Turret damage

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EVE rely on math to determine the outcome of events, like all computer games. A player can develop a feel for the workings of the game by experience and practice over time. Such knowledge is often powerful and can be used to tip the balance in pvp encounters, either by fitting a ship in a special way or by piloting in a certain way. Some, but not all, of this practical knowledge stems from the intuitive understanding of the math behind it. This article will look into the math that controls how guns and turrets work, hopefully this will give the reader insights into damage dealing that could otherwise take many weeks or months of playing to develop. The information here is all about numbers and theory though. So don't think that practical experience won't be needed, a pvp:er needs more than just an understanding of the math. A pvp:er need to be able to read and predict an opponents behaviour, to have a knowledge of various ships and fittings as well as to develop the nerves to stay calm in a battle that can go either way.

(is this text just silly?)

Governing Equation

This equation can look a bit intimidating at first, but its very central to how turrets deal damage and as such its a good idea to develop an understanding for it. Its purpose is to calculate the chance to hit with turrets.

To paraphrase Oli Geist, this can be abstracted to:

Chance to hit = 0.5 ^ (tracking term + range term).

Those who know their math recognizes that x^(a+b) is the same as (x^a)*(x^b), so this can also be written as:

Chance to hit = 0.5^(tracking term) * 0.5^(range term)

From this we can see that tracking and range are treated entirely separetly and are then multiplied. So anything that effects range does not effect the tracking. In other words, excellent tracking can not make up for a lack in range.

General

Remember that anything to the 0th power = 1, so you'll always hit if (tracking+range) = 0. (1 is 100%.) Also remember that for any number 'x' < 1, the value of x^y approaches zero as y increases. Since 0.5 is less than one, we'd like the value of (tracking + range) to be as low as possible for better hits.

Range

So what can we do to maximize hits? Well, the range term becomes zero if we're at or under our turret's optimal range. It goes to 1 at (optimal + falloff), to 4 at (optimal + 2* falloff), and to 9 at (optimal + 3* falloff). 0.5^4 is 0.0625, or a 6.25% chance of hitting a large stationary target at that range. 0.5^9 is 0.002, or a 2/1000 chance of hitting. Good luck with that. So we can say that any range over (optimal + falloff) is very sketchy, and staying at or near optimal is good. No surprises here.

Tracking

What about tracking? Well, if the transversal speed is 0, the tracking term evaluates to zero (multiply with a zero and the answer is always zero). Since 0.5^0 = 1, this means that the tracking part of the equation will give you a 100% hit chance, as long as the target is also inside the optimal range the chance to hit will remain at 100%. So getting your target to fly straight to you, or straight away from you, is great. Because you will hit every time as long as the target remains inside optimal range. This is what makes the tactic known as kiting so effective. If the target makes sure to keep up the transversal speed, the best way to minimize this term is some combination of:

• Keep the target at longer range
• Lower the transversal speed
• Increase your turret's tracking speed
• Increase the target's signature radius

Modules that can help with these are AB's/MWD's, webs, tracking computers, and target painters. You can also choose 'keep at range' on a target to minimize the traversal -- though bear in mind that your target will be able to hit you more easily too. Note also that if the range should ever become zero, the equation makes a forbidden division with zero. This results in an error and the turrets will no longer be able to hit.

Turret damage dealing

Random damage distribution

At the heart of a turrets damage output is a single randomly generated value between 0 and 1 that is several digits long, something like 0.317226. This random number is used both to determine if the turret hits the target and also to determine how much damage the hit actually does. Should the randomly generated number be less than 0.01 (1% chance), it will be a perfect hit (aka wrecking), this kind of hit always deal exactly three times the base damage. The thing about perfect hits is that they always occur as long as that random number was lower than 0.01 and at the same time lower than the hit chance. So perfect hits are not scored by 1% of the shots that hits, its scored 1% of all hits and misses taken together. This means that if your chance to hit is 1% or below, you can only do perfect hits. Most shots will be misses of course, but those that do hit, they will be perfect.

A turrets base damage is a fixed number that is calculated by multiplying the turrets damage multiplier and the sum of the damage types from the ammo. This value and the randomly generated number will be used to calculate the raw damage that a turret does, damage before any resistances that is. Its also this number that EFT uses to calculate the theoretical volley damage.

The raw damage dealt by a turret is calculated by taking the randomly generated number that resulted in a hit, adding 0.49, and multiplying this sum with the turrets base damage. Since the first 0.01% of the random value is used for perfect hits, the random values that hit normally are those between 0.01 and up to 1 at most (1 if the chance to hit is 100%). So if the chance to hit was 100%, this means that the normal damage will be between 50% (0.01+0.49) and 149% (1+0.49) of the base damage, or in the case of perfect hits always exactly 300% of the base damage. This number will then be reduced accordingly by the targets damage resistances in order to obtain the final damage number.

