UNIT-1

 Kepler's laws of planetary motion

In astronomy,  Kepler's laws give an approximate description of the motion  ofplanets around the  Sun.

Kepler's laws are:

1. The orbit of every planet is an ellipse with the Sun at one of the two foci.

2. A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.[1]
3. The square of the orbital period of a planet is directly proportional to the cubeof the semi-major
axis of its orbit.

Figure 1: Illustration of Kepler's three laws with two planetary orbits. (1) The orbits are ellipses, with focal
pointsƒ1 and ƒ2 for the first planet and ƒ1 and ƒ3 for the second planet. The Sun is placed in focal point ƒ1.
(2) The two shaded sectors A1 and A2 have the same surface area and the time for planet 1 to cover
segment A1 is equal to the time to cover segment A2. (3) The total orbit times for planet 1 and planet 2
have a ratio a13/2 : a23/2

.
 Kepler and the Elliptical Orbits
 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.
 It fell to Kepler to provide the final piece of the puzzle: after a long
struggle, in which he tried mightily to avoid his eventual
conclusion, Kepler was forced finally to the realization that the
orbits of the planets were not the circles demanded by Aristotle
and assumed implicitly by Copernicus, but were instead the
"flattened circles" that geometers call ellipses (See adjacent
figure; the planetary orbits are only slightly elliptical and are not as flattened as in this example.)
 The irony noted above lies in the realization that the difficulties with the Martian orbit
derive precisely from the fact that the orbit of Mars was the most elliptical of the planets for which Brahe
had extensive data. Thus Brahe had unwittingly given Kepler the very part of his data that would allow
Kepler to eventually formulate the correct theory of the Solar System and
thereby to banish Brahe's own theory!

 Some Properties of Ellipses

Since the orbits of the planets are ellipses, let us review a few basic properties of ellipses.

 For an ellipse there are two points called foci (singular: focus) such that the sum of the distances to
the foci from any point on the ellipse is a constant. In terms of the diagram shown to the left, with
"x" marking the location of the foci, we have the equation

a + b = constant

that defines the ellipse in terms of the distances a and b.

  The amount of "flattening" of the ellipse is termed the eccentricity. Thus, in the following figure the
ellipses become more eccentric from left to right. A circle may be viewed as a special case of an
ellipse with zero eccentricity, while as the ellipse becomes more flattened the eccentricity approaches
one. Thus, all ellipses have eccentricities lying between zero and one.

The orbits of the planets are ellipses but the eccentricities are so small for most of the planets that they look
circular at first glance. For most of the planets one must measure the geometry carefully to determine that they
are not circles, but ellipses of small eccentricity. Pluto and Mercury are exceptions: their orbits are sufficiently
eccentric that they can be seen by inspection to not be circles.

 The long axis of the ellipse is called the major axis, while the short axis is called the minor
axis (adjacent figure). Half of the major axis is termed a semimajor axis. The length of a semimajor
axis is often termed the size of the ellipse. It can be shown that the average separation of a planet
from the Sun as it goes around its elliptical orbit is equal to the length of the semimajor axis. Thus,
by the "radius" of a planet's orbit one usually means the length of the semimajor axis. For a more
detailed investigation of the properties of ellipses, see this java applet 
  The Laws of Planetary Motion

Kepler's First Law:

I. 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. 

 Kepler's Second Law:

II. The line joining the planet to the Sun sweeps out
equal areas in equal times as the planet travels around
the ellipse.
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. 

 Kepler's Third Law:

III. The ratio of the squares of the revolutionary periods

for two planets is equal to the ratio of the cubes of
their semimajor axes:

 In this equation P represents the period of revolution for a planet and R represents the length of its
semimajor axis. The subscripts "1" and "2" distinguish quantities for planet 1 and 2 respectively.
The periods for the two planets are assumed to be in the same time units and the lengths of the
semimajor axes for the two planets are assumed to be in the same distance units.

 Kepler's Third Law implies that the period for a planet to orbit the Sun increases rapidly with the
radius of its orbit. Thus, we find that Mercury, the innermost planet, takes only 88 days to orbit the
Sun but the outermost planet (Pluto) requires 248 years to do the same
Calculations Using Kepler's Third Law
A convenient unit of measurement for periods is in Earth years, and a convenient unit of measurement for
distances is the average separation of the Earth from the Sun, which is termed an astronomical unit and is
abbreviated as AU. If these units are used in Kepler's 3rd Law, the denominators in the preceding equation
are numerically equal to unity and it may be written in the simple form

This equation may then be solved for the period P of the planet, given the length of the semimajor axis,

 
or for the length of the semimajor axis, given the period of the planet,

As an example of using Kepler's 3rd Law, let's calculate the "radius" of the orbit of Mars (that is, the length
of the semimajor axis of the orbit) from the orbital period. The time for Mars to orbit the Sun is observed to
be 1.88 Earth years. Thus, by Kepler's 3rd Law the length of the semimajor axis for the Martian orbit is

which is exactly the measured average distance of Mars from the Sun. As a second example, let us calculate
the orbital period for Pluto, given that its observed average separation from the Sun is 39.44 astronomical
units. From Kepler's 3rd Law

which is indeed the observed orbital period for the planet Pluto.
  Mathematics of the three laws

 First law

"The orbit of every planet is an ellipse with the Sun at a focus."

