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.
What is your longitude?
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?
