FF&Sv1 claims to base its calculations and design system on 1 displacement ton being equal to 14 cubic meters, the geometric properties of a sphere and a 1 cm thick shell.
From any geometry textbook, we get the following formula for a sphere
[Eq.1] Volume (V) = 4/3 * pi * r^3
[Eq.2] Surface (s) = 4 * pi * r^2
And rearrange equation [1] to solve for radius
[Eq.3] Radius (r) = (V * 3/4 * 1/pi)^1/3 = 0.620350490929 * (V)^1/3
From FF&Sv1 and basic geometry:
[Eq.4] Length (L) = Diameter = 2 * r
Let us examine the 200 dT hull from Fire Fusion & Steel v1, pg 11.:
Rate = 200 displacement tons (dT)
Volume (V) = 2800 cubic meters
Material Volume of 1 cm thick shell (MV) = 9 cubic meters
Length or Diameter (L) = 17 meters
Surface Area (s)= 100 * MV = 900 square meters
Check the math:
Volume (V) = 200 dT * 14 cu.m. per dT = 2800 cubic meters [OK]
Radius = 0.620350490929 * (2800)^1/3 = 8.7436
Length (L) = Diameter = 2 * 8.7436 = 17.4872 [-3% error]
Surface Area (s) = 4 * pi * 8.7436^2 = 960.7037 sq.m. [-6% error]
Material Volume (MV) = s/100 = 9.6070 cu.m. [-6% error]
Let us examine the 2 dT hull from Fire Fusion & Steel v1, pg 11.:
Rate = 2 displacement tons (dT)
Volume (V) = 28 cubic meters
Material Volume of 1 cm thick shell (MV) = 0.5 cubic meters
Length or Diameter (L) = 4 meters
Surface Area (s)= 100 * MV = 50 square meters
Check the math:
Volume (V) = 2 dT * 14 cu.m. per dT = 28 cubic meters [OK]
Radius = 0.620350490929 * (28)^1/3 = 1.8837
Length (L) = Diameter = 2 * 1.8837 = 3.7675 [+6% error]
Surface Area (s) = 4 * pi * 1.8837^2 = 44.5919 sq.m. [+12% error]
Material Volume (MV) = s/100 = 0.4459 cu.m. [+12% error]
There is no reason for a design system as detailed and math oriented as FF&S to improperly calculate the surface area and diameter of a sphere. Here is the corrected table for Fire Fusion & Steel v1, pg 11 (note that all values in a collumn have been rounded to the same number of significant figures)
I leave it to any other gearheads and FF&S fans to decide whether the errors are large enough to warrant inclusion as errata.
TNE-0304 Fire Fusion & Steel v1, pg 10:
First select a basic hull size from the Hull Size table(page 11). All hulls have a rate (or "tonnage") which is the displacement of the hull in tonnes of liquid hydrogen (the most common starship fuel). Therefore, a ship with a rate of 7 could hold 7 tonnes of liquid hydrogen (provided nothing else was put in the hull). Thus, Traveller uses the term "ton" as a unit of volume where 1 ton equals 14 cubic meters (m3) of internal volume (also measured as 14 kiloliters). The Hull Size table indicates the volume (in cubic meters), the material volume of the shell (assuming a uniform thickness of 1 centimeter) in cubic meters of hull material, and the length of the hull (actually its diameter, as all hull ratings are based on the characteristics of a sphere) in meters.
TNE-0304 Fire Fusion & Steel v1, pg 11:
Rate: The hull rating is a standard measure of the volume of the hull expressed in tonnes of liquid hydrogen. Each tonne of liquid hydrogen displaces 14 cubic meters of volume (and is referred to as a "displacement ton").
Vol: The enclosed volume of the hull in cubic meters (m3). Each cubic meter contains 1000 liters of volume, and so the term kiloliter is sometimes used interchangeably with cubic meter.
MV: Material volume, the volume (in cubic meters) of material required to enclose the hull in a shell 1 centimeter thick.
L: The length of the hull in meters. For spherical hulls, this is also the diameter.
Surface Area: Surface area in square meters is the hull material volume multiplied by 100.
