Geocaching pioneers Lewis and Clark navigated by the sun and the moon and the stars. They used a sextant to measure the angle above the horizon of various celestial bodies. They also used a new-fangled artificial horizon gadget since the horizon is much more difficult to determine out in the woods versus out at sea. They used a scientific pocket watch, called a “Gold Chronometer” to tell time. Their search areas ended up being fairly large, however, which is why geocaching was slow to catch on.
To find this cache, you have to compute the location from a couple of measurements that are the equivalent of (corrected and very accurate) sextant readings. You can narrow your location on the earth to somewhere along a circle by measuring the altitude (angle above the local horizon) of a star. If you measure the altitude of the pole star, for instance, that circle is fairly close to being your latitude parallel. This is slightly complicated by the fact that the earth is rotating, so you also have to know what time it is and have a model for earth rotation. If you measure the altitude of two stars, that defines two circles, which usually intersect in two places. One of those intersections will be your location.
Below I have listed the altitudes of two stars from the cache location and the time at which those measurements are valid. The measurements were not taken with a sextant. They were computed by the Cartes du Ciel (Sky Chart) application and checked against the U.S. Naval Observatory Astronomical Applications Department web site . Also, they are not exactly the altitudes that you would observe with a sextant. They have been corrected (primarily for refraction) so that you can plug them directly into the formulas below.
Date: October 13, 2004
Time: 8 pm PDT
Julian Date: 2453292.625
Altair: altitude 60° 32.927’
Alpheratz: altitude 42° 58.732’
The right ascension and declination of these stars can be obtained from many sources, but I list them here so that there is no ambiguity in my computational model. These are the right ascension and declination of the date listed above.
Altair: Ra: 19h 51m 0.83s, Decl: 8° 52’ 50.6”
Alpheratz: Ra: 0h 08m 38.14s, Decl: 29° 07’ 01.8”
That’s enough information to solve the puzzle, but I am providing some additional help:
First, figure the right ascension of the Greenwich meridian at the time of the observations.
From Meeus, “Astronomical Algorithms”, first edition, p. 83-84:
Time in centuries from the year 2000 epoch:
T = (JD – 2451545.0)/ 36525
The right ascension of Greenwich in degrees:
Rag = 280.46061837 + 360.98564736629 * (JD – 2451545.0)
+ 0.000387933*T*T - T*T*T/38710000
Rag = 68.00260° (remainder after dividing by 360°)
If the (unknown) location is (Lat, Lon), the altitude for an object at (Ra, Decl) can be computed as follows:
alt = arcsin(sin(Lat)*sin(Decl) + cos(Lat)*cos(Decl)*cos(Rag + Lon - Ra))
We can now plug in the data for the two stars and have two equations with two unknowns, latitude and longitude. There are lots of ways to solve this and lots of mathematical tools. One way to attack it is an automated “guess and check”. Set up the formulas so you can specify a latitude and longitude and compute the resulting altitude for each star. The sum of the differences between the computed altitudes and the true altitudes (after taking the absolute value of the difference) is a measure of how close you are. You can determine which way to go by taking a step in some direction and seeing if the error goes up or down.
Disclaimer: This is not how Lewis and Clark navigated, and not how celestial navigation is normally done. Some of the finer points of the corrections that must be made to achieve accuracy sufficient for geocaching have been glossed over. This is a puzzle cache that illustrates a little astronomy.