Mathematical Laws of KSP (updated 7 July)

[Gaarst]

Yesterday, I made a joke on this thread about the fact that Kraken probability followed an exponential law when using service bays.

As several users found it funny, I had this idea of making more of this type of mathematical laws applied to KSP.

Though they are based on real observed KSP behaviours enhanced with very serious statistical studies, I don't recommend using those to plan any space program. You've been warned !

Here they are:

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The Mathematical Laws of KSP

1st Law: The booster number theorem

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The ideal number of boosters (N_B) for any rocket is given by:

N_B = N_C + 1

Where N_C is the current number of boosters in the rocket.
Note that N_C has to be recalculated each time you add anything to the rocket (boosters included).

Sidenote: Moar boosters !

 

2nd Law: The structural integrity law

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The structural integrity factor (I) determines the resistance of a rocket to structural loads of any sort: aerodynamics, mass, internal forces... It varies between 0 and 1, 1 being a virtually indestructible rocket, while 0 appoximately corresponds to the structural integrity of a cooked spaghetti.

I = N_s / N_tot

Where:

• N_s is the number of struts in the rocket
• N_tot is the total number of parts in the rocket

Sidenote: Moar struts !

 

3rd Law: The Δv problem

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The Δv problem comes from the discrepancy observed between theoretical Δv predictions and the actual amount of Δv used during a mission. The fact that it is present during any kind of orbital manoeuvre suggests that there is an undiscovered phenomenon which accounts for this difference.
The Δv problem can be stated very simply as:

Δv_M > Δv_M'

Where:

• Δv_M is the Δv used during any manoeuvre M
• Δv_M' is the theoretical Δv required for this manoeuvre M

Sidenote: Always remember to take more Δv than you should need

 

4th Law: The Kraken-service bay distribution

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The probability of a Kraken attack in function of the mission time (P_K,t) on your ship can be described by an exponential law. The corresponding probability density function is given by:

P_K,t = ne^{-(nt / √ N)} / √ N

Where:

• n is the total number of parts inside service bays
• N is the total number of service bays in your ship
• t is the time elapsed since the beginning of the mission (or the duration of the mission), in hours

Intergrating the above formula over t gives us the probability of a Kraken attack (P_K) on your ship after a mission time t:

P_K = 1 - e^{-nt / √ N}

Sidenote: Do not use service bays.

 

5th Law: The patched conics uncertainty principle

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You cannot know exactly either your velocity or your heading: there is limited accuracy on one of these two values. As a result your trajectory will always have a fraction of random fluctuations that will become more significant the more accuracy is needed on that trajectory.
In other words, you won't know if you're actually going to Eeloo until you're more than halfway there, and it is impossible to have a perfectly stable circular orbit as in that case, the uncertainty of position reaches several metres.

σ_α σ_v > k / 2

Where:

• σ_α is the uncertainty over the heading in °
• σ_v is the uncertainty over the velocity in m/s
• k is the reduced Kerman constant, whose exact value is still to determine

Using properties of the ellipse and Keplerian laws, it is possible to derive a similar formula for almost circular orbits (e close to 1):

σ_rσ_t > ʞ / 2

Where:

• σ_r is the uncertainty over the distance between the ellipse focus and a point A on the ellipse
• σ_t is the uncertainty over the time to reach that point A
• ʞ is the elliptic reduced Kerman constant, whose exact value is still to determine

 

6th Law: Properties of the memory usage function

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While memory usage is very difficult to predict as it depends on the usage of the game: number of launches, mods, rockets, settings...; one can easily establish basic properties of the derivative of such a function.
Define the function M(t) as the memory usage in bytes in terms of time, then dM/dt has the following properties:

1. lim_∞ (M(t)) > 3.7*2³⁰
2. dM/dt > 0
3. dM/dt ∝  (N_mod, N_load, N_vess, N_part)
4. There exists an unpredictable, non-constant and partially random factor K > 0 varying non-linearly with t such that M(t) = ∫ (dM/dt)dt + K
5. There exists a factor K_min > 0 such that: K > K_min and K_min ∝ T

Where:

• N_mod is the number of mods installed in the game
• N_load is the number of loading executed in the current session, including reverts and launches
• N_vess is the number of vessels in the current save
• ^WN_part is the number of parts in the current active vessel
• T is the total time spent on the current save

 

7th Law: The lag number

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There exists a lag number L describing the lag experienced by the player. While it doesn't not depend on t and is not directly related to M(t), one can establish the following relation of proportionality:

L ∝ (N_mod, N_load, N_vess, N_part)

 

8th Law: On mission durations, the Crown law (proposed by Red Iron Crown)

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Similarly to the Δv Problem, the theoretical and actual mission times are different, yet linked by the following formula:

T_a = T_e²

Where T_a is actual real life time to complete a mission and T_e is the estimated real life time to complete a mission.
One can notice that this theorem is actually a more precise version of the Δv problem, applied to time durations. Eventual relations between the two have been sporadically conjectured but are yet to be proven.

Sidenote: You're not going to bed before 3am if you decide to go to Jool at 8pm.

