keepar law

1. WELL COME

2. 
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12.  Johannes Kepler was a German astronomer and
mathematician of the late sixteenth and early
seventeenth centuries. Kepler was born in
Wurttemberg, Germany in 1571. His parents were poor,
but he was sent to the University of Tubingen on a
scholarship in recognition of his abilities in
mathematics.

13.  Unlike Brahe, Kepler believed firmly in the Copernican
system. In retrospect, the reason that the orbit of Mars
was particularly difficult was that Copernicus had
correctly placed the Sun at the center of the Solar
System, but had erred in assuming the orbits of the
planets to be circles. Thus, in the Copernican theory
epicycles were still required to explain the details of
planetary motion.

14.  Kepler's laws give a description of the motion of
planets around the Sun.

15.  The orbit of every planet is an ellipse with the Sun at
one of the two foci.
 The radius vector of a planet sweeps over equal areas
in equal intervals of time.
 The square of the orbital period of a planet is directly
proportional to the cube of the semi-major axis of its
orbit.

16.  Isaac Newton solidified Kepler's laws by showing that
they were a natural consequence of his inverse square
law of gravity with the limits set in the previous
paragraph. Further, Newton extended Kepler's laws in
a number of important ways such as allowing the
calculation of orbits around other celestial bodies.
 The solutions to the two-body problem, where there
are only two particles involved, say, the sun and one
planet, can be expressed analytically. These solutions
include the elliptical Kepler orbits, but motions along
other conic section (parabolas, hyperbolas and straight
lines) also satisfy Newton's differential equations.

17.  The focus of the conic section is at the center of mass
of the two bodies, rather than at the center of the Sun
itself.
 The period of the orbit depends a little on the mass of
the planet.

18.  "The orbit of every planet is an ellipse with the Sun at
one of the two foci.“

19.  An ellipse is a particular class of mathematical shapes
that resemble a stretched out circle. Ellipses have two
focal points neither of which are in the center of the
ellipse (except for the one special case of the ellipse
being a circle). Circles are a special case of an ellipse
that are not stretched out and in which both focal
points coincide at the center.

20.  Eccentricity of an ellipse is defined as the ratio of
distance between foci to length of major axis. In the
figure below, if eccentricity is denoted as “e”, the
distance between the foci as “c” and the lengths of the
major axis as “a”, the formula is:
 e = c
a

23.  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.

24.  When c is nearly to a, the ellipse appears flattened.
 When c is smaller than a, the ellipse appears rounded.
 When c is zero, the ellipse becomes a circle.

25.  The distance between the foci, c, the major axis, a, and
the minor axis, b, can be solved using the Pythagorean
Theorem:
 c2 = a2 - b2

26.  “The radius vector of a planet sweeps over equal areas
in equal intervals of time.”

27.  Radius Vector is an imaginary line connecting the
sun and the planet.

29.  Kepler's second law is illustrated in the preceding
figure. The line joining the Sun and planet sweeps out
equal areas in equal times, so the planet moves faster
when it is nearer the Sun. Thus, a planet executes
elliptical motion with constantly changing angular
speed as it moves about its orbit. The point of nearest
approach of the planet to the Sun is termed perihelion;
the point of greatest separation is termed aphelion.
Hence, by Kepler's second law, the planet moves
fastest when it is near perihelion and slowest when it is
near aphelion.

30.  Kepler's second law is equivalent to the fact that the
force perpendicular to the radius vector is zero. The
"areal velocity" is proportional to angular momentum,
and so for the same reasons, Kepler's second law is also
in effect a statement of the conservation of angular
momentum.

31.  About January 4th, by about 1.5%, not enough to
make the Sun look different.

32.  "The square of the orbital period of a planet is directly
proportional to the cube of the semi-major axis of its
orbit."

33.  This third law used to be known as the harmonic law,
because Kepler enunciated it in a laborious attempt to
determine what he viewed as the "music of the
spheres" according to precise laws, and express it in
terms of musical notation.

34.  P2
planet P2
earth
A
3
planet A
3
earth