Why Correct Altitude
As you have already learnt that the altitudes of the sextant are observed above the visible horizon. Eventually that altitude needs to be converted as an altitude above the rational horizon, which is a line parallel to the sensible horizon, passing through the centre of the earth.
When the altitude is so converted the line joining the centre of the object and the rational horizon can be considered as an arc of a great circle and the triangle PZX can be solved.
Definitions
Even though the definitions are familiar to you let us just revise them to ensure that there is no confusion.
Visible Horizon is the circular line limiting the observer’s view at sea.
Sensible horizon is the imaginary plane passing through the observer’s eye. Both the planes of the visible and the sensible horizon are not parallel to each other, as the height of eye Is some distance above the surface.
When observing the stars or the planets they appear to be points but when we observe the sun or the moon we observe the disc and we need to determine the centre of the moon or the sun.
In order to do that we observe either the lower limb or the upper limb and apply a correction to arrive at the altitude of the object.
Lower limb is the arc of the visible disc, closest to the horizon.
Upper limb on the other hand is the arc, which is farthest from the horizon.
Can we now say whether the diameter is greater at aphelion or perihelion?
Now let us look at the actual correction parameters.
We have seen that the sextant altitude is the angle subtended at the observer between the visible horizon and the object.
Index error is the error on the sextant that needs to be applied to ensure our process further.
Just to recapitulate the sextant has various errors and they together are termed Index error.
The index error is one of the adjustable errors and affects the observations. The other adjustable errors are the error of perpendicularity (Index glass not being perpendicular to the plane) and the side error. (Horizon glass not being perpendicular to the plane)
Index error is defined as the error caused by both the horizon and the index glass not being parallel to each other.
Whether you actually correct a sextant or not it would be advisable to check the index error by observing the true and reflected horizon with the sextant clamped at zero. If they are not in line turn the micrometer screw to bring them in line and the reading shall be the index error. The error shall be On the arc if the reading is positive and Off the arc if the reading is negative. On the arc errors are subtracted and off arc errors added.
Even though sextant is not used for normal navigation, you are reminded that it is one of the best fall back instrument and in any case it is included as a skill that you should acquire as per STCW Convention 2010.
In the figure, C represents the Centre of the earth, O the observer, Z the zenith, and X the body.
► The circle bounding the observer’s view at sea is called the visible horizon
► The plane which passes through the centre of the celestial sphere, and is perpendicular to the observers vertical ZOC, is called the Rational horizon
► A circle whose plane is parallel to the plane of rational horizon but passes through the observer’s eye is called the sensible horizon
► The angular depression of the visible horizon below the sensible horizon is called as the Dip.
in the fig, ∠SOV, the angle of dip, is greater as the height of the observer’s eye is bigger. Values of dip for heights of eye up to 100 feet are given in Nautical Tables. We can work it out by formula
Dip = 1.77 h², where h= height of observer above sea level in meters.
The navigator on the ship’s bridge when observing any celestial body brings the body down to the line of sea horizon and measures the angle of one of its limb. This angle measured is called as the Sextant altitude of the body.
The sextant used by the observer for the observation should indicate the correct altitude. However, being a mechanical aid it is possible that the sextant may not indicate the correct altitude. This zero error is referred to as the index error and is On the arc’ if the correction is to be subtracted & ‘Off the arc if the correction is to be added to the sextant altitude.
The sextant altitude corrected for index error, if any is called is Observed altitude.
The angular height of a celestial body above the sensible horizon is called the apparent altitude. ∠XOS is the apparent altitude.
From figure it is evident ∠ XOV- ∠ SOS = ∠ XOS
Observed Altitude – Dip = Apparent altitude.
True Altitude ∠ XCR is the angular height of the centre of a body above the observer’s rational horizon
Celestial triangle or calculations of position lines are based on ∠XCR. The observation of any body can never be carried out from centre of earth or the Celestial Sphere, but will always be made from Surface of the earth.
Parallax
Parallax is the angle at the centre of the body between the centre of the earth and the observer at the surface. The parallax like the refraction is maximum when the attitude is zero and nil when the altitude is 90° Parallax is also dependent on the distance of the body from the earth. Parallax is maximum in the case of the moon because of its proximity, less in the case of the sun and nil in the case of the stars. The nautical almanac gives the value of horizontal parallax. The parallax is obtained as follows.
Parallax in altitude = Horizontal parallax X Cos apparent altitude.
