Question E3A01:
What is the approximate maximum separation measured along the surface of the Earth between two stations communicating by EME?  

Choices:  
A: 2,000 miles, if the moon is at perigee  
B: 2,000 miles, if the moon is at apogee  
C: 5,000 miles, if the moon is at perigee  
D: 12,000 miles, if the moon is “visible” by both stations    

Correct Answer is  D: 12,000 miles, if the moon is “visible” by both stations  

To communicate by EME, both stations must have the Moon above their local horizon at the same time. The region where the Moon is visible is essentially one hemisphere of Earth. The maximum separation occurs when both stations are just on opposite edges of that hemisphere (the Moon is on the horizon for both). Those two points are 180° apart along a great circle.

Key geometry:
- Great-circle distance:  
  $$s = R_E \cdot \alpha$$  
  where $R_E$ is Earth’s radius and $\alpha$ is the central angle in radians.
- For opposite edges of the visible hemisphere, $\alpha = \pi$ (180°), so:  
  $$s_{\max} = \pi R_E \approx \pi \times 3960\ \text{mi} \approx 12{,}400\ \text{mi}$$
- This is typically rounded to “about 12,000 miles.”

Why D is correct:
- “Visible by both stations” implies both are within the Moon-visible hemisphere.
- The farthest two points within one hemisphere are opposite points on its boundary, giving about half Earth’s circumference:
  $$\tfrac{1}{2} C_E = \tfrac{1}{2} (2\pi R_E) = \pi R_E \approx 12{,}400\ \text{mi} \approx 12{,}000\ \text{mi}$$

Why the others are incorrect:
- A and B (2,000 miles) and C (5,000 miles) are far below the geometric limit of roughly 12,000 miles.
- References to perigee or apogee are red herrings; the size of the Moon-visible footprint on Earth is determined by Earth geometry, not the Moon’s slightly varying distance. Any parallax/refraction effects are small and do not change the order of magnitude.

Rule of thumb:
- Earth circumference: $$C_E \approx 24{,}900\ \text{mi}$$
- Maximum EME station separation (surface distance):  
  $$\approx \tfrac{1}{2} C_E \approx 12{,}450\ \text{mi} \Rightarrow \text{memorize as } 12{,}000\ \text{mi}.$$