Sail Away

## Notes on Interpolation of GHA and Dec values

The nautical almanac pages available from this web site are built up according to the following scheme:

 ``` UT GHA ddGHA Dec dDec ° ' '/h ° ' '/h ... 13:00 19 04.5 +00.1 S 13 39.4 +00.8 14:00 34 04.6 +00.0 S 13 40.2 +00.8 15:00 49 04.6 +00.1 S 13 41.0 +00.9 ...```

The values for the Greenwich Hour Angle (GHA) and Declination (Dec) are given for the integral hours of Universal Time (UT). In the columns ddGHA and dDec the increment (+) or decrement (-) for the next one hour of time for the GHA and Dec are given.

For the Greenwich Hour Angle (GHA) this increment/decrement is not the complete variation since each hour angle increases with about 15° per hour. The value ddGHA is only the increment/decrement additional to this fixed increment of 15° per hour.

These increment/decrement values are given in (decimal) minutes-of-arc per hour ('/h).

### Linear Interpolation of the integral-hour GHA and Dec values

Altitude measurements for celestial navigation are valid only in conjunction with the exact time of observation. This time must be known down to the second. Also the values for GHA and Dec must be calculated for exactly this time of observation. So the values from the tables for the integral hours of UT are can not readily be used for the purpose of determining the geographical position of the observer.

The process of determining these "in-between-values" for GHA and Dec is called interpolation. More precisely we use "linear" interpolation for calculating the required GHA and Dec values.

The next example described how this interpolation is done as well as how to use the "Interpolation and Conversion Tables for Celestial Navigation" with the Nautical Almanac data.

### Example

On June 7. 2001 the altitude of the Moon at UT 15h 18m 23s is measured. For working out a Line of Position (LoP) the GHA and Dec of the Moon for exactly this time of observation is required.

The Nautical Almanac has the following entries for the Moon on June 7. 2001:

 ``` UT GHA ddGHA Dec dDec ° ' '/h ° ' ' ... 13:00 176 41.8 -31.3 S 23 09.6 +01.8 14:00 191 10.5 -31.3 S 23 11.4 +01.8 15:00 205 39.2 -31.3 S 23 13.2 +01.6 15:18:23 ? ? ? ? 16:00 220 07.9 -31.3 S 23 14.8 +01.5 17:00 234 36.6 -31.3 S 23 16.3 +01.4 ...```

First, the procedure to obtain the value of GHA will be explained.

Looking at the table above, it is clear that the value for 15:18:23 will be between 205° 39.2' (the value for 15:00:00) and 220° 07.9' (the value for 16:00:00). The increment from 205° 39.2' to 220° 07.9'. is 15° -31.3' (or 14° 29.7').

The average increment of the GHA for the Sun over the year is exactly 15° per hour (due to the rotation of the Earth measured in UTC). If the celestial objects used for Celestial Navigation would not move (around the Sun) the increment of their GHA would also be 15°/h (the motion of the Moon and the planets around the Sun as seen from the Earth, makes that their GHA increment is slightly different from 15°/h). So this value of 15°/h is closely related to the definition of the UTC time scale that is used. Therefore you will find interpolated values for 15°/h in the appendix of almost any Nautical Almanac publication. Here, these 15°/h interpolated values are included in the "Interpolation and Conversion Tables for Celestial Navigation". From these tables the interpolated value for dHA18:23 are found on the "18Min" page in the row "23Sec" from the third column labeled "dGHA", which reads: 04° 35.7'.
This is the "raw" increment for the GHA if the increment would be exactly 15°/h:
dHA18:23 = 04° 35.7'.

In the above extract from the Nautical Almanac, the value ddGHA indicates how much the GHA increases additionally to the 15° in one hour. Notice, that this value may be positive as well as negative.

On June 7. 2001 at 15:00 UT this additional increment is -31.3 minutes of arc. Now we want to know this additional increment not for one hour (60m 00s) but only for 18m 23s.
To obtain this fraction of one hour you would calculate the following:
18 / 60 + 23 / 3600 = 0.3064
This result says that 18m 23s corresponds to 0.3064 hour.

So the additional increment for the GHA corresponding to the -31.3'/h for 18m 23s is:
0.3064 * -31.3' = -09.59032'.
Since minutes of arc values are rounded to one tenth of a minute of arc, the result is -09.6'.

Notice that these interpolated additional increment/decrement values are very small for the Sun (usually ±00.0' or ±00.1') but for the Moon or the planets they may be not be neglectable.