From the above figure, it is clear that:
a) ∠CXO is called as the Parallax in altitude.
b) Parallax increases with the nearness of the body to the earth.
c) The moon is the nearest Celestial body and hence has the largest Parallax, varying from 58′ to 60′.
d) The Sun is further away and has a smaller Parallax that never exceeds 9″ of arc.
e) The planets have each a very small Parallax depending on their distance from the earth,
f) Stars have no Parallax, as they are so far away, the radius of the earth does not subtend an angle at such a great distance.
The angle of parallax is maximum when the body is on the observers horizon and zero when the body id at the observers zenith.
Maximum value of parallax occurs when the body is on the sensible horizon or rational horizon of the observer. This is called ‘Horizontal Parallax‘. The horizontal parallax depends on the radius of the earth and the distance of the body from the earth’s centre.
The variation in the value of the Suns horizontal Parallax can be considered negligible, and a constant figure used to calculate the Parallax.
In case of moon, the value is variable to a certain extent. This could be considered as obvious. The moon is closer and though smaller in real diameter, its apparent diameter varies as its distance from the earth. The moon also moves in an ellipse and therefore is sometimes closer and sometimes farther away
Did you know that the Ancients in India without many normal aids had a very good idea as to the movement of the earth and the moon?
The apparent disc of the Sun and Moon are approximately ½ degree in diameter. We need the altitude of the Centre of the disc. As it is difficult to judge the position of the Centre, it is a practice to take the.altitude of the lower edge / lower limb or upper edge / upper limb & apply correction to obtain the altitude of the Centre.
The size of the Sun & moon is of course, constant, so the only variable is the distance. The values of semi diameter SD of Sun & moon are given in the Nautical Almanac. The value given here is calculated using the distance of the body from the earth’s centre, which gives the SD for observer with moon on the rational horizon. In case of the Sun the earth’s radius is negligible compared with the earth-Sun distance & no correction is necessary. In case of the moon, the variation in SD is significant & an additional correction, which increases with SD, must be applied. This correction is called the augmentation of the SD.
Refraction: All rays of light passing through atmosphere bend towards the normal as they approach the earth. This bending or refraction of rays affects the measurement of correct altitude.
Refraction
We have seen the laws of reflection and refraction in physics. That who has forgotten, the following is a revision.
Light Is assumed to travel in a straight line at uniform speed if the medium in which it is travelling is uniform. However, If light enters a medium of different density the direction of travel changes. This change in direction of motion is called refraction.
If light travels or enters a denser medium it is refracted towards the normal (line perpendicular to the medium surface) and if it enters a less dense medium it is refracted away from normal.
The amount of change in direction is directly proportional to the angle between the direction of travel and the normal. Therefore if the incident ray is nearly parallel to the surface of the medium at which refraction takes place, relatively large amount of refraction occur.
Snell’s Law States that Sine of the angle of incidence and sine of the angle of refraction are inversely proportionate to the indices of the refraction of the substance in which they occur.
If the index of refraction change suddenly then the change in direction of travel of light is also sudden. (As when the light passes through water) However, if a ray of light travels through a medium of gradually changing index of refraction, the path travelled by the ray of light would be curved. This is the situation encountered by the ray of light emanating from the celestial body, which enters through atmosphere of varying refractive index. The bending of a ray of light from terrestrial body to the eye of the observer is called terrestrial refraction. (e.g. sighting of a light at a larger distance than calculated). This affects the dip of the horizon. A ray of light, from celestial body undergoing a similar bending is called astronomical refraction.
The effect of astronomical bending is to make a celestial body appear higher in the sky. If the body is at the observers Zenith, there is no refraction, but as the Zenith distance increases, the refraction also increases.
Because of this refraction when an observer takes a sight, he should remember the Position of the body in the sky as observed by him is not its true location and similarly the horizon is also not true. Thus his observations are not true. He has to apply certain corrections to the refraction effect and these corrections are available in the nautical almanac for standard average conditions and are termed as Mean Refraction’. The nautical almanac tabulates refraction for a standard atmosphere and a correction is provided when abnormal atmospheric conditions exist. This additional correction is either negative or positive.
Summary
The corrections, which are applicable to the observations of various Celestial bodies used in navigation, are as follows:
| Sun | Moon | Planet | Star |
|---|---|---|---|
|
Dip |
Dip |
Dip |
Dip |
|
Refraction |
Refraction |
Refraction |
Refraction |
|
Semi Diameter |
Semi Diameter after Parallax and augmentation |
NA |
NA |
|
Parallax |
Parallax |
NA |
NA |
The index error of the sextant must be applied to all the observations.
Correction of Sextant Altitude (SUN)
|
Sextant Altitude SUN |
|
40°00' |
|
I.E (on the arc) |
- ve |
00°03' |
|
Observed Alt |
|
39°57' |
|
Dip (12 m) |
- ve |
00°6.1' |
|
App Alt |
|
39°50.9' |
|
Refraction |
- ve |
00°01.1' |
|
App Alt |
|
39°49.8' |
|
Semi Diameter |
+ ve |
00°16' |
|
Parallax |
+ ve |
00°00.1' |
|
True Altitude |
|
40°05.9' |
In practical problems, using tables, which combine the corrections, saves time. Answers obtained from these tables do vary slightly from those obtained using individual corrections. The accuracy of total corrections is well within the limits required for practical navigation.
The argument used in the tables in the almanac is the apparent altitude, so that the dip correction must be applied to the observed altitude. The Sun’s total correction tables are given for summer & winter months. This allows two values for SD to be used, a mean value for summer months & winter months.
Two values are given in each table, the one in bold type to be used for lower limb & the lighter one for upper limb observations. The corrections allow for refraction,SD & the Parallax.
Correction by use of total corrections
Thus for observation of Sun if the tables are used the corrections applied are :
|
Sextant Altitude |
|
I.E +/- |
|
Observed Altitude |
|
Dip |
|
Apparent Altitude |
|
Total Correction |
|
True Altitude |
Correction of star altitude by total correction
The star total correction table is a table of refraction, as after the dip correction this is the only remaining correction.
The correction of stars attitude =
|
Sextant Altitude Star |
|
I.E +/- |
|
Observed Altitude |
|
Dip |
|
Apparent Altitude |
|
Total Correction |
|
True Altitude |
Correction of altitudes of planets
The planets also show phases just like the moon.We need to take into account this factor when correcting the altitude of planets in order therefore to allow for parallax and the effect of phase shown by the planet additional corrections for planets is to be allowed in addition to the total correction.
Planet correction Summary:
|
Sextant Altitude Planet |
|
I.E +/- |
|
Observed Altitude |
|
Dip |
|
Apparent Altitude |
|
Total Correction |
|
True Altitude |
Augmentation of the Moon's semi diameter.
The moon or sun’s semi diameter is tabulated as seen from the centre of the earth. Moon being comparatively closer its semi diameter about the same as that of the sun. However as the observer is at the surface of the earth the semi diameter observed by him shall be greater than that tabulated in the almanac. When the moon is on the horizon the distance of the observer and the moon and the centre of the earth is about the same. However as the moon rises and reaches the nadir the distance becomes less by about 4000 miles. This causes an apparent increase in the moon’s semi diameter, which is called augmentation of the moon’s semi diameter. The distance between the sun and the observer being very large there is no need to take into account such augmentation for the sun. Horizontal parallax is the parallax when the body is on the observer’s sensible horizon.
Dip or depression
Dip is the angle of depression between the sensible horizon and the visible horizon. As the height of eye is always above the earth it follows that the angle of depression shall always be negative i.e. the dip is subtracted from the observed altitude to obtain the apparent altitude, which is the altitude above the sensible horizon.
The corrections to be applied to celestial bodies are summarised as follows.
| Stars | Planets | Sun | Moon |
|---|---|---|---|
|
Index error |
Index error |
Index error |
Index error |
|
Refraction |
Refraction |
Refraction |
Refraction |
|
|
|
Semidiameter |
Augmented Semidiameter |
|
|
Correction for Parallax and phase for venus and Mars |
Parallax in Altitude |
Parallax in Altitude |
|
Total Corrn. For Refraction Only |
Total Corrn for refraction and a small corr for venus and Mars |
Total corr includes refraction, Parallax and SD either LL or UL. |
Total corr includes refraction, parallax andSD. Given in two parts. |
Total correction
The corrections given as refraction, correction for parallax and parallax in altitude where specified is given as total correction in the nautical almanac. When applying the correction keep a good eye for its sign. An inspection of the nautical almanac shall show that total corrections are tabulated separately for stars, planets, sun and for the moon. Work out any two of the following in your journal