Optimizing the Conquest: a Mathematical Model of Space CombatAuthors: CapnHector, intrinsic_parity, Thaago, Vanshilar*
IntroductionThe Conquest-class battlecruiser is a ship designed for a combination of speed and firepower, with a specific goal of winning engagements by rapid deployment of overwhelming firepower while sacrificing defense. Therefore, correct weapon choice is paramount to success. Simulations show that the Conquest is an effective ship for combating high-level threats [1], but the optimal weapon layout remains unknown. While simulation can demonstrate the effectiveness of a specific layout [1], it is unrealistic to simulate all possible combinations of weapons against all the diverse threats that spacefaring fleets face in the Sector. Furthermore, human bias, such as for symmetry, and excessive focus on particular threats may guide unsystematic simulation experimentation. Thus, effective combinations may exist that are so far untested. To provide an unbiased analysis of weapon effectiveness and layout effectiveness for the Conquest, we developed a mathematical model of space combat, calibrating against simulation data, and used this model to forecast the time to kill (TTK) of weapons and combinations of weapons when mounted on the Conquest-class battlecruiser. To our knowledge, this is the first complete mathematical model of spaceborne weapons and their effects in the Persean sector.
MethodsWeapons and shipsThe weapons studied were the Squall MLRS, Locust SRM, Hurricane MIRV, Harpoon MRM, Sabot SRM, Gauss Cannon, Mark IX Autocannon, Mjolnir Cannon, Hephaestus Assault Gun, Storm Needler, Arbalest Autocannon, Heavy Mauler, Hypervelocity Driver, Heavy Needler, and Heavy Mortar. A Conquest layout was considered to include 2 choices of large missile, 2 choices of medium missile, 2 choices of large gun, and 2 choices of medium gun. The target ships were Glimmer, Brawler (LP), Vanguard, Tempest, Medusa, Hammerhead, Enforcer, Dominator, Fulgent, Brilliant, Radiant, Onslaught, Aurora, Paragon and Champion-class vessels. Damper Fields and Fortress Shields were not modeled. For the weapons, damage per shot, damage type, firing pattern over time, accuracy, and recoil (accuracy loss per shot) data were used. For target ships, the data used were ship width and length, number of armor cells, baseline flux dissipation capacity, maximum flux capacity, shield effectiveness, hull integrity score, and armor score. All variables were imported from Starsector version 0.95.1a files, except weapon firing pattern was constructed based on firing duration and delay data, and armor cell number was calculated based on a previously published and verified algorithm [2].
Mathematical modelMathematical model
For the mathematical model, the following assumptions were made:
1. The target vessel is in the center of the weapon's field of fire, but with an error term that follows the normal distribution. We define f=SD(error)/range(px). The target vessel is facing the weapons.
2. Range to target is 1000 units.
3. All weapons target the center of the target vessel.
4. Weapons firing angle is random with an angle range determined by the weapon's known maximum angle range. The weapons are mounted on turrets, not hardpoints.
5. The weapons fire continuously for the testing period.
6. The target vessel has no flux capacitors and only enough flux vents to offset shield upkeep cost.
7. The target vessel has no officer or hullmods.
8. The target vessel will raise shields to block any shots that it can block without overloading. If it were overloaded by blocking the next shot, the target vessel will lower shields instead.
A single shot's firing angle is modeled as drawn from a uniform distribution [-maxangle/2, maxangle/2]. The target vessel's visual angle is calculated based on ship width and engagement range, ignoring curvature. These are used to compute the expected hit location in pixels, to which is added a random error described by a normal distribution such that mu=0 and SD=f*1000 pixels. The result is used to determine whether the target vessel was hit and if so, which armor cell was hit, based on the pixel ranges for the target vessel's hull width and each armor cell's width. The hit is considered to be immediate with no travel time.
