Fill in the blanks:
$(a)$ The value of gravitational intensity at the center of the Earth is ..... .
$(b)$ The potential energy of a satellite is $-8 \times 10^9 \, J$,then its binding energy is ............ .
$(c)$ Kepler's second law regarding the constancy of the areal velocity of a planet is a consequence of the law of conservation of .......... .

(N/A) At the center of the Earth,the gravitational field intensity is $0$ because the mass enclosed within a sphere of radius $r=0$ is zero.
$(b)$ The binding energy $(BE)$ of a satellite is defined as the negative of its total mechanical energy. Since the total energy $E = -|PE|/2$ is not given,we use the relation $BE = -E$. For a satellite,$PE = 2E$. Given $PE = -8 \times 10^9 \, J$,then $E = -4 \times 10^9 \, J$. Thus,$BE = -(-4 \times 10^9 \, J) = 4 \times 10^9 \, J$. However,if the question implies the total energy is $-8 \times 10^9 \, J$,then $BE = 8 \times 10^9 \, J$. Assuming the standard convention where $PE$ is given,the binding energy is $4 \times 10^9 \, J$. If the value provided is the total energy,it is $8 \times 10^9 \, J$.
$(c)$ Kepler's second law (law of areas) is a direct consequence of the law of conservation of angular momentum.