 

9th Law: the Kythagorean theorem (proposed by CliftonM)

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Linked in a way to the memory usage function but more specific, the Kythagorean theorem links the number of loadings done in a game to the number of Kraken attacks that will occur as a consequence:

L² + Q² = K²

With L the number of times spent loading, Q the amount of quicksaves-quickloads (also denominated as F5-F9 by some) done during the game and K the number of Kraken attacks induced by the two previous factors.
While it is still not sure if one loading and two quickloads will give you √5 Kraken attacks, this theorem is definitely based upon observable facts.

 

10th Law: The FSPEHTWP probability law (proposed by Hcube)

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The FSPEHTWP (Forgetting Solar Panels when going to Eeloo at a Hundred Thousand times Warp with a Probe) probability is a very specific formula applicable in the only context of going to Eeloo with a probe while at 100 000x time warp: it describes the probability of forgetting to open the solar panels before doing the aforementioned operation, and hence depelt the electric charge available, losing all authority on the probe resulting in the tragic failure of the mission (unless quicksaves were done, then refer to the law above for consequences):

P_FSPEHTWP = 1 / √(N+1)

Where N is number of times that mistake was already made by the operator of the probe.
Due to the extreme specificity of this law, the situations in which it applies are rare, but efforts to find a general law describing the same problem in other cases, while not strictly speaking successful, tend to show that using the FSPEHTWP law outside its original context yields a reasonable appoximation of the actual probability.

Sidenote: If you have never made this mistake yet, then PFSPEHTWP = 1. *everyone* forgets them.

 

11th Law: The spaceplane collider paradox (proposed by GoSlash27)
 

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Due to quantum fluctuations derived from the Heisenberg Uncertainty principle, it is possible that some objects may be created for extremely short lived amounts of time without violating conservation rules. A notable example of this, and one of the only observation of quantum proerties applied to macroscopic scale is the "spaceplane collider event": a virtual KSC might appear at any location spontaneously. Because of the size of the KSC, the lifetime of this virtual KSC is immensly short, making it invisible and undetectable by the most precise intruments.
Should your spaceplane be at the location of this apparition, it will collide with the KSC instantaneously, inducing rapid unplanned disassembly: this is the spaceplane collider event.
While the detailed properties of this event are way too complicated to fit inside the observable universe, a probability for this event has been given by the highly renowned KSP expert Slashy:

P_c = r / (r+d)

Where:

• P_c is the probability of a collision with a structure on KSC grounds
• r is your spaceplane's current distance from KSC
• d is Kerbin's diameter

It is to be noted that this probability increases with the disance to the original KSC. This comes from the fact that the KSC influences space-time around it, reducing the probability of another KSC appearing when close.
This rule can be summed as: "The farther away you are from KSC, the more likely it is that you will crash into it.", hence the "paradox" denomination.

 

12th Law: The criticality conundrum (proposed by GoSlash27)
 

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Murphy's Law (or Sod's law for our British mates) is an established and observed property of the Universe which states that a catastrophic failure will occur, presumably at the worst time possible. A derivate of this law for the specific context of rocket science can be formulated:

P_f = Γ/e^π

With P_f the probability of a failure of a part or operation during a mission and Γ the criticality of this part or operation to mission success.
While hard to precisely define, the criticality number states how important an event or part is for the mission.
This presumably counter-intuitive statement, that is still to be formally explained and proven, is known and a determining factor of the mission design; it has also led to the popular idiom amongst the KSC staff: "The more you need something to go right, the more likely it is to go wrong".

 

13th Law: The Hatch-Tensor equation (proposed by GoSlash27)
 

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The Hatch-Tensor equation describes the interaction between a Kerbal on EVA and the hatch of the nearest capsule. Orginially formulated by Hatch and Tensor Kerman (hence the name), this equation states that the ability of a Kerbal to move himself is proportionnal to his/her distance to the closest hatch:

A ∝ r

Where A is the Kerbal's agility and r the distance to the closest hatch.
The agility of a Kerbal is a formal definition of how able a Kerbal is to realise an EVA task (should it be plant a flag, climb a ladder...). It depends on the Kerbal itself, the gravity of the body and surprisingly the sex of the Kerbal (the agility factor of female Kerbals is so that it can give rise to a mathematical singularity therefore making the Kerbal completely uncontrollable).
Note that the agility is sometimes replaced by the clumsiness which is the inverse of the agility. The clumsiness of a Kerbal is therefore inversely proportional to the distance: the closer a Kerbal is to a hatch, the clumsier it gets.

 

14th Law: The memory problem (proposed by *Aqua*)
 

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After rocket builders refined their crafts more and more they sent on a mission to Tylo for the first time. Right after landing, the astronauts got out of their landers and upon touching the ground realised they forgot the flag and therefore tried to get back into the craft to get it - just to discover the designers forgot to add ladders and that they can't get back in.
This tragic event (not that tragic actually, reports mention that the Kerbal are still waiting for a rescue mission on Tylo) led to research on the memory capacities of the involved designers. The study showed that the builders have memory leaks (not to be confused with the Unity memory leaks), and further statistical analysis have quantified the probability for them to forget a crucial part of the mission vessel at a given time into the mission:

P(M) = 1 - t_c / t_e

Where:

• P(M) is the probability an important part is missing
• t_c is the current mission time in seconds
• t_e is the estimated total mission time in seconds

Research on this topic show some possible connections with the critciality conundrum, but further analyses are required for definitive conclusions.

