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## Time - Hour-Angle Conversion Table

Longitude as part of our geographical coordinate system may be given both as

• the intersecting angle between the Local Meridian and the Prime Meridian of Greenwich expressed in Degrees (°) and Minutes-of-Arc (')
• or as a time difference between the Local Time and the Time at the Prime Meridian of Greenwich expressed in Hours (h), Minutes (m) and Seconds (s).

Time is closely linked to the rotation of the Earth. Over the time span of 24 hours, the Greenwich Hour Angle of the Sun increases by 360°. This implies an increase of 15° per hour. Since local time is basically based on mean solar time, locations having 15° of Longitude difference, will have exactly 1 hour of local time difference.

### Purpose and Scope

The Time - Hour-Angle Conversion Table tabulates the relationship between Time and Hour-Angle as used as part of the celestial coordinate system.

 ``` HA Time 15° ~ 1h 1° ~ 4m 15' ~ 1m 1' ~ 4s ``` This relationship allows to translate local time differences into Longitude differences and vice versa. It also shows that on the spring and autumn equinoxes, when the Geographical Position of the Sun is located on the Equator, the Geographical Position is "moving" at a speed of 15 nautical miles / minute or 1 nautical mile / 4 seconds.

### Table

Table The Time - Hour-Angle Conversion Table gives the relation between Time and Hour-Angle for integral values of the Hour-Angle. Time is given in 4-minute intervals. The information for the fractional values of the Hour-Angle can be obtained from the first pages of the Interpolation Tables for Celestial Navigation (Pages "0Min" - "3Min": the column "dHA" gives the increment of Hour-Angle for the mean 15°/h GHA increment of the Sun corresponding to the Time entry).

The table for the Time - Hour-Angle Conversion is also available from the pre-compiled Interpolation Tables for Celestial Navigation (page 35).

### Application of the Time - Hour-Angle Conversion Table

The standard application of this Time - Hour-Angle conversion is the translation of the observed local noon time into the corresponding Longitude. However, the table can also be useful for the determination of Rise Time and Set Time of celestial objects.

#### Example 1: finding Longitude from Culmination Time

On Aug 23. 2001 at a location with unknown Longitude, it is observed that local noon occurs at 03h 08m UT. At this moment the Sun attains the highest altitude above the horizon and thus the Geographical Position of the Sun is located on the local Meridian.

From the Nautical Almanac it can be read that on Aug 23. 2001 the Greenwich Culmination Time is:
Tculmination = 12h 03m (UT).

At the unknown position the Culmination Time is about 9 hours earlier, so this location must be East of Greenwich. The exact time difference is:
Tdlocal = ( 03h 08m ) - ( 12h 03m ) = - ( 08h 55m ).

The Time - Hour-Angle Conversion Table (take the nearest entry lower than 08h 55m : "08:52") gives the following equivalence:
08:52 ~ 133° 00.0'

The result for the remaining 03 minutes has to be looked up in the Interpolation Tables for Celestial Navigation (entry "03 Min / 00 Sec", data column "dHA" on page 11):
00:03 ~ 45.0'

08:55 ~ 133° 45.0'

So the Longitude corresponding to 03h 08m UT local Culmination Time on Aug 23. 2001 is:
133° 45.0'

Notices:
• Since the time differences between local Culmination Time and Greenwich Culmination Time cannot be simply determined more accurately than 1 minute of time, the longitude accuracy obtained in this way is limited to 15' (corresponding to 15 nautical miles on the Equator or less on higher Latitudes).

• The above method of finding Longitude with the time of local culmination, can be performed with other celestial objects (Moon or Planets) of which the Greenwich Culmination Time is known.

#### Example 2: finding local rise- or set times

In the next example the (local) time of sunrise on Aug 23. 2001 at location S 24° 37.4' E 079° 41.8' will be elaborated. The Greenwich Culmination Time on this date is found in the Nautical Almanac:
Tculmination = 12h 03m (UT).
Also the Declination of the Sun at noon time (12:00 UT) can be found here:
DecSun = N 11° 19'4

Next the Longitude 079° 41.8' is translated into a time difference against the Greenwich local time. The Time - Hour-Angle Conversion Table gives the time equivalent of 079°:
079° ~ 05h 16m 00s.

The rest of the time, corresponding to the remaining 41.8' of longitude, can be found in the Interpolation Tables for Celestial Navigation. Look for the dHA entry nearest to 41.8' (in the "2Min" column at page 10):
41.8' ~ 00h 02m 47s.

So the Longitude of E 079° 41.8' corresponds to a time lead (Eastern Longitude = negative "Tdlocal") of:
Tdlocal = - ( 05h 16m 00s + 00h 02m 47s ) = - ( 05h 18m 47s ) compared to Greenwich time.

So at the given location, local noon will occur a little bit more than 5¼ hours before Greenwich noon. In order to find the time of sunrise, half the "length-of-the-day" must be subtracted from this time. This "half-the-length-of-the-day" value is obtained from the Sight Reduction Tables. Look in the "CONTRARY" pages at the entry for S 24° Latitude (given Latitude = S 24° 37.4') and N 11° Declination (for DecSun = N 11° 19'4) and then search for the line at which the Altitude is zero (H=0). The Local Hour-Angle (LHA) for this combination is:
LHAH=0 = 85°

According to the Time - Hour-Angle Conversion Table these 85° correspond to a time difference of:
Tdh = 05h 40m

Adding the appropriate values as in the following scheme, gives a first iteration value of the (local) time of sunrise:
Trise = Tculmination - Tdh + Tdlocal
Trise = ( 11h 46m 00s ) - ( 05h 40m 00s ) - ( 05h 18m 47s )
Trise = ( 11h 46m 00s ) - ( 10h 58m 47s )
Trise = ( 10h 105m 60s ) - ( 10h 58m 47s )
Trise = ( 00h 47m 13s )

This is only a first approximation because the Declination value for 12:00 UT was used and this will be slightly different from the value for 00:47 that should have been used. However these changes are small and for normal use, this first approximation will be sufficiently accurate.

So sunrise on Aug 23. 2001 the given position (S 24° 37.4' E 079° 41.8') can be expected at: 00h 47m (UT).

As mentioned, this is only a first iteration because integer values for Declination and Latitude were used in the Sight Reduction Tables. Also the Declination value for 12:00 UT was used and this will be slightly different from the value for 00:47 that should have been used. However these changes are small and for normal use, this first approximation will be sufficiently accurate.
For a second iteration, the Declination value of the Sun at 00h 47m 12s must be interpolated from the Nautical Almanac and used as entry in the Sight Reduction Tables for determining the value of Tdh ("half-the-length-of-the-day" period), which also requires some interpolation, since the Sight Reduction Tables have only integer values for Declination and Latitude.

Similar to the time of sunrise, the time of sunset can be determined by:
Tset = Tculmination + Tdh + Tdlocal

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