|Fig. 1: the Sun. The Death Star would require the power of 600,000 of these to completely destroy the Earth with a 1-second laser pulse. (Source: Wikimedia Commons)|
Few machines in science fiction match the raw destructive power of the Death Star, a moon-sized space station capable of turning an entire planet into an asteroid field with its enormous laser weapon. However, despite its effectiveness at quashing rebellions, the Death Star would be an absurdly impractical device to construct, even if its exhaust vent were not vulnerable to proton torpedoes. This analysis discusses the energy required to overcome the gravitational cohesion of an Earth-sized planet.
To calculate the energy needed to destroy a planet, we must first operationalize the term "destroy". Here, destroy will mean to overcome the planet's gravitational binding energy. This means that the energy must, at minimum, be equal to that needed to move every bit of mass contained in the planet out to infinity (i.e. outside the gravitational influence of every other part of the planet). For a sphere of uniform density, the gravitational binding energy is given by Ω=(3/5)GM2/R, where G is the universal gravitational constant, M is the mass of the sphere, and R is its radius.  Though a planet does not have uniform density (the core is much denser), we will approximate it as such for mathematical simplicity.
Under this model, the energy required to destroy the Earth would be 2.24 × 1032 Joules, approximately equal to the total energy output by the sun over an entire week (The sun's mean radiative luminosity is 3.83 × 1026 W).  Additionally, if the laser beam only fires for less than a second, during this time the weapon's power output would be roughly that of 600,000 suns (see Fig. 1).
To achieve these numbers would of course be completely ludicrous, even in the realm of science fiction. Lasers require electric power to operate, meaning that the battle station would need to contain either batteries or some sort of generator. We need not do a detailed calculation to see that batteries, or indeed any form of chemical energy storage, would be grossly inadequate--hydrocarbon fuels represent the upper limit for stably storing chemical energy, and it would take a gas tank larger than the Earth to store the required amount.
The sheer scale and impracticality of such a powerful weapon begs the question of whether it is necessary, even for an evil galactic empire. The answer: probably not. Other than intimidation, there is no military justification to completely obliterate a planet into dust. A much weaker laser could achieve a similarly destructive effect by vaporizing a portion of the surface, sending an inward-propagating shockwave through the planet that would amount to a worldwide cataclysmic earthquake. This would bypass the need for the considerable amount of energy needed to overcome the gravitational binding. As another option, it would require orders of magnitude less energy to cook off the atmosphere, and many orders of magnitude less than that to render the surface toxic via nuclear fallout. As a one-off weapon, even just crashing the Death Star (or a less expensive moon of similar size) into a planet would achieve comparable destruction.
Given the energy challenges involved, it was wise of the White House to reject a 2013 petition to construct a Death Star as a means of economic stimulus.  Besides, humanity has plenty of perfectly adequate means of self-destruction as things currently stand.
© Mason Black. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
 K. R. Lang, Astrophysical Formulae, Vol. I (Springer, 1999).
 R. Weiner, "White House Rejects 'Death Star' Petition," Washington Post, 12 Jan 13.