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Here's a shipload of hindsight: The optimal anti-air artillery calibres of World War Two.
7.62...7.92 mm machineguns were rather ineffective. There were some aircraft with thin armour plating during WW1 already. Such armour plating provided good protection at as little as 300 m altitude, for the bullets quickly lost energy by climbing against gravity and drag. The use of twin or quad mountings didn't help much.
12.7...13.2 mm machineguns were little better. Their effective range was around 500 m, but they had the target in range for so little time that they depended rather on lucky hits to achieve much. The use of quad mountings didn't help much. It's irritating that the Soviets nevertheless introduced 14.5 mm anti-air machineguns based on their anti-tank rifle calibre after WW2, including multiple barrel mounts. Their usefulness was mostly limited to anti-helicopter work and the slightly bigger 23 mm twin gun (a Cold War design) was vastly better.
A single Czechoslovakian 15 mm gun design was of marginal relevance. It was too much gun and mounting for the modest effectiveness of the calibre.
The first really effective AAA calibre was 20 mm. It's widely recognised as the smallest calibre for a useful high explosive round with a proper fuse. 20 mm weapons weighed about double what 12.7 mm weapons weighed during WW2, but their weight was still bearable. 20 mm quad mounts were very heavy, but it was possible to devise a 20 mm light AAA pack gun. The calibre was also very effective in ground support and against merely bulletproofed AFVs (most early WW2 tanks and late in WW2 still the scout vehicles and halftracks). This was the best calibre to equip infantry regiments or even infantry battalions, but this was hardly ever done. The result was that the infantry was largely without proper air defences, but at least the infantry was fairly good at using camouflage, concealment and cover. This happens to have been the reason why 20 mm would have been sufficient for an infantry AAA capability: Its up to 2 km effective range covered approx. the distance at which aviators could still spot infantry targets save for road marches and field fortifications on open fields.
There were some 25 and 28 mm AA guns in France, Japan, Soviet Union and the U.S., but they did not earn good reputations. Japanese 25 mm suffered from sluggish mountings and a ceiling that permitted very accurate level bombing against anchored or moored ships from a mere 7,000 ft altitude.
The next really convincing calibre was 30 mm. The German 30 mm HEIT (high explosive incendiary tracer) Minengeschosse were designed for a giant explosive charge with added incendiary side effect. One hit usually sufficed to shoot down a single engine aircraft. Even the rugged 4-engine B-17 was said to usually be doomed after four 30 mm HEIT hits. This was in part because of the huge blast effect an in part because the fuse was designed to explode just after penetrating the skin, so it would explode inside a wing or stabiliser. 30 mm wasn't only a sweet spot for its destructive power leap over 20 mm. It was also a sweet spot because the late WW2 Mk 103 showed that a huge rate of fire (for a light AAA at the time) and a high muzzle velocity was doable with this calibre without excessive weight of the whole system. The effective range was greater than 2 km in the AAA role with its muzzle velocity without needing a muzzle velocity at 900+ m/s, which would have worn out the barrel too quickly. So IMO a 3 cm AAA of about the Mk 103's weapon design (with a suitable mount) would have been the perfect light AAA for divisional air defence and SPAAGs. It would have reached higher than 2 cm guns, pushing the threats farther up to altitudes (8,000 ft and more) from where the rear elements of a division (esp. artillery positions) would be much less recognisable. Hardly any 3 cm AAA was used during WW2.
Next, there's the 37...40 mm group. These guns had vastly inferior rate of fire to 3 cm Mk 103, but the destructive power of late war HE-rich grenades left almost no chance to single engine aircraft with even only one hit and approached one-shot kill power against medium bombers. Moreover, they had increased reach than smaller calibres. The altitude of 10,000 ft was considered to be safe from them during WW2. Some of their fuse designs self-destructing at about that distance anyway, so greater theoretical ballistic ceilings didn't matter. This calibre was the best fit for naval guns. It pushed level bombers so high that they became too inaccurate even against anchored or moored individual ships. They made twin-engine skip and torpedo bombing inefficient or (if they got close) suicidal. It was a powerful-enough calibre for motor torpedo boats to fend off motor gun boats. The effective firepower against ground attack aircraft at short distances was certainly vastly inferior to the much higher practical rate of fire of 3 cm Mk 103, though. I'd like to add that the British 2 pounder went back to WW1 and this explains its poor muzzle velocity, which rendered it rather unimpressive against aircraft passing by. The 4 cm Bofors of the late 1920's was the best and earliest really good gun in this calibre and would still (even with 1980's munitions) be a very good AAA in face of the cheap cruise missiles used in the Russo-Ukrainian War.*
The 45...47 mm calibre group was a bust. The Soviets used it on ships, the Czechs had some developed, but these guns had a terrible rate of fire and marginal advantages over 4 cm.
