Blowing up REALLY big things is hard because, well, they are REALLY big.

Though we are going to try anyway I guess.

The first step is quantifying the minimum effort it is going to take to destroy our REALLY big thing

Keep in mind that REALLY REALLY big things are MUCH harder to destroy than regular REALLY big things.

We got a few ways to do this. Which one we use depends on how exactly we are planning on destroying this REALLY big thing.

Firstly if we want to know the minimum energy to blow it up with an instantaneous application of energy. We can calculate its gravitational binding energy in joules, which uses the following equation,

        3*(gravitational constant)*(Mass in Kilograms)^2
       ----------------------------------------------------
                     5*(Radius in Meters)

Where the gravitational constant equals 6.67408e-11 m³ kg^-1 s^-2

Here is the equation in Wolframalpha if you want to play around with it.

Some example GBEs for reference,

• Luna: 1.245e29 joules

• Earth: 2.239e32 joules

• Sol: 2.276e41 joules

Realize that his number is not the amount of "hitpoints" the REALLY big thing has. Any inefficiency in your method of blowing it up needs to be accounted for.

For instance if you wanted to blow up the Sol with TNT, you would need 5.4e34 kilograms of TNT. Which means the TNT would actually out mass Sol by four orders of magnitude. Thus you would need to recalculate the GBE for Sol mass of TNT along with adjusting the radius of the body to account for the inefficiency in your method.

On second thought lets just not use conventional explosives to do this because most of the time you will end up creating a REALLY REALLY big thing made out of TNT before you actually have enough to blow up a normal REALLY big thing. Perhaps ironically REALLY REALLY big things made out of TNT don't have the total energy needed to completely blow itself up.

There is also a minimum amount of work you must be doing in order to permanently damage a REALLY big thing.

For instance if you are on the Luna and your right arm is super strong for some reason... throwing a one kilogram Moon rock at 500 m/s will just result in the rock falling back to Luna eventually. The rock MUST be thrown at least at 2.38 kps (escape velocity) in order to avoid this. Note: As a REALLY big thing is destroyed it's escape velocity will go down making this easier.

Similarly if you are trying to get rid of the REALLY big thing with some kind of heat based weapon. Said weapon either must be able to cause impulsive shock or output such a tremendous amount of energy the REALLY big thing can't radiate the energy away faster than your energy weapon pumps heat into it.

Keep in mind that simply melting then vaporizing small amounts material will not cut it. The vaporized material will cool and freeze almost instantly after it leaves your beam, then it will fall back to the surface of the REALLY big thing undestroying it.

Method two is trying to destroy REALLY big things, by taking the REALLY big thing and crashing it into a REALLY REALLY big thing.

Keep in mind that in general the easiest way to do this will almost always be to crash it into the nearest REALLY REALLY big thing.

If we are trying to deorbit Luna you only need to change it's velocity by ~1 kps to make it get extra cozy with Earth. If you want to drop it into Sol you need to change it's velocity by at least ~29 kps.

So any way in order to quantify this we are going to need to use the ideal rocket equation to find out how much we can change the REALLY big thing's velocity.

The ideal rocket equation is as follows, (effective exhaust velocity or Isp)*(the natural log of (wet mass/dry mass).

Some examples of rocket Isp are,

• Space Shuttle Solid Rocket Booster: 250

• Saturn V Stage 1 Engine: 263 (Sea level)

• Space Shuttle Main Engine: 366-452.3 (depending on atmospheric pressure)

As a rule of thumb in order of increasing Isp, solid rocket fuels (~250 or less) < mono-propellants < bi-propellants (~300-450) < Lithium-Florine-Hydrogen tri-propellant (~530) < Nuclear Thermal Rockets (>1000) < Orion Drives (>10,000) < Ion/Magnetoplasmadynamic Engines (>10,000) <<<<<<<<<<<<<<<<<<< Ideal Light Pressure Drives (~30,000,000), to get above this point you need to break some important laws of physics.

Keep in mind this is a very general rule of thumb. Everything is in the right place but the numbers are hardly exact. Don't cite me on a term paper.

Now if we are using reasonable engines (chemical ones) it will take masses of fuel on the order of the all of Earth's ocean to even change a REALLY big thing's velocity measurably.

Another thing to keep in mind Ion/Magnetoplasmadynamic Engines and Ideal Light Pressure Drives produce extremely tiny amounts of thrust. So really the only engines that might be able to accomplish moving a REALLY big thing in a reasonable time frame are either a shitload of NTRs and/or Orion Drives.

OK so now that you figured out what kind of engine you are going to use we need to figure out exactly how much delta-v it will take to turn our REALLY big thing into a cosmic billiard ball.

For those that do not know delta-v in this context is the amount your REALLY big thing needs to change its velocity in order to smack into another really big thing.

The easy way is if your REALLY big thing is just a moon/satellite, generally the answer will be equal to slightly less than it's orbital velocity.

I.e. for Luna which has an orbital velocity of ~ 1 kps the delta-v needed to drop it onto the Earth is ~ 1 kps.

But for planets dropping them into the Star they are orbiting is generally going to cost more delta-v than smashing them into a REALLY big thing that is nearby (relatively speaking).

Do do that we are going to use a Hohmann Transfer Calculator.

Hohmann transfers are pretty much the minimum amount of delta-v needed to get into a different orbit.

Alright so if we want to smash Earth with Mars, we plug in

227900000 km for the initial height and 149597870 km for the final height.

The gravitational parameter = gravitational constant*mass

For Sol that is 1.33*10²⁰. It says "planet radius" on the calculator but really it means whatever the object is orbiting. The Sol's radius is 695,700 km.

Now this silly calculator thinks we want to circularize our orbit (which we in fact don't want to do) so for our purposes ignore the "Total DeltaV" number and only focus on "DeltaV1".

The number I got is 2,637 m/s which looks about right to me.

Now that we have the DV number we can plug that into the ideal rocket equation along with the Isp of whatever rocket we are using and figure out exactly how much fuel mass we are going to need.

2,637 = 450*ln(x/6.41693e23)

Now I could calculate that out but that looks annoying so lets just use Wolframalpha to solve it for us.

So the total mass of fuel we need to cause Mars to collide with Earth is 2.25*10²⁶ kg or about half the mass of Saturn.

You might be thinking "isn't this impossible because moving a Saturn mass of fuel to Mars would probably take like a solar mass of fuel" and you would be right.

For this kind of thing to work you practically need to be using the Light Pressure Drive.

Also remember if your REALLY big thing has an atmosphere Isp drops by close to 100% unless you are using the Light Pressure Drive.

Anyway I think that is it for PART 1: Finding Out How Much Energy/Work it Takes to Destroy A REALLY Big Thing And Other Things Too. If you actually read this far sorry for being so wordy at times I didn't edit this. If there is any interest in me going into more details about the problems/different methods I might make more parts.

If you have any questions feel free to ask.