• Damage generation
  • Calculating variables
    • Turret Base Damage = Turret Damage Multiplier * Sum of Ammo Damages
    • Hit Chance = ((calculated by the big equation at the top of this page))
    • X = Random (0 to 1)
  • Calculating raw damage
    • If (X < 0.01 and X < Hit Chance) Then RAW DAMAGE = 3 * Turret Base Damage (these are called perfect or wrecking hits)
    • Else If (X < Hit Chance) Then RAW DAMAGE = (X + 0.49) * Turret Base Damage (these are normal hits)
    • Else RAW DAMAGE = 0 (this is a miss)
  • Calculating damage after resistance
    • Resistance modified EM Damage = Ammo EM damage * (100% - Target EM Resistance)
    • Resistance modified Thermal Damage = Ammo Thermal damage * (100% - Target Thermal Resistance)
    • Resistance modified Kinetic Damage = Ammo Kinetic damage * (100% - Target Kinetic Resistance)
    • Resistance modified Explosive Damage = Ammo Explosive damage * (100% - Target Explosive Resistance)
    • Damage dealt = RAW DAMAGE * (Sum of Resistance modified damages) / (Sum of Ammo Damages)

The quality of the hit will be described by the value of the random number + 0.49, ranging from barely scratching (least damage) to excellent (highest damage) for regular hits and there may also perfect hits and misses. On the normal hits this number is multiplied with the turrets average damage. By looking carefully at the damage generation above, it becomes clear that the Hit Chance not only means that a turret can miss but also places a cap on the maximum damage number on the non-perfect hits, the next section looks into how this effects the turrets DPS.

Hit description	Random damage modifier
Perfectly	0.490 to 0.500 (special case, 3x the average damage instead)
Barely scratches	0.500 to 0.625
Hits lightly	0.625 to 0.750
Hits	0.750 to 1.000
Well aimed	1.000 to 1.250
Excellent	1.250 to 1.490

A turret with a 100% hit chance will see a natural and unavoidable damage spread between 50-149% of its base damage for normal hits, and always do exactly 300% of its base damage on perfect hits.

Example: A turret fires on target. The chance to hit is 0.8981, the EVE server rolls a random number between 0 and 1, and gets 0.6573, its less than the chance to hit so the target is struck. At this point 0.49 is added to the random number which then becomes 1.1473. The turret had a damage multiplier of x2.1 and the ammo does 4 EM and 2 Thermal, so the base damage is 2.1 multiplied with 6 (4+2), which is 12.6. After multiplying this with the random number we get the raw damage, which is 1.1473 x 12.6 = 14.456. In this damage the raw EM part is 1.1473 x 2.1 x 4 = 9.6373 and the raw Thermal part is 4.8187. Assuming the targets shield was hit, and that only the normal 20% resistance to Thermal is in effect, the final damage then becomes 9.6373 + (4.8187*0.8) = 13.492 points, in the log the hit will be described as a well aimed one and be rounded off to one decimal place.

Damage reduction due to a decreased hit chance

When a turret has less than 100% chance to hit the turrets damage dealing capabilites are reduced in two different ways. The first and obvious one is that it sometimes misses, the other is that the max damage on normal hits (e.g. not perfect ones) are reduced as well. Both of these effects will decrease the DPS output. If you read the previous section, you may remember that the a random number between 0 and 1 is generated to see if a turret hits. If this value is lower than the chance to hit, the turret hits, this number is then used further by adding 0.49, this sum is then multiplied with the turrets base damage to obtain the raw damage done (damage before resistances). What this all means is that targets that are tricky to hit also take less damage since high and juicy rolls now are discarded as misses. This can be seen in the damage log as well, a hard to hit target never recieves excellent or well aimed hits, sometimes barely scratching is the highest (perfect hits still happen). The practical effect is that the DPS decreases more than the chance to hit does.

Example: (please note that perfect hits are not considered in this example to make the numbers easier to follow) A turret has 50% chance to hit at optimal+falloff. The highest randomly rolled number that can result in a hit is thus 0.5, higher numbers means a miss. This will shrink the damage interval down to 0.5 to 0.99 for normal hits, which on average is 0.745 ((0.5+0.99)/2). Compare that with a case inside optimal range where the chance to hit is 100%, where the damage interval is 0.5 to 1.49 and the average is 0.995. At optimal+falloff (and ignoring perfect hits) the DPS from normal hits are hence reduced to 50% x 0.745=37.25% of normal. (If perfect hits are taken into consideration for a full DPS value it becomes 39.5% instead, see table below)

The table below shows how DPS goes down as a target becomes harder to hit. Do note that the reason for the hit chance reduction doesn't matter, be it because of falloff or just tracking issues, the DPS goes down identically. The table can be used to see both how the damage declines as one goes deeper into falloff, and how the damage declines as a result of a lowered hit chance due to tracking reasons. If you wish to combine the effects of tracking and falloff, you can do this by first picking the hit chance from falloff and then the hit chance from tracking, multiply the values to get the combined hit chance, look in the hit chance column until you find a hit chance that is closest to the product you got, and read your relative DPS from that line.

