# Energy/Momentum required to deflect earth from its orbit

This question occurred to me when thinking about the firepower people have on earth.

How hard is it to change the characteristics of the orbit of the earth when an explosion happens on its surface, using conservation of momentum?

I know already that the orbit of the earth is not so accurate already and fluctuates a lot, causing the so-called "leap second".

So my question is: How could such a thing be measured/quantified?

Can we say something like: The explosion energy required to deflect the eccentricity of the earth's orbit by 1%. And how much would that be?

Any ideas?

• Related: physics.stackexchange.com/q/56245/2451 and links therein. – Qmechanic Mar 15 '14 at 22:49
• This question strikes me as purely calculational and should be closed as homework. It's probably better suited for XKCD's What-If than Physics.SE. – Brandon Enright Mar 15 '14 at 23:43

So if we consider the 1% change to be directly inward (toward the sun) we can look at the gravitational potential energy difference between these two potential levels. This would be equal to 1% of the Earth's current gravitational potential energy. The gravitational potential energy is given by the equation

\begin{equation} U = -\frac{GmM}{r} \end{equation}

If we substitute the mass of earth, $5.97\times10^{24}~\mathrm{kg}$, and the mass of the sun, $1.99\times10^{30}~\mathrm{kg}$, and use the semi-major axis of the earth's orbit, $149,598,261~\mathrm{km}$ as the orbital radius we can find that the energy of the earth in its current orbit is approximated as

\begin{equation} U = 5.30\times10^{33}~\mathrm{Joules} \end{equation}

1% of this energy is

\begin{equation} 5.30\times10^{31}~\mathrm{J} \end{equation}

This is a lot of energy, and I find it sometimes difficult to comprehend very large orders of magnitude. To better understand this energy we can first compare this figure to the energy of the largest nuclear weapon every detonated by humankind, the Tsar Bomba, a nuclear weapon with a yield of approximately $225~\mathrm{PJ}$ or $2.25\times10^{18}~\mathrm{J}$. In order to change the energy of the earth's orbit by 1% using these weapons we would need to direct the equivalent energy of 22.6 trillion Tsar Bombas on the earth inward with respect to its orbit around the sun. This is still a very large number and I wanted to see if I could think of a better comparison.

The meteor that impacted with the earth approximately 65 million years ago, initiating the mass extinction of the dinosaurs is thought to have had a impact energy equivalent to $4.23\times10^{23}~\mathrm{J}$. In order to decrease the earth's orbital energy by 1% we would need the energy equivalent to approximately 126 Million of the Chicxulub impactor.

Based on the figures above, and assuming my math is correct, it would seem that none of the actions taken by man thus far would have any appreciable effect on the earth's orbit.