author Star navigation Leif Roschier, Tapani Makkonen (Theory and background)

## Contents

### Theory and background

A definition of celestial navigation from http://www.celnav.de:

"Celestial navigation is he art and science of finding one's geographic position by means of astronomical observations, particularly by measuring altitudes of celestial objects - sun, moon, planets, or stars."

From the above web page you will also find "A Short Guide to Celestial Navigation" in pdf format written by Henning Umland.

Here we demonstrate nomograms in the context of celetial navigation by presenting a nomogram for the formula used to compute altitude azimuth. How this formula is used in the celestial navigation is not explained here, the interested reader may refer, e.g., to the above mentioned document for further details. Note also that in practice the nomogram presented here is virtually useless, as the graphical accuracy is not sufficient. The angles should be determined to accuracy of 0.1 arcseconds (arcsecond is 1/60 degrees) to achieve a reasonable position accuracy. This clearly cannot be obtained with the nomogram presented here. However, the nomogram could be constructed to cover a smaller range of angles, with corresponding increase in accuracy.

The formula for altitude azimuth is

$\cos A_z=\frac{\sin Dec - \sin H \sin Lat}{\cos H \cos Lat}\ .$

Above, $Dec$ is the declination of the body, i.e. the angle measured from the equator to the geographic position of the respective body (e.g. a star) along an arc extending from the equator to the north pole. Here, by geographic position of the body (GP) one means location of the projection of the body on the surface of the earth, i.e. the point where the line from the center of the earth to the body intersects the surface of the earth. Declination is positive from 0${}^{\rm o}$ to 90${}^{\rm o}$ when GP is located on the northern hemisphere negative from 0${}^{\rm o}$ to -90${}^{\rm o}$ when GP is located on the southern hemisphere.

$H$ is the altitude of the respective body (e.g. star). It is the angle between the line of sight from the observer to the respective body and the celestial horizon. Celestial horizon is a plane passing through the center of the earth whose normal is in the direction of the line from the center of earth to the observer on the surface of the earth. On northern hemisphere altitude is positive from 0${}^{\rm o}$ to 90${}^{\rm o}$ and on southern hemisphere it is negative from 0${}^{\rm o}$ to -90${}^{\rm o}$.

Above, $Lat$ is the latitude of the observer, i.e. the angle measured from the equator to the position of the observer along an arc extending from the equator to the north pole (along the meridian on which the observer is positioned). On northern hemisphere latitude is positive from 0${}^{\rm o}$ to 90${}^{\rm o}$ and on southern hemisphere it is negative from 0${}^{\rm o}$ to -90${}^{\rm o}$.

Azimuth is the angle between the meridian on which the observer is located and the arc from the position of the observer to GP. Azimuth is measured from 0${}^{\rm o}$ to 180${}^{\rm o}$.

True directions (here denoted by $t_s$) in navigation are measured from the true north. It is the angle between the arc from the observer to the north pole (the meridian on which the observer is located) and the arc from the observer to the body measured from 0${}^{\rm o}$ to 360${}^{\rm o}$. For example, if the observer is looking directly to the north and the body is on observer's left hand side, the true direction is between 180${}^{\rm o}$ and 360${}^{\rm o}$.

Hence, to obtain the true direction of GP from the azimuth, one needs to take into account the local hour angle (LHA) of the body. The reference line from which the local hour angle of the body is measured is the the meridian on which the observer is located. LHA is measured from 0${}^{\rm o}$ to 360${}^{\rm o}$. Hence, when the body has just passed the reference line, LHA is slightly larger than 0${}^{\rm o}$ and when it is just about to pass the reference line LHA is slightly less than 360${}^{\rm o}$. The formula for computing the true direction of the body from the azimuth and LHA is then $t_s = \begin{cases} 360^\circ-A_z, & \mbox{if }0^\circ\le LHA \lt 180^\circ \\ A_z, & \mbox{if }180^\circ\le LHA \lt 360^\circ \end{cases}$

Below, one denotes $a=A_z$, $d=Dec$, $b=Lat$, and $h=H$.

### Nomograph construction

Equation

$\cos(a)=\frac{\sin(d)-\sin(b)\sin(h)}{\cos(b)\cos(h)}$

may be transformed into constructional determinant by using variables x and y:

$\cos(a)\equiv x$

$\sin(d)\equiv y$

$x \cos(b)\cos(h) - y + \sin(b)\sin(h) = 0 \, .$

Last three equations are in matrix form:

$\begin{pmatrix} 1 & 0 & -\cos(a) \\ 0 & 1 & -\sin(d) \\ \cos(b)\cos(h) &-1 & \sin(b)\sin(h) \end{pmatrix} \begin{pmatrix} x \\ y \\ 1 \end{pmatrix} = 0.$

For this equation to be true in with any x and y, we require

$\begin{vmatrix} 1 & 0 & -\cos(a) \\ 0 & 1 & -\sin(d) \\ \cos(b)\cos(h) &-1 & \sin(b)\sin(h) \end{vmatrix} =0,$

and this is our constructional determinant. We may express the determinant in form (exchange 2nd and 3ed row, and 2nd and 3rd column) :

