Last quarter is a funny phase. It has the same shape as an evening half-moon but flipped. Instead of the right half lit by sunlight, the left half is. A first quarter moon has completed a quarter of its orbit around the Earth, while a last quarter moon has completed 3/4 of the trip. For that reason it's also known as the third quarter moon.

Most skywatchers are active during the evening hours, so we routinely catch sight of the crescent, half-moon, waxing gibbous (the egg-shaped moon between half and full) and full moon. If you're a night owl you'll also see the waning gibbous moon just after full. Schoolkids and early-rising workers are familiar with the morning crescent in the brightening dawn sky, especially from mid-fall to early spring when the sun rises late.
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The last quarter moon inhabits the wee hours, rising around midnight standard time (1 a.m. daylight time). If my neighborhood is any gauge, nearly everyone's in bed by then. When I'm out with the telescope observing comets and other deep-sky wonders I'll often lose track of time. Not when the moon's at last quarter. At its rising my parental subconscious tells me it's late and time for bed. If I don't pack up soon, I'll pay for my pleasure.
That said, it's also true that the last quarter moon is visible at dawn just like the crescent, standing due south at sunrise. OK, so maybe I'm exaggerating at little at how difficult it is to see moon's last half. But I can vouch that nighttime sightings are uncommon unless you're a fox, an owl or an amateur astronomer.

Before full phase the moon rises before sunset, so it's automatically visible as soon as the sky gets dark. At full, the moon lies directly opposite the sun and rises at sunset. After full, the moon rises after sunset. Depending on the angle the moon's path makes to the horizon the delay between successive moonrises can be as short as 20 minutes and as long as 90. Forty minutes to an hour is typical.
Two days after full, the moon rises about 1.5 hours after sunset. When it reaches third quarter on July 31st — seven days past full — it comes up around 12:30 a.m. local daylight time. The moon rises later because it's orbiting the Earth. Each night it moves about one outstretched fist to the left or east.
One night after full, the moon has traveled about a fist farther east. To see it rise again, the Earth has to rotate an additional 45 minutes. Two days after full, the Earth must rotate 90 minutes (2 x 45) to return the moon to view. By the time of last quarter, it has to spin for more than 4 1/2 hours for the moon to rise! No wonder that phase gets so little attention.
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While first quarter and last quarter phases are both halves, one is waxing or increasing in phase towards full, while the other is waning toward crescent. The line of advancing sunrise on the waxing moon is called the terminator. Post-full moon, the terminator becomes the line of advancing sunset. It's essentially the boundary between day and night on the moon.
Craters and mountain peaks along the terminator cast dramatic shadows because the sun is either just rising (first quarter) or setting (last quarter) here. The contrast between shadow and low-angled sunlight highlights every tiny bump and irregularity, making the day-night boundary the best place to observe craters and other lunar features.
This is especially true at the quarter phases, when the terminator cuts a straight line across the moon's face, highlighting craters directly in our line of sight. There's no foreshortening. A round crater looks round, not squeezed into a narrow oval the way we see them at other phases.

The moon is a sphere. When we face it squarely, everything is fully open to view. But when we look off towards its edge, features "crunch" together and narrow, hiding details. The gibbous or crescent moon terminator will still expose dramatically-lit craters (above), but they're all smooshed together. That's why first and last quarter phases are the best for viewing the moon in a telescope or pair of binoculars.
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Tonight, the moon — one day before last quarter — will rise right around midnight. I hope you'll get the opportunity to become better acquainted with this reclusive, nocturnal light.
"Astro" Bob King is a freelance writer for the Duluth News Tribune. Read more of his work at duluthnewstribune.com/astrobob .