An amateur astronomy club received a small grant to image a list of deep-sky targets over one season. Each target can be imaged only on a night that satisfies several constraints at once, and the club must decide which targets are imageable from its site this season. For a target to be imageable on a given night, all of the following must hold:
Altitude: the target must rise at least 30 degrees above the horizon (low targets are blurred by atmosphere). Moon: the night must be a dark night (Moon illumination at most 20 percent) for faint targets; bright targets (magnitude brighter than 6.0) tolerate any Moon. Window: the target's nightly observable window must be at least 90 minutes long to collect enough exposure.
The club images at most one target per usable night and has 24 usable nights this season.
Dark nights available counts how many of the season's usable nights are dark (Moon at most 20 percent) and the target is above 30 degrees during its window. For bright targets (magnitude below 6.0), this column counts all usable nights the target is above 30 degrees, dark or not.
| target | peak altitude (degrees) | magnitude | observable window (minutes) | dark nights available |
|---|---|---|---|---|
| Nebula-A | 55 degrees | magnitude 7.4 | 120 minutes | 9 dark nights |
| Cluster-B | 40 degrees | magnitude 5.2 | 75 minutes | 14 dark nights |
| Galaxy-C | 28 degrees | magnitude 8.1 | 110 minutes | 0 dark nights |
| Nebula-D | 62 degrees | magnitude 9.0 | 95 minutes | 6 dark nights |
| Cluster-E | 48 degrees | magnitude 4.8 | 130 minutes | 16 dark nights |
| Galaxy-F | 35 degrees | magnitude 8.8 | 140 minutes | 5 dark nights |
The grant rewards completed targets, and a faint target needs at least 3 separate dark, high-altitude nights to stack enough exposure; a bright target needs only 1 qualifying night. So when we decide whether a target is completable this season, the test is not just is it imageable on one night but do we have enough qualifying nights for it.
Remember the magnitude split: faint means magnitude 6.0 or dimmer (a larger magnitude number is fainter). Bright targets ignore the Moon constraint entirely. And the altitude and window gates apply to every target regardless of brightness.
The club wants to complete as many targets as possible this season. A target is completed only if it is imaged on enough qualifying nights, at least 3 dark, high-altitude nights for a faint target, or 1 qualifying night for a bright target, and the club has 24 usable nights, imaging at most one target per night. Using the sources, what is the largest number of targets the club can complete this season?
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