$\begin{vmatrix} 1 & -\cos(a) & 0\\ \cos(b)\cos(h) & \sin(b)\sin(h) &-1\\ 0 & -\sin(d) &1 \\ \end{vmatrix} =0,$

and further (multiply second row by -1)

$\begin{vmatrix} 1 & -\cos(a) & 0\\ -\cos(b)\cos(h) & -\sin(b)\sin(h) &1\\ 0 & -\sin(d) &1 \\ \end{vmatrix} =0,$

and (add to 3rd column -1 times first column)

$\begin{vmatrix} 1 & -\cos(a) & -1\\ -\cos(b)\cos(h) & -\sin(b)\sin(h) &1+\cos(b)\cos(h)\\ 0 & -\sin(d) &1 \\ \end{vmatrix} =0.$

### Generated nomograph

Generated portable document file (pdf): File:Ex star navi.pdf

### Source code

"""
ex_star_navi.py

Equation: cos(a)=(sin(d)-sin(b)sin(h))/(cos(b)cos(h))

This program is free software: you can redistribute it and/or modify
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program.  If not, see <http://www.gnu.org/licenses/>.
"""
from pynomo.nomographer import *

# for testing
d=40.0*pi/180.0
h=30.0*pi/180.0
b=60.0*pi/180.0
#print acos((sin(d)-sin(b)*sin(h))/(cos(b)*cos(h)))*180.0/pi
#print arange(0.0,40.0,1.0,dtype=double).tolist()

a_params={
'u_min':0.0,
'u_max':90.0,
'f':lambda u:1,
'g':lambda u:-cos(u*pi/180.0),
'h':lambda u:-1.0,
'title':'a',
'title_x_shift':0.0,
'title_y_shift':0.25,
'scale_type':'linear smart',
'tick_levels':5,
'tick_text_levels':5,
'tick_side':'right',
'tag':'none',
'grid':False,
#        'extra_params':[{
#            'u_min':40.0,
#            'u_max':90.0,
#            'scale_type':'linear',
#            'tick_levels':4,
#            'tick_text_levels':3,
#            'tick_side':'right'},
#            {
#            'u_min':10.0,
#            'u_max':40.0,
#            'scale_type':'linear',
#            'tick_levels':3,
#            'tick_text_levels':2,
#            'tick_side':'right'}]
}

d_params={
'u_min':0.0,
'u_max':90.0,
'f':lambda u:0.0,
'g':lambda u:-sin(u*pi/180.0),
'h':lambda u:1.0,
'title':'d',
'title_x_shift':0.0,
'title_y_shift':0.25,
'scale_type':'linear smart',
'tick_levels':5,
'tick_text_levels':5,
'tick_side':'right',
'tag':'none',
'grid':False,
#        'extra_params':[{
#            'u_min':0.0,
#            'u_max':40.0,
#            'scale_type':'linear',
#            'tick_levels':4,
#            'tick_text_levels':3,
#            'tick_side':'right'},
#            {
#            'u_min':40.0,
#            'u_max':80.0,
#            'scale_type':'linear',
#            'tick_levels':3,
#            'tick_text_levels':2,
#            'tick_side':'right'}]
}

bh_params={
'ID':'none', # to identify the axis
'tag':'none', # for aligning block wrt others
'title':'Grid',
'title_x_shift':0.0,
'title_y_shift':0.25,
'title_distance_center':0.5,
'title_opposite_tick':True,
'u_min':0.0, # for alignment
'u_max':1.0,  # for alignment
'f_grid':lambda u,v:-cos(u*pi/180.0)*cos(v*pi/180.0),
'g_grid':lambda u,v:-sin(u*pi/180.0)*sin(v*pi/180.0),
'h_grid':lambda u,v:1.0+cos(u*pi/180.0)*cos(v*pi/180.0),
'u_start':0.0,
'u_stop':90.0,
'v_start':0.0,
'v_stop':90.0,
'u_values':[0.0,15.0,30.0,40.0,45.0,50.0,55.0,60.0,65.0,70.0,75.0,80.0,85.0],
'v_values':[0.0,15.0,30.0,45.0,60.0,75.0,85.0],
'grid':True,
'text_prefix_u':r'',
'text_prefix_v':r'',
'text_distance':0.5,
'v_texts_u_start':True,
'v_texts_u_stop':False,
'u_texts_v_start':True,
'u_texts_v_stop':False,
'extra_params':[{
'u_start':0.0,
'u_stop':40.0,
'v_start':60.0,
'v_stop':89.9,
'u_values':arange(0.0,41.0,1.0,dtype=double).tolist(),
'v_values':arange(60.0,91.0,1.0,dtype=double).tolist(),
'v_texts_u_start':False,
'v_texts_u_stop':False,
'u_texts_v_start':False,
'u_texts_v_stop':False,
'u_line_color':color.cmyk.Sepia,
'v_line_color':color.cmyk.Sepia,
}
]
}

block_params={
'block_type':'type_9',
'f1_params':a_params,
'f2_params':bh_params,
'f3_params':d_params,
'transform_ini':False,
'isopleth_values':[[50,[75,15],'x']]
}

main_params={
'filename':'ex_star_navi.pdf',
'paper_height':20.0,
'paper_width':20.0,
'block_params':[block_params],
'transformations':[('rotate',0.01),('polygon',),('scale paper',)]
}
b=Nomographer(main_params)