The VLS cells can hold ESSM...Evolved Sea Sparrow Missiles. These have a range of about 7 miles, a bit longer than SeaRAM. They are also meant as Point Defense for the ship, but--like SeaRAM--can defend other ships inside this radius from the firing ship. Each VLS cell can hold 4 ESSM.
In the same attack, HMS Diamond used a Sea Viper (Aster) SAM. Also a costly missile. In the case of the RN destroyer, it is specified in the sources I read that she was defending a nearby merchant ship.
AA specialist ships like the USS Carney and HMS Diamond are designed to provide Area Air Defense to groups of ships. That is done with these sorts of large missiles. They have ranges over 100 miles, thus they can defend a group of ships under their "umbrella" for a large patch of ocean around them. It takes these kinds of missiles to provide this ability.
For the Carney (I am not up to speed on the Diamond) she has a layered defense ability. Long Range Area Defense is provided by the Standard Missile family of missiles, with a version of the SM-2 being most common, but now SM-6 is also "coming on the scene" more.
The next layer comes from her SeaRAM missiles. These are Point Defense weapons, having a range of only about 5 miles, and can only defend other ships if they are within that proximity to the firing vessel. Otherwise, they are meant to defend Carney from attacks directed specifically at her, and which got past her outer Standard Missile defense. Normally, Burkes do not carry SeaRAM, but Carney is one of a group of destroyers assigned to ballistic missile defense work. Her radars are then frequently pointed up into space, and she is then vulnerable to an undetected attack launched from a surface source. SeaRAM is self contained, and has its own surface scanning radar, so it can scan, detect, and fire independently from the ship's AEGIS system. That is why all the ballistic missile assigned destroyers got backfitted with a SeaRAM (and you can tell which ships are assigned to the duty by which have this weapon.)
The last two layers are gun systems, both the Phalanx 20 MM gun, and the Mk 38 MOD 2 25 MM gun. These have a range of a few miles, and are "last ditch," engaging whatever has gotten really close.
All these have been mentioned below already.
Nothing I have read about Carney specifies whether she was performing Area Defense or not. I have read that it is not clear whether she was the target. As earlier noted, HMS Diamond was not defending herself, and was performing Area Defense. It is likely this drone swarm was rather spread over a large area, headed in a general direction towards Carney, but also other nearby ships which Carney was put there to defend. And it is almost certain she detected them well over 5 miles away, which is SeaRAM range, and wanted them destroyed as far away as possible. That is precisely what the Standard Missile is for, expensive as it is.
There are labs (like DARPA) working on cheaper classified alternatives precisely because of how expensive it has become. Until they come out of the lab into operation, you're looking at this expensive scenario.
Yes, they can certainly be overwhelmed. Normally they operate in groups, so have a collective capability, which helps, but an enemy with enough resources can still overwhelm them.
Read each drone was shot down with a SAM. Also that these drones are throwaway one way type programmed sorties.
If both of the above is correct, does the USS Carney and similar vessels have other means to shoot down drones other than a SAM? Also read it is $2million per SAM and storage is limited. Wondering if they can be overwhelmed by a mass attack.
Some merchant ships were struck by drone armaments so assuming they are low warhead type of munitions.
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