In order to create a probabilistic model of armor damage permitting cell by cell operations, ship armor is described as a [5, armorcells+4] matrix which is damaged according to cell by cell calculations with each hit on the central [1,armorcells] cells damaging the [5,5] matrix around it, in an inverse of the Starsector armor pooling function, such that damage is spread over a [5,5] matrix with corner cells taking 0 damage, the central cells and adjacent cells taking 1/15 damage, and their neighbor cells taking 1/30 damage, to a total of 100% of damage spread over the hit armor matrix, with central cells absorbing double the damage of peripheral cells. By contrast, Starsector is thought to pool armor strength from the cells, with peripheral cells contributing half the armor contribution of the central cells. While formal proof that the method is exactly equivalent to Starsector's handling of armor has proven elusive, we have previously noted that, based on experimentation, any error is expected to be small. Elementary algebra shows that the two methods of calculating damage are at least equivalent when it is not the case that armor %u2248 5% of starting armor. For an illustration of armor calculations, see Figure 1.
Weapon hits were modeled as the expected value of the hit (i.e. an expected value wave over the armor), an approximation previously found to be very close to the true value found by simulating individual shots [3]. Shot damage could only be reduced by a maximum of 85% by armor. When armor was lower than 5% of maximum, armor was counted as 5% of maximum for purposes of determining damage reduction. Damage was first applied to shields, then to armor if shields were not used to block, and then the excess to hull if armor could not absorb the entirety of the hit strength. Damage modifiers were used as in Starsector, that is, all weapon types contribute their full hit strength to hull damage, but adjusted by damage type for shield and armor damage. The mathematical model in its entirety is included in the appendix.
The factor f was determined to be 0.05 based on previously published [4] empirical data (Figure 2), corresponding to an error in assumption 1 such that 95% of the time the enemy ship is within 100 pixels of the center of the weapon's firing arc.
The maximum permitted combat time was 500 seconds, which was exceeded by some combinations with Hurricane-Hurricane or Hellbore-Hellbore vs. Paragon or Radiant. These outliers were excluded from the analysis.
Statistical analysis
The distribution of hits for each weapon, ship and timepoint was generated based on the mathematical model using 100 000 samples. Because the model was deterministic, it was run once for every studied combination of weapons for every ship. The distribution for each weapon, ship and timepoint combination was calculated once, then re-used based on a lookup table. The total number of combat models run was 16 times the number of possible unique layouts, that is, 16 x 7938 = 127 008 combats. For each combat model, the time to kill and weapons used were recorded. Time to kill was averaged across all ships for each weapon separately, for pairs of weapons from each category, and for combinations of large missiles and large guns. The results represent the reduction / increase in time to kill that results from using a specific combination, all other equipment being equal, on average, with flux and range combinations ignored. All analyses were performed using RStudio 2022.07.1+554. While analysis of variance was not performed, the sample size was at least 48 combats for each reported result. Further sub-analyses were performed for capital ships and Remnants as meriting special consideration.
Figure 2. Effect of positional error (f-factor) on TTK
ResultsThe results of the main analysis including pairs and combinations of weapons are presented in Table 1. Scores for individual weapons are presented in Table 2. Mean time to kill each type of ship in the second analysis which included medium guns is presented in Table 4. Preliminary analyses (reported separately) have shown that the model yields vastly different results if weapon accuracy is not simulated. Finally, Figure 3 demonstrates different strategies against the hardiest ship tested, the Radiant-class drone battleship.
Table 1. Combinations of weapons
"Large missiles" "Avg. time to kill" "TTK speed score"
"Squall Locust" 21.8 "4.0%"
"Locust Locust" 22.1 "2.7%"
"Squall Hurricane" 22.5 "0.9%"
"Squall Squall" 22.9 "-0.8%"
"Locust Hurricane" 23 "-1.3%"
"Hurricane Hurricane" 23.9 "-5.0%"
"Medium missiles" "Avg. time to kill" "TTK speed score"
"Harpoon Harpoon" 20.1 "12.8%"
"Sabot Harpoon" 22.1 "2.6%"
"Sabot Sabot" 25.8 "-12.2%"
"Large guns" "Avg. time to kill" "TTK speed score"
"Mjolnir Mjolnir" 18.8 "20.7%"
"Gauss Mjolnir" 19.4 "17.1%"
"Gauss Gauss" 20 "13.2%"
"Hephaestus Mjolnir" 20.6 "10.3%"
"Mjolnir Storm Needler" 21 "8.3%"
"Mark IX Mjolnir" 21.1 "7.7%"
"Gauss Hephaestus" 21.1 "7.6%"
"Hellbore Mjolnir" 21.8 "4.0%"
"Gauss Storm Needler" 21.9 "3.5%"
"Gauss Mark IX" 22 "3.1%"
"Hephaestus Hephaestus" 22.7 "0.1%"
"Hephaestus Storm Needler" 22.9 "-0.8%"
"Gauss Hellbore" 23.1 "-1.8%"
"Hephaestus Mark IX" 23.2 "-2.3%"
"Storm Needler Storm Needler" 23.7 "-4.1%"
"Mark IX Storm Needler" 24.1 "-6.0%"
"Hellbore Hephaestus" 24.2 "-6.4%"
"Mark IX Mark IX" 25 "-9.3%"
"Hellbore Storm Needler" 25.1 "-9.7%"
"Hellbore Mark IX" 26.1 "-12.9%"
"Hellbore Hellbore" 28.7 "-21.0%"
"Medium guns" "Avg. time to kill" "TTK speed score"
"Arbalest Heavy Needler" 21.6 "5.2%"
"Heavy Mortar Heavy Needler" 21.8 "3.9%"
"Heavy Autocannon Heavy Needler" 22 "3.3%"
"Arbalest Arbalest" 22 "3.1%"
"Arbalest Heavy Autocannon" 22.1 "2.8%"
"Heavy Needler Heavy Needler" 22.1 "2.5%"
"Arbalest Heavy Mortar" 22.1 "2.5%"
"Heavy Autocannon Heavy Mortar" 22.2 "2.1%"
"Heavy Autocannon Heavy Autocannon" 22.3 "1.8%"
"Arbalest Heavy Mauler" 22.3 "1.7%"
"Heavy Autocannon Heavy Mauler" 22.6 "0.4%"
"Arbalest Hypervelocity Driver" 22.7 "0.0%"
"Heavy Mauler Heavy Mortar" 22.8 "-0.6%"
"Heavy Mortar Hypervelocity Driver" 23 "-1.2%"
"Heavy Mauler Heavy Needler" 23 "-1.2%"
"Heavy Autocannon Hypervelocity Driver" 23 "-1.3%"
"Heavy Needler Hypervelocity Driver" 23 "-1.4%"
"Heavy Mortar Heavy Mortar" 23.1 "-1.6%"
"Heavy Mauler Hypervelocity Driver" 24.2 "-6.3%"
"Hypervelocity Driver Hypervelocity Driver" 24.3 "-6.5%"
"Heavy Mauler Heavy Mauler" 24.4 "-7.0%"
Large missiles x Large guns x Medium guns table is too large to post on the forum, available here: https://pastebin.com/Yvrftsw1
Table 2. Single weapon effectiveness.
"Large missile" "Avg. time to kill" "TTK speed score"
"Squall" 22.4 "2.5%"
"Locust" 22.5 "1.8%"
"Hurricane" 23.9 "-4.0%"
"Medium missiles" "Avg. time to kill" "TTK speed score"
"Harpoon" 21.1 "11.2%"
"Sabot" 25.8 "-9.2%"
"Large gun" "Avg. time to kill" "TTK speed score"
"Mjolnir" 20.5 "12.1%"
"Gauss" 21.3 "8.3%"
"Hephaestus" 22.8 "1.1%"
"Storm Needler" 23.7 "-2.7%"
"Mark IX" 23.9 "-3.7%"
"Hellbore" 26 "-11.5%"
"Medium gun" "Avg. time to kill" "TTK speed score"
"Arbalest" 22 "2.8%"
"Heavy Autocannon" 22.2 "2.0%"
"Heavy Needler" 22.3 "1.3%"
"Heavy Mortar" 22.5 "0.5%"
"Hypervelocity Driver" 23.3 "-3.0%"
"Heavy Mauler" 23.4 "-3.3%"
Table 3. Target ship durability.
