Kepler’s Law of Planetary Motion

Kepler’s Law of Planetary Motion : 

Kepler’s Law of Planetary Motion

History of the Laws : 

History of the Laws Johannes Kepler was a German astronomer and mathematician of the late sixteenth and early seventeenth centuries. His work was largely based on the work of his mentor, Tycho Brahe. Kepler was able to use Bahe's precise measurements (made before telescopes) to determine, mostly by trial and error, three laws that described the motion of the five planets then known. First Law: Kepler's Elliptical Orbit Law: Each planet moves in an elliptical orbit, with the sun at one focus of the ellipse. Second Law: Kepler's Equal-Area Law: A line from the sun to each planet sweeps out equal areas in equal time. The picture to the right depicts this law. Third Law: Kepler's Law of Periods: The periods of the planets (T) are proportional to the 3/2 powers of the major axis lengths of their orbits (L). Expressed mathematically, then, this says that: T is proportional to L3/2 or T2 is proportional to L3

Mathematics of the First Law : 

Mathematics of the First Law Symbolically: where (r, θ) are heliocentric polar coordinates for the planet, p is the semi-latus rectum, and ε is the eccentricity.

Slide 4: 

At θ = 0°, perihelion, the distance is minimum At θ = 90°, the distance is P.

Slide 5: 

At θ = 180°, aphelion, the distance is maximum

Slide 6: 

The semi-major axis a is the arithmetic mean between rmin and rmax: So

Slide 7: 

The semi-minor axis b is the geometric mean between rmin and rmax: So

Slide 8: 

The semi-latus rectum p is the harmonic mean between rmin and rmax: The eccentricity ε is the coefficient of variation between rmin and rmax: The area of the ellipse is The special case of a circle is ε = 0, resulting in r = p = rmin = rmax = a = b and A = π r2.

Slide 9: 

Aphelion -- the point on its orbit when the Earth is farthest from the sun. Perihelion -- the point on its orbit when the Earth is closest to the sun.

Slide 10: 

The Ellipse in Polar Coordinates     Again, if all the values of (r,f) of a curve are related by some equation which can be symbolically written r = r(f) then the function r(f) is said to be the equation of the line, in polar coordinates. The simplest function is a constant number a, giving the line r = a    The value of r equals a for any value of f. That gives a circle around the origin, its radius equal to a, shown in the drawing on the right above. The Ellipse Consider next the curve whose equation is r = a(1– e2)/(1+ e cos f) where the eccentricity e is a number between 0 and 1. If e = 0, this is clearly the circle encountered earlier.

Slide 11: 

Kepler's First Law: The orbits of the planets are ellipses, with the Sun at one focus of the ellipse. Kepler's First Law is illustrated in the image shown above. The Sun is not at the center of the ellipse, but is instead at one focus (generally there is nothing at the other focus of the ellipse). The planet then follows the ellipse in its orbit, which means that the Earth-Sun distance is constantly changing as the planet goes around its orbit. For purpose of illustration we have shown the orbit as rather eccentric; remember that the actual orbits are much less eccentric than this.