At what time would you look to see a first quarter Moon at its highest position in the sky

Additional Reading at www.astronomynotes.com:

• Astronomy without a telescope

The changing Moon is a familiar sight to all of us, since it is the second brightest object in the sky after the Sun. The images of the Moon below were created by taking images of the Moon over approximately one month. 

One cycle of Moon phases as seen from Earth beginning with the New phase, followed by the First Quarter, Full, and Third Quarter phases.

Looking more closely at these images, you can notice a few characteristics in the appearance of the Moon:

• It always keeps the same face pointed towards the Earth.
• It goes from completely dark to completely illuminated and back again.

Let’s start with the phases. First, we need to again talk about the layout of the Sun, Earth, and Moon system. We already have learned that the Earth is orbiting the Sun, and it is rotating on its tilted axis as it orbits. At the same time, the Moon is orbiting the Earth, and as it orbits around the Earth, it is rotating, too. Just like the Earth, the half of the Moon that is pointed towards the Sun is illuminated, and the half of the Moon that points away from the Sun is dark. So, the simple explanation for the phases of the Moon is that, at any one time, half of the Moon is light and half is dark, and the appearance of the half of the Moon that we see changes as it orbits the Earth. For a listing of the individual phases and a description of their appearance, see "Phases of the Moon and Percent of the Moon Illuminated."

For this lesson, we begin by referring to the following animation.  Note that the animation is 6 minutes, 4 seconds long, and that there is much to interpret.

Connecting the Moon's changing appearance for an Earth observer with the relative motions between the Earth, Sun, and Moon

Once the movie begins, you should see that the Earth begins to rotate and the Moon begins to orbit the Earth. The Moon is also rotating, but that is less obvious (we will discuss this more later). Please keep this animation in mind when we discuss the phases—remember that all of these phenomena happen simultaneously (Earth's rotation and orbit, Moon's rotation and orbit).

Next visit Lunar Phase Simulator developed by the University of Nebraska-Lincoln.  This is an image that requires some interpretation, but if you can figure it out, you will have mastered an understanding of the phases of the Moon. The Sun is far off to the left on the diagram, so sunlight is illuminating the left side of both the Earth and the Moon. The point on the Earth immediately underneath the Sun experiences noon, and if you imagine the stationary Earth in the diagram rotating counterclockwise, that point will experience sunset when the Earth has rotated 1/4 of the way around, then midnight, then sunrise, then back to noon. Those times are labeled. In the same diagram, the Moon is shown in eight different locations along its orbit around the Earth. For example, the full Moon occurs when the Sun, Earth, and Moon are in a line in that order (labeled as position 4). The sunlight that is hitting the Moon illuminates the half of the Moon that points towards the Earth. So, when the Moon is full, its entire illuminated half is pointed directly at the Earth. The next question that you can answer from this diagram is: What time is the full Moon visible on Earth? To answer this question, we can integrate our knowledge about the rotation of the Earth. Picture yourself on the daylight part of the Earth pointing directly at the Sun. Remember, it is noon for you when you are on the part of the Earth pointing directly at the Sun, which is when the Sun is transiting your meridian. Six hours later, when the Earth has rotated one quarter of the way around, you would have to look to the western horizon to see the Sun, and to the eastern horizon to see the Full Moon. So, at sunset (about 6:00 PM), the full Moon is rising. Six hours later, the Earth has rotated an additional one quarter of the way around. Now, the Moon is directly in front of you (that is, this time it is transiting your meridian). So, the full Moon transits at midnight. Six hours later, at about 6:00 AM, the Moon will now be on your western horizon (setting), and the Sun will be on the eastern horizon (rising). The New Moon phase occurs when the Sun, Moon, and Earth are in a line in that order (unlabeled, but it is position 8). In this case, the unilluminated side of the Moon faces the Earth. Thus, during New Moon, we do not see the Moon in our sky at all. Using the logic from the paragraph above, even though we can't directly see the New Moon, we know that the New Moon transits at 12 PM (noon), sets at sunset (about 6 PM), and rises at sunrise (6 AM).

The other phases fall in between these two extremes. For example, at First Quarter, the side of the Moon facing the Earth is half illuminated and half dark. It will rise at noon, transit at 6 PM, and set at midnight.