Sidenote: The further you are into a mission the more likely it is you forget to add a part that's essential for mission success.

 

15th Law: The contract conundrum (proposed by Snark)

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The probability of a contract being successfully completed depends on the number of criteria of the contract and, in a similar fashion to the critcality conundrum, the importance of that contract for your space program. The greatest the stakes, the less likely your are to succeed it.
A more mathematical approach to the problem defines the probability that the contract is doable after mission start:

P_C = (1 - f / f)ⁿ

Where:

• P_C is the probability that you've included everything you need to complete the current contract
• f is the financial reward for completing the contract
• f is your total net worth in your current career
• n is the number of criteria for the contract

In other words, the more vital it is that you complete a contract, the higher the likelihood that you've forgotten something.

 

16th Law: EladDv's first law of Kraken drives: the loss rate rule (proposed by EladDv)

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The probability of a Kerbal to be killed on a craft that contains a Kraken drive increases exponentially with the time elapsed since the mission started and the time since a Kerbal was last killed by the Kraken.
This rule took many experiments to confirm and unfortunately no Kerbal has survived to tell about their encounter with the Kraken so far, may they rest in peace within the great void between dimensions.

P_K = 1 - e^{- N / (t+t_K)}

With:

• P_K the probably of a Kerbal to be killed at a given time
• N the number of Kraken drives on the vessel
• t the time elapsed since the start of the mission
• t_K the time since the last kerbal was killed by the Kraken

 

17th Law: EladDv's second law of Kraken drives: the reactivation breakdown rule (proposed by EladDv)

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A special case of the first law for a single drive on its own: a Kraken drive that has been activated has a chance to fail on the next activation, the more times you activate it, the more likely it is to fail. The failure probability is also related to the acceleration the drive produces and the time elapsed since the mission started.  This case was developed by Schrödinger Kerman after many "thought" experiments he conducted which involved a Kerbal and a Kraken drive inside a box.

P_f = 1 - e^{-1 / ((N + 1) / I_fr + (a * t))}

With:

• P_f the probability of a failure
• N the number of times the device was activated
• I_fr the intrinsic failure rate of the drives which has universally been confirmed to be [deprecated]
• a the acceleration in m/s²
• t the time since the start of the mission

 

18th Law: EladDv's third law of Kraken drives: the critical minimal velocity rule (proposed by EladDv)

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If a drive is active while it's orbital speed drops below 800 m/s it will destroy the vessel. This was observed after the first few kerbals not to go insane from the Kraken or plainly die were blown to bits after they tried to land back on Kerbin: they accelerated from orbital speeds down the atmosphere, and then blew off the radar. Since they were never found, rumors say their bodies are still out there near the Kraken's lair to fend off intruders and curious Kerbals.

{ O * (a / |a|) * (800 - |v - a.dt|) * ((800 - v) / |800 - v|) < 0 => P_d = 1
{ ELSE                                                                                 => P_d = 0

With:

• P_d the failure probability
• v your current orbital speed
• a the acceleration relative to the direction of orbital motion
• O a binary variable: if the drive is on then O = 1, if not then O = 0

(Note that |x| represents the absolute value of a variable)
While the mathematical formulation looks very complex, it is only to represent the data in the most accurate way; this can logically be broken down to a much simpler version as a chain of binary variables which correspond to different states. 

 

19th Law: The landing fuel usage law (proposed by mhoram)
 

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Let a be the altitude of a ship S situated at the surface of a planet.
And let a₁ be the altitude of a ship S₁ trying to land at the same location as S.
Then the amount of delta-V Δv needed to perform the landing maneuver depends on the altitude of both ships and a constant positive scalar ε > 0 which depends on the skill of the pilot:

Δv = (|a₁ - a| + ε) / 1s      (19A)

 

Corollary 19bis : You did not pack enough fuel to land safely (proposed by mhoram)
 

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Assumption 1: The ship you are trying to land has a finite amount of delta-V available. (Proof by stating a fact ... or experimentation ... your choice. Please notify me if you found a way around this in real life or make a bugreport to Squad if you found a way around this in KSP without using cheats)

For the proof of the corollary we investigate the three cases: a > a₁ (A), a = a₁ (B) and  a < a₁ (C), and prove that in each case the corollary is true.

Case A: a > a₁
In this case the ship is not landed, so you will need more fuel for landing

Case B: a = a₁
In this case the amount of delta-V needed to land is, according to (19A):
Δv = ε > 0 m/s
Let f be the amount of Delta-V the ship has left:

• if f < ε, then you don't have enough fuel to land;
• if f >= ε, then you will need need at least ε m/s for landing and you will need to repeat this step with (f - ε) < f fuel left.

Since, according to Assumption 1, f is finite, after f / ε iterations you run into the case f < ε, where you do not have enough fuel.

Case C: a < a₁
You crashed and did not land safely.

Q.E.D.

 

20th Law: The cheesy perseverance rule (proposed by RocketSquid)

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Explosions are fun. Unplanned explosions are funner funnier funn... more fun. But the latter can compromise the mission, frustrating whoever is in control the craft (Mission Control or the wierd pink things controlling all Kerbals from above).
Cheese is good. Unplanned cheese is gooder more good better. Thus it is common, that after a series of unplanned explosions, people just leave and go eat cheese.
The probability of giving up on a mission and leaving to eat cheese is then given by:

P_cheese = dx / dt ≈ x / t

With x the number of unplanned explosions that occured during the mission and t the elapsed mission time (not accounting for the time-dilation caused by the explosions or complex crafts).

According to this statement, at a rate of 1 explosion per second, most reasonable people will view cheese as the gooder more good better option.
Note, however, that the above formula is simplified and does not account for the explosion hilarity constant (0.2) or the first law of unreasonability; a complexer more complex version yet still simpler more simple simpler than the actual fomula can be written by simply adding a global constant to the above formula (though the value of this constant is not known).
Note as well that the above formula only applies when the alternative is cheese, as cake, pie, bacon, etc. have slightly different equations involving constantser more constants that have yet to be determined.

 

21st Law: The Grand Kraken case [DEPRECATED] (proposed by CliftonM)

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The Grand Kraken has always been one of the greatest fears design considerations of Kerbal engineers. Naturally, in order to evaluate the risks of an attack and thus better protect the spacecrafts Kerbals on board, it was extensively studied by mathematicians. A definitive formula was found after various hypotheses, but unfortunately, by the time it was found, the Grand Kraken had crashed on Bop, thus making the following formula useless.

K = √(x/500 + y/500 + z/500) / 100

Where:

• K is the chances of a Kraken attack
• x is the distance to the KSC on the x-axis
• y is the distance to the KSC on the y-axis
• z is the distance to the KSC on the z-axis

Note that the formula doesn't use the direct distance to the KSC per se, but a compound of its different components. Critics have stated that, therefore, the formula could yield different results if the reference point was held upside down or rotated, but no studies have ever been made to prove this statement.

Sidenote: A distance in any of the axis of more than 5000km will yield a probability greater than 1. Some believe that this mathematical incoherence could attract the Grand Kraken, thus triggering the attack.

 

22nd Law: On time dilation in strong partal fields

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Any vessel made of at least one part generates a partal field around it. The strength of this field is proportional to the number of parts contained in the vessel and has an intrinsic constant called the "partal constant" denoted by P. The partal constant is essentially made of two parts: a universal constant part which is defined to get consistent results when using different units, and a special part which depends on the characteristics of the install of the person running the experiment (mods, specs of the simulator (computer)...).

Note that a parallel between the partal field and the lag number described in law 7 can easily be drawn, although the lag number is a simplification that allows not to consider the whole partal field.

The partal field has different physical implications, one of which is time dilation. Familiarly called "physics lag", this dilation is caused by the action of the partal field on the very fabric of space-time, and literally slows down time and hence the laws of physics (this is where the idiomatic name comes from).

The effect of the partal field has been studied by many scientists, starting from Albert Kerman's field equations, to the precise metrics defined by Blackshield Kerman and Roy Kerman that will be discussed here. The following is a quick introduction to the effects of these fields on time in common situations.

 

22A: At a fixed distance to a non-rotating spherical spacecraft

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The simplest case is that of a standing (or floating if you are in space) observer near a non-rotating spherical spacecraft. Non-spherical spacecrafts use boring and complicated integrals describing the parts repartition inside the spacecraft in order to accurately model the partal field, but the result is close enough for any of the usual practical applications.

The effect of a partal field in this case is described by the following equation, derived from Blackshield Kerman's metric and based upon Albert Kerman's basic principles.

t = t_c * √(1 - 2PN / rc²)

Using the Blackshield Kerman radius defined as: r_b = 2PN / c² ; the above equation can be written:

t = t_c * √(1 - r_b / r)

Where:

• t is the proper time for an observer (call it Alice Kerman) at a distance r from a vessel composed of N parts
• t_c is the coordinate time for an observer (call it Bob Kerman) arbitrarily far from any vessel
• P is the partal constant for the partal field
• c is the speed of light

 

22B: Combination of case A and movement near the spacecraft

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A more useful case is the time dilation caused by an observer getting closer to a strong partal field, which has direct applications in creating more precise docking methods with motherships. In this case, we have to account for the partal field itself, as described in part A, but also the time-dilation caused by the movement of the observer. Indeed, while not generating a proper partal field, a moving observer undergoes a slight time-dilation solely caused by its velocity, as described by Albert Kerman's special postulates.

