## HEIGHTS OF FEATURES

(This observing exercise is geared for high school students.)

Even though we are not on the **Moon**, we can determine the heights of some of its mountains. The procedure is to measure the length of the mountain's shadow. If we know (or can compute) the height of the **Sun**, then by using a little geometry, or trigonometry if necessary, we can calculate the feature's height. The procedure gives the height of a mountain, but the same process can be used with the rim of a crater wall.

The diameter of the Moon is 2200 miles (3500 km), and this can be used to compute the true lengths of the shadows. All you have to do is (a) measure the shadow length with a ruler with a ruler and (b) measure the full diameter of the Moon with the same ruler. Measure the feature's dimensions as precisely as you can. This exercise works best if the image is projected onto a board or wall, but it can also be performed on a computer screen.

1. Measuring the full diameter of the Moon may be tricky for two reasons. (A) Unless the phase is a Full Moon, then the illuminated disk of the Moon is not a circle. You will need to measure from the north pole to the south pole. If the Moon is a crescent, this would be from one “horn” to the other one. (B) The other difficulty is that your view is of only part of the Moon. You will need to move the telescope to one pole, and then measure the full size of the displayed region of the Moon. Next, move the telescope to the edge of the previous view. Repeat the measurements and movement of the telescope until you have moved across and measured the full diameter.

2. You can now compute the **Scale** of your image of the Moon. The full diameter is 3500 km (or 2200 miles) and you have measured the image in cm, mm, or inches. The Scale is equal to the full diameter divided by the measured size of the image.

Scale (km/cm) = 3500 km / _________________ cm = __________________ km/cm

3. Measure the shadows of interest using the same ruler. Make certain that the same units, for example centimeters, are used for both the Moon's diameter and the shadows' sizes. This works best if you are projecting the image on a board or wall, but it can be done off of a computer monitor. Knowing the scale, you can convert this length to kilometers.

Shadow Length (km) = Scale (km/cm) x shadow length (cm)

#### Feature Scale (km/cm) Measured Size (cm) True Size (km)

1. | | | | |

2. | | | | |

3. | | | | |

4. | | | | |

4. Calculating the **Angle of the Sun (θ)** is a three-step process.

A) You need to determine the **Phase of the Moon**. The “Phase-Day” was given for the time of your observation.

If the Phase-Day is less than 15, then

Sun-Moon-Earth (SME) angle = Phase-Day / 29 x 360° = __________________ degrees

If the phase-day is greater than 15, then

Sun-Moon-Earth (SME) angle = (29 – Phase-Day) / 29 x 360° = ___________________ degrees

B) However, you will have to perform an angular offset. The figure below shows how the Moon always keeps the same face toward the Earth. Notice that when the Moon is at Third Quarter, the Sun is on the horizon, casting a long shadow.

Sun's Angle = Sun-Moon-Earth angle – 90° = _____________________ degrees

C) Finally, you need to correct for the feature's longitude on the Moon. In the Table below, the longitudes for several lunar features are given, and find the value for your feature.

#### Feature Longitude

**Piton Mountain **1° West

**Tycho Crater** 11° West

**Archimedes Crater** 1° West

If the phase-day is less than 15, then the Longitude value is considered to be negative. If the phase-day is greater than 15, then the Longitude value is positive.

Corrected Sun's Angle (θ) = Sun's angle + Longitude = ____________________ degrees

6. Compute the height of the feature using the trigonometric relationship

**A = B tan (θ) **

Height (km) = Shadow Length (km) x tan (Corrected Sun's Angle)

Height (km) = _________________ x ______________________

Height (km) = _________________________

### Additional Exercise

Not all craters were formed at the same time. See if you can find two craters of equal size. Use the procedure to determine the heights of the crater rims. Then compute the ratio of the heights to the diameters for both craters. Can you think of reasons why these values could be different?

*James Sowell, 2013*