Following three laws were presented by Kepler:
1. Law of Orbits: All the planets revolve around the Sun in an elliptical orbits and Sun is situated at one focus of the orbit.
(This law is different from the copernican model according to which planet can revolve around the sun only in circular orbits.)
Here θ is the center of the elliptical orbit and PO = AO is called the half major axis. Sun is at its one focus S and planet is shown at P.
2. Law of Areas: According to this law, the line joining planet and Sun sweeps equal area in equal time. It is clear from this law, as the distance of planet increases from sun, its orbital velocity goes on decreasing and vice-versa.
Derivation: Suppose at any time, the position vector and linear moment of the planet are \(\vec{r}\) and \(\vec{p}\) respectively.
∴ Distance travelled by the planet on its orbit in time interval ∆t will be \(\vec{v}\) x ∆t, where is orbital velocity of the planet.
Therefore area sweep by line joining the planet and Sun in time interval ∆t,
(Because angular momentum remains constant under influence of central force)
i.e., Areal velocity of the planet = Constant
This is the law of areas.
3. Law of Time Periods: Square of time period of planet is directly proportional to cube of radius of semi-major axis of an ellipse traced by the planet, i.e.,
T2 ∝ a2
Derivation: Suppose radius of major axis of ellipse is ‘a’ and radius of minor axis is 'b', then the area of the ellipse will be πab.
∴ Time period of the planet
From the definition of an ellipse, it latus rectum be l, then,
\(I=\frac{b^2}{a}\Rightarrow b^2=la\)
∴ Substituting this value of b2 in equation (1), we get
If average distance between Sun and planet be r, then
r ≈ a
\(\therefore\) \(T^2\propto r^3\)
According to Newton, mostly the planets revolve around the Sun in nearly circular orbits and according to Kepler’s second law, the areal velocity of radius vector of planet remains constant, therefore the orbital velocity v of the planet and its angular velocity ω will remain constant. If r be the average distance between the planet and the Sun and m its mass, then the centripetal force acting on the planet towards the Sun,
\(F = mr\omega^2 =mr\frac{4\pi^2}{T^2}\)
Where, T is the time period of the planet around the Sun.
According to Kepler's third law,
\(T^2\propto r^3\)
or T2 = Kr3
Thus Newton obtained following three conclusions from Kepler’s law:
(i) A force acts on the planet towards the Sun.
(ii) This force is directly proportional to the mass of the planet.
(iii) This force is inversely proportional to the square of the average distance between the planet and the Sun.
On the basis of above conclusions, Newton presented his law of gravitational attraction as, “The gravitation applied by a body on another body is:
(i) Directly proportional to the mass of that body, and
(ii) Inversely proportional to the square of the distance between them.”
