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
... ...
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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").
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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.
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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".
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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:
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.
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( ) ___________
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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
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With the Interpolation Tables the calculation scheme gives the following result:
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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
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Since the increment "d" for the hour interval is positive, also the
resulting fractional increase "c" is positive.
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