alternate_calculations

By Richard Maw

delta-v = 20 × delta-v-per-tank × ln(20 ÷ (20-tanks))

to get number of tanks needed:

tanks = (20-(20 ÷ e^(delta-v ÷ (20 × delta-v-per-tank))))

Table:

Tanks | SS1:17 | derived delta-v increase |
---|---|---|

1 | 1 | 1.025865888 |

2 | 1 | 1.053605157 |

3 | 1 | 1.08345953 |

4 | 1 | 1.115717757 |

5 | 1 | 1.15072829 |

6 | 1.2 | 1.18891648 |

7 | 1.2 | 1.230808332 |

8 | 1.2 | 1.277064059 |

9 | 1.4 | 1.328526668 |

10 | 1.4 | 1.386294361 |

11 | 1.4 | 1.451832175 |

12 | 1.4 | 1.52715122 |

13 | 1.6 | 1.615110961 |

14 | 1.6 | 1.719961149 |

15 | 1.8 | 1.848392481 |

16 | 2 | 2.011797391 |

17 | 2.2 | 2.231905865 |

18 | 2.5 | 2.558427881 |

19 | 3 | 3.153402393 |

By Richard Maw

The relative volume of the spaceship for how much of it is armour is 1 - (1 - armour relative volume) × (number of armour systems ÷ 20).

The relative length is the cube root of that, which makes the relative surface area the square of the relative length, so we have a power of ⅔.

Since we've the same mass of armour, but it's covering less surface area we can work out how much armour it would be by multiplying the DR by the unmodified surface area it would apply to then dividing by the surface area we actually have. Since we've got units relative to our standard surface area that turns into divide by relative surface area.

Since our table has a multiplier, we'd instead take 1 ÷ relative surface area. It boils down to (1 - [1 - armour relative volume] × [number of armour systems ÷ 20])^¯⅔. I guessed at a relative armour volume of 0.05 since other things were a twentieth, and the multipliers line up pretty close.

Number of Armor Systems | P3/34 dDR | Derived dDR |
---|---|---|

1 | ×1 | ×1.03297547465021 |

2 | ×1 | ×1.06881107965281 |

3 | ×1 | ×1.10792555779384 |

4 | ×1 | ×1.15082685405698 |

5 | ×1.2 | ×1.1981377946646 |

6 | ×1.2 | ×1.25063133380997 |

7 | ×1.3 | ×1.30927985773568 |

8 | ×1.4 | ×1.37532564246477 |

9 | ×1.4 | ×1.45038401945126 |

10 | ×1.5 | ×1.53659871049957 |

11 | ×1.6 | ×1.63688341862429 |

12 | ×1.8 | ×1.75531216611339 |

13 | ×1.8 | ×1.89777930761406 |

14 | ×2 | ×2.07317897254703 |

15 | ×2.3 | ×2.29566282051463 |

16 | ×2.6 | ×2.58936042162822 |

17 | ×3 | ×2.99948142932869 |

18 | ×3.6 | ×3.6231640499951 |

19 | ×3.6 | ×4.72059696539276 |

20 | ×3.6 | ×7.36806299728077 |

The size modifier adjustment is less obvious. We can calculate relative length as before.

Number of Armor Systems P3/34 Target SM Relative Length

0 | 0 | 1 |

1 | 0 | 0.9839091396 |

2 | 0 | 0.9672740271 |

3 | 0 | 0.9500461658 |

4 | 0 | 0.9321697518 |

5 | 0 | 0.9135800684 |

6 | 0 | 0.8942014037 |

7 | 0 | 0.873944305 |

8 | 0 | 0.8527018983 |

9 | 0 | 0.8303448511 |

10 | 0 | 0.806714323 |

11 | 0 | 0.7816118324 |

12 | -1 | 0.7547842314 |

13 | -1 | 0.7259005852 |

14 | -1 | 0.6945149558 |

15 | -1 | 0.6600030609 |

16 | -1 | 0.6214465012 |

17 | -1 | 0.5774001751 |

18 | -2 | 0.5253587872 |

19 | -2 | 0.4602582041 |

20 | -2 | 0.3684031499 |

One thing that stands out is that the average value in the ranges of assigned SM is pretty close to the continuous size modifier length multiplier.

SM | Average | Continuous SM length multiplier |
---|---|---|

0 | 0.9881366799 | 1 |

-1 | 0.6723415849 | 0.6812920691 |

-2 | 0.4513400471 | 0.4641588834 |

This would imply that some form of rounding of the size modifier is involved, and we can turn a length multiplier into a Size Modifier with 6×log₁₀(multiplier), but just rounding ends up with:

Number of Armor Systems | P3/34 Target SM | Relative Length | Continuous SM | Rounded |
---|---|---|---|---|

0 | 0 | 1 | 0 | 0 |

1 | 0 | 0.9839091396 | -0.0422700313 | 0 |

2 | 0 | 0.9672740271 | -0.08670284159 | 0 |

3 | 0 | 0.9500461658 | -0.1335317426 | 0 |

4 | 0 | 0.9321697518 | -0.1830299622 | 0 |

5 | 0 | 0.9135800684 | -0.235520304 | 0 |

6 | 0 | 0.8942014037 | -0.2913879164 | 0 |

7 | 0 | 0.873944305 | -0.3510974599 | 0 |

8 | 0 | 0.8527018983 | -0.415216621 | 0 |

9 | 0 | 0.8303448511 | -0.484449018 | 0 |

10 | 0 | 0.806714323 | -0.5596813932 | -1 |

11 | 0 | 0.7816118324 | -0.6420532482 | -1 |

12 | -1 | 0.7547842314 | -0.7330630888 | -1 |

13 | -1 | 0.7259005852 | -0.834737121 | -1 |

14 | -1 | 0.6945149558 | -0.9499103859 | -1 |

15 | -1 | 0.6600030609 | -1.082724302 | -1 |

16 | -1 | 0.6214465012 | -1.239577517 | -1 |

17 | -1 | 0.5774001751 | -1.431138532 | -1 |

18 | -2 | 0.5253587872 | -1.677263996 | -2 |

19 | -2 | 0.4602582041 | -2.021990769 | -2 |

20 | -2 | 0.3684031499 | -2.602059991 | -3 |

Adding between 0.143 and 0.177 to the SM before rounding will make it line up with the declared SM value. The chosen value would be significant for determining the appropriate SM adjustment when using smaller armour systems. It's possible to calculate the maximum number of systems for a SM adjustment by reversing the formula as 20*(1-[10^{SM-0.5-adjustment÷6}^3])÷(1-armour relative volume).

Using an adjustment in the middle of 0.16, the thresholds for the SM adjustment to the nearest two sizes smaller armour system are: SM Max Systems -2 20.1 -1 17.9 0 11.2 The notable distinction being that you can have the -1 SM with 11.3 armour systems.

If you'd instead prefer a house rule based on mathematical principles rather than an arbitrary table, the thresholds are: SM Max Systems -2 19.9 -1 17.3 0 9.2 So 9.3 armour systems will give -1 SM, 17.4 will give -2 and a solid lump of armour would have SM -3.

You could in principle use these formulae to handle edge cases where uniform density of spaceship systems is incongruous, e.g. Gasbag, Open Space, Solar Panel or Space Sail systems, but I don't relish that complication.

alternate_calculations.txt · Last modified: 2021/01/09 06:36 by weby