The 50...57 mm group began with a German 50 mm gun meant to close a gap between 10,000 and 15,000 ft altitude at which 37 mm was ineffective and heavy AAA (88 mm) not yet effective-enough because of high angular speeds and rapid change of necessary time fuse settings. This was a sweet spot for level bombing during WW2 and also relevant over the Reich because of the poor service ceiling of the Short Stirling night bomber. Eventually, the 50 mm calibre proved to be too weak and Germany advanced to develop a 55 mm gun, but only completed prototypes. Its combination of calibre and muzzle velocity indicates that it would likely not have covered the 10,000 to 15,000 ft gap. 55 mm was considered to be capable of reliably one-hit kill even a 4engined bomber such as a B-17 or Lancaster.** A properly-developed 55 to 57 mm gun would have proved very useful for protecting railheads, railway stations, bridges and (against night bombers) cities during WW2. Basically, it would have been the perfect defence against light and medium bombers and heavy bombers used in typical medium bomber mission profiles (= level bombing mostly between 10,000 and 15,000 ft).
75...76.2 mm guns were often leftovers from the First World War, especially in France and the UK. Even the best guns in this calibre group lacked the ceiling to defend against American-style B-17/B-24 attacks due to the bombers' turbochargers permitting attack altitudes of about 27,000 ft. The calibre had such a low lethality with a time fused fragmentation shell that all the optical tracking, searchlight, radar rangefinding, computing and automated fuse-setting equipment of a heavy AAA battery would have been wasted on it. That is, unless one uses point detonation fuses and hopes for direct hits (then you need no automated fuse setters). There would still have been the ceiling issue, but the guns would have been deadly against large aircraft (medium bomber and bigger) within their reach at much lower expense than the next group. Everything changed for Western Allies' 76.2 mm AAA at about the end of WW2 when VT (proximity) fusing suddenly turned a 76.2 mm twin into a more effective counter-kamikaze defence than a quad-40 mm mount.
85...90 mm guns were in another sweet spot that gave just enough ceiling (great ceiling if very high barrel wear was accepted) against almost every threat. The munition expenditure was terribly high per kill until late in war when Germany used impact fusing against large bombers*** and Americans created incomparably more expensive and sophisticated 90 mm proximity fuses that made the calibre effective against any target in range provided the fuse was not jammed and no ground features or waves triggered the fuses early. The calibre was used on some warships, but its usefulness against warships was limited to shooting illumination rounds at night. The shell was simply too weak against ships.
100...128 mm calibres were powerful with their fragmentation effect, but very expensive, bulky and heavy. The saving grace for this calibre group was the dual purpose potential; you could equip warships with these guns and plausibly use them both against aircraft and against any unarmoured ships at useful distances. Even 127 and 128 mm calibres were disappointing against unarmoured ships, but still accepted as barely good-enough for a naval DP gun. The Americans successfully used 127 mm guns as DP guns in the secondary artillery (against small ships and up to light cruisers) and tertiary artillery (heavy AAA) role, whereas Germany stuck with a 150 mm secondary and 105 mm tertiary/heavy AAA separation that yielded less guns for either role. The American 127 mm DP guns were unusually quick-loading and the first calibre to receive the new VT (radio proximity) fuses. Naval heavy AAA and DP guns were hardly worth their effort as AAA guns if not equipped with such VT fuses.
The Royal Navy used 133 mm (5.25 inch) guns as DP guns, but they were much less successful than the American 127 mm guns due to more sluggish mounts, smaller maximum elevation and lower rate of fire.
