You ask a lot of open ended questions that require too much work to answer in detail …
Mostly because I feel that the answers are pretty simple ("I dunno, but it must be quite low") and I know that the moment I put figures on them, the discussion is going to go off along a tangent (dissecting the specific figures).
Nothing on the ‘Jolly Roger’ is illegal. She may have legitimate papers for every piece of equipment. Inspect to your heart’s content. She still has more cargo space than the average Far Trader, she can just protect it a lot better. Perhaps she operates as a Q-ship when there is money to be made at it.
But her construction tells everyone who inspects her what her purpose is, which means that up until she pulls her first illegal act, she'll be scrutinized by the naval authorities. Once she does pull her first act of piracy, she'll be identified in weeks rather than the usual months.
The Patrol Ship responded immediately, how much faster than that do you want them to react?
Well, if they weren't doing anything really important, I think they should have been escorting the departing merchant

.
As a side observation, your interceptors are going to be very busy if you scramble fighters every time a Merchant approaches a little off course.
It's not just a little wobble in the course, the JR is making a bee-line for another ship. Even if she isn't trying to capture the other merchant, she's in extreme violation of the traffic rules.
Surface Fighters are three turns from extreme range and five turns from an intercept – the Patrol was their best bet.
As I tried to suggest, I think the situation is a very low-probability one.
You seem to dislike all of my assumptions about the existence of interstellar traffic, but offer nothing but generalities in response. If one ship per year visits each starport then I agree that Piracy is impractical – as are the starports themselves. So you tell me, how many ships visit the Class A starport of an Industrial World?
I'd be more than happy to tell you, if you'll allow me to drag in evidence from post-CT sources -- specifically
Far Trader. The highest figures there are 100-500 million tons cargo per year and 5-10 million passengers per year. I'm not sure how much that helps, though, since we'd still have to decide how big the ships that carry those are. If we believe that they all travel by small 200T ships, there could be a quarter of a million ships just for the passengers. Which, if we assume they make a round trip every 20 days and are approaching the jump limit just close enough for the
Jolly Rodger's scheme for one hour of that time, means that at any given hour 520 of them are approaching a circumference 8 million km long, which means that the nearest one is only 15,000 km from the JR when it shows up (assuming, of course, that none of them goes three-dimensional on us). But if we think that they travel by Tukera Longliner, the figure goes down by a factor 2.5, and if we think some of them may go by 5,000 or 10,000T ships, we're down by one or two orders of magnitude. So I guess I have to put another vague question to you: How many of the ships that trade with such a world are small enough for the JR to tackle?
However, I realize that I overlooked one aspect of your original post: That the JR was trying its luck at a high-population world. Such a world would have more than enough ships to station pickets at the jump limit close enough to each other for their weapons to overlap. Especially if they're not required to go three-dimensional :devil:
The Jolly Roger specifically preys on high traffic worlds with patrols. Since I just came from the next world, I would imagine that the odds of a departing ship approaching in the direction of my arrival point is about 50% (the other 50% heads to the next world in the other direction).
Most worlds would have more than two neighbors within reach. On the average, three within jump-1, six more within jump-2, nine more within jump-3, 12 more within jump-4, 15 more within jump-5, and 18 more within jump-6. There are 46 worlds within jump-6 of Regina, for instance. Not all of them will have ships jumping to them, of course -- few will have jump-5 traffic and very few will have jump-6 traffic.
But if we just look at the ships that are going to the world you're coming from, very few will be coming straight towards you. Or rather, a lot of the time you will not be coming straight towards them. That's because your arrival time is subject to jump variation. If your jump takes 168 hours, you will indeed be coming straight towards them. But if it takes more or less time than that, you'll arrive at a point on the jump limit off to the side.
True, the most likely time for a jump to take is 168 hours, so the first picket a system defense CO would place would be on the direct line between the world and the neighbor with the most traffic. But most of the jumps will not take 168 hours.
In this example, the ‘Jolly Roger’ was forced to flee with a Free Trader or Subsidized Merchant and was not able to deliver it’s cargo. After wilderness refueling and multiple jumps to the Sword Worlds to sell the captured ship and the cargo, the JR needs new papers and a new transponder before heading to a new sector to seek and deliver ‘protected cargo’ (in up to 100 dTon lots and at a loss) until another opportunity presents itself.
But now the word is out that a ship like the JR is a pirate. Since earlier intense examination of its papers didn't find anything wrong, it has probably been traced back to the shipyard where it was built, which furnishes the warning notice with all sorts of forensic evidence that'll help identify it. Even if it can't be traced back, all those suspicious customs officials and navy patrols have taken reams of readings, measurements, and pictures of it. And any customs officer or navy captain lucky enough to identify her will be richer to the tune of millions of credits, so don't you think that all ships that correspond to the JRs specifics will be examined carefully for years to come? And not just inside the Imperium. Why should a Sword Worlder captain eschew a multi-million credit windfall?
Hans