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# Alternate Delta-v for tanks

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:

TanksSS1:17derived 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

## Alternate Spaceships Armor and Volume values

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 0.983909 2 0 0.967274 3 0 0.950046 4 0 0.93217 5 0 0.91358 6 0 0.894201 7 0 0.873944 8 0 0.852702 9 0 0.830345 10 0 0.806714 11 0 0.781612 12 -1 0.754784 13 -1 0.725901 14 -1 0.694515 15 -1 0.660003 16 -1 0.621447 17 -1 0.5774 18 -2 0.525359 19 -2 0.460258 20 -2 0.368403

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.