You also argue that since tracking speed and target speed (sV/tV) are fixed data points and not variable they can be functionally ignored. I stipulate that you are wrong.
Hah! And I stipulate that
you are wrong!
/Tongue-in-cheeckOk, seriously though, which bit do you disagree with?
a) Target speed and tracking speed are the same for the final-fire and area-defence variant. The tracking bonus is likely to be the same, or only very marginally different.
b) When all these three input-variables have the same respective value in both scenarios, then the factor computed from these inputs also takes the same value in both scenarios.
b) When the factor is a constant multiplier it does not bear influence
on the comparison between the two systems.
For
a) I would think we can readily agree that the
target speed will be the same, because we should only really compare situations in which we are attacked by the same weapon.
Tracking speed is also the same because that is the way both of our examples are constructed. (In another part of your post you wrongfully assert that I choose different tracking speeds for both examples, but more on that later). So the only source of disagreement I can find is the
tracking bonus. Now I think I have demonstrated in my last example that it is very conceivable that the tracking bonus is maxed out (and thus equal for both). Even if its not, it will not take a drastically different value for both scenarios, as there is at most a 10s difference in tracking time leading to a 4 percentage point difference in the tracking bonus.
For
b) I would hope that this is relatively obvious. It’s the same formula, it’s the same inputs, so it would be same output. It is fair to request an analysis on the effect of a small change in the tracking bonus on the result of the factor, but again I approximate that is not big.
For
c) let me repeat that I stipulate that we can functionally ignore the common factor for a relative comparison between the two systems and
only for this purpose. If we want to compare X to Y and X = a * s, while Y = b * s, then X/Y= (a*s) / (b*s)=a/b. I.e. for the relative comparison the common constant factor “s” becomes irrelevant. I fully agree that only focusing on “a” or “b” will not give you an answer to the absolute value of X and Y, but that is what I have repeatedly said all along, even before your criticism. I explicitly wrote “
this is not the final hitchance, which would include other effects (most importantly tracking speed), but it’s the separable factor due to the range of the BFC.”
Maybe the last bit is the most controversial here, because you write:
But since we’re actually talking about a modifier that can substantially reduce the hit probability it must be included.
Now I could equally assert that you “forgot” the crew grade factor. That can easily be +30%, so its very relevant right? Yes, of course its relevant, but not for the question posed, as both systems would have the same factor.
Now we plug-in weapon and fire control values for the area defense example:
4*P=(90/(1(5*24)))(((0.01)+(1-10/192))/2)(12/24+0.2) = 1.01 missiles intercepted not 4.6.
The final defense example becomes:
8*P=(1-10/48)(12/24+0.2) = 4.43 missiles intercepted not 6.3.
Well you went through great length through about the tracking speed and was it really worthwhile? We could have saved ourselves the trouble and just abbreviated the factors as “A” and thereby have arrived at the same relative result. Oh wait ;-)
Also, and I want to be quite clear on this: I have never claimed the final-fire defence variant could intercept
6.3 missiles. I have claimed that it can intercept
6.3 * A (yes I did put that in bold) missiles, where “A” is a factor that encompasses other variables that are not down to range. Like for instance tracking speed, which I mentioned as the most important. And if you evaluate this “A” Factor to (12/24+0.2)=0.7 you get 6.3 * 0.7 =
4.41, i.e. your result (minor difference due to rounding).
Anyway, down into the tedious bit of who claimed what why that is or is not correct:
Especially with you own examples have fire controls with significantly different tracking speeds against the same target speed(3000 vs 12000).
Actually I am a bit surprised by this statement. Not only is it
not true, because my example entails the same exact tracking speed in both scenarios, but also because you also then use the same tracking speed of 12,000 for both and you must be aware of the firecontroll’s speed rating because it enters your calculations on the weight as well. I do not understand how you could interpret my post as to say the firecontrolls have different tracking speeds:
-
So for the area-defence version we would have to spend 800t on the firecontroll (size 16, 4x range, 4x tracking ). […]
Conversely, what happens for the final-fire version? The firecontroll is only 200t (size 4, 1x range, 4x tracking ),
The last point of contention is that your “density function” falls short, it only accounts for a single 5 second impulse not the multiples that you assert.
That is not quite true. We can actually compute the number of interceptions by integrating over the probability density function alone: N=2*integral_(0)^(r) dx 1 / (5v) =2r/(5v). Plug in the numbers v=24k, r=90k, so N=1.5. So my formula assumed that you get an expected 1.5 shots versus missiles that stream by. Of course that is an average number; in actual combat you will either get 1 or 2 shots versus any particular missile salvo. Because I integrate over the entire range of the area-defence variant, I
do account for multiple interceptions.
Since the area defense analysis needs variable intercept range accounted for substitute ((1-bR/fR)+(1-mR/fR))/2 for (1-r/fR). This is a crude model, but serves.
Well, you use a crude approximation, while I use an exact formula. When these two do not arrive at the same final value I would not take this as a sign that
my formula is broken.
Specifically, the way you account for multiple shots is insufficient. You basically treat it as though you would have multiple shots, all with the hit-probability of the middle of the lasers range. While that may serve as a first approximation it does not account properly for the fact that multiple shots will never occur at an average distance. The example missile flew 120kkm/5s, while the laser had a range of 90kkm (=180kkm in both directions). So you can only get a second shot at the missile if it ended up more than 30kkm from the area-defence variant during the first 5s interval- and the further away it was in the first interval, the closed it will be in the second.
Oh, and somewhat more importantly, you seem to be missing a factor of 2. Remember, the assumption was that the area-defence variant serves as advance picket, i.e. it is at a certain distance of the protected task group and can engage missiles that are streaming towards it, as well as those that have passed the ship and are flying onwards to the task group. So you get to use the full range of the ship
twice.
Something else to note: The above figures assume that the lasers are turret mounted with turret tracking at least 12,000kps. If not the tracking speed drops to 3,000kps, even with the advanced fire control , and the numbers tank (.47 vs 1.01 and 2.06 vs 4.43||.18 vs .72 and .79 vs 3.17). Nor are the area defense and final defense examples equal in hs/tonnage. Area defense FC and Turret total 32.32 hull spaces while the final defense FC and Turrets total 36.64 hull spaces (assuming quad turrets). This does not take into account the required power plants. If the PP tech is Pebblebed each laser needs a 1hs reactor as well taking the area defense suite to 36.32hs and the final to 42.64hs.
Yeah true, I ignored the gearing and power plant requirements, both of which should have been considered on a closer look. One rather subtle point though: You do not need such a large reactor for the final fire variant. Implicitly the area-defence calculation assumed that enemy salvos were at least 10s apart (so that you could actually shot at one salvo more than once when the opportunity was there). So it would be sufficient if the final fire variant fired every 10s (or even less frequent) to match that. Hence you could get away with half the powerplants/laser for the final fire variant. ( I recognize you have used 6 PP for the 8 Lasers on the final fire variant, but I am not sure whether this is the reason). This does not invalidate your point at all though, even if it decreases the difference in mass slightly. As a funny coincidence when gearing and powerplants are considered both setups have pretty much the same costs though, so in a way it is still a fair comparison (to some extend)