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I admit I'm incredibly naive on this subject, but what makes it so hard to track an object as large as an aircraft carrier when starting from a known position such as a naval port?
As described above the issue would be continuous observation, not how to follow it assuming you never lose sight of it.
You certainly can't do continuous observation but even just with commercial satellite offerings you can get pretty close.

For example nowadays Planet Labs [1] offers 30-50cm resolution imaging at a rate of one image or 120sec video stream every 90 minutes over a given 500 km^2 region. There is no situation where an aircraft carrier is going to be capable of evading a commercial satellite offering with that frequency and resolution. Once you know approximately where it is or even where it was in the semi-recent past, it's fairly trivial to narrow in and build a track off the location and course.

1. https://www.planet.com/products/satellite-monitoring/

Commercial operations like Planet Labs currently cover most of the Earth multiple times a day.
Clouds occasionally happen
SAR is not blocked by clouds.
What would you track them with? Follow them with helicopters and/or boats?
Break out the pocket book and pay Planet Labs to do it. You could do it with much less frequent visits than this probably the search area for it every 2 hours isn't very large and image recognition systems are pretty good. The big threat is cloud cover.

https://www.planet.com/pulse/12x-rapid-revisit-announcement/

Note that that article is from 2020. Nowadays the frequency is actually down to 90 minutes/1.5hr. The resolution is up as well and they can do massive image capture (~500km^2) and video (120sec stream) from their passes.

Also nowadays they provide multi-spectal capture as well which can mostly see through cloud cover even if it takes a bit more bandwidth and postprocessing.

What if US government bans US-based companies from selling pictures within area where carrier operates?

(of all "national security" reasons these is one of more reasonable ones)

The problem then is the black out zones themselves reveal a lot as well if adversaries can find their bounds. That narrows the search area for their own observation satellites immensely even if it's too large to respond to IRL.
Well in that case congratulations. You've just made it easier. Now you don't even have to track them. You just have to look for the blacked out box, the "error we can't show you this", reused imagery from their long running historical imagery dataset, or improperly fused/healed imagery after alteration.

So now you don't have to do the tracking, just find the hole.

And then you can use a non-US provider to get direct imagery now that you know exactly where to look.

If the restricted area is large, a carrier is regionally disabling for an imagery provider. If it's smaller (and therefore must move over time to follow the carrier group) as soon as the imagery provider starts refusing sales in an area, any customer can test and learn its perimeter with trial purchases, find a coarse center, and learn its course and speed. You don't care about anything else until there's actual hostilities.
It would make tracking impossible, as no other country operates satellites.
...literally yes (to the latter)? Is that not exactly why modern warships have to implement things like measures to reduce their radar cross section? If you could actually just rely on "ocean too big" then there would be no need for that.
It is in part for small crafts (frigates and corvettes) but for pretty much anything larger there's no concealing those ships.

The primary reason however for minimizing radar cross section and increasing radar scatter is to harden protections against radar based weapon systems during a conflict.

Even if the ship is still visible in peacetime operations, once electronic countermeasures/ECM are engaged, it gets an order of magnitude harder for guided missiles to still "see" the ship.

Depending on the kit, once missiles are in the air the ship and all of their friends in their strike group/squadron is going to start jamming radar, popping decoys, and trying to dazzle the missiles effectively enough for RIM-174/SM-6, RIM-66/SM-1, and RIM-67/SM-2s to intercept it without the missiles evading. And should the missile make it to close-in range then it's just praying that the phalanx/CIWS takes care of it.

And if everything fails then all that jamming and dazzling + the reduced radar cross section is going to hopefully result in the missiles being slightly off target/not a complete kill on the vessel.

So they still serve a purpose. Just not for stealth. Instead serving as compounding increases to survival odds in engagement scenarios.

But what you're describing is stealth. "Stealth" doesn't mean "invisible". Humans wearing combat fatigues aren't literally invisible either especially when moving, they're just harder to track/get a visual lock on to aim at.

The point still stands that you cannot rely on "ocean is too big for anyone to find me" because it very much is not.

You don't even need a free account on flightradar24 to track its planes, at least two launch from it and pattern circle around it almost daily.
That relies on transponders which can be switched of if decision is taken to do so.
Sure, and they don't decide to do that in many cases.