Correcting for Sextant Parallax Error

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If you think adventure is dangerous, try routine; it is lethal.

— Paulo Coelho


Introduction

Figure 1: Common Example of Parallax.

Figure 1: Common Example of Parallax (source).

Navigators use the altitudes of solar system objects to assist them with determining their positions. The most commonly used solar system objects are the Sun, Moon, Venus, and Mars. There is a small error caused by the fact that navigators are making their sextant measurements from the surface of the ocean and not from the center of the Earth, which is the reference point used by nautical almanacs. Parallax has no practical significance when measuring the positions of stars because they are so far away relative to the radius of the Earth.

Solar system objects have been used by navigators for other purposes. For example, Galileo recommended using the Jovian moons to determine Greenwich Mean Time (GMT). Before Harrison and the marine chronometer, this was one of the few means by which navigators could determine GMT at far distant points around the world.

Background

parallax
Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines.
Parallax Error (P)
The very small angle (fraction of an arcmin) between the observer and the center of the Earth (see Figure 3), as seen from the celestial object of interest (Sun, Moon, Venus, or Mars). This angle is maximum when the celestial object is low on the horizon and nil when it is overhead.
Horizontal Parallax (HP)
This is the maximum possible parallax at a specific time for a given lunar distance from Earth. It is computed assuming that the center of the Moon is on the horizon. Almanac's list HP at hourly intervals. The actual parallax reduces as the altitude of Moon moves above the horizon.

Analysis

Parallax Equation

Equation 1 shows a commonly used expression for the parallax error present in a celestial body measurement.

Eq. 1 \displaystyle P=\arcsin \left( {\sin \left( {{{H}_{P}}} \right)\cdot \cos \left( {{{H}_{T}}} \right)} \right)

where

  • P is angle of parallax correction.
  • HP is maximum parallax, which occurs when the celestial object is on the horizon.
  • Ha is apparent altitude of the celestial object.

The value of HP is given by Equation 2.

Eq. 2 \displaystyle H_P=\arcsin \left( {\frac{{{{r}_{E}}}}{R}} \right)

where

  • rE is radius of the Earth.
  • R is distance from the center of the Earth to the center of the celestial body.

The closest navigational celestial object is the Moon and it has the greatest parallax variation. Stars are so far away that we cannot measure their parallax using navigational instruments – stellar parallax does exist, however. For more on stellar parallax and its use in determining the distances to nearby stars, see this Wikipedia article.

The dynamic range of the Moon's HP is fairly limited, as I will show here.  The radius of the Earth is constant, but the distance between the Earth and the Moon varies because the Moon's orbit is an ellipse. The following  Wikipedia quote gives the range of variation.

The actual distance varies over the course of the orbit of the moon, from 363,104 km (225,622 mi) at the perigee and 405,696 km (252,088 mi) at apogee, resulting in a differential range of 42,592 km (26,465 mi).

Figure 2 shows that range of values that the Moon's HP can take as its distance from Earth varies.

Figure 3: Range of HP.

Figure 2: Range of Lunar HP Values.

Derivation

Figure 3 shows how you can derive the Equation 1 using the Law of Sines. The derivation assume that their is negligible difference in the distance to the celestial object from the observer (R′) and the center of the Earth (R).

Figure 2: Tutorial Image on Parallax (<a href="http://mathscinotes.com/wp-content/uploads/2015/10/Chapter2-1.pdf" target="_blank">source</a>).

Figure 3: Tutorial Image on Parallax (source).

As shown in Equation 3, Equation 1 is often simplified to P\doteq \cos \left( {{{H}_{T}}} \right)\cdot {{H}_{P}} by assuming that \sin \left( {{{H}_{P}}} \right)\doteq {{H}_{P}} for HP small.

Eq. 3 P\doteq \cos \left( {{{H}_{T}}} \right)\cdot {{H}_{P}}

Conclusion

Now that I understand how to correct for parallax errors, I will move on to index and refraction errors. Once I have reviewed all the errors, I will then show how to apply them using several examples.

 
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