Calculating the Distance

On January 13, 2025 we were treated with a rare occurrence of an occultation of Mars by the Moon.  An occultation occurs when an object, in this case Mars is obstructed by another, in this case the Moon.   Hence, on January 13^th across the world people saw the bright red star-like Mars slip behind the Moon and then about an hour later reappear on the other side, as so brilliantly captured by member, Mauri Rosenthal in this video: Mars Occultation. The AAA has access to remote telescopes far away and so for this event I decided to practice some citizen science using our Association’s telescope in Texas and my own telescope in upstate Connecticut.

The Mars-Moon occultation, observed from Fort Davis, Texas, and Sherman, Connecticut, provides an opportunity to calculate the distance to the Moon using the principle of parallax. Parallax is the apparent shift in the position of a celestial body when viewed from two different locations on Earth. By carefully analyzing the timing of the occultation at these two sites and understanding the geometry of the Earth-Moon-Mars system, we can derive Mars’s distance with surprising accuracy.  

During the occultation, observing from Fort Davis in Texas I noted that the Moon began to obscure Mars at 19:50:40 CST and that Mars reappeared at 20:48:40 CST. In Sherman, Connecticut, the occultation began later, at 21:26:28 EST, with Mars emerging from behind the Moon at 22:35:28 EST. These time differences occur because the Moon and Mars appear in slightly different positions relative to the background stars when viewed from the two locations, which are separated by approximately 2,800 kilometers. This separation forms the baseline for calculating the parallax.

The critical step in determining parallax lies in understanding how the apparent position of Mars relative to the Moon changes as seen from the two locations. The occultation timing provides the necessary data because the position of the Moon relative to Mars changes over time as the Moon moves across the sky.

1. Moon’s Motion Across the Sky: The Moon moves across the sky at an angular speed of approximately 0.549 degrees per hour (or 30 arcminutes per hour). This motion translates to about 0.5 arcseconds per second of time.

2. Time Differences and Angular Displacement: From the observations:
   • At Fort Davis, the Moon obscured Mars at 1:50:40 UTC and Mars reappeared at 2:47:00 UTC. In Sherman, Connecticut, the occultation began at 2:26:28 UTC and ended at 3:34:00 UTC. The time difference from when Mars was observed to be emerging from behind the moon in Fort Davis was 2:47:00 while it was 3:34:00 in Sherman, CT for a time difference between the two locations of: 03:34:00−2:47:00 = 47  mins ~ 2,820 seconds
   • During this interval, the Moon moved approximately:
     
     
   • Next we determine the distance between the locations ~ 2,807 km
   • Now we can determine the angular displacement as observed from the two locations by multiplying the angular speed by the time difference: 
     
     
     This difference represents the angular separation between Mars as seen from the two locations, also known as the parallax angle.
     
     
     
     Calculating Parallax: To calculate the distance to the Moon (D), we use the formula:
     
     The baseline is the distance between the two observation sites, approximately 2,807 kilometers. The parallax formula relates the baseline and the angular displacement:
     
     
     Wow! Not bad! The actual distance is closer to 384,400 km – so we are off by 2.9%! Nice!