OK
I did at least half of job so I start summery ( this post will be modified so it is a part so far )
assumtpions
PD is part of bigger ship not a dedicated PD ship
I carve out enough space to put FC+ 2 quad gauss turrets and compare to similar space using railgun setup
it is worth mentioning while gausess increases rof , railguns hits max for in tier 3 tech in 10cm v3/ c3 with 4 shots per gun
it is worth mentioning that it is difficult to calculate total cost for rail system as plant costs are usually neglected as needed for engines
it is worth noting that increasing plant efficiency allows to mount more railguns pre tech level
ships could be made a bit faster but I assumed some typical average designs -ie 4km for magneto era ships
all ship speed fits adequate gun system level - it differs in game but i wanted to do as equal conditions as possible
there is no diffrence in size of PD Fire control as gausses uses a bit bigger - you can add 1 more gun to railgun comparison if you want ot cover it
techs
1. gauss r-1 system 2quad = 8shots every 5 sec with 8km/s tracking
eq rail 10cm v1/c1 system can fit 15 guns so 60shots every 15 sec with ship speed = tracking at 1km/s tech
tech cost very similar
2. gauss r -2 system 2 quad = 16 shots every 5 sec 12 km/s tracking
eq rail 10cm v2/c2 can fit 16 guns with 64 hot every 10 sec with ship speed = tracking at 2km/s
3. gauss r-3 system 2quad = 24shots every 5 sec with 16km/s tracking
eq rail 10cm v3/c3 system can fit 17 guns so 68shots every 5 sec with ship speed = tracking at 3km/s tech (ion eng)
in this place railguns hits its maximum efficiency a they have 5 sec rof - no further tech can change number of shot or size of guns that is critical
till this place railguns are cheaper than gauss - however it is not budget ruining diffrence
things start getting interesting in megneto plasma era
4. gauss r-3 system 2quad = 24shots every 5 sec with 16km/s tracking - gaus stay with old design in this tech level as their jump is huge and next level guns fits by cost level 5 tech - so no new gauses but better tracking so 20km/s trasking on this level
eq rail 10cm v4/c4 system can fit 18 guns so 72shots every 5 sec with ship speed = tracking at 4km/s tech (magneto eng )
interesting situation , isn't it ?
railguns offers 3 times shots volume vs 1/5 tracking - all will depend on missile types ( it should be diffrent analysis later )
5. gauss r-4 system 2quad = 32 shots every 5 sec with 25km/s tracking
eq rail 10cm v5/c5 system can fit 19-20 guns so 76-80shots every 5 sec with ship speed = tracking at 5km/s tech (early fusion tech )
at this point technology cost between gausses and railguns is pretty similar
railguns offer twice the volume for 1/5 tracking - do you see the trend ?- it is because rails can not lower size or increase rof
from this point further analysis is pointless as gausses still have 2 rof upgrades ahead and relation on similar tech turrets to average ship speeds is like 4:1 so keep constant in terms of relative tracking and accurancy
of course you may go for faster ships when using railguns
so you can pump tracking possibly up to 30% for a cost of lower comparable efficiency somewhere else
10cm railguns are much better weapon when other then PD situation - sheer volume of shots guarantee this
but as a long term choice for PD - gausses are winner
I can only guess ( thinking of doing some calculations later) that non perfect in terms of power multiplier missile engines means that pre magneto / early fuzion you face relatively slow missiles so rails got advantage os some kind and later vs modern really fast missiles tracking will be a killing factor for this kind of PD
wall of text
I'd like to check how this tech level PD system would behave vs missiles of its era to check PD umbrella efficiency
but this will be another time
sorry for spelling - I had very limited time when running this and I'm not native