JEE Main & Advanced Physics Gravitation / गुरुत्वाकर्षण Kepler's Laws of Planetary Motion

Planets are large natural bodies rotating around a star in definite orbits. The planetary system of the star sun called solar system consists of nine planets, viz., Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto. Out of these planets Mercury is the smallest and closest to the sun and so hottest. Jupiter is largest and has maximum moons (12). Venus is closest to Earth and brightest. Kepler after a life time study, work out three empirical laws which govern the motion of these planets and are known as Kepler?s laws of planetary motion. These are,

(1) The law of Orbits : Every planet moves around the sun in an elliptical orbit with sun at one of the foci.

(2) The law of Area : The line joining the sun to the planet sweeps out equal areas in equal interval of time. i.e. areal velocity is constant. According to this law planet will move slowly when it is farthest from sun and more rapidly when it is nearest to sun. It is similar to law of conservation of angular momentum.

Areal velocity \[=\frac{dA}{dt}=\frac{1}{2}\,\,\frac{r(vdt)}{dt}=\frac{1}{2}rv\]

\[\therefore \] \[\frac{dA}{dt}=\frac{L}{2\,m}\]                           

[As \[L=mvr\]; \[rv=\frac{L}{m}\]]  

(3) The law of periods : The square of period of revolution \[(T)\] of any planet around sun is directly proportional to the cube of the semi-major axis of the orbit.

\[{{T}^{2}}\,\propto \,{{a}^{3}}\] or \[{{T}^{2}}\propto {{\left( \frac{{{r}_{1}}+{{r}_{2}}}{2} \right)}^{3}}\]

Proof :  From the figure\[AB=AF+FB\]

\[2a={{r}_{1}}+{{r}_{2}}\]   \[\therefore \]\[a=\frac{{{r}_{1}}+{{r}_{2}}}{2}\]        

where  a = semi-major axis

\[{{r}_{1}}=\] Shortest distance of planet from sun (perigee).

\[{{r}_{2}}=\] Largest distance of planet from sun (apogee).

Important data  

Note :

• Kepler's laws are valid for satellites also.