The best way to determine longitude using only the Sun is by doing a "time sight". You measure the altitude of the Sun several hours before or after local noon and then use tables (or a calculator) to find your longitude.
If you want to use tables, a good set is available for free
here.
If you want to use a calculator (which is quicker and, IMO, easier), you can use the following formula:
cos(LHA) = (sin(Ho) - sin(Dec.) · sin(Lat.)) / (cos(Dec.) · cos(Lat.))
... where:
- "LHA" is the local hour angle of the Sun
- "Ho" is the measured altitude of the Sun, corrected for index error, height of eye, refraction and semi-diameter
- "Dec." is the declination of the Sun at the time of the sight
- "Lat." is the latitude of the boat at the time of the sight (usually determined from dead reckoning or "run up" from a noon sight or sight of Polaris
Once you find the local hour angle of the Sun, you can compare it to the Greenwich hour angle (GHA) and determine your longitude.
*Note: the tables linked above do this in a roundabout way by using the "apparent time at ship" and comparing that to Greenwich Mean Time (GMT). But this is really the same concept, because the difference in time is directly linked to the difference in longitude.