"Ship" "Avg. time to kill" "Avg. TTK speed score"
"glimmer" 8.8 "158.1%"
"tempest" 9.2 "147.9%"
"brawlerlp" 10.2 "121.4%"
"medusa" 10.8 "109.1%"
"vanguard" 12.1 "87.6%"
"fulgent" 13 "74.7%"
"hammerhead" 14.6 "55.0%"
"enforcer" 16 "41.4%"
"brilliant" 24.1 "-6.0%"
"aurora" 25.1 "-9.5%"
"champion" 25.4 "-10.7%"
"conquest" 27 "-16.1%"
"dominator" 32.2 "-29.6%"
"onslaught" 35.1 "-35.4%"
"paragon" 47.5 "-52.3%"
"radiant" 51.6 "-56.1%"
To our knowledge, this was the first complete mathematical model describing weapon damage in space combat in the Persean sector. Our results demonstrate that it is critical to consider all factors in the simulation. For example, our preliminary results from models that did not consider the different shot distributions and accuracy of different weapons diverged markedly from those produced by the complete model. A model without a positional error factor (f-factor) also significantly underestimated TTK compared to simulations, proving the need to include a measure of uncertainty in mathematical models of space combat.
Our results with regard to target vessel durability display linearity: ships are approximately as durable as can be expected from their deployment costs. Generally, older technology stands better to the rigors of space combat. An interesting special case, however, is the Conquest-class battlecruiser itself, which clearly has the defensive attributes of a cruiser, being in line with the defensive attributes of the Aurora- and Champion-class vessels. Another outlier is the Dominator-class heavy cruiser, which was found to have defensive attributes on par with capital ships.
The weapons of the Sector are surprisingly balanced, with no clearly dominating choices among the categories of single weapons. While certain weapons such as the Mjolnir appear to be generally advantageous in TTK, this comes at the cost of flux efficiency and / or range. The exception to this general rule is the category of medium missiles, where Harpoon MRMs should be the preferred choice, as they have a longer range, equal ammunition, and result in quicker times to kill than Sabot SRMs or combinations including Sabot SRMs.
It is interesting to note that among medium guns, older Collapse-era technology has a surprising advantage, as the Arbalest resulted in slightly faster kills than newer weapons, despite accuracy problems that were included in the model. However, we note that this comes at a range disadvantage that may not be justified by the slightly faster TTK.
The picture is significantly more nuanced when considering combinations of weapons, where carefully chosen combinations yield significantly larger effects than those of single weapons. It is especially interesting to note that while the Locust SRM is by itself inferior to the Squall MLRS, the combination of one Squall MLRS and one Locust SRM was the strongest combination of large missiles tested, although it should be noted that the ensuing range limitation likely means this combination might be inferior to two Squall MLRS in a fleet setting. Even more important than careful choice of weapon combinations, however, is avoiding combinations with limited potential against a variety of targets. For example, a Hellbore Hellbore combination was associated with -21.0% speed to achieve kill, and was notably one of the only combinations that could not kill all targets.
Overall, our results highlight the potential in systematic study and simulation of space combat, and the importance of commanders' personal skill in designing compatible weapon layouts holistically, considering the excellent balance among single weapons.
Author statementCapnHector created the mathematical models and wrote the report. Vanshilar, intrinsic_parity and Thaago made significant contributions to developing the model.
References[1] Vanshilar 2022,
https://fractalsoftworks.com/forum/index.php?topic=25459.msg379575#msg379575[2] Vanshilar 2022,
https://fractalsoftworks.com/forum/index.php?topic=25459.msg379452#msg379452[3] Vanshilar 2022,
https://fractalsoftworks.com/forum/index.php?topic=25459.msg379708#msg379708[4] Vanshilar 2022,
https://fractalsoftworks.com/forum/index.php?topic=25459.msg379103#msg379103Note: tables updated on 22-11-04. Errata in:
https://fractalsoftworks.com/forum/index.php?topic=25536.msg380632#msg380632
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