Why do modern fighter jets use 20mm guns? Is the armor of a modern jet that thick? Does, for example, .50 (12.7mm) ammunition lack velocity to be effective?
I'm just curious on why is that 20mm is so widely used in air-to-air guns.
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20mm is the smallest caliber that is practical to load with a bursting charge. Some fighter jets use 23mm, 27mm, or 30mm guns.
The reason 20mm has become one of the most common aircraft gun calibers is that it produces the lightest gun that meets the minimum requirements to be useful in air combat: enough rate of fire to hit a maneuvering jet, good enough ballistics for meaningful range, enough bursting charge capacity for meaningful damage.
The majority of modern combat jets do not carry armor, but they are built with considerable redundancy and internal subdivision. A bursting charge (explosive payload) is needed to do meaningful damage to a modern combat aircraft.
Smaller calibers can cause some damage that will require repairs, but not enough to rely on them to bring down a jet in combat. With missiles prioritized, most air forces want the lightest gun that's still effective.
Larger calibers have a cost in design. A large caliber rapid-firing gun will be much heavier, its propellant gases (muzzle exhaust) can cause flameouts in the engine, and firing will throw the aircraft's aim off center. A slower-firing one can be suitably light, but still has the same recoil force. Another issue with large calibers is collateral damage - 30mm+ rounds send fragments a couple hundred meters, far enough to damage friendly aircraft in air combat maneuvering.
The 20mm rounds used by combat aircraft have limited armor-piercing capability anyway (comparable to .50 BMG). Armored aircraft are rare, and some exceptions include CAS aircraft such as the A10 and to a lesser extent the Su-25 and the Su-34. This armor is mostly designed to protect them against ground fire, not enemy fighters.
If you want to know why you put a caliber usually between 20 and 30mm on a fighter plane (if you do any at all) you have to consider a few things.
With that in mind think about what a gun can achieve:
Typical weight for a gun system of calibres 30mm and higher is in the range of several tons. The GAU-8/A Avenger (the A-10 gun) has a (dry) weight of 1800kg. That is without ammunition, which also adds considerable weight. Thus it limits the payload for other ordnance available. Anything larger is fairly impractical due to weight considerations.
You are fighting at high velocities and at long ranges. You need a gun that has a high range of fire and a high projectile velocity. Thus anything below a .50 caliber gun has troubles considering the ranges involved (projectiles slow down considerably if fired at longer ranges and need to be fired in a higher arc).
Even with a high rate of fire (say 6000 rounds per minute) you end up with a space of 10 meters in between every round at the respective muzzle velocities (~ 1100 m/s). Your target is also moving at a considerable speed, thus even if you fire a long burst at a passing plane, it is unlikely to hit with more than a handful of rounds - if you hit at all.
Thus you want to maximize the effect you have, which means you need a projectile with a bursting charge. Otherwise it might just pass through the enemy plane without dealing real damage. Control surfaces, tanks etc. are all not so much a problem with just a small hole in it. This means that you need a bursting charge - which needs a larger projectile, thus anything below 20mm is almost impractical, because the charge would be too small.
This leaves us with guns in (usually) the 20mm - 30mm range.
Almost all fighter plane and fighter bomber guns are in that range. A few examples:
One should remember also "ballistic drop" because that greatly complicates shooting a moving target at distance. And fighter jets are already moving fast. Narrowing the "3D-ness" of where the round should end up makes it easier to be far more accurate, one imagines, and gives more time for other considerations a pilot may have...
Just going on the assumption a round is the same length, whether 20mm or 30mm, which does not seem to be so, resulting in ONLY the increased diameter of perhaps similarly weighted materials to be important, which does not seem to be so (but both "not so's" seem to be conservative), a 30mm round would weigh 2.25 times what a 20mm round weighs. That would mean, if the energies at firing were able to be equal, unless one chose to make them different, that the "velocity squared" term would be 2.25 times greater, or in other words, that velocity would be 1.5 times greater. Then, ignoring that a smaller round ought to confront less wind resistance and therefore maintain its velocity longer (in time) than a bigger round, the 20mm round will be faster at all points along its flight path so it will reach whatever target in 2/3 the time the bigger one will.
Remembering Galileo, both dropping due to gravity at the same rate, the shorter flight time will result in less drop, accentuated by the fact that drop accelerates as time increases. In fact, a small (small) amount of research indicates it might be about 4/7, call it 55% of the larger round's drop.
Given the conservatism and ignoring how independent choices might make some of it moot anyway, I'm comfortable moving on with the idea of the drop being 1/2. So your round might fall, say, 200 feet below the other aircraft instead of 400 feet. Both can be planned for and a gun can be mounted to make it look like a laser shot for a pilot (at a single, chosen range), but it seems targeting a target might be a LOT easier to be accurate about when allowing for the much smaller drop. Also, the greater the drop, the higher above the target one must aim and therefore the longer the parabolic path the round must take increasing the loss of velocity vs. the smaller round, and on and on...
Easier to at any given accuracy means quicker, less time spent on its mechanics, and so, the extra second here and there. Seems like those seconds could matter a lot on their own.
Penetrating power would strongly depend upon energy content at impact for any given material/geometry. The round that takes about 2/3 the time to "get there" seems, by the same small (small) amount of research, to have 4/3 the energy at impact making it likelier that it would penetrate so that the bursting would be internal where it would have far greater effect: more ripping up flesh, electronics, flesh, fuel, flesh, hydraulics, flesh, radar profile, flesh... and less just messing up the paint.
Using the quick example in another Answer, I get a 36 foot separation between rounds. It seems to me that tracking the enemy so as to impact it with more than one round would be easier having a radically smaller drop to factor in. But I will say that it seems unlikely the designers planned for more than one round to hit as surely 2-3 guns firing at the same instant, a foot or two apart perhaps would be wonderful if they thought mor ethan one round was incredibly worthwhile and that just is not how the fighters are built so... Still, missing a few and tracking in on the enemy would seem easier as well with the smaller drop envelope.
Finally, the shorter time from firing to impact would have some smaller effects, like leaving less time for the enemy to maneuver out of the shot ("dodge")(realistically, more time for whatever instant thing he did to work, if it ever could... fewer of those extra seconds I mention above), like being able to take a meaningful shot on less time in frame (so taking a shot vs. "D*mn, too late to even take a shot"), like being able to fire in aid of a team member rather than worrying it might hit him, and other things that are even less quantifiable.