It's never been in doubt.
It's also not relevant, because six times the radiative cooling rate is going to be dozens, if not hundreds, of times below that you could get with liquid cooling.
That's not true at all, if liquid-cooled autocannons were that efficient they could maintain much higher rates of fire than air-cooled gatling autocannons, which I've already demonstrated is not the case. Liquid cooling is not magical, it still has to obey thermodynamic laws and running liquid next to a hot barrel is not "orders of magnitude" better than a gatling design.
Irrelevant.
The energy needed to heat the barrels several hundred kelvin in a vacuum is not great.
You can't just ignore a nearly 300 degrees Kelvin temperature differential compared to what you experience under Earth-normal conditions. That is a massive difference.
If that single barrel is one sixth the mass (extremely unlikely), the same heat energy would come close to making it six times the absolute temp.
Except that is not how gatling barrels are designed. They are generally very similar in size and weight to the barrels of single-barreled autocannons, you can easily see this by examining a gatling design.
However, even if you can get six times the shots before the barrel becomes unusably hot, that's still not going to be many shots, in the absence of active cooling.
Of course that is going to be a massive difference, that is why you have ROF of upwards of 6000 rpm for six-barreled gatling weapons vs. only around 1000 rpm for a liquid-cooled single-barrel weapon.
You're still not getting the concept. A gatling weapon primarily maintains a high ROF due to spreading the thermal load among multiple barrels, not by conductive cooling. This is especially true for high ROF burst firing where you are firing literally 100 rounds every second for a 6000 rpm gatling weapon.
On a scale of 1 to ten, if a good water cooling system is a 10 and forced air convection is an 8, radiative cooling is going to be a 1.
Citation needed, especially for a coolant loop that would need to operate in space from near absolute zero. You are literally making up numbers that don't have any connection to reality. Your "rating scale" doesn't even have units. Sorry, you need to do better than that.
There is an enormous gulf in cooling rates between spinning a set of barrels through air and through a vacuum.
Not at the high burst ROF of a modern gatling autocannon, the more significant factor will be the number of rounds going through the barrel per second and the only way to reduce that is reduce your ROF (i.e., reduce the rotation speed) or to use a design that has more barrels. Take a look at the picture I posted after 3000 rounds were fired through that weapon, that would have occurred in 30 seconds at a full ROF of 6000 rpm. Are you seriously suggesting that during only 30 seconds of fire in a vacuum vs. Earth-normal atmospheric conditions that there would have been a dramatic difference during that time?
I am not overstating the effectiveness of water cooling. Water has the highest volumetric heat capacity of any material in existence and nothing without a phase change component (which may very well be boiling the water) can compete with it's ability to remove heat from an object.
You are very much overstating the effectiveness of a water-cooled autocannon because you don't understand that the barrel is not fully immersed in a large quantity of water, there is simply a small coolant loop along the outside of the barrel that helps reduce the heat load. The effectiveness of the coolant loop is still limited by conductive heat transfer through the metal of the barrel to the coolant itself and then by the quantity of liquid coolant available. You also don't seem to understand that several designs for water-cooled machineguns and autocannons have existed historically and have almost all been entirely replaced by gatling weapons, with some historical examples such as the 2A7 having water cooling added specifically due to problems with ammunition cook-off in that particular design. Modern air-defence and CIWS systems are all gatling weapons.
You are greatly understating the effectiveness of air cooling and/or exaggerating the effectiveness of radiative cooling.
You are not understanding that a liquid-cooled single-barrel autocannon is still nowhere near competitive with a six-barreled gatling weapon.
Not their only advantage.
Their only meaningful advantage compared to gatling weapons.
Which is a small weight penalty compared to the overall weight of the weapon and aircraft.
Again, quite marginal compared to an aircraft's overall power output, and not an issue with a recoil-operated dual barrel autocannon (GSh-23) or a gas-operated gatling design (GSh-6-23).
are significant advantages vs. rotary cannon.
No, they are marginal advantages. That is why rotary cannons are quite uncommon compared to the widespread use of gatling designs.
Higher rates of fire and resistance to jamming/increased reliability are advantages vs. gas/recoil powered single-barrel guns.
The sustained ROF for a rotary cannon is not competitive with that of gatling weapons, it's simply the shorter spin-up time during a 0.5-1 second engagement window where a rotary cannon offers any significant combat advantage.
It will become more relevant as temperatures get higher, but unless you have a weapon that can withstand phenomenal temps, it's never going to be more than a small fraction of the cooling of a weapon that isn't used in a vacuum.
The overwhelming majority of heat leaving the minigun in your image isn't being radiated away, it's being conducted into the air that comes into physical contact with it.
As I mentioned above, you are arguing that a liquid-cooled autocannon is going to outperform the gatling weapon. Liquid-cooled weapons are simply not as good as you claim.
Which wouldn't be the case if the heat had no where to go.
The heat very much has somewhere to go, it is being radiated into space.
If you can come up with a figure for the amount of heat per shot and the material and mass of the barrel, it should be easy to show how a a good water loop could keep a fairly high sustained RoF weapon within a few 10s of C of water temp and the water within a few C of it's freezing point.
Really? Then show me your calculations. The information you claim to require is easily available if you want to back up your claim.
I already know that liquid-cooled single-barreled autocannons are nowhere near as efficient as you claim they are, otherwise they would predominate over gatling weapons, and they clearly don't. If you have "math" that you think can "prove" over 100 years of military history is somehow "wrong", please show your work or stop making baseless claims. I am pretty certain here that the history of warfare during the entire 20th century and the laws of thermodynamics have already answered this question for you by actually designing coolant loops for autocannon barrels and have found they were less efficient than a gatling design.
You'd also be able to see how a radiatively cooled weapon would rapidly rise to far higher temperatures, no matter how many barrels you gave it (within reason), making it only useful for short bursts, with protracted cooldown periods in-between.
Again, you need to look at real-world firearm designs here which you are trying very hard to ignore despite having been given several examples in my posts.
Anyway, you wouldn't want to fire a cannon made out of anything I know of at near absolute zero anyway. This is another advantage of having it attached to a loop...it will never be outside a narrow temperature range, meaning it won't be brittle from cold, suffer excessive wear from heat, have to deal with wild swings of expansion/contraction, nor have any lubricants operate outside their optimal viscosity ranges.
How exactly do you think your liquid coolant loop is going to work at those temperatures if it's "never outside a narrow temperature range"? That just makes no sense, it has to deal with the same extremes of temperature that the barrels do, otherwise it's going to be counter-productively heating a cold barrel when you pass warm coolant past it.