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## Interpolation and Conversion Tables for Celestial Navigation

The Hour-Minute-Seconds and Degree-ArcMinute-ArcSeconds system used to express values of time and angle goes back to the Babylonian sexagesimal number system (base 60). Being used to the Indian-Arabic decimal number system, this sexagesimal system is not very well suited for doing even simple multiplication. This multiplication is needed for example while doing linear interpolation on the integral-hour GHA and Declination values from Nautical Almanac Tables. Also for determining a calculated altitude in the table-based sight reduction process such an interpolation is needed.

Although the underlying calculations are quite simple and can be done easily with a simple electronic calculator it is nevertheless more useful and less error prone to use look-up tables for (parts) of these operations.

### Purpose and Scope

The Interpolation and Conversion Tables presented here are designed to be used for both the interpolation of the Nautical Almanac data (GHA and Declination) as well as for the interpolation of the data from the Sight Reduction Tables (Altitude).
In order to obtain the highest possible accuracy logarithmic tables are used for the linear interpolation. As described below, linear interpolation with these tables involves only three "table look-ups" and one addition. The interpolated values have an accuracy better than ±0.05 arcminutes compared to the same value calculated with full precision.

### Arrangement

The tables have an entry for each Minute / Second combination (from 00Min/00Sec to 59Min/59 Sec). Each entry represents a fraction of an hour. The complete Table consists of 3600 entries: one for each second of the hour. For each entry four data values are given arranged in two sub tables (the example shown below gives the first entries of the "16 Min" data):

16 Min
```    Sec  fMin   dHA
'    °   '
00  16.0  04 00.0
01  .     04 00.2
02  .     04 00.5
03  .     04 00.8
04  .     04 01.0
05  .     04 01.2

06  16.1  04 01.5
07  .     04 01.8
08  .     04 02.0
09  .     04 02.2
10  .     04 02.5
11  .     04 02.8

12  16.2  04 03.0
13  .     04 03.2
14  .     04 03.5
...
```
```     p       s

29823   65386
29827   65390
29832   65395
29836   65399
29841   65404
29845   65408

29850   65413
29854   65417
29859   65422
29863   65426
29868   65431
29872   65435

29877   65440
29881   65444
29886   65449
...     ...
```
• The first data column ( "fMin" ) is the decimal (fractional) Minutes representation of the Minute / Second combination: Minutes+Seconds/60. These values are recorded with a precision of 1/10th of a Minute.

• The second data column "dHA" contains the interpolation values for the "mean" hourly increase of the Greenwich Hour Angle of 15°/Hour corresponding to the Minutes / Seconds entry. This is the principle increase of the GHA of the celestial objects corresponding to the Minutes and Seconds of the time of observation.

• The first data column of the second sub table("p") is the logarithm of the hour fraction expressed in Seconds: log10(Minutes*60+Seconds).
This value is used to enter the logarithmic part of the interpolation.

• The second data column of the second sub table("s") is the value returned by the logarithmic part of the interpolation and is used to convert the result back into a "Minutes" value (from column "s" to column "fMin").

### Tables

The complete Interpolation Tables consists of 20 pages each containing the data for a 3-Minute span. The fraction of the hour is recorded both in integral Minutes and Seconds and in decimal Minutes.

 The complete interpolation manual contains some extra pages with explanations and is available in a ready-to-print PDF format. PDF files can be viewed and printed e.g. with the free tools "xpdf", "Evince" or "Acrobat Reader".

### Use and Application of the Interpolation Tables

The tables presented here are designed for the following two problems typically occurring in the process of elaborating a Line-of-Position from a Sextant measurement:

1. For the fraction of the hour of the time of observation, find interpolated increment of the GHA corresponding to the principle 15°/hour increase of the GHA of celestial objects.

2. Given the increment on a GHA-, Declination- or Altitude value for the next hour, find the correct fraction of this increment corresponding to the given fraction of the hour of the time of observation.

The first problem, finding the principle interpolation correction for the GHA, is straight forward with the table, since this value can be directly read form the second data column in the first sub table (dHA).

The second problem is the typical (linear) interpolation problem of determining a new data value between two known values. In order to solve this with a reasonable accuracy, the logarithmic tables are used with the following calculation scheme:

 ``` d ____'__ --> p(d) ___________ f ___m __s f ____'__ --> p(f) ___________ ______________ c=d*f/60 ____'__ <-- s( ) ___________ ```

In the first line the increment/decrement for one hour interval (e.g. from the Nautical Almanac) or one degree of Declination (e.g. from the Sight Reduction Tables) is recorded (d). In both, the Nautical Almanac and the Sight Reduction Tables this value is given in decimal Minutes.
In the second line the fraction of the hour or degree is entered as Minutes / Seconds combination in the first column or directly in decimal minutes (less accurate) in the second column.

Both "d" and "f" are converted to "log(d)" and "log(f)" using the "p" column of the Interpolation Tables. Then both logarithmic values are added and finally the resulting logarithmic value is converted back to the fraction c=d*f/60 expressed in Minutes or Arcminutes using column "s" of the Interpolation Tables.

Notice: the increment/decrement value "d" has an algebraic sign. This sign is not considered in the logarithmic part of the calculation. The sign of the interpolation result "c" is the same as the sign of "d". The value "f" is always positive.

Example
The increase of e.g. Declination for the next hour interval d= 44.3' and the fraction of the hour f=34m 56s.
With an electronic calculator the result would be:

 ``` c= (44.3) * (34 + 56/60) / 60 = 25.8 ``` With the Interpolation Tables the calculation scheme gives the following result: ``` d 44'3 --> p(d) 34246 from the 44 Min page/ 20 Sec line f 34m 56s f ____'__ --> p(f) 33214 from the 34 Min page/ 56 Sec line _________ c=d*f/60 25'8 <-- s( ) 67460 from the 25 Min page/ 50 Sec line ```

Since the increment "d" for the hour interval is positive, also the resulting fractional increase "c" is positive.

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