This training will introduce navigators to the fundamentals of the celestial navigation. After completing the course, the navigator will be able to:

- Name the coordinates that define a location on Earth, a location on the sky, and a location as viewed from the rotating Earth.
- Define "geographic position" with respect to the position of a celestial object.
- Define zenith and zenith distance.
- Describe how one measures zenith distance.
- Describe the relationship between zenith distance and geographic position.
- Define lines of position (LOPs), and what intersecting LOPs determine.
- List the steps that a navigator performs as he/she plots lines of position; describe how to plot LOPs on a chart.
- Summarize the accuracy of positions determined using celestial navigation.
- List some advantages and limitations of celestial navigation.
- List the information required to obtain a celestial navigation fix.
- List the instruments required to make an observation.
- List the necessary corrections to a sextant measurement, also known as sextant altitude or Hs, to produce an observed altitude or Ho.
- Describe the Intercept Method.
- Describe why an "assumed position" is used.
- Describe how to calculate the intercept distance using Ho and Hc.
- Describe how the intercept distance, azimuth, and the moniker HoMoTo are used to plot a LOP.
- Describe the special publications used to determine Hc and Zn of an object.
- Describe the contents of The Nautical Almanac and Pub 249, and how they are used in sight planning.
- List the different navigational sights taken during a 24-hour period on a naval passage.
- Define the concept of a running fix and explain how to plot it.
- Describe how mid-morning and mid-afternoon Sun sights can constrain a running fix.
- Summarize the concept of a Local Apparent Noon Sun sight, and how it can yield a fix.

The main idea behind celestial navigation is that Earth is a sphere (nearly), and the sky is a sphere—called the celestial sphere—that surrounds it (it appears that way, anyway).

So...if we know what star is directly overhead, we know where we are on Earth. Although there is often no star directly overhead, using stars’ positions relative to ours on Earth provides information about our location. Let's dive in a little deeper.

Most people understand that a location on Earth is described by a set of coordinates, the location's **longitude** and **latitude**. (Height is often included, but we will ignore it for now). The **geographic coordinates** are unique. That is, there are no two places on Earth that have the same longitude and latitude.

Our longitude is an angular measure along Earth's equator of how far East or West we are from the **prime meridian** that goes through Greenwich, England. For example, if we are standing in Greenwich, our longitude is zero. If we are in Warsaw, Poland our longitude is 21 degrees East. Lima, Peru's longitude is 77 degrees West.

Our latitude is an angular measure of how far North or South from the equator we are. For example, if we are exactly on the equator, our latitude is zero. At the North Pole, our latitude is 90 degrees North; at the South Pole, our latitude is 90 degrees South.

For each of the locations on the map, use the grid lines to obtain an approximate latitude and longitude. Enter the coordinates in whole degrees in the text fields below, then click Done to view the answers.

Buenos Aires, Argentina

Nuuk, Greenland

San Francisco, California

Mumbai, India

Sydney, Australia

Buenos Aires, Argentina 35° S, 58° W

Nuuk, Greenland 64° N, 52° W

San Francisco, California 38° N, 122° W

Mumbai, India 19° N, 73° E

Sydney, Australia 34° S, 151° E

Similar to the **geographic coordinate system** of longitude and latitude, there is a coordinate system for the celestial sphere. In navigation, the **celestial coordinate system **uses **Greenwich hour angle** and **declination** to uniquely describe the positions of bright stars, the Sun, Moon, and planets.

There is a **celestial equator,** which is an imaginary line on the sky directly above the Earth's equator.

Given a star in the sky, its **Greenwich hour angle**, often called **GHA**, is an angular measure along the celestial equator of how far West it is from the meridian at Greenwich, England. Its **declination**, often just called **D****ec**, is an angular measure of how far North or South it is from the celestial equator. A star directly on the celestial equator has a declination of 0 degrees; one directly above the North pole is at declination 90 degrees North.

Based on the coordinates shown in the image, what is the declination of the star Arcturus?

The correct answer is b.

The declination of Arcturus is 19 degrees North. Interestingly, 19 degrees North is also the latitude of Hawaii. Why is this important? Find out by continuing through the pages ahead.

As you progress through the content, notice that the conventional format to express Earth coordinates is to write the zonal (north-south) value first: (latitude, longitude). Celestial coordinates typically express the meridional (west-east) value first: (GHA, declination).

At this point you may see a link between the geographic coordinate system on Earth—that is longitude and latitude, and that of the celestial sphere—Greenwich hour angle and declination. And there is a connection. For every coordinate on Earth, there is exactly one corresponding point in the sky that is directly overhead. The point directly overhead is called our **zenith**. Conversely, for every star in the sky, there is exactly one location on Earth directly under the star. We call this the star's** geographic position, **abbreviated **“GP”.**

It is worth noting that in this module, we often use the word "star" for any navigational object in the sky. In reality, we can (and do) use the Sun, Moon, and bright planets as navigational objects. The concepts are the same so we will simply use the word ``star" as a general term for all of these objects.

So how are celestial objects used in navigation? If we look up and there is a star at our zenith (that is, directly overhead), then we are at the star’s geographic position.