Symbolically:

where (r, θ) are heliocentric polar coordinates for the planet, p is the semi-latus rectum, and ε is the
eccentricity.

perihelion, the distance is minimum

At θ = 90°, the distance is 

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

The semi-major axis a is the arithmetic meanbetween rmin and rmax:

so

The semi-minor axis b is the geometric meanbetween rmin and rmax:

so

The semi-latus rectum p is the harmonic meanbetween rmin and rmax:

The eccentricity ε is the coefficient of variationbetween 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.
  Second law

Figure 3: Illustration of Kepler's second law. The planet moves faster near the Sun, so the same area is
swept out in a given time as at larger distances, where the planet moves more slowly.

The line joining a planet and the Sun sweeps out equal areas during equal intervals of time."

Symbolically :

where   is the "areal velocity".

This is also known as the law of equal areas. It also applies for parabolic trajectories andhyperbolic
trajectories.
  Third law

"The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of

its orbit."

Symbolically:

where P is the orbital period of planet and a is the semimajor axis of the orbit.

The proportionality constant is the same for any planet around the Sun.

So the constant is 1 (sidereal year)2(astronomical unit)−3 or 2.97472505×10−19 s2m−3. See the

actual figures: attributes of major planets.

See also scaling in gravity.

  Orbital elements

Keplerian elements

In this diagram, the orbital plane (yellow) intersects a reference plane (gray). For earth-orbiting satellites, the reference plane
is usually the Earth's equatorial plane, and for satellites in solar orbits it is the ecliptic plane. The intersection is called
the line of nodes, as it connects the center of mass with the ascending and descending nodes. This plane, together with
the Vernal Point, (♈) establishes a reference frame.

The traditional orbital elements are the six Keplerian elements, after Johannes Kepler and his laws of
planetary motion.
When viewed from an inertial frame, two orbiting bodies trace out distinct trajectories. Each of these
trajectories has it's focus at the common center of mass. When viewed from the non-inertial frame of one
body only the trajectory of the opposite body is apparent; Keplerian elements describe these non-inertial
trajectories. An orbit has two sets of Keplerian elements depending on which body used as the point of
reference. The reference body is called the primary, the other body is called thesecondary. The primary is
not necessarily more massive than the secondary, even when the bodies are of equal mass, the orbital
elements depend on the choice of the primary.

The main two elements that define the shape and size of the ellipse:

 Eccentricity ( ) - shape of the ellipse, describing how flattened it is compared with a circle. (not
marked in diagram)
 Semimajor axis ( ) - the sum of the periapsis and apoapsis distances divided by two. For circular
orbits the semimajor axis is the distance between the bodies, not the distance of the bodies to the
center of mass.

Two elements define the orientation of the orbital plane in which the ellipse is embedded:

 Inclination - vertical tilt of the ellipse with respect to the reference plane, measured at
the ascending node (where the orbit passes upward through the reference plane) (green angle i in
diagram).
 Longitude of the ascending node - horizontally orients the ascending node of the ellipse (where
the orbit passes upward through the reference plane) with respect to the reference frame's vernal
point (green angle Ω in diagram).

And finally:

 Argument of periapsis defines the orientation of the ellipse (in which direction it is flattened
compared to a circle) in the orbital plane, as an angle measured from the ascending node to the
semimajor axis. (violet angle   in diagram)
 Mean anomaly at epoch ( ) defines the position of the orbiting body along the ellipse at a
specific time (the "epoch").

The mean anomaly is a mathematically convenient "angle" which varies linearly with time, but which does
not correspond to a real geometric angle. It can be converted into the true anomaly  , which does
represent the real geometric angle in the plane of the ellipse, between periapsis (closest approach to the
central body) and the position of the orbiting object at any given time. Thus, the true anomaly is shown as
the red angle   in the diagram, and the mean anomaly is not shown.

The angles of inclination, longitude of the ascending node, and argument of periapsis can also be
described as the Euler angles defining the orientation of the orbit relative to the reference coordinate
system.
Note that non-elliptic orbits also exist; If the eccentricity is greater than one, the orbit is a hyperbola. If the
eccentricity is equal to one and the angular momentum is zero, the orbit is radial. If the eccentricity is one
and there is angular momentum, the orbit is a parabola.

Derivation from Newton's laws

 Deriving Kepler's first law

To derive Kepler's first law, define:

where the constant

has the dimension of length. Then

and

Differentiation with respect to time is transformed into differentiation with respect to

angle:

Differentiate

twice:

Substitute into the radial equation of motion

and get

Divide by the right hand side to get a simple non-

homogeneous linear differential equation for the
orbit of the planet:
 An obvious solution to this equation is the
circular orbit

Other solutions are obtained by adding

solutions to the homogeneous linear
differential equation with constant
coefficients

These solutions are

where   and   are arbitrary

constants of integration. So
the result is

Choosing the axis of

the coordinate
system such that  ,
and inserting  ,
gives:

If   this

the equation
of an ellipse
and illustrat

Kepler's first

 Deriving Kepler's second law

Only the tangential acceleration equation is needed to derive Kepler's second law.

The magnitude of the specific angular momentum

is a constant of motion, even if both the distance  , and the angular speed  , and the tangential
velocity  , vary, because

where the expression in the last parentheses vanishes due to the tangential acceleration equation.

The area swept out from time t1 to time t2,

depends only on the duration t2−t1. This is Kepler's second law.

 Deriving Kepler's third law

In the special case of circular orbits, which are ellipses with zero eccentricity, the relation between the
radius a of the orbit and its period Pcan be derived relatively easily. The centripetal force of circular
motion is proportional to a/P2, and it is provided by the gravitational force, which is proportional to 1/a2.
Hence,

which is Kepler's third law for the special case.

In the general case of elliptical orbits, the derivation is more complicated.

The area of the planetary orbit ellipse is

The areal speed of the radius vector sweeping the orbit area is

where

The period of the orbit is

satisfying
implying Kepler's third law