The following equation allows to describe the time-dilation for an observer moving inside a strong partal field.

dt² = (1 - 2PN / rc²) * dt_c²  -  (1 - 2PN / rc²)^-1 * (dx² + dy² + dz²) / c²

Note that the velocity of the moving observer relative to the spacecraft is: v² = (dx² + dy² + dz²) / dt_c² ; so the above formula can be rewritten as:

dt² = (1 - 2PN / rc²) * dt_c²  -  (1 - 2PN / rc²)^-1 * v² * dt_c² / c²

Using the Blackshield radius r_b, the equation simplifies to:

dt² = (1 - r_b / r) * dt_c²  -  (1 - r_b / r)^-1 * (dx² + dy² + dz²) / c²

Where:

• dt is a small increment in proper time t for an observer (call it Alice Kerman) at a distance r from a vessel composed of N parts
• dt_c is a small increment in coordinate time t_c for an observer (call it Bob Kerman) arbitrarily far from any vessel
• dx, dy and dz are small increments in the three coordinates (x,y,z) of Alice
• P is the partal constant for the partal field
• c is the speed of light

 

22C: At a fixed distance to a rotating spacecraft

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The case of a rotating spacecraft is more complex and has to be described using the Roy Kerman metric. Though this case is less common, it is always useful to have an accurate description of time dilation.

The equation describing the time dilation of an observer near a partal field emitted by a rotating spacecraft is:

dt² = (1 - r_br / ρ²) * dt_c²

Where:

• dt is a small increment in proper time t for an observer (call it Alice Kerman) at a distance r from a vessel composed of N parts
• dt_c is a small increment in coordinate time t_c for an observer (call it Bob Kerman) arbitrarily far from any vessel
• r_b is the Blackshield radius of the vessel
• ρ is a value defined by: ρ² = r² + (J_p / Nc)² * cos²θ
• J_p is the angular partal momentum of the spacecraft given by: J_p = ω * ∫_N r² dn ; with ω the angular velocity of the rotating spacecraft
• c is the speed of light

Note that here, spherical coordinates about the rotating spacecraft are used. θ is therefore the polar angle of the observer (and φ would be the azimuth)

 

22D: Combination of case C and movement near the spacecraft

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Finally, the most complex case to be studied here is the case of an observer moving relative to a rotating spacecraft. The time dilation in this case is also described by the Roy Kerman metric, but the equation is far more cumbersome. A good application of this case is shown in the movie Interplanetary where a scene shows a spacecraft docking with a rotating station due to a symmetry error in the station's RCS system; although the movie overly simplifies the phenomenons involved and does not show the real beauty of the mathematics involved.

The following equation, even though it is a bit complex, is the accurate description of this case, and all of the other cases above can be derived from this unique equation:

c²dt² = (1 - r_br / ρ²) * c²dt_c²  -  ρ² / Δ * dr  -  ρ²dθ²  -  (r² + α² + r_brα²sin²θ / ρ²) * sin²θ * dφ²  +  2r_brα²sin²θ / ρ² * c dt_cdφ

Where:

• dt is a small increment in proper time t for an observer (call it Alice Kerman) at a distance r from a vessel composed of N parts
• dt_c is a small increment in coordinate time t_c for an observer (call it Bob Kerman) arbitrarily far from any vessel
• dr, dθ and dφ are small increments in the spherical coordinates (r,θ,φ) of Alice
• r_b is the Blackshield radius of the vessel
• α is a value defined by: α = J_p / Nc
• ρ is a value defined by: ρ² = r² + α²cos²θ
• Δ is a value defined by: Δ = r² - r_br + α²
• c is the speed of light

The above formula can be used to derive the cases A, B and C:

• setting dr = dθ = dφ = 0 (no movement) gives case C
• setting J_p = 0 (no rotation) yields α = 0, ρ = r, Δ = r² - r_br and converting to cartesian coordinates gives case B
• setting dr = dθ = dφ = 0 and J_p = 0 gives case A

 

 

23rd Law: The Hatch obstruction theorem (proposed by msasterisk)

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The Hatch obstruction theorem is another important mathematical formulated by Hatch Kerman that describes the probability of a hatch being obstructed. After Hatch and Tensor Kerman published their results known today as the Hatch-Tensor equation (13th law), engineers thought that the understanding of hatches mechanics was sufficient to allow safe EVAs for Kerbals during their missions. They soon realisted that the Hatch-Tensor equation was but an incomplete description of complex phenomenons. Following further studies focussed on the phenomenon of hatches obstruction, Hatch Kerman published the Hatch obstruction theorem: the probability of a hatch being obstructed is given by:

P_h = E * (τN_partsI / 3600N_h)

Where:

• E is a binary value (1 or 0) depending on the answer to the question: "Did you read the engineers' report ?"
• τ is the time spent on the mission
• N_parts is the number of parts attached to the command pod used
• I is the importance coefficient of the hatch
• N_h is the total number of hatches on the ship

Note that if P_h comes out to be greater than one, a Kraken attack is inevitable before you can try to EVA.

 

24th Law: The explosion reciprocity axiom (proposed by Hodari)

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The explosion reciprocity axiom is a very simple, yet extremely important in the field of manned space exploration, and can be written as follows:

"For every explosion, there is an equal and opposite revert/reload."

 

25th Law: The CTD exponentiality rule (proposed by nosirrbro)

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The CTD exponentiality rule gives the probability of a CTD happening during a given launch, and is formulated as such:

P_CTD = s*e^sl + r

With:

• P_CTD the probability of a CTD at some point during the current launch
• s is the stability constant, a value that cannot be directly measured but that can be reasonably approximated by reversing the equation in a known situation. It is constant during a single game session and cannot equal 0
• l is the number of launches operated during the current game session
• r is the RAM limit, and is a binary value than can either equal 0 when there is usable RAM available for the game, and 1 where there is not

Note that due to the presence of the r factor, a CTD is guaranteed to happen when the game runs out of RAM. It is present as such in the equation as the mathematician who first wrote this formula was too lazy to add the case of a memory leak and just added a binary value that automatically increased to probability to 1 or more when such an event happened. All observations to this day report that a memory leak does lead to a crash in every case, so the equation was left in this state.

Sidenote: the above formula is a simplification of the actual equation describing the CTD probability with a modded install. As a result it is only valid on a stock install.

 

26th Law: The graphical incidence formula (proposed by Robotengineer)

  Reveal hidden contents

Mods decrease FPS. Better graphics decrease FPS. Mods that improve graphics should then decrease FPS a lot, following this logic. This fallacy has always been present in Kerbal lore and several thinkers (2 actually because there were only 2 Kerbal thinkers ever) tried to prove that this fallacy was indeed a fallacy and therefore wrong. After their failed efforts, mathematicians decided to take a look at this problem, and ended up with the following result:

FPS' ∝ FPS^{1 / Σ_N I_M}

Where:

• FPS' is the resulting FPS
• FPS is the base FPS that would be obtained without any graphical mods
• N is the number of graphic mods
• I_M is the impact factor of a given mod (related to its beauty factor) that becomes greater if the mod has a great impact on the game graphics (here summed over N)

Therefore, your resulting FPS is proportional to the base FPS but decreases with the number of mods and their incidence on the game.
Note that this formula works only with graphic mods, other types of mods decrease FPS following other formulae.

 

27th Law: The biome jump law (proposed by REMD)

  Reveal hidden contents

The biome jump law is an easy and handy law that is used by engineers to quickly estimate the delta-V that they should put on a lander that will have to do several biome jumps in order to harvest science.

Σ_N Δv_B + Δv_LO + Δv_EVA * K = Δv_R < Δv_S

Where:

• Δv_B is the Δv needed for a given biome jump
• N_B is the number of biome jumps planned
• Δv_LO is the Δv needed to get to orbit from the planetary surface
• Δv_EVA is the Δv provided by a Kerbal pushing the vessel
• K is the number of EVAs that the crew is willing to operate (increases with the science potential gains)
• Δv_R is the total Δv that is required to perform the mission
• Δv_S is the Δv in the ship that should be greater than the Δv needed to perform the mission

As a consequence of the Δv problem (3rd law), the estimated required Δv needed for completion of the mission is always smaller than the actual Δv that will be needed. Therefore, apprentice engineers are taught to keep a safety margin in order to account for this issue. The fact that apprentice engineers don't learn the meaning of the word safety before the very end of their formation might be contributing to the fact that most of them forget to allow more Δv than needed.

 

28th Law: The abandonment theorem (proposed by Andem)

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One of the greatest ethical dilemmas of space exploration by manned vessels has always been whether to rescue or not a stranded Kerbal: leaving it in orbit is sure unethical, but spending funds on rescuing him for limited science/funds rewards is unethical in a scientific productivity perspective (note that the definition of ethics differs depending on the field it is applied to, but I'm not going to discuss this here as it is not a dissertation on the vocabulary and technical terms used in the Kerbal space industry, go create that thread if you wish before I do it and get all the likes ). Note that the dilemma stated above does not apply to rescue contracts, as the reward is always greater than the investment, otherwise, just skip the contract.
In order to solve this dilemma, some mathematicians have tried to convert this ethics problem into a mathematics problem (because whatever you may think, maths is easier) and failed to find a solution. On the other hand, they have come up with a nice equation describing the probability of setting up and launching a rescue mission for a stranded Kerbal:

P_R = B / T_R + √F - D_R

With:

• B the belovedness of the stranded Kerbal (0 if you can't remember the Kerbal's name)
• T_R the estimated time spent on the rescue, note that the actual time may differ (see the Crown law)
• F the funness of the recovery mission (hard to predict, as random stowed events can cut this to 0 instantly)
• D_R the difficulty of a recovery mission

Note that the probability that a Kerbal is abandoned is simply given by 1 - P_R

 

29th Law: The Kerblams equality principle (inspired by this thread) 

  Reveal hidden contents

Stuff explodes.
This observation is one of the founding principle of the space exploration and all Kerbalkind's learnings come from this simple statement.
Despite the apparent simplicity of it, and its ubiquity (not sure it means what I think it does but it sounds clever), the field that is in charge of studying explosions (Kerblamistics or explosionistics) is a hard one: to predict an explosion sequence, one must know all the parameters that have to be derived from other fields, and therefore any basic prediction requires knowledge in a lot of domains. While we shall not go into the depth of Kerblamistics here, following is a very basic principle in this domain, showing both the complexity and beauty of it:

"The size and energy released by an explosion does not depend on the exploding part size or mass"

Although counterintuitive at first, this statement can be observed and empirically proven very easily, but the formal demonstration this principle is still not established and Kerblamists still debate over it.

