Planetary Motion, which was derived from the assumption that the Sun is stationary. In actuality, the Sun feels a gravitational force due to the pull of the planet, and the planet feels a gravitational force due to the pull of the Sun. From the Second Law, acceleration = force / mass. The smaller mass (the planet) experiences the greater acceleration, hence the greater resultant velocity, and therefore the larger orbit. The Earth/Moon mass ratio is 81/1, thus the Moon's orbit is 81 times that of Earth's orbit around the common center of mass, the balance point along a line joining two objects, of the Earth/Moon system. The Earth/Sun mass ratio is 1/330,000; thus, Earth's orbit is 330,000 times bigger than the Sun's orbit. Letting the mass of any two objects in orbit about each other be M ₁ and M ₂, then the common center of mass is determined by M ₁a ₁ = M ₂a ₂ where a ₁ + a ₂ = a, the relative semi‐major axis as used by Kepler. Because of Newton's law of reciprocity, Kepler's Third Law must be rewritten: P²(M ₁ + M ₂) = a ³ where it is assumed that masses are measured in solar masses, orbital periods in years, and the relative orbital semi‐major axis in astronomical units (expressed in other units, the general form of Kepler's Third Law is P ² G(M ₁ + M ₂) = 4 p ²a ³ where the gravitational constant (G) must be given an appropriate value). As modified by Newton, Kepler's Third Law becomes an indispensable tool for determining the masses of other objects in the universe whose orbital motions may be measured.

Newton's Law of Gravitation states that between any two objects there exists a force of attraction proportional to the product of their masses and inversely proportional to the square of the distance between them. Therefore, F = G m ₁m ₂/r ². The Law of Gravity is a universal law that applies to everything including all phenomena on Earth, the motions of the planets, motions of the stars in the Galaxy, motions of the galaxies in the great clusters of galaxies, and everything else in the universe. Gravity is the dominant force in the macroscopic (large‐scale) universe, but it is actually the weakest of the four known forces in nature (the other three are the electromagnetic, the strong nuclear, and the weak nuclear forces). Although it is a mathematically simple law, the fact that the strength of gravity depends on distance makes application of the law to most real circumstances exceedingly difficult to apply; even a small change in position results in a change in force, thus a change in the acceleration or rate of change of velocity. This means that velocity and position cannot be written as simple algebraic expressions, but must be expressed as summations of many small, ever different, changes. Newton was forced to invent calculus in order to compute orbits; but with this new mathematical formulism, he was able to show that orbits are indeed described by Kepler's Laws.