Yeah, in addition to the fact that they seem to be starving, I question the 10 year lifespan for two reasons:
1 - there aren't any proper studies I can find on their actual lifespan in the wild (so the 10 year thing seems to be a guess).
2 - while they grow fast as juveniles/larvae, they seem to only grow ~2.4 mm per year as adults; given that they become adults (i.e. sexually mature) at ~50 mm at ~2 years old, that would seem to indicate to me that they would live closer to 23 years in the wild.* [Edit: I should say here, that 23 years would be until the stars grow to the average adult size of ~96.6 mm; the largest star found in the study was 133.0 mm, so they could potentially live for ~38 years at the max size I've seen listed.]
To add to this, even if they did only live a decade, keeping one for just 1 year - very few people are able to keep them for more than ~13 months - (and having it slowly starve to death that whole time) when it comes in at 5-7 years old and has 3-5 years left in the wild seems like something I personally wouldn't call successful.
Regardless, as mentioned, the ones that survive shipping (which a lot die before they ever reach our tanks) are slowly starving to death in our tanks and are definitely not thriving - if they were thriving, we'd see spawning behavior and many stars living more than a year - so I can't personally view it as humane to keep them at the moment.
Personally, I currently recommend against trying to keep a tropical, true starfish (Asteroid) species, particularly biofilm-feeding species, except for Aquilonastra stars.
That said, if you're really determined to try keeping them anyway, I'd
strongly suggest setting up an Aquilonastra farm, a cryptic refugium to farm sponges, and possibly a colonial tunicate farm as well (all of these should be separate from the Linckia's tank so they can't get to the farm and ruin the "crop") - the Linckias feed on these (they presumably emulate biofilms to some degree), and the most successful Linckia keeping I've seen have
always involved at least one of these aspects (typically the Aquilonastra stars).
With Aquilonastra stars alone, I've seen people keep them 2-3 years; with the cryptic refugium sponges involved too, the longest I've heard is ~4 years. Still not good by my calculations, but arguably acceptable if they really only live a decade.
For those who are truly determined to keep true starfish against my recommendation, here are some signs to watch for to help track starfish health:
Some ways to potentially gauge the health of the starfish that may be useful:
1 - Visual: is anything visually wrong with the specimen? (From what I know of starfish, these cues are generally pretty obvious if they're there at all - if the starfish is disintegrating, there's something wrong).
2 - Weight: is the specimen putting on or maintaining weight (generally healthy), or is it losing weight (under most circumstances, this would be unhealthy)?
3 - Size/Length: if it's not already full size, is the specimen growing? Is it shrinking (either from weight loss, disease, limb loss, etc.)?
4 - Reproduction: is the specimen engaging in reproductive behaviors/activities (i.e. courting, nesting, spawning, etc.)? (I recognize this one is not as common of an indicator in captive starfish at this point, but there are a few instances of starfish attempting to spawn in captivity).
5 - Longevity: is the specimen relatively close to meeting, actually meeting, or exceeding their expected wild lifespan (or at least surviving for a few years - healthy), or is the specimen dead/dying prior to doing so (unhealthy)?
6 - Speed: how quickly can the specimen right itself when it's flipped over?
7 - Level of activity: how much is the specimen moving around? (Generally speaking, low activity indicates poor health; moderate activity indicates good health; and high activity indicates good health but probable stress - it could be searching for food, oxygen, etc., but it's probably not in terrible health when moving a lot).
8 - Grip strength: does the specimen have a strong hold/grip with its tube feet? (A strong grip indicates good health, a weak grip indicates poor health).
9 - Willingness to eat: is the specimen eating? (Seems obvious, but can be really hard to tell with some stars - some stars leave little trails called feeding scars through the things they're eating; sometimes you can see their everted stomachs; other times, like if they're feeding on biofilm, you may not be able to tell at all - them crawling onto visible food is a good sign they're willing to eat, but - as mentioned above - the food may or may not meet their nutritional needs).
And a final note on starfish keeping:
One important thing to keep in mind with foods for these guys - just because they eat it, doesn't mean it's meeting their nutritional needs; it might be, but it might not be. With how long it seems to take these guys to starve, unless you're monitoring their health closely, you might not notice if it's not meeting their needs until it's too late.
*Source: