### Sample Calculation

Below is a sample calculation for how you could obtain your longitude from the relative position of a star in the sky on a particular date at a particular time. It is not the same as the computations that Fred Noonan was completing in the Electra during the World Flight, but it does show the magnitude of error in longitude when using an accurate time, but changing the date by one day, as suggested in the Date Line Theory.

PART I – What is your longitude?

On July 2nd, you are at sea and you observe the star Antares as it transits and it has a right ascension of 16 hours 46 minutes (RA = 16h46m). At this same moment, the ship’s GMT clock reads 17:28.

1. The hour angle (HA) of Antares is 0 because the star is transiting - it is directly overhead.

HA = 0

2. The sidereal time (ST) is the right ascension (RA) plus the hour angle (HA), so for Antares it is 16 hours 46 minutes.

ST= RA+HA
ST = 16h46m + 0
ST = 16h46m

3. It is July 2nd, so the right ascension of the sun (RAs) is 6 hour 44 minutes. The sun's right ascension changes 2 hours per month, or 1 hour every 2 weeks, or 4 minutes a day.

RAs = 0 on March 21st
RAs = 2 on April 21st
RAs = 4 on May 21st
RAs = 6 on June 21st
RAs = 6 + 4 minutes x 11 days later  = 6h44m on July 2nd
RAs = 6h44m

4. The hour angle of the sun (HAs) is the local sidereal time (ST) minus the right ascension of the sun (RAs), so 16h46m – 6h44m, so HAs equals 10h2m.

HAs + RAs = ST
HAs + 6h44m = 16h46m
HAs  = 10h2m

5. Local Solar Time (LST) is the hour angle of the sun (HAs) plus 12h, so 10h2m plus 12 means the LST is 22h2m.

LST = HAs + 12h
LST = 10h2m +12h
LST = 22h2m

6. Longitude is the degree difference from Greenwich (Longitude = 0). Change from Greenwich Mean Time (GMT) to local solar time (LST) equals the change in longitude.

GMT longitude = 0 degrees
GMT = 17h28m

LST longitude (local longitude) = x degrees
LST (local time) = 22h2m

GMT – LST = GMT longitude – LST longitude
17h28m – 22h2m = 0 – x

x = 4h 34m

In degrees of longitude, 1h equals 15 degrees and 4m equals 1 degree, so:

x = 4(15 degrees) + 34/4 degrees
x = 60 + 8.5
x = 68.5 degrees West longitude (West because local time is earlier than GMT)

Solution I: Our longitude is 68.5 degrees West

PART II - What is your longitude if you use the next day?

On July 2nd, you are at sea and you observe the star Antares as it transits and it has a right ascension of 16 hours 46 minutes (RA = 16h46m). At this same moment, the ship’s GMT clock reads 17:28.
What is your longitude if you use July 3rd instead of July 2nd?

1. The hour angle (HA) of Antares is 0 because the star is transiting - it is directly overhead.

HA = 0

2. The sidereal time (ST) is the right ascension (RA) plus the hour angle (HA), so for Antares it is 16 hours 46 minutes.

ST= RA+HA
ST = 16h46m + 0
ST = 16h46m

3. On July 2nd, the right ascension of the sun (RAs) is 6 hours 44 minutes, but the sun's right ascension changes 2 hours per month, or 1 hour every 2 weeks, or 4 minutes a day. So, on July 3rd, one day later, the RA of sun has changed by 4 minutes.

RAs on July 2nd = 6 hours 44 minutes
RAs on July 3rd = 6 hours 48 minutes
RAs = 6h48m.

4. The hour angle of the sun (HAs) is the local sidereal time (ST) minus the right ascension of the sun (RAs), so 16h46m – 6h48m, so HAs now equals 9h58m.

HAs + RAs = ST
HAs + 6h48m = 16h46m
HAs  = 9h58m

5. Local Solar Time (LST) is the hour angle of the sun (HAs) plus 12h, so 9h58m plus 12 means the LST is 21h58m.
LST = HAs + 12h
LST = 9h58m +12h
LST = 21h58m

6. Longitude is the degree difference from Greenwich (Longitude = 0). Change from Greenwich Mean Time (GMT) to local solar time (LST) equals the change in longitude.

GMT longitude = 0 degrees
GMT = 17h28m

LST longitude (local longitude) = x degrees
LST (local time) = 21h58m

GMT – LST = GMT longitude – LST longitude
17h28m – 21h58m = 0 – x

x = 4h 30m

In degrees of longitude, 1h equals 15 degrees and 4m equals 1 degree, so:

x = 4(15 degrees) + 30/4 degrees
x = 60 + 7.5
x = 67.5 degrees West longitude (West because local time is earlier than GMT)

Solution II: Our longitude is 67.5 degrees West, if we use July 3rd

PART III – What is the actual mileage difference caused by a change of dates?

1. What is the difference in longitude found when the date changes from July 2 to July 3?

The longitude found for July 2nd is 68.5 degrees West, and the longitude found for July 3rd is 67.5 degrees West.

68.5 – 67.5 = 1 degree of difference in longitude for a day’s change in RA of the sun

2. How many miles does 1 degree in longitude account for near the equator?

1 degree of Longitude in miles = cosine (latitude) * length of degree (miles) at equator

At the equator, latitude = 0, so:

1 degree Longitude = cos (0) * 69.172 mi = 69.172 miles

1 degree difference in longitude at the equator = 69.172 statute miles or 60.108825 nm.

Solution III: Changing the date by 1 day near the Equator amounts to ~60 nm

NEXT: How does this relate to Amelia Earhart’s final flight?