Kepler proposed three laws of planetary motion based on Tycho Brahe's observations: 1) Planets orbit the Sun in ellipses, with the Sun located at one focus. 2) A line connecting a planet to the Sun sweeps out equal areas in equal times. 3) The square of a planet's orbital period is proportional to the cube of its average distance from the Sun.

1. Kepler’s Laws
Introduction:
In the early 1600s, Johannes Kepler proposed three laws of planetary motion. Kepler was able to
summarize the carefully collected data of his mentor - Tycho Brahe - with three statements that
described the motion of planets in a sun-centered solar system. Kepler's efforts to explain the
underlying reasons for such motions are no longer accepted; nonetheless, the actual laws themselves
are still considered an accurate description of the motion of any planet and any satellite.
Ist Law (The Law of Ellipses):
Statement: The orbit of every planet is an ellipse with the Sun at one of the two foci.
Key Points
 An ellipse is a closed plane curve that resembles a stretched out circle (The Sun is at one focus
while the other focus has no physical significance. A circle is a special case of an ellipse where
both focal points coincide.
 How stretched out an ellipse is from a perfect circle is known as its eccentricity: a parameter
that can take any value greater than or equal to 0 (a circle) and less than 1 (as the eccentricity
tends to 1, the ellipse tends to a parabola).
 Symbolically, an ellipse can be represented in polar coordinates as:
[latex]text{r}=frac{text{p}}{1+ epsilon cos theta }[/latex], where [latex](text{r},
theta)[/latex] are the polar coordinates (from the focus) for the ellipse, [latex]text{p}[/latex] is
the semi-latus rectum, and [latex]epsilon[/latex] is the eccentricity of the ellipse.
 Perihelion is minimum distance from the Sun a planet achieves in its orbit and is given by
[latex]text{r}_{text{min}}=frac{text{p}}{1+epsilon}[/latex]. Aphelion is the largest distance
from the Sun a planet reaches in his orbit and is given by
[latex]text{r}_{text{max}}=frac{text{p}}{1-epsilon}[/latex].
Key Terms
 Eccentricity: The coefficient of variation between [latex]text{r}_{text{min}}[/latex] and
[latex]text{r}_{text{max}}[/latex]: [latex]epsilon=frac{text{r}_{text{max}}-
text{r}_{text{min}}}{text{r}_{text{max}}+text{r}_{text{min}}}[/latex]. The further appart the
foci are, the stronger the eccentricity.
 Perihelion: The point in the elliptical orbit of a planet or comet etc. where it is nearest to the
Sun. The point farthest from the Sun is called aphelion.
 Semi-latus rectum: The latus rectum is a chord perpendicular to the major axis and passing
through the focus. The semi-latus rectum is half the latus rectrum. See distance p in.

2. Second Law (The Law of Equal Areas):
Statement: A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
Key Points
 In a small time the planet sweeps out a small triangle having base line and height. The area of
this triangle is given by [latex]text{dA}=frac{1}{2} cdot text{r} cdot text{rd} theta[/latex].
and so the constant areal velocity is
[latex]frac{text{dA}}{text{dt}}=frac{1}{2}text{r}^{2}frac{text{d} theta}{text{dt}}[/latex].
 The period [latex]text{P}[/latex] satisfies [latex]pi text{a} text{b}=text{P} cdot
frac{1}{2}text{r}^{2} dot theta[/latex]. One can see that the product of
[latex]text{r}^2[/latex] and must be constant, so that when the planet is further from the Sun it
travels at a slower rate and vise versa.
 A planet travels fastest at perihelion and slowest at aphelion.
Key Terms
 Angular velocity: A vector quantity describing an object in circular motion; its magnitude is
equal to the speed of the particle and the direction is perpendicular to the plane of its circular
motion.
 Mean motion: An angle of [latex]2pi[/latex] (radians) divided by the orbital period (of a
celestial body in an elliptic orbit).

3. Third Law (The Law of Harmonies):
Statement: The square of the orbital period of a planet is directly proportional to the cube of the semi-
major axis of its orbit. P2
𝛼 a3
where 𝛼 (
𝑦2
𝐴𝑈3)
In this section we will consider the special case of the planets going around the Sun. If we choose to
measure the length of the semi-major axis of an orbit in astronomical units (abbreviated AU, where 1 AU
is the distance from the Earth to the Sun) and we measure the orbital period in years (abbreviated as 𝑦 ),
then we can express Kepler's Third Law as where P is measured in years and is measured in
astronomical units.
Key Points
 Kepler’s third law can be represented symbolically as [latex]text{P}^{2} propto
text{a}^{3}[/latex], where P is the orbital period of the planet and a is the semi-major axis of
the orbit (see.
 The constant of proportionality is
[latex]frac{text{P}_{text{planet}}^{2}}{text{a}_{text{planet}}^{3}}=frac{text{P}_{text{earth
}}^{2}}{text{a}_{text{earth}}^{3}}=1frac{text{yr}^{2}}{text{AU}^{3}}[/latex] for a sidereal year
(yr), and astronomical unit (AU).
 Kepler’s third law can be derived from Newton’s laws of motion and the universal law of
gravitation. Set the force of gravity equal to the centripetal force. After substituting an
expression for the velocity of the planet, one can obtain: [latex]text{G} frac{text{M}}{text{r}}
= frac{4 pi text{r}^{2}}{text{P}^{2}}[/latex] which can also be written
[latex]text{P}^{2}=frac{4 pi ^{2} text{a}^{3}}{text{GM}}[/latex].
 Using the expression above we can obtain the mass of the parent body from the orbits of its
satellites: [latex]text{M}=frac{4 pi^{2} text{r}^{3}}{text{G} text{P}^{2}}[/latex].
Key Terms
 Astronomical unit: The mean distance from the Earth to the Sun (the semi-major axis of Earth’s
orbit), approximately 149,600,000 kilometres (symbol AU), used to measure distances in the
solar system.
 Sidereal year: The orbital period of the Earth; a measure of the time it takes for the Sun to
return to the same position with respect to the stars of the celestial sphere. A sidereal year is
about 20.4 minutes longer than the tropical year due to precession of the equinoxes.