2005-06-05 by
qntm

Know your enemy.

- Names: Earth, Terra, "the world"
- Age: 4,550,000,000 years
- Mass: 5,974,200,000,000,000,000,000 metric tonnes; this increases by 10,000 to 100,000 tonnes daily due to the accumulation of meteorites and interplanetary dust
- Radius: 6,378.1 km
- Surface gravity: 9.798 m s
^{-2} - Escape velocity: 11,186 m s
^{-1}

Crust |
0 to 35km | Rock, hard and soft sediments, ice, miscellaneous | 0 to 1000°C |

Mantle |
35 to 2900km | Oxides of silicon, magnesium, iron and aluminium | 1000 to 3700°C |

Core |
2900 to 6371km | Iron (liquid, turning to solid at 5150km) | 3700 to ~5000°C |

Iron |
34.6% |

Oxygen |
29.5% |

Silicon |
15.2% |

Magnesium |
12.7% |

Nickel |
2.4% |

miscellaneous |
5.6% |

The rest are all in the standard SI kilogram-metre-second system since it makes the working infinitely easier. You can most likely convert to kilometres and tonnes in your head, and converting to tons and miles is a simple job for the Google Calculator.

- Mass: 5.9742 * 10
^{24}kg - Distance from Sun: 149,595,000,000 m
- Orbital period: 31,556,926 s (365 days, 5 hours, 48 minutes, 46 seconds)
- Orbital velocity: 29,780 m s
^{-1} - Orbital momentum: 1.7789 * 10
^{29}kg m s^{-1} - Orbital kinetic energy: 2.6488 * 10
^{33}kg m^{2}s^{-2}

- Radius: 6,378,100 m
- Sidereal rotation period: 86164.1 seconds (23 hours, 56 minutes, 4.1 seconds - this is not to be confused with the length of a day, which is precisely 24 hours)
- Angular velocity: 0.0000729211 rad s
^{-1} - Moment of inertia: 9.6987 * 10
^{37}kg m^{2} - Angular momentum: 7.07236 * 10
^{33}kg m^{2}s^{-1} - Rotational kinetic energy: 2.5786 * 10
^{29}kg m^{2}s^{-2}

The energy required to destroy the Earth varies by method. It also varies depending on what existing "free" energy you are using. For example, burning the Earth up in the heart of the Sun takes a lot of energy, but since all that energy is freely given out by the Sun, we don't need to figure out how to generate it - just the energy needed to get it there in the first place.

These calculations are all exceedingly approximate.

- Surface area of Earth: 5.112 * 10
^{14}m^{2} - Black-body temperature of Earth: ~287 K
- Power radiated by Earth: ~1.966 * 10
^{17}kg m^{2}s^{-3} - Black-body temperature of Earth after being completely boiled: ~3134K
- Power radiated by Earth at this temperature: ~2.796 * 10
^{21}kg m^{2}s^{-3} - Ratio of these two powers: 1 to ~14,000

So if the surface area of Earth were increased by roughly 14,000 times, it would be completely reduced to a gas.

- Cross-sectional area of Earth: 1.278 * 10
^{14}m^{2} - Necessary area of mirror: ~1.80 * 10
^{18}m^{2}

The minimum amount of energy required to directly destroy the Earth in situ is equal to the Earth's gravitational binding energy:

- Gravitational binding energy of the Earth: 2.2405 * 10
^{32}kg m^{2}s^{-2}

In the case of the antimatter method all that energy will be generated from the lossless conversion of equal parts of matter and antimatter to energy.

- Mass of matter and antimatter required to blast the Earth apart: 2.4928 * 10
^{15}kg - Mass of antimatter: 1.2464 * 10
^{15}kg

To do this we simply need to cancel out all of the Earth's current orbital kinetic energy. This is already calculated above...

- Orbital kinetic energy = 2.6488 * 10
^{33}kg m^{2}s^{-2} - Energy output of the Sun, for reference = 3.86 * 10
^{26}kg m^{2}s^{-3}

- Mass of the Sun: 1.98892 * 10
^{30}kg - Distance of Earth from the Sun: 149,595,000,000 m
- Gravitational potential energy of the Earth at this point: -5.3003 * 10
^{33}kg m^{2}s^{-2} - Distance of Jupiter from the Sun: 778,570,000,000 m
- Gravitational potential energy of the Earth at Jupiter: -1.0184 * 10
^{33}kg m^{2}s^{-2} - Potential difference: 4.2819 * 10
^{33}kg m^{2}s^{-2} - The Earth's existing kinetic energy = 2.6488 * 10
^{33}kg m^{2}s^{-2} - The difference which we need to make up = 1.6331 * 10
^{33}kg m^{2}s^{-2}

Practically speaking there may be ways to cut down the figures from the previous two methods by judicious use of things like gravity assists (get free energy from a passing planet e.g. Venus, Mars).

- The ratio of a circle's circumference to its diameter, π = 3.14159265...
- The gravitational constant, G = 6.67300 * 10
^{-11}m^{3}kg^{-1}s^{-2} - The speed of light, c = 299,792,458 m s
^{-1} - The Stefan-Boltzmann constant, σ = 5.670400 * 10
^{-8}kg s^{-3}K^{-4} - Escape velocity = sqrt(2GM/r), where G is the gravitational constant, M is the Earth's mass and r is its radius
- Orbital velocity = 2πr/T, where r is the orbit's radius and T is the period of one orbit
- Momentum = Mv, where M is the Earth's mass and v is its velocity
- Kinetic energy = (1/2)Mv
^{2}, where M is the Earth's mass and v is its velocity - Angular velocity = 2π/t, where t is the period of one rotation
- Moment of inertia = (2/5)Mr
^{2}, where M is the Earth's mass and r is its radius (assumes the Earth is a sphere) - Angular momentum = Iω, where I is the Earth's moment of inertia and ω is its angular velocity
- Rotational kinetic energy = (1/2)Iω
^{2}, where I is the Earth's moment of inertia and ω is its angular velocity - Gravitational potential energy = -GMm/r, where G is the gravitational constant, m is the mass of the object whose potential we are determining, M is the mass of the object whose gravitational field we are sitting in, and r is the distance between them
- Gravitational binding energy = (3/5)GM
^{2}/r, where G is the gravitational constant, M is the Earth's mass and r is its radius - Einstein's mass-energy equation, E = mc
^{2}, where E is energy, m is mass and c is the speed of light - The Stefan-Boltzmann law: P = σAT
^{4}, where P is the power radiated by a perfect black body, A is its surface area and T is its temperature