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The following is a set of calculation notes of the physical properties of Bane by Anthony Hadley, approximating the correct values for its in-game description based on information taken from the Battlezone II website.


The following are the calculations made to determine the correct values for bane.des,
which previously used values copied directly from rend.des. Where possible I've taken
variables from the BZII website; where this isn't possible I've derived estimates from
said variables, and where that also isn't possible I've substituted values derived from
appropriate comparison with Earth, Mire and/or Rend.

GLOSSARY OF SYMBOLS USED:
Ø - Diameter
R - Radius
P - Polar radius
δ - Percentage difference
|n| - Absolute value
~ - Approximation
ξ - Random from set
A - Surface gravity (acceleration due to gravity)
D - Arbitrary scale factor
E - Expected value (average of a possible range)
<0, ..., n> - Average from set
T - Temperature
H - Solar irradiance
r - Substituted radius
d - Orbital radius
h - Solar irradiance via substitution

POLAR RADIUS [P]
Here we find the percentage difference between Rend's equatorial and polar radii. Since
Bane is in a less extreme situation than Rend, we're using those values as limits to add
a random error to Bane's equatorial radius to stand in as a new polar radius.
    Ø[rend] = 12000km [game_worlds_rend.htm]
    R[rend] = Ø[rend]/2 = 6000km
    P[rend] = 7654km [rend.inf]
    δ[rend] = |21.670|%
    Ø[bane] = 7400km [game_worlds_bane.htm]
    R[bane] = Ø[bane]/2 = 3700km
    δ[bane] = ~δ[rend] = |20|%
    P[bane] = ξ((3700+20%), ..., (3700-20%)) = (3700+7.6486%) = 3983km

SURFACE GRAVITY [A]
Here we use both Earth and Rend's radii and surface gravities to calculate arbitrary
scale factors. Averaging these scale factors produces a stand-in for Bane, which we can
use to produce a stand-in for its surface gravity.
    Ø[earth] = 12742km
    A[earth] = 9.81ms^-2
    D[earth] = Ø[earth]/A[earth] = 1298.879
    Ø[rend] = 12000km
    A[rend] = 8.75ms^-2 [rend.inf]
    D[rend] = Ø[rend]/A(rend) = 1371.429
    Ø[bane] = 7400km
    E(D[bane]) = <D[earth], D[rend]> = <1298.879, 1371.429> = 1335.154
    A[bane] = Ø[bane]/E(D[bane]) = 7400/1335.154 = 5.54ms^-2

BLACK-BODY TEMPERATURE [T]
Having already been provided a temperature for Bane, we can simply convert it form one
scale to another.
    T = -9C [game_worlds_bane.htm]
    T = 264.15K

SOLAR IRRADIANCE [H]
Here we begin with the standard formula for solar irradiance, substituting our sun in for
the binary suns of the Core system and using scale factors derived from Mire and Rend's
solar irradiance to draw the values back in line to the correct stars (essentially moving
Bane to our solar system for the purposes of the calculation, then moving it back).
    H[bane] = (R[sun]^2/d[bane]^2)H[sun]

    H[sun] and R[sun] are unknown and cannot be calculated due to Core system's binary
nature. Substituting for Sol, using h for substituted irradiance:

    d[bane] = 2528.204e9 [game_worlds_bane.htm]
    h[bane] = (695700^2/2528.204e9^2)5.19e7 = 4.53

    Comparison with Mire and Rend's actual values required to draw Bane's value back in
line for correct system:
    d[mire] = 399.426e9 [game_worlds_mire.htm]
    h[mire] = (R[sun]^2/d[mire]^2)H[sun] = 181.41
    H[mire] = 1367.6 [mire.inf]
    D[mire] = H[mire]/h[mire] = 7.539
    d[rend] = 189.990e9 [game_worlds_mire.htm]
    h[rend] = (R[sun]^2/d[rend]^2)H[sun] = 801.81
    H[rend] = 7533 [rend.inf]
    D[rend] = H[rend]/h[rend] = 9.395
    E(D[bane]) = <D[mire], D[rend]> = <7.539, 9.395> = 8.467
    H[bane] = h[bane]*E(D[bane]) = 4.53*8.467 = 38.36

MASS [M]
Here we have rearranged the gravitational field equation to solve for mass, using values
provided and calculated above.
    M = (AR^2)/G = ((5.54x3700)^2)/6.67408e-11 = 1.137e24kg

ESCAPE VELOCITY [V(esc)]
This is the standard calculation for escape valocity, so we can simply plug in above
values.
    V(esc) = sqrt(2GM/R) = sqrt(2x6.67408e-11x1.137e24/3700) = 6.57km/s
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