From any geometry textbook, we get the following formula for a sphere
[Eq.1] Volume (V) = 4/3 * pi * r^3
[Eq.2] Surface (s) = 4 * pi * r^2
And rearrange equation [1] to solve for radius
[Eq.3] Radius (r) = (V * 3/4 * 1/pi)^1/3 = 0.620350490929 * (V)^1/3
From FF&Sv1 and basic geometry:
[Eq.4] Length (L) = Diameter = 2 * r
Let us examine the 200 dT hull from Fire Fusion & Steel v1, pg 11.:
Rate = 200 displacement tons (dT)
Volume (V) = 2800 cubic meters
Material Volume of 1 cm thick shell (MV) = 9 cubic meters
Length or Diameter (L) = 17 meters
Surface Area (s)= 100 * MV = 900 square meters
Check the math:
Volume (V) = 200 dT * 14 cu.m. per dT = 2800 cubic meters [OK]
Radius = 0.620350490929 * (2800)^1/3 = 8.7436
Length (L) = Diameter = 2 * 8.7436 = 17.4872 [-3% error]
Surface Area (s) = 4 * pi * 8.7436^2 = 960.7037 sq.m. [-6% error]
Material Volume (MV) = s/100 = 9.6070 cu.m. [-6% error]
Let us examine the 2 dT hull from Fire Fusion & Steel v1, pg 11.:
Rate = 2 displacement tons (dT)
Volume (V) = 28 cubic meters
Material Volume of 1 cm thick shell (MV) = 0.5 cubic meters
Length or Diameter (L) = 4 meters
Surface Area (s)= 100 * MV = 50 square meters
Check the math:
Volume (V) = 2 dT * 14 cu.m. per dT = 28 cubic meters [OK]
Radius = 0.620350490929 * (28)^1/3 = 1.8837
Length (L) = Diameter = 2 * 1.8837 = 3.7675 [+6% error]
Surface Area (s) = 4 * pi * 1.8837^2 = 44.5919 sq.m. [+12% error]
Material Volume (MV) = s/100 = 0.4459 cu.m. [+12% error]
There is no reason for a design system as detailed and math oriented as FF&S to improperly calculate the surface area and diameter of a sphere. Here is the corrected table for Fire Fusion & Steel v1, pg 11 (note that all values in a collumn have been rounded to the same number of significant figures)
Code:
[FONT="Courier New"]
RATE VOL Surface MV L
1 14 28.0911 0.2809 2.99
2 28 44.5919 0.4459 3.77
3 42 58.4319 0.5843 4.31
4 56 70.7853 0.7079 4.75
5 70 82.1390 0.8214 5.11
6 84 92.7549 0.9275 5.43
7 98 102.794 1.028 5.72
8 112 112.365 1.124 5.98
9 126 121.543 1.215 6.22
10 140 130.388 1.304 6.44
15 210 170.856 1.709 7.37
20 280 206.977 2.070 8.12
25 350 240.176 2.402 8.74
30 420 271.217 2.712 9.29
35 490 300.572 3.006 9.78
40 560 328.556 3.286 10.2
45 630 355.395 3.554 10.6
50 700 381.256 3.813 11.0
55 770 406.267 4.063 11.4
60 840 430.530 4.305 11.7
65 910 454.128 4.541 12.0
70 980 477.128 4.771 12.3
75 1,050 499.586 4.996 12.6
80 1,120 521.550 5.216 12.9
85 1,190 543.061 5.431 13.1
90 1,260 564.154 5.642 13.4
95 1,330 584.860 5.849 13.6
100 1,400 605.205 6.052 13.9
200 2,800 960.704 9.607 17.5
300 4,200 1,258.88 12.59 20.0
400 5,600 1,525.02 15.25 22.0
500 7,000 1,769.63 17.70 23.7
600 8,400 1,998.34 19.98 25.2
700 9,800 2,214.63 22.15 26.6
800 11,200 2,420.82 24.21 27.8
900 12,600 2,618.57 26.19 28.9
1,000 14,000 2,809.11 28.09 29.9
2,000 28,000 4,459.19 44.59 37.7
3,000 42,000 5,843.19 58.43 43.1
4,000 56,000 7,078.53 70.79 47.5
5,000 70,000 8,213.90 82.14 51.1
6,000 84,000 9,275.49 92.75 54.3
7,000 98,000 10,279.4 102.8 57.2
8,000 112,000 11,236.5 112.4 59.8
9,000 126,000 12,154.3 121.5 62.2
10,000 140,000 13,038.8 130.4 64.4
20,000 280,000 20,697.7 207.0 81.2
30,000 420,000 27,121.7 271.2 92.9
40,000 560,000 32,855.6 328.6 102
50,000 700,000 38,125.6 381.3 110
60,000 840,000 43,053.0 430.5 117
70,000 980,000 47,712.8 477.1 123
80,000 1,120,000 52,155.0 521.6 129
90,000 1,260,000 56,415.4 564.2 134
100,000 1,400,000 60,520.5 605.2 139
200,000 2,800,000 96,070.4 960.7 175
300,000 4,200,000 125,888 1,259 200
400,000 5,600,000 152,502 1,525 220
500,000 7,000,000 176,963 1,770 237
600,000 8,400,000 199,834 1,998 252
700,000 9,800,000 221,463 2,215 266
800,000 11,200,000 242,082 2,421 278
900,000 12,600,000 261,857 2,619 289
1,000,000 14,000,000 280,911 2,809 299
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I leave it to any other gearheads and FF&S fans to decide whether the errors are large enough to warrant inclusion as errata.
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