Comment: The relative DPS above is over 1 at the start, this is not an error. Its because its relative to the base damage of the turret. What puts it above 1 are the perfect hits since they do extra high damage.

Parts into falloff	Hit chance	Relative DPS	Reduction in DPS by %
0	1,0000	1,0151	0,0%
0,1	0,9931	1,0048	-1,0%
0,2	0,9727	0,9747	-4,0%
0,3	0,9395	0,9268	-8,7%
0,4	0,8950	0,8641	-14,9%
0,5	0,8409	0,7906	-22,1%
0,6	0,7792	0,7104	-30,0%
0,7	0,7120	0,6274	-38,2%
0,8	0,6417	0,5454	-46,3%
0,9	0,5704	0,4672	-54,0%
1	0,5000	0,3951	-61,1%
1,1	0,4323	0,3303	-67,5%
1,2	0,3686	0,2736	-73,0%
1,3	0,3099	0,2249	-77,8%
1,4	0,2570	0,1840	-81,9%
1,5	0,2102	0,1502	-85,2%
1,6	0,1696	0,1225	-87,9%
1,7	0,1349	0,1003	-90,1%
1,8	0,1058	0,0825	-91,9%
1,9	0,0819	0,0685	-93,2%
2	0,0625	0,0576	-94,3%
2,1	0,0470	0,0492	-95,2%
2,2	0,0349	0,0428	-95,8%
2,3	0,0256	0,0379	-96,3%
2,4	0,0185	0,0343	-96,6%
2,5	0,0131	0,0316	-96,9%
2,6	0,0092	0,0277	-97,3%
2,7	0,0064	0,0192	-98,1%
2,8	0,0044	0,0131	-98,7%

The formulas used to calculate this table were:

• Chance to Hit: 0.5^(0+(Falloff parts / 1)^2)
• Relative DPS: if(HitChance>0.01 then (HitChance-0.01)*((0.50)+(HitChance+0.49))/2+0.01*3 else HitChance*3)
• Reduction in DPS: (RelativeDPSat100%hit/RelativeDPScurrent)*100%-100%

Answers to some questions

Do small targets take less damage from big guns?

Usually they do. If they have zero transversal speed, they will take full damage from any attacks in optimal range. If they have a transversal speed however this will result in two things, first of all they become harder to hit and secondly the maximum damage they can receive goes down as well. A hit will never deal less than half the guns base damage, and half of something very high still hurts.

Since big guns have large values in signature resolution and low values in turret tracking, it becomes harder to hit a small target and the hits will have a lower maximum damage. Unless of course the small ship pilot drops his transversal to 0, at which point the big guns will hit easily and deal full damage, the award for such piloting is a brand new noob ship.

Can a turret hit a target with a larger angular velocity than its own tracking?

Yes it can. From the chance to hit equation it can be deduced that the tracking part can be broken down into:

Chance to hit (tracking part) = 0.5 ^ (((Angular velocity / Turret tracking) * (Size ratio between gun and target))^2)

From this it can be seen that the angular velocity and the turret tracking is compared as a ratio, its not an upper limit. So turrets with less tracking than a targets angular velocity can still hit, the only thing that happens is that the chance to hit is reduced. If we consider the case where the angular velocity is equal to the turret tracking, and assuming that the size ratio between gun and target is 1 (the guns signal resolution is equal to the targets signature radius), we get Chance to hit = 0.5^((1/1*1)^2) = 0.5^1 = 0.5, or 50%. So roughly speaking a gun will hit 50% of the time when angular velocity and turret tracking are the same on same sized targets. If the angular velocity is twice as high as the tracking (and gun and ship still have identical sizes), the chance to hit becomes 0.5^4 = 6.25%. If the tracking is twice as big as the angular velocity (and again, size ratio is 1) then the chance to hit will be 0.5^((0.5)^2) = 0.5^0.25 = 84.1%. In theory small and fast ships could benefit more from tracking bonuses than they would from raw damage bonuses, at least if nobody tries any clever manual piloting moves.

Are frigates with a smaller signature radius better than larger frigates in frigate pvp?