150 mm and bigger calibre heavy AAA and super-heavy AAA existed in marginal quantities with examples in Germany and in Japan. IMO such calibres should be limited to firing shrapnel rounds against incoming skip bombers and torpedo bombers from a super destroyer's or light cruiser's main guns (138...155 mm calibres, not real DP guns). These guns were so big, bulky and expensive that there was simply no realistic case for them being cost-efficient in a land-base AAA role. They were unsuitable for 85...90° maximum elevation naval turret mounting, so their AAA niche was a secondary capability of shooting at incoming aircraft within their elevation range (~30°) that was commonly provided for anti-ship fires. The USN introduced a class of light cruisers which had true 155 mm DP guns, but that proved to be inefficient and unsuccessful.
Surface-to-air missiles: Let's face it. The missile tech, guidance and fusing was simply not mature enough by WW2, but the experiments and the experience with the leap in aircraft speeds possible with turbojets showed that surface-to-air missiles would replace heavy AAA after WW2.
So all in all, I see for the WW2 period three different sweet spots for light AAA calibres (20, 30 and 37...40 mm), each dependent on tactical niches and two sweet spots for heavy AAA calibres (88...90 and mostly 127/128 mm with 100...105 mm being a very AAA-biased DP compromise), separated in land-based heavy AAA and naval heavy AAA/DP guns. Without hindsight, I'd likely have preferred 20 mm, 40 mm and 75 mm plus cruisers using their main batteries to shoot shrapnel at skip bombers and torpedo bombers at pre-set distances.
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Fast forward to today:
Shahed 136-class cruise missiles can fly at 15,000 ft, above effective 40 mm gun ceiling. They are barely in reach of 57 mm guns. A modern 76.2 anti-aircraft gun based on the widely-employed naval gun and its munitions can easily reach Shahed 136 at 15,000 ft, but would struggle to reach and hit Shaheed 238 (more expensive, faster, higher, shorter-ranged) at their 30,000 ft service ceiling. Warships can self-defend against such cheap missiles even with small calibres (20...57 mm), but bigger calibres (76.2 up to 130 mm) are much more useful for protecting escorted ships due to their greater effective range.
We need defences against flying munitions after land-based air defences obsessed about shooting at platforms (except in training) for generations. Something small calibre (up to 2 cm) makes sense for very short lines of sight (around treetop altitude), fairly compact autocannons can wipe clean the sky up to about 10000 ft. We need cheap guided missiles or the (unmanned) fighter approach to deal with higher-than-15,000 ft munitions without breaking the bank (which is what Germany did trying to defend against rugged bombers at 27,000 ft with 8.8, 10.5 and 12.8 cm guns).
A use of 155 mm (gun-)howitzers (which are equivalent to WW2 15 cm heavy anti-air artillery muzzle energy) for shooting down some munitions with special proximity-fused HE shells (normal PROX fuses won't work) should be possible, but this should be a mere secondary task for 155 mm guns.
S O
*: No better gun in the 37...40 mm calibre range was deployed after the Bofors other than its own evolution, the somewhat longer-barrelled 40 mm Bofors of the Cold War. 35 mm guns appeared around 1970 and excelled with much better rates of fire, but 40 mm remained the smallest calibre for proximity fusing until at least the 1970's. I don't know when 35 mm became viable for proximity fusing.
**: In the end, the Soviets brought a successful 57 mm gun into service soon after WW2. This gun has a much higher muzzle velocity than the German 55 mm prototypes. The Soviets quickly gave it radar and computer fire control and it's now again in wartime use in the Russo-Ukrainian War, presumably without the radar and computer fire control. It's mostly used as artillery, but would no doubt prove highly effective against cruise missiles if equipped with functioning automatic fire control and proximity-fused munitions. The West uses a Swedish 57 mm gun in naval context.
***: Germany understood this very, very late in the war. All that fuse-setting equipment was wasted because 88 mm AAA was about triple as deadly against 4-engine bombers with impact fusing, 105 mm about double as deadly and the 128 mm calibre gained very little from it.
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