If we know the star's Declination and Greenwich Hour Angle, or GHA, we know our latitude and longitude. Our latitude is simply the star's Declination. And our longitude, measured westerly, is the star's GHA.

You will recall that the declination of Arcturus was 19 degrees north. When Arcturus is at zenith over Hawaii, its geographic position is located at a precise coordinate on the island, with latitude and longitude corresponding to the declination and GHA of Arcturus. Thus, Hawaii’s latitude is 19 degrees north. We can easily verify this on a map.

Here is an important concept to remember. A star's geographic position, that is, its (west longitude, latitude), is its (GHA, Declination).

This is the basic idea of how a star’s position can be used to find our location on Earth.

What is the Geographic Position (lon, lat) of Star A?

The correct answer is b.

The GP of Star A is 70 degrees W, 45 degrees N. A star's geographic position (GP) = its (west longitude, latitude) = its (GHA, Declination). Note that the order of these coordinates is reversed from the typical (latitude, longitude) values denoting cities, landmarks, or other locations on Earth.

Identify the position (GHA, Dec) of each celestial object shown in the image.

Star B GHA Dec

Star C GHA Dec

Star D GHA Dec

Star B GHA: 90° Dec: 15°N

Star C GHA: 40° Dec: 12°S

Star D GHA: 20° Dec: 30°N

Note that because GHA is measured westward from 0 to 360 degrees, the abbreviation "W" to specify direction is not included.

Let's recap. At this point we know that for each star, there is only one place on Earth where it is directly overhead; that is, the star is at the** zenith**. This location on the surface of the Earth is known as the star's **geographic position**. Therefore, if a star is at our zenith, we are at its geographic position. If we know the star's GHA and Dec, we know our longitude and latitude through the relationship: Geographic position of a star = its (west longitude, latitude) = its (GHA, Dec).

There are two main problems that we have to overcome. The first problem is that due to Earth's rotation, the GHA of every object changes continuously. (Declination changes also, but much, much more slowly). So how do we know the star's GHA and Dec?

The second problem is that there usually is not a star directly overhead.

The first problem is solved with a good watch and an almanac. Today, good watches are inexpensive, so this is not an issue (Note: this was not true 200 years ago). And the United States Naval Observatory in the US, along with Her Majesty's Nautical Almanac Office in the United Kingdom, produce the book, "**The Nautical Almanac**” each year. Using The Nautical Almanac, or some trustworthy apps or web sites, it is easy to find the GHA and Dec for bright objects throughout the year. We will come back to The Nautical Almanac later in this module.

The second problem—of there rarely being a star directly overhead—is solved by understanding that if a star is not quite directly overhead, then we are not quite at its geographic position. For example, if the star is one degree away from our zenith—called **zenith distance**—then we are one degree away from its geographic position. If the star is two degrees away from our zenith—that is, its zenith distance equals two degrees—then we are two degrees away from its geographic position. And so on.

But what does "being one degree away from its geographic position" really mean in terms of miles? On Earth, one degree of latitude equals 60 nautical miles. (A nautical mile is just a little bit larger than a normal, "statute" mile: 1 nautical mile (nm) = 1.15 miles). So if we measure a star's zenith distance, and it is one degree, then we are 60 nautical miles from its geographic position. If the zenith distance is two degrees, we are 120 nautical miles from its geographic position. And so on.

In navigation, angular measurements—such as zenith distance—are typically not whole degrees, but also fractions of a degree. Similar to how an hour is made up of 60 minutes of time, each degree is made up of 60 minutes of arc, otherwise known as arcminutes. Since we just learned that on Earth one degree of latitude equals 60 nautical miles, then one minute of arc equals one nautical mile. An easy way to remember this is the saying "**a mile a minute**."

This is an important point to remember: each minute of arc equals one mile (that is, "a mile a minute"), and each degree equals 60 miles.

Select the correct term for each of the following definitions:

We have learned that if we measure a star's zenith distance—that is, how many degrees it is away from being directly above us—then we know how far away we are from its geographic position.

For example, let's say we measure a star's zenith distance as exactly 43 degrees. Because each degree is 60 nautical miles, we are 2580 miles from its geographic position. But, we don't really know exactly which direction the geographic position is. In other words, we are somewhere on a circle with a radius of 2580 miles centered at the geographic position. This is called a **circle of position**, because we are located somewhere on the circle.

So how do we find out where on the circle we are? Simple: measure more than one star! Each star will have a different sized circle, centered on a different geographic position. For example, let's say we measure the zenith distance of star number two, and it is 21 degrees. We then know that we are 1260 miles from star number two's geographic position (1260 = 21 x 60). So our second **circle of position** has a radius of 1260 miles centered at star number two's geographic position.

Our position must be at the intersection of all of the circles.

When looking at a large area of Earth, it is easy to see why the arcs centered at a geographic position are called "circles of position".

Finding our position using traditional celestial navigation relies on graphing our position on a chart. And using a large area, such as a globe, is not very good for graphing precise positions. Therefore we tend to look at a much smaller region. Within a small area of maybe a few hundred miles, we’ll often see only a small arc of the "circles of position", which tend to look more like lines than circles. In fact, this is very useful because dealing with lines is much easier, mathematically and graphically, than dealing with circles. If we can limit