When comparing small signature radius, say 35m and 43m, the differance is only noticeable on short distances (up to 1 or 2 km). The one with the smallest radius get a small advantage from this factor. However, other factors come into play as well, and this advantage is many times offset by the fact that the larger signature frigates uses guns that generally has higher tracking values (the trend is: projectiles have the worst tracking, lasers are in between, hybrids have the best; when compared on an equal basis). So it really doesn't seem to matter. What is clear is that a high tracking can be more important than the damage multiplier on the gun, especially since the max damage drops if the gun can't track well enough.

Are target painters useful for turrets?

Yes, but only if the chance to hit is below 100% due to poor tracking or by using large guns against a small target, if the decreased hit chance is because of falloff then a target painter will not help at all. A regular target painter I increases the signature radius of a ship by +25%. Since all tracking terms are multiplied with another in the hit chance equation, the number can be moved around inside it to see how it effects the other factors. So another way to look at it is this: the signature radius increase is identical to what would happen if the target instead had reduced its angular velocity to 80% (100%/125%) of its current or as if the turrets had gotten +25% more tracking against the target. Keep in mind that this last part was only a comparison, a target painter doesn't actually reduce anyones speed or improves anyones tracking.

If I group my guns, does EVE pick the same random number for all of them?

No. Even if the guns are grouped on your screen, they are still treated as separate turrets. This can be seen by looking at the turret groups average damage distribution. Its also quite easy to see by looking at the damage output when shooting deep into falloff (or against hard to track targets).

+0.49? It says +0.50 on the EVEonline wiki

Yes it does. But some tests that were made to ensure the validity of this article says otherwise. The data is too big to present here, so results and the method to collect it will be presented instead. Should you wish to check for yourself feel free to follow this procedure. A frigate was named 'Ouch' and abandoned at a safespot. An Osprey was fitted with lasers (infinite ammo, perfect for afk:ing), a remote shield transfer and shield transfer drones. The guns and the ammo was chosen so that the damage would never go below 10.0, this assures that all data points collected will have 3 accurate digits and improve the quality of the data. The damage was also only done to the shields, they where never allowed to drop below 25% since a bleed through into armor can happen that can mess with the observed damage. Finally, the ships were positioned within optimal range and the speed set to zero to ensure that the chance to hit is 0.5^0 = 100% and nothing less. After 10,656 shots on the poor frigate, enough data was collected to make some conclusions about how the random damage distribution looks like. The data ends up in the My Documents\EVE\gamelogs on windows, and was easily copied into a prepared Excel sheet for fast (well, it took a few minutes before the data was processed) results.

• Base damage
  • The ammo type dealt: 7 EM and 5 Thermal
  • The base damage on the lasers were: 24.9063
  • Compensating for resistance (20% thermal) the modified base damage is: 22.8308

Of the 10,656 shots the lowest recorded damage was 11.4 (recorded 15 times) and the highest non-perfect was 34.0 (recorded 33 times), perfect hits dealt 68.5 damage (recorded 101 times). On average, each damage number (anything between 11.5 to 33.9) was recorded 46.7 times (standard deviation = 7.02). The reason for the lower occurances of the min and max results on normal hits comes from rounding effects. Any damage in between has an interval of 0.1 units (22.2500 to 22.3499 both produce the 22.3 in the log). However the min and max values do not have that span. The lowest theoretical number is Base Damage x 0.5 = 11.415, hence the interval to get 11.4 in the log is between 11.415 and 11.4499, that is only 0.0345 differance. So the expected number of occurances of the value 11.4 is only 34.6% of the average number, 15 recorded values / 34.6% = 43.4, close to average and inside the standard deviation. The upper interval is 67.8%, 33 times / 67.8% = 48.7, also close to average and inside the standard deviation. (Note: 34.6%+67.8%=102.4%, which is impossible ofc, the error comes from rounding errors in the 4th decimal of the basedamage, awesome precision isn't needed for this comparative calculation since the natural random deviation is much larger anyhow, so good enough).

• Lowest damage random multiple
  • Modified base damage x 0.51 = 11.6
  • Modified base damage x 0.50 = 11.4
  • Modified base damage x 0.49 = 11.2
  • The lowest observed damage is 50%
• Highest non-perfect random multiple
  • Modified base damage x 1.50 = 34.2
  • Modified base damage x 1.49 = 34.0
  • Modified base damage x 1.48 = 33.8
  • The highest non-perfect damage is 149%
• Perfect hits deal 68.5 damage
  • Modified base damage x 3 = 68.5

The collected data shows that the normal damage is distributed within 50%-149%. Since the first 1% unit is used for critical rolls, this means that the constant has to be 0.49.

Tracking Examples

coming soon

References:

http://wiki.eveonline.com/wiki/Turret_damage

http://www.eve-ivy.com/forum/viewtopic.php?p=201888#201888

http://www.eve-ivy.com/forums/viewtopic.php?p=216525#p216525

http://www.hostile.dk/files/eve/eve-tracking101.swf