Lets summarize what we know up to now:
- the GHA at 15:00:00 UT is GHA15:00:00 = 205° 39.2'
- the default increment for 18m 23s corresponding to the 15°/h is dHA18:23 = 04° 35.7'.
- the interpolated additional increment for 18m 23s is ddGHA18:23 = -09.6'

Adding these increments/decrements dHA18:23 and ddGHA18:23 to the GHA15:00:00 gives us the GHA for 18m 23s :

 GHA15:18:23 = (205° 39.2' ) + ( 4° 35.7' ) - ( 0° 09.6' ) = 210° 05.3'

The same procedure can be applied for the Declination:
The increment for Dec at 15:00:00 is +1.6'. The fraction of the hour is the same as before: 18m 23s corresponds to 0.3064 hour. The increment for 0.3064 hours:
- dDec18:23 = 0.3064 hours * +01.6'/hour = +00.5'.

Add this increment dDec18:23 of 00.5' to the Dec15:00:00 of S 23° 13.2':

 Dec15:18:23 = ( S 23° 13.2' ) + ( 00.5' ) = S 23° 13.7'

Notice that the procedure for the Declination is a little bit simpler than for the GHA because the increment/decrement values are always small (but not neglectable!) and there is no "default" increment for the Declination.

The GHA and Declination of the Moon on June 7. 2001 at 15h 18m 23s UT is:

 ``` UT GHA Dec ° ' ° ' 15:18:23 210 18.6 S 23 13.7 ```

### Example using Interpolation with Logarithmic Arithmetic

The drawback of the method described in the previous example is that the hour fraction must be calculated (with division and addition) and the subsequent calculation of the interpolated values require multiplication of decimal numbers.
This drawback can be avoided using the second part of the "Interpolation and Conversion Tables for Celestial Navigation" (the second subtable of each Minute table with columns labeled "p" and "s").

Using the same example as above the mathematical procedure is as follows:

The 18Min 23Sec entry of the Interpolation shows the following information:

 ``` 18Min : 23 18.3 04 35.7 | 3.0426 6.5989```

The second data column (labeled "Min") says that 18Min and 23Sec represent 18.3Min (rounded). The first column of the second sub table reading 3.0426 (labeled "p") is the logarithmic representation of 18Min 23Sec expressed in seconds:
log(18*60 + 23) = 3.0426.

What can we do with this information?
For the interpolation of the GHA in the example above, the following calculation had to be performed:

`                 (18/60 + 23/3600) * -31.3' = -09.6' `
yielding the interpolated increment for the GHA. This operation can be rewritten as:

```1/216000 * (18 * 60 + 23) * (-31.3' * 60)   =      -09.6'             or
(18 * 60 + 23) * (-31.3' * 60)   =      -09.6' * 216000    or
log( (18 * 60 + 23) * ( 31.3  * 60) ) =  log( 09.6  * 216000 )  or
log(18 * 60 + 23) + log(31.3 * 60)  =  log( 09.6  * 216000 )```

The value log(18 * 60 + 23) was found in the Interpolation Tables at the 18Min 23Sec entry. for the value log(31.3' * 60) look in the second column of the Interpolation Tables for the value 31.3 (found at the 31Min 20Min entry):

 ``` 31Min : 20 31.3 07 49.5 | 3.2742 6.8305```

The first entry of the logarithmic sub table (labeled "p") gives the value of log(31.3' * 60) = 3.2742.
So the calculation above can be continued:

```
log(18 * 60 + 23) + log(31.3 * 60)  =  log( 09.6  * 216000 )  or
3.0426        +    3.2742       =  log( 09.6  * 216000 )  or
6.3168                   =  log( 09.6  * 216000 ) ```

The value we are interested in is the 09.6 minutes of arc. Now the second entry of the logarithmic sub table (labeled "s") has the values log( Min * 216000) compiled. With this column we can find (except for the sign) the interpolation result 09.6' from the calculated sum value 6.3168.

The value closest to 6.3168 in the column labeled "s" is found at 09Min 36Sec:

 ``` 09Min : 35 . 02 23.7 | 2.7597 6.3160 09Min : 09Min : 36 . 02 23.9 | 2.7604 6.3167 09Min : 37 . 02 24.2 | 2.7612 6.3175 09Min : 38 09.6 02 24.4 | 2.7619 6.3182```

The decimal value for the corresponding minutes ("Min") is found at the 38Sec line: 09.6 minutes of arc.

Finally the sign of the increment (which was lost by converting to logarithmic numbers) must be considered: the increment for one hour was negative (-31.3'), so the resulting interpolated increment is also negative: -09.6'.

In practice the interpolation scheme is quite simple. It consists of:

• two table lookups yielding the logarithmic representation of the hour fraction and the increment/decrement of GHA or Dec,
• an addition of the two logarithmic numbers found in the previous step
• and another table lookup to convert the sum back into a degree value.

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