5) In this activity, you found some “angular” sizes of celestial objects. a) You drew a scale model of the sun, earth, and moon with the earth represented by a circle 1.28 cm in diameter, and found the angular size of the moon was about 0.5°. If you doubled the distance to the moon, how would this angular measurement change?
Finding an exact answer to this would require trigonometry, which we do n’t require for this class. It is sufficient to give an approximation, here. It is obvious that the further away the object is, the smaller its angular size (that is, the amount of space it takes up in the sky). Approximately speaking, doubling the distance would cut the angular measurement of a faraway object in half.
b) If you cut this measurement in half, how would the angular measurement change?
Similarly, moving an object closer will increase its angular size. Again, approximately speaking, cutting the distance in half (without changing the moon’s size) would double the angular measurement.
c) In your diagram, the moon was 38 cm from the earth, and the sun would be 150 m away (about 1½ football fields away). The furthest true planet, Neptune, is
4,515,000,000 km from the sun. How far would this distance be in your model?
Our scale is 12800 km : 1.28 cm, or 10000 km : 1.00 cm; so we just chop 4 zeroes from the number of kilometers to get the number of centimeters.
Thus, we get 4,515,000,000 km : 451500 cm. 451500 cm = 4,515 m. (There are about 1600 m in a mile, so this is close to 3 miles away!) d) The nearest star (Alpha Centauri), is about 4,230,000,000,000 km away from earth.
How far is this in our model with the 1.28 cm earth?
Using the same method we get 4,230,000,000,000 km : 423,000,000 cm =
4,230,000 m. (This is about 2600 miles, which is about the distance from
Los Angeles CA to Washington DC, on the scale where the earth is 1.28 cm.
Stars are really, really far away.......)

6) One observation made from moon observations was that the lunar day is longer than a solar day, which makes the time of the moon rise and set change daily. In this problem, you will slightly alter the behavior of our model, in order to better understand the effects of the motion of the moon. a) What is the difference between rotation and revolution ?
Rotation is the movement of one body about its own axis, while revolution is the motion of one object moving around another. (The earth undergoes rotation on its axis, and the moon undergoes revolution about the earth.) b) If the rate of revolution of the moon around the earth in our model increased , what would happen to the length of a lunar day? Sketch a diagram to express this.
Here we see the moon moving around the earth. If the moon travels only a short distance in a fixed time, it is observed only from the narrow range of positions on earth (fine dotted lines). If it travels a further distance, though, the moon will, be visible from a wider range as the earth rotates along with it (see coarse dotted line).
Therefore, if the moon’s revolution was a bit faster , the lunar day would be longer.
c) If the rate of revolution of the moon around the earth decreased , what would happen to the length of a lunar day?
If the revolution of the moon slowed down, the lunar day would be shorter , but not shorter than a solar day. d) If the rate of revolution of the moon exactly equalled the rotation of the earth, what would happen to the length of a lunar day? How would this affect observations of the moon from the earth?

If the moon’s revolution rate equalled earth’s rotation rate, the moon would just appear to hover in one place over the sky, not moving! The moon would only ever be visible from half the earth, and would undergo a complete change of phases every day . e) If the direction of revolution of the moon reversed , would a lunar day be longer or shorter than a solar day?
If the moon revolved in a direction opposite the earth’s rotation, a lunar day would be shorter than a solar day. f) It takes about 27
⅓
days for the moon to revolve once around the earth, yet a complete change of phases, viewed from earth, takes 29 ½ days. Make a brief hypothesis as to what could cause this discrepancy.
The earth is also undergoing a revolution around the sun - so that the angle of the sun’s rays is actually slowly changing, as well! (We don’t account for this complication in our simple model.) This causes a lag in the change of phases.

7) During your observations of the moon throughout the semester, you estimated the angle between the sun and the moon. We then investigated a model of the moon’s motion that correlated to our observations. a) Draw below how the moon looks to us during each of 4 phases: first quarter, third quarter, new, and full. Arrange these in the sequence we observe them occurring, starting with the new moon. b) Find, in your moon observations, an observation that is close to a first quarter moon, that includes the direction and angle of the sun. Duplicate this observation below.
Answers will vary, but they should look like the 2nd diagram above, be sometime in late afternoon/evening, and the sun should be somewhere around the neighborhood of 90° in the western direction. c) Find, in your moon observations, an observation that is close to a third quarter moon, that includes the direction and angle of the sun. Duplicate this observation below.
Answers will vary, but they should look like the 4th diagram above, be sometime in early morning, and the sun should be somewhere around the neighborhood of 90° in the eastern direction.

d) How does the direction toward the sun compare for parts b) and c) ?
It is in opposite directions. e) The diagram below shows the moon at 1st and 3rd quarter in our model. (Assume the north pole of the earth is on the TOP of the page.) Shade in the dark halves of the moon and earth, and label the actual angle between the sun and moon when it is in these positions. Is this diagram consistent with your previous answers?

f) You could never actually observe the new moon during your observations. Why not?
Draw in the new moon in the diagram above, with the dark half shaded. In our model, what is the angle between the sun and moon during the new moon phase?
The new moon is near the sun, and we see only the shaded side of the moon, making it very difficult to see. The angle between the new moon and the sun in our model is 0°. (In actuality it is not really quite 0°, except during a solar eclipse - there is a “tilt” to the plane of the lunar orbit that we do not include in our model - this “tilt” is the reason we don’t get a solar eclipse every time there is a new moon.) g) When you saw the full moon, the sun was never in view at the same time. Why not?
Draw in the full moon in the diagram above, with the dark half shaded. In our model, what is the angle between the sun and moon during the full moon phase?
The full moon is only out at night, because the moon faces the shaded side of the earth when it is full. The angle between the full moon and the sun in our model is 180°. (Again, this is not exact in reality, due to the “tilt” of the lunar orbit - for this same reason, we don’t get a lunar eclipse every time there is a full moon.) h) During what exact point in the moon’s phases does the direction from the moon toward the sun change? Does it change from east to west, or from west to east?
The direction changes from west to east when it reaches it largest angle at the full moon.
8) This year (2012) there will be 13, not 12, full moons. (There will be 2 in August.)
Considering there are 29.5 days in a lunar month determine the next year after this that there will be 13 full moons in a calendar year.

There’s 29.5 days in a lunar month, and 366 days in 2012 (it’s a leap year!), so there’s 366 days from Jan. 9th 2012 to Jan. 9th 2013.
366/29.5 = 12.4 lunar months, so it’s either 12 or 13 lunar cycles until the
Jan 2013 full moon.
29.5
✕
12 = 354 days; 366 – 354 = 12; this isn ’ t 2013 yet
29.5
✕
13 = 383.5 days; 383.5 – 366 = 17.5; Jan 9 + 17.5 days = Jan 26 or 27
This is the closest you can get without knowing the time the moon is fullest. (The actual answer is 5:38 am on Jan 27, 2012.)