 

---

 

If you have any comment or correction to these laws, please make them !
If you have any suggestions for more of these, please share them !
If you want to add your own laws, please do so ! I'll add them to the OP if you wish.

If you like laws that were proposed by other members and added to the OP, remember to show them some love !

Edited July 6, 2016 by Gaarst
Added 28th and 29th laws

[lukethecoder64]

Very accurate.

Rep'd.

[Tex]

I am NOT a math person in any sense of the word, but this made me chuckle in the middle of class. Very very true and hilarious.

[lajoswinkler]

This should be taught in KSP schools.

[theJesuit]

Very good! A detailed and thorough treatise.

Peace.

[viktor19]

  lajoswinkler said:

This should be taught in KSP schools.

If we had any...

[KAL 9000]

The probability of your vessel randomly exploding is given by P_ve, and is equal to:

P_ve = max(100, (1/(N(mods) X (1/N(Fps)))) X N(part))Percent

N(Fps) is equal to the number of frames per second

[GoSlash27]

I realize that this is intended to be tongue-in-cheek, but a lot of people *actually* try to run their space programs in accordance with rules #1 and #2 and then wonder why their rockets are bloated laggy uncontrollable messes.

So for those who are struggling, I would like to point out: "Moar boosters" and "moar struts" is just a running joke, not actual advice.

Best,

-Slashy

[Red Iron Crown]

^^ Would be amusing to watch someone try #1.

I'll add:

T_a = T_e²

Where T_a is actual real life time to complete a mission and T_e is the estimated real life time to complete a mission.

[Superfluous J]

  Red Iron Crown said:

^^ Would be amusing to watch someone try #1.

Just watch "episode 1" of any random Let's Play out there and you've got a decent shot of seeing it.

[Gaarst]

Thanks to everyone for their positive feedback

I updated OP and added Red Iron Crown's law on mission durations.

@KAL 9000

I'm not sure I correctly understand you equation, but from what I got, the "1/N_mods" term means that the more mods you have, the less likely your craft is to explode. Am I right ? Is that what you meant ?

I'll add your law to OP (if you wish, of course) as soon as I'm sure I understood it correctly.

[Herr Doktor Strangemember]

Great post have +(e^0) rep

[KAL 9000]

  On 10/25/2015 at 8:06 PM, Gaarst said:

Thanks to everyone for their positive feedback

I updated OP and added Red Iron Crown's law on mission durations.

@KAL 9000

I'm not sure I correctly understand you equation, but from what I got, the "1/N_mods" term means that the more mods you have, the less likely your craft is to explode. Am I right ? Is that what you meant ?

I'll add your law to OP (if you wish, of course) as soon as I'm sure I understood it correctly.

Expand  

No, because it's supposed to be (1/N(parts))/(1/N(mods)), which would be a fraction divided by a fraction, which usually a large number. Note: If you are using stock, N(mods) is 0, so it's 1/infinity, which is basically 0. Note: N(mods) if using CKAN is 1, because of CKAN's stability. This is a probability of your vessel RANDOMLY exploding, not exploding if it's poorly designed (i.e. not enough boosters).

[Guest]

Krythagorean Theorem:

L^₂ + Q² = K²

L = number of times spent loading

Q = Amount of quick saves*

K = # of Kraken attacks

*Thi means only the amount of times reverted.  There can be multiple per actual quick save.

[Hcube]

Probability of forgetting to open the solar panels before time accelerating 100.000x on your way to eeloo and losing power and control of the probe :

P=1/(√(k))

Where k=number of times you already made that mistake.

If you have never made this mistake yet, then k=1. *everyone* forgets them. 

[Gaarst]

Updated OP: added @CliftonM and @Hcube's laws.

[GoSlash27]

The Spaceplane Kraken Paradox:

P_c = 2r/(r+d)

where

Pc= probability of a collision with a structure on KSC grounds

r= your spaceplane's current distance from KSC

d= Kerbin's diameter

"The farther away you are from KSC, the more likely it is that you will crash into it."

The criticality conundrum:

P_f = Γ/e^π

Where

P_f = Probability of failure

Γ = criticality of a part or maneuver to mission success.

"The more you need something to go right, the more likely it is to go wrong".

 

 

 

[GoSlash27]

And one more:

 Hatch Tensor Equation

A = 1/r

where

A= a Kerbal's agility

r= a kerbal's distance from a hatch.

"The closer a Kerbal is to a hatch, the clumsier it gets".

[*Aqua*]

The missing problem

After rocket builders refined their crafts more and more they send them on a mission to Tylo. Right after landing they embark and upon touching the ground first they remember to get back into the craft to get the flag - just to discover they forgot to add ladders and can't get back in.

P(M) = 1 - t_c / t_e

Where:

• P(M) is the probability an important part is missing.
• t_c is the current mission time in seconds.
• t_e is the estimated total mission time in seconds.

Side note: The further you are into a mission the more likely it is you forget to add a part that's essential for mission success.

 

Edited January 17, 2016 by *Aqua*

[Gaarst]

Added @GoSlash27 and @*Aqua*'s laws to the OP.

Edited a few things to make them fit better in the post: noticeably titles (to avoid generalities or repetitions) and descriptions.

For the spaceplane Kraken paradox (or spaceplane collider paradox), I removed the factor of 2 to avoid a probability greater than 1. For the Hatch tensor equation, I changed the 1/r term to r, to get an increased agility with distance. Added a +1 to @Hcube's law to include the 0th term.

Edited January 17, 2016 by Gaarst

[Snark]

The contract conundrum:

P(C) = (1 - f/f₀)ⁿ

where

• P(C) is the probability that you've included everything you need to complete the current contract
• f is the financial reward for completing the contract
• f₀ is your total net worth in your current career
• n is the number of criteria for the contract

 

"The more vital it is that you complete a contract, the higher the likelihood that you've forgotten something."

[Gaarst]

Added @Snark's law. Title is subject to change, as conundrum is already used for 12th law.

[EladDv]

yeah i have a few new laws-
 

The First law of Kraken Drives:

the probability of a kerbal to be killed on a craft that contains a kraken drives increases exponentially in proportion to the time since the mission has started and the time since the last kerbal was killed by the kraken, this took many experiments to confirm and sadly no one kerbal has survived to tell about their encounter with the kraken so far, may they rest in peace within the great void between dimensions.

P_k= 1-e^{-(N_k/(T+Tk))}

P_k probably of a kerbal to be killed.

N is the number of kraken drives on his vessel.

T time since the start of the mission.

T_k time since the last kerbal was killed by the kraken.

 

The Second law of Kraken drives:

a special case of the first law for single drive on it's own, a Kraken drive that has been activated has a certain chance to fail on the next activation, the more times you activate it the more likely it is to fail,and this is also proportional to the acceleration that the drive produces and the time elapsed since the mission started.  This case was developed by schrodinger kerman after many "thought" experiments he conducted which involved a kerbal and a kraken drive inside a box.

P_f =1-e ^{-1/((N+1)/I_fr   + (A_h *T))}

where: P_f probably to fail.

N is the number of times the device was activated.

I_fr the intrinsic failure rate of the drives which has universally been confirmed to be [deprecated].

A_h acceleration in m/s².

T time since the start of the mission.

 

The Third law of Kraken Drives:

if a drive is active while it's orbital speed drops below 800 it will destroy the vessel. This was observes after the first few kerbals to not go insane from the kraken or just plainly die were blown to bits after they tried to land back at kerbin, they accelerated from orbital speeds back down and then blew off the radar and since weren't recovered, rumors say their bodies are still out there near the kraken's lair to fend off intruders and curious kerbals.

P_d = { O*(A_h/|A_h|) *(800-|V-A_h |)*((800- V)/|800- V|)<0 P_d=1

        { Else                                 P_d=0

 

P_d the failure probability.

V your current orbital speed.

A_h the acceleration in relation to the direction of your current orbital speed.

O a binary variable- if the drive is on then the value is 1, if not the value is 0.

Please note that |X| represents an absolute value of a variable

While the mathematical sentence looks very complex it's only to represent the data in the most accurate way, this can logically be broken down to a much simpler version as a chain of binary variables which correspond to different states. 

Edited January 20, 2016 by EladDv

[Gaarst]

Added @EladDv's three laws of Kraken drives, and put spoilers in OP: it was starting to get messy.

[mhoram]

The Landing Fuel Usage Law:

Let a₀ be the altitude of a ship S₀ situated at the surface of a planet.

And let a₁ be the altitude of of a ship S₁ trying to land at the same location as S₀.
Then the amount of Delta-V v needed to perform the landing maneuver depends on the altitude of both ships and a constant positive scalar ε > 0 which depends on the skill of the pilot:

v(a₀, a₁) := (abs(a₁-a₀) + ε * 1m) / 1s      (A)

 

Corollary: "You did not pack enough fuel to land safely"

Assumption 1: The ship you are trying to land has a finite amount of Delta-V available. (Proof by stating a fact ... or experimentation ... your choice. Please notify me if you found a way around this in real life or make a bugreport to Squad if you found a way around this in KSP without using cheats)

For the proof of the corollary we investigate the three cases a₀ > a₁, a₀ = a₁ and  a₀ < a₁. and prove that in each case the corollary is true.

Case a₀ < a₁: In this case the ship is not landed, so you will need more fuel for landing

Case a₀ = a₁: In this case the amount of Delta-V needed to land, is according to (A):
v(a₀, a₁) = ε ^m/_s > 0 ^m/_s
Let f be the amount of Delta-V the ship has left.
If f < ε ^m/_s, then you don't have enough fuel to land.
If f >= ε ^m/_s, then you will need need at least ε ^m/_s for landing and you will need to repeat this step with (f - ε ^m/_s) < f fuel left.
Since according to Assumption 1, f is finite, after f / (ε ^m/_s) iterations you run into the case f < ε ^m/_s, where you do not have enough fuel.

Case a₀ > a₁:
You crashed and did not land safely.

Q.E.D.

Edited January 20, 2016 by mhoram