> almost all of these Mars robots seem to do useful science well beyond the end of the stated mission

How are we defining "useful science" here? They all do some kind of scientific investigations, but it's hard to say if it's useful or not. If you look at coverage or discussions about the lander (including here), people get very excited about the landings, and somewhat excited about the pictures. The science almost never gets discussed. Even when NASA tries to hype it up, it seems the most important stuff they have is continually telling us they found Mars has some water and used to be wet, and that there's the possibility there once was life on Mars.

People seem enamored with the idea that we're sending things to Mars and doing "something." But there doesn't seem to be a lot of concern about what that "something" actually is.

I'd go as far as saying "useful science" doesn't have much meaning. The misunderstanding comes from the conflation of "science" and "engineering".

The reason engineering exists is to be useful: to create artifacts and technology to make our lives better, easier, etc.

But science only has three uses: to inform engineering; to satisfy our human curiosity; and to beget more science. Any "usefulness" here is of a very narrow scope. I don't mean to say that it's not important, but that's not the same as being useful.

And even whether or not it's important is hard to say. Whether the experiments being done were important or completely unimportant, we'd get the same thing - hyped up results from NASA talking about how these things were a big deal, vague comments from lander fans about how the landers are doing important scientific work. You're not going to get NASA to come out and say, "actually, we can't think of a good reason to send anything to Mars anymore. Maybe in a couple of decades when our technology improves." Similarly, you're not going to get them to say that the shuttle or SLS were misfires. If you look at the stuff NASA releases, these were all enormous successes.

If I try to look at what the Insight lander accomplished, for instance, I find articles claiming that Insight found that the core of the Mars was much larger than previously thought. Let's leave aside whether the exact size of Mars' core is important or not. If I look for estimates of the size of the core prior to Insight, they're all in line with the estimates that came from the Insight data. From what I can see Insight brought the range of estimates closer together, but it seems to be about what people were expecting beforehand.

>and to beget more science. <snip> but that's not the same as being useful.

For those looking to keep a job in sciences, begetting more science sounds pretty darn useful

It is a bit like asking how exactly does the XGB matrix build in stable diffusion create images. We know the end result that is produced, but we can't pick a single number and talk about what it contributes to the overall algorithm. I would even guess that you could replace any single number with another random value within the appropriate range and you wouldn't be able to tell which AI was before and after. The links each item serves is too complex to work out.

In the same way, any single 'act of science' can be hard to measure. They enable some experiments, which allow us to test some hypothesis. In some cases it is simple enough to point out when it is testing hypothesis that end up benefitting humanity, but that's only part of the effect. Many times a hypothesis is either rejected or fails to be rejected and slightly influences a larger theory which in turn is used to create new hypothesis. This creates a circular reference until some time later we end up with a theory that is useful for improving humanity. But which hypothesis were actually pivotal to forming the theory? Which acts of science allowed testing a hypothesis and receiving a clear enough failure that future effort was reallocated to other, eventually more fruitful, areas? Other than trivial cases, the cause and effect becomes such a tangled web we can't be sure.

You even have the publicity side. Where sometimes the effect isn't directly contributing to the theory, but engaging more attention to a field which leads to more children choosing the route of becoming a scientist.

Math has a similar issue. Much of pure mathematics has no known or foreseeable application. Yet, we can look back at what math does have very useful applications and see how many times the math predated the application being found, sometimes by long enough gaps that those who discovered the tools never saw them to be applied.

Someone is making these value judgements. There's a reason why certain projects get funded while others don't.

If someone has no idea what research will yield results, and has no preference for what research gets funded, that's fine. But if all research is the same to them, they should probably let those who view certain research as more important than others to make the decisions about what gets funded.

There are people who are choosing, and there are reasons behind which might end up being better than others, but those are often half science and half politics. Sometimes the end up being done for less than justifiable reasons. One factor is looking at scientists that seem to have 'scienced' well in the past and favoring them over new scientists. This is more conservative in that less money is sent to random ideas that won't go anywhere, but also means that funding is trapped in more classical theories lead to incorrect models and theories sticking around for longer than they should.

Every part needs to last 3 months. If even one part breaks after 2 months the whole thing failed. Each part increases the odds that one will break after 2 months. The only way to work around this is to design each part to last for 5 years, that way the odds are high that all parts survive for the required 3 months. It also means that odds are reasonable the rover will work for much longer than 3 months, and in many cases you can operate in a degraded state for a even longer.