Question

How much energy would be required to move the earth into a circular orbit with a radius $1.0km$ larger than its current radius?

Short answer

Total of$1.89\times {10}^{25}J$energy would be required to move the earth into a circular orbit with a radius$1.0km$.

Step-by-step solution

Given Information

New radius of the circular orbit is $1.0km$larger than the current radius.

Formula used

Excess energy $∆E=\frac{GMm}{2R}-\frac{GMm}{2\left(R+r\right)}$

Here, gravitational constant $G=6.67\times {10}^{-11}N·m^2/k{g}^2$.

Mass of the sun, localid="1648185717314" $M=1.989\times {10}^{30}kg$.

Mass of the earth, $m=5.97\times {10}^{24}kg$.

Initial radius of earth's orbit, localid="1648185813557" $R=1.5\times {10}^{11}m$.

New radius of earth's orbit,$R+r=\left(1.5\times {10}^{11}+{10}^3\right)m$.

:  Calculation

Substitute the values and obtain$∆E$.

$$\begin{gathered} ∆E=\frac{GMm}{2R}-\frac{GMm}{2\left(R+r\right)} \\ ∆E=\frac{GMm}{2}\left[\frac{1}{R}-\frac{1}{R+r}\right] \\ ∆E=\frac{6.67\times {10}^{-11}N·m^2/k{g}^2\times 1.989\times {10}^{30}kg\times 5.97\times {10}^{24}kg}{2}\left[\frac{1}{1.5\times {10}^{11}m}-\frac{1}{1.5\times {10}^{11}m+{10}^{3}m}\right] \\ ∆E=39.6\times {10}^{43}\times 4.78\times {10}^{-20}J \\ ∆E=1.89\times {10}^{25}J \end{gathered}$$

Final Answer

Total of$1.89\times {10}^{25}J$energy would be required to move the earth into a circular orbit with a radius$1.0km$ .