THE PRINCIPLES of the celestial sun sight: Understanding the expression LOP (line of position) in four steps.

Please remember that the entire context of this website originates from the La Rochelle exercise. This exercise serves as the gateway leading you to an interactive worksheet that offers comprehensive, step-by-step guidance for both calculations and theoretical aspects.

### Introduction:

The St. Marcq Hilaire method involves comparing two entirely different methods:

1) Using the sextant to measure the true altitude circle, also known as the observed altitude circle (Ho).

2) Calculating, from our estimated position, the appropriate altitude circle (Hc).

The required conditions are as follows: the estimated position must not have an uncertainty of more than 30 miles. The time of the measurement with the sextant must be equal to the time integrated into the calculation.

In this case, the two circles are close together with a horizontal difference on the Earth called the intercept. The estimated position serves as the reference point for plotting the true altitude circle, locally associated with a straight line, called the line of position.

The method, which isn’t purely mathematical, involves two independent circles situated closely to each other with nearly identical characteristics; it’s also known as the intercept method.

Of course, it’s all about the observed altitude circle, taken with the sextant but it lacks a usable reference because the geographical position (GP) point, which lies in the center of your observed circle, is too far away and cannot be found on your nautical chart.

### The line of position

A Line of Position (LOP) is a marked straight line on a nautical chart that represents the potential location of a vessel, determined using a sextant and relevant calculations.

Essentially, it signifies that the vessel is positioned somewhere along this line, **yet it doesn’t provide an exact fix on its location.**

### the solution

To accurately determine a vessel’s position, it is necessary to have the intersections of at least two LOPs. The process involves taking an initial observation with a sextant to measure the sun’s height above the horizon and calculate the first LOP.

Subsequently, a second observation is taken some time later as the sun changes its azimuth angle during its east-to-west movement. The second LOP is then calculated, and the intersection of these two lines reveals the vessel’s precise position.

THE PRINCIPLES of the celestial sun sight in 4 steps:

##### THE PRINCIPLES of the celestial sun sight: step 1

While a thread stretched between the Sun center and the Earth center strikes the Earth’s crust at the point GP.

GP stands for Geographical Position (Sometimes referred to as ‘PG’.

Observer 1, 2 and 3 see the sun under the same height so they are on the same position circle (red circle)

The sextant turned towards the point GP during the observation.

The (LOP) : line of* *position is a portion of the circle that is:

locally assimilated as a straight line!

The L.O.P is therefore perpendicular to the direction of the sun.

Note that the coordinates of point GP can be determined using the Sun’s declination and the Greenwich Hour Angle, as found in the Nautical Almanac.

However, these coordinates are only useful in the context of our spherical solution for the formulas.

##### THE PRINCIPLES of the celestial sun sight: step 2

Ho (observed height)

On our unknown but true position, we measure the height of the sun above the horizon with a sextant.

After making all necessary sextant corrections (which will be explained later), we modify this height as if we were at the center of the earth and call it the observed altitude.(Ho)

With this observed height (Ho), there is a position circle.

Remember locally assimilated as a straight line.

However, we do not have a reference to draw it on the map.The point GP is probably too far away to use it in our navigation map.

##### THE PRINCIPLES of the celestial sun sight: step 3 :

Hc (calculated height)

At our dead reckoning position, we use the exact time of the observation, the nautical almanac, and the spherical trigonometry formulas to calculate the height of the sun, which is referred to as the calculated height (Hc).

That is to say Center Earth, Center Sun.

To summarize these spherical trigonometry formulas will always calculate from Center Earth, Center Sun, or Center Star, Moon, or Planets.

At the same time we also calculate with the spherical trigonometry formulas the direction of the sun. (azimuth)

The calculated height (Hc) also has its own position circle.

##### THE PRINCIPLES of the celestial sun sight: step 4

compare Ho et Hc:

The Ho and Hc values are completely independent of each other.

Naturally, the calculated height includes the exact time of the observation with the sextant

While our dead reckoning (estimated position) is not too far off from our true position, we can compare the circle of positions Ho with Hc.

To clarify we can estimate that the circle of position Ho is parallel to the circle of position Hc, since the two circles are close.

It is preferable that the dead reckoning uncertainty does not exceed more than 30 miles.

In other words using dead reckoning on the map and the direction of the sun, we can draw a Line of Position Hc perpendicular to the direction of the sun that passes over our dead reckoning position.

That’s our reference!

The distance between the two lines of positions is called the intercept and we adopt Ho – Hc = intercept.

In practice, neither the Line of Position (L.O.P.) for our Dead Reckoning (D.R.) position nor the circles are plotted or drawn.

##### THE PRINCIPLES of the celestial sun sight: step 4 …

Imagine that your estimated position based on dead reckoning is actually your true position.

At this moment, you use a sextant to measure the height of the sun and calculate its height using your formulas. In this scenario, the observed height will be equal to the calculated height.

Now, the intercept will be zero, and the two lines of positions join as they pass over the dead reckoning position perpendicular to the sun’s direction.

##### Firstly how to calculate the intercept?

THE PRINCIPLES of the celestial sun sight

##### To clarify if the intercept is positive or negative:

##### Intercept (**+**) Ho> Hc

+ “towards the sun”

##### Intercept( -) Hc > Ho

– “away from the sun”

##### THE PRINCIPLES of the celestial sun sight: How to draw the LOP on the map:

In conclusion four examples

+ “towards the sun” (positif)

– “away from the sun” (negatif)

In fact we will learn how to construct a map ourself.

### THE PRINCIPLES of the celestial sun sight: Additional Knowledge:

Concerning the time interval between two Lines of Position (LOPs), the process of transferring a LOP, and understanding the uncertainty area associated with a LOP

Sun’s LOP running fix and the uncertainty zone.

Day planning with the two sun LOP’s

Using the local apparent noon sight latitude with the sun is a widely adopted technique in maritime navigation.

It offers a straightforward and swift approach for determining latitude.