PARTICLE BEAM POWER OUTPUT
EARTH, AEGIS SATELITE

The weapon under consideration is a Particle Beam Cannon, and its theoretical destructive capability and the possible colatoral damage this weapon might have caused when Earth Alliance President William, Morgan Clark aimed this weapon system at his own home world - a final act of vengence when it had become clear that he was about to be defeated and held responsible for his crimes against humanity.

The composition of the Particle Beam and its role in the weapon is unclear; the role can be anything from polluting the target area with radiation to enhancing the way the energy is delivered. One constant remains: however the energy reaches its target, the delivered amount of it, and hence the effective power output of the weapon can be calculated.

The starting information for the calculation is as follows: the Particle Beam Cannon is able to MELT, "turn into glass" an area of the Earth's crust, 400,000 square kilometers (this is 4*10¹¹ square meters), to the depth of 4 centimeters (this is 0.04 meters) in 30 seconds sustained fire at maximum power.

Earth's crust is composed of between 60 - 65% of silica by weight, also known as silicon dioxide, its chemical formula is SiO2.  Silica occurs in the form of sand as well as other common minerals such as quartz, flint agate, and a lot of others. For our purposes let's take the 65% as the value for silica in Earth's crust.

Alumina, or aluminum oxide, its chemical formula is Al2O3, is the other most common mineral in Earth's crust, it is the component of a variety of minerals. The 3 elements' oxygen, silicon, and aluminum account for 84% of Earth's crust by weight. The other major components of the crust are calcium, potassium, sodium, and magnesium with other elements occurring in smaller quantities. By these numbers most of the crust is composed of silica/ (silicon and oxygen) and Alumina/ (aluminum and oxygen), with only about a fifth left for the compounds of the other elements.  A lot of the compounds outside the major two, will have similar melting properties on average to the two. All this considered it is a valid and relatively small approximation, to say that Earth's crust is composed of 65% of silica and 35% Alumina, both by weight - basically we substitute the properties of Alumina for the minority of the remaining compounds and retain the value for silica. Since the density of silica and Alumina are 2.6 g/m³ and 4.0 g/m³ respectively, it is a straightforward calculation to show that 65% of silica and 35% of Alumina, both by weight, is the same as 74% of silica and 26% of Alumina, both by volume.

The whole process of melting a substance involves two distinct steps, which take up energy.  First the substance has to be raised to the temperature at which it begins to melt.  Second the melting substance takes up energy it needs for the process but its temperature does not rise until all of it is melted, any hot pockets in the already melted part are reabsorbed into melting the solid.  The two steps require differing amounts of energy described in differing ways because of the specifics of each step, and so, the energy for each step has to be calculated separately.

Let's deal with silica first:

One mole (a mole is a standard number in chemistry, used not unlike a "dozen", except that a mole is very large. One mole is 6.022*10²3.) of SiO2, silica/silicon dioxide, molecules has mass of 60.09 g (g for grams).

The density of silicon dioxide is 2.6*10⁶ g/m³.  The area under question, 4*10¹¹ m² (square meters), to the depth of 0.04 m (meter), gives a volume of 1.6*10¹⁰ m³ (cubic meters).

Since silica makes up 74% of crust by volume (see above), the volume of silica here is 0.74 * 1.6*10¹⁰ m³ = 1.2*10¹⁰ m³, which is 1.2*10¹⁰ m³ * 2.6*10⁶ g/m³ = 3.1*10¹⁶ g, which is 3.1*10¹⁶ g/60.09 g = 5.1*10¹⁴ moles.

The molar heat capacity, the amount of energy required to raise the temperature of one mole of a substance by one K (Kelvin), of silicon dioxide is 44 J/Kmol (J stands for Joule, a unit of energy, K is unit of temperature, mol stands for mole).

The melting point for silicon dioxide is 1700 degrees C (Celsius). Let's take the temperature before the weapon hits as 25 degrees C, a pleasant, cool summer day, or a normal room temperature.  Hence, to reach its melting point the temperature of the silicon dioxide has to be raised by 1675 degrees C, or 1675 K, as one Kelvin equals one degree Celsius.

So the energy required to raise the silica in our volume to its melting point temperature is 1675 K * 5.1*10¹⁴ mol * 44 J/Kmol = 3.8*10¹⁹ J.

The molar heat of melting for silica is 8 kJ/mol (kJ stands for kilojoule, one thousand joules), so the energy required to melt the silica in our volume once it reached its melting temperature is 5.1*10¹⁴ mol * 8*10³ J/mol = 4.1*10¹⁸ J.

In total the amount of energy required to melt the silica in the volume being melted by the Particle Beam Cannon is 4.2*10¹⁹ J.

Now to the Alumina component:

Alumina has the following properties: melting point of 2045 degrees Celsius; density of 4.0*10⁶ g/m³; molar heat capacity of 79 J/Kmol; molar heat of melting of 109 kJ/mol; one mole of Alumina has mass of 101.96 g.

Hence similar to above, since the Alumina makes up 26% of the volume of 1.6*10¹⁰ m³, from its density it follows the mass of the Alumina to be melted is 1.7*10¹⁶ g, which is 1.6*10¹⁴ moles. Starting from 25 degrees Celsius, the Alumina has to be raised through 2020 K to its melting temperature.

Following the same procedure in calculations as above for silica, the energy required to raise the Alumina in our volume to its melting point temperature is 2.6*10¹⁹ J, and the energy required to melt the Alumina in our volume at it melting point temperature is 1.8*10¹⁹ J.

In total the amount of energy required to melt the Alumina in the volume being melted by the Particle Beam Cannon is 4.4*10¹⁹ J.

Therefore, the total amount of energy required to melt Earth's crust over an area of 400,000 square kilometers down to the depth of 4 centimeters is 8.6*10¹⁹ J. Since this is to happen in 30 s (seconds), the power of the Particle Beam Cannon is 8.6*10¹⁹ J/30 s = 2.7*10¹⁸ J/s = 2.7*10¹⁸ W (watts), as one J per second is one watt.  In more familiar terms the power of 2.7*10¹⁸ W is 2.7 million Terawatts.

The above result still has to be corrected for one more thing.  Mixtures of chemical compounds, that is impure compounds, tend to melt at lower temperatures than the pure compounds in them. The mantle, the part of Earth's surface directly under the crust, starts to melt at around 1500 degrees Celsius. The comparison of this to silica's and alumina's melting points of around 2000 degree Celsius would indicate that the melting points are lowered by around 20 - 25% due to the compounds making themselves mutually impure. Say it is 25% on average. Assuming that it takes about the same amount of energy to melt the impure compounds once they are heated to their melting points, the energy required to reach the melting points is reduced by 25%. Reducing the energies above for reaching melting points gives a value for energy required to melt the silica in our volume of 3.3*10¹⁹ J, and 3.8*10¹⁹ J for the Alumina in our volume.

Hence the total energy required to melt Earth's crust over an area of 400,000 square kilometers (this would be an area of radius of 357 kilometers), corrected for being an impure substance, is 7.1*10¹⁹ J. Since this happens in 30 s, the power is 7.1*10¹⁹ J/30 s = 2.4*10¹⁸ W which is 2.4 million Terawatts.

CONCLUSION: Assuming that the Particle Beam Cannon can melt Earth's crust over an area of 400,000 square kilometers, down to the depth of 4 centimeters, in 30 seconds, it means that the power output of the Particle Beam Cannon is 2.4 million Terawatts.

~Written by Stilgar