AY 101 Lecture Notes - Lecture 5: Semi-Major And Semi-Minor Axes, Orbital Period, Physical Law
Basic Physics
In the sixteenth and seventeenth centuries, scientists discovered the laws of the motion
of material objects. These laws help scientists to explain and predict the motions of
celestial bodies.
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)
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
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Document Summary
In the sixteenth and seventeenth centuries, scientists discovered the laws of the motion of material objects. These laws help scientists to explain and predict the motions of celestial bodies. 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. 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).
