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How to Destroy a Planet

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Overview

This document is treats the explosion of planets and planetary scale objects in detail. It is unapologetically technical. The discussion below is intended as a reference for readers who are proficient in basic physics or mathematics, who may want to reproduce results stated in other commentary pages on this site. The information below can also be used as the basis for similar calculations applied to planetary-scale explosions in other works of science-fiction.

Special thanks are due to, in alphabetic order:

• Frank Gerratana for the images of Alderaan.

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Binding energy.

An object propelled outwards from the surface of a planet does not necessarily fall to ground again. If it is hurled with more than a certain threshold speed, the escape velocity, it has sufficient energy to fly out indefinitely without being turned by the planet's gravity. For worlds habitable to human beings the escape velocity at the surface is of the order of 11km/s.

In order to completely destroy a planet a sufficient quantity of energy must be supplied for all of the planet's material to be ejected at or above escape velocity. We shall call this minimal amount the total gravitational binding energy. However this is not simply the kinetic energy of a planet-size mass moving at the surface escape velocity. Removal of the outer layers of the planet means that the new "surface" is closer to the centre of gravity (tending to give stronger surface gravity) but the gravity also tends to be weakened because there is less mass left. The real value of the total binding energy depends greatly on how the planet's bulk is distributed throughout its interior.

Assuming that the planet is spherical (a very good approximation for any body capable of sustaining human or other life), the proper way to do the calculation involves the consideration of the density as a function of radial distance r from the centre,

and the mass contained within a certain radius, These can be applied in the standard equation for the Newtonian two-body gravitational potential, to give an equation for the binding energy contribution of a thin shell of material at radius r within the planet:

To evaluate the total binding energy U from this differential equation it is necessary to know a functional form for the density profile of the planet. Planets tend to be divided into several layers, each of which has distinctive chemical characteristics. In the case of the typical terrestrial planet there will at least be a distinct crust, mantle and core. The density of material can change suddenly at the boundaries between layers. For this reason I find it useful to treat the density function in a piecewise fashion for each layer (l) , as polynomials of r. In each layer the polynomial coefficients are chosen for a fit to a geological density profile.

The total mass within a layer l and the layers below it is given by:

Equipped with these expressions it becomes possible to solve the differential equation for the binding energy. For the sake of expressing the final solution in a simple and succinct form it is useful to define the following three properties for each layer:

Note that the first of these is simply proportional to the internal mass of a layer used above:

Then the total gravitational binding energy of the planet (U) can be expressed in the equation:

The only thing remaining to be done is to determine the radii of layer boundaries and the polynomial coefficients of the density function within each layer of a particular planet, and then simply substitute the values into the above summation.

It must be emphasised that the gravitational binding energy is merely the absolute minimum energy required to blow a planet apart. It assumes complete efficiency in converting the injected energy into kinetic energy of the blast. In a real planetary demolition a substantial fraction of the input energy would be inadvertently wasted in heating some of the material. Also, a planet exploding at just above the threshold energy would take minutes or hours to waft apart completely. Although the demolition would be incomprehensibly violent to living beings on the surface of the doomed world, such a weak explosion would seem inelegant and might offer enough time for the escape of lucky starships already prepared for launch. Fortunately the Death Stars have been designed to yield many thousands of times more energy per blast than the minimum required. The planet Alderaan vanished spectacularly in under two seconds rather than hours. (For further discussion, refer to Death Stars.)

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Habitable planets.

Habitable planets tend to be on the order of 6400km in radius and composed of a mainly silicate crust and mantle, and a metallic core. Average density over the whole sphere tends to be around four to six times that of water. Using the density profiles for Earth and Venus below, the binding energy for a habitable world with mass about 5.9 x 10²⁴kg turns out to be:

U = 2.4 x 10³² joules.

For use in the equations of the previous section, the radii for the layer boundaries were taken to be:

	r0 = 0.000 x 100 m

	r2 = 3.474 x 106 m
	r3 = 5.578 x 106 m
	r4 = 6.098 x 106 m
	r5 = 6.378 x 106 m

The density profile in each section was treated as a quadratic function (ie. the coefficients went from zeroth to second order). In the various layers, the 0th, 1st and 2nd coefficients in SI metric units were respectively:

	1.24 x 104	-1.16 x 10-4	-2.48 x 10-10
	6.81 x 103	-3.85 x 10-4	 0.0
	1.39 x 104	-1.69 x 10-3	 0.0
	8.01 x 103	-7.86 x 10-4	 0.0

Radial density models of Venus and Earth. Dashed profile is Earth. The interior of Venus and tectonic implications, Roger J. Philips & Michael C. Malin, in Venus. Diagram originally from: Basaltic Volcanism Study Project 1981, Basaltic Volcanism on the Terrestrial Planets, New York Pergamon, pp.633-699.

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Gas-giant planets.

Gas-giant planets are usually composed primarily of hydrogen, helium and other lighter elements. This gives them very low density. The planet's nett density is comparable to water. Gas-giants range in mass from about a dozen to several hundreds of times the standard mass of a habitable world. In radius the greatest of gas-giants are over a hundred thousand kilometres.

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Alderaan

Assuming that PAL video preserves every frame of the movie and does not repeat any frames, the characteristic speed of the outermost portions of the debris cloud is about 1.8x10^{^7}m/s. This is 6% of lightspeed, or about 1600 times escape velocity.

It follows that the energy of the explosion is less than 2.6 million times the threshold. The exact number depends on how the energy is distributed throughout the mass. This is only an upper limit because it's based on the most rapidly expanding (highest velocity) particles.

There were two equatorial rings expanding outwards in the equatorial plane. They moved at different speeds, but both were highly relativistic. A direct Newtonian interpretation of the motion of the rings is invalid; relativistic corrections are needed. The first ring moves at about 0.29c, and the second ring that overtakes and consumes it consumes it has a velocity of at least 0.91c. There is even an irritating possibility that it might be mildly superluminal.

The planet Alderaan, seen in the instants immediately before its demolition. The image of the initial condition of the planet is used as a yardstick to determine the scale of the explosion in subsequent frames. The frames of film are set 1/24 seconds apart.

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Rings

Flaming circular rings expanding from the center of the exploding object at velocities which can be highly relativistic. It is difficult to explain why it is planar, rather than expanding in a spherical front.

[Indepdently suggested by Robert Brown(1) and Brian Young.] Perhaps it really is spherical, but the emission is not isotropic. The beamed emission may only reach an observer from only an annular region. Put another way, from a given position and angle, light is received only from a ring because of the angles at which the light is emitted from particular parts of the shell. However this might not be consistent with the symmetry of the explosion; it may imply that different sides of the shell beam in different directions. Perhaps the beaming is concial rather than unidirectional? Perhaps a more subtle physical effect is at work, like a rainbow.

Alternatively, it may be necessary to determine why there is a preferred plane in the explosion for the ring but not for the other debris. It may be related to the spin axis of the object. In the case of the Death Stars, the special plane may be determined by the path travelled by the chain reactions leading to the final detonation. Alternatively, the preferred plane may be determined by the nett angular momentum of the planetary or Death Star deflector shield, or some other shield characteristic.

Incandescent rings emerging from the explosions of Alderaan and the first Death Star.

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