For various planets in the solar system, a graph is plotted with the logarithm of the orbital time period ( T ) on the y-axis and the logarithm of the orbital radius ( r ) on the x-axis. What will be the slope of the resulting straight line?

Step-by-step solution

According to Kepler's Third Law of planetary motion, the square of the time period ( T ) of a planet is directly proportional to the cube of its orbital radius ( r ). T^2 = k r^3 Taking the logarithm on both sides: (T^2) = (k r^3) 2 T = 3 r + k Rearranging the equation to match the form of a straight line ( y = mx + c ): T = 3 2 r + 1 2 k Here, y = T and x = r . The slope of the graph ( m ) is the coefficient of r , which is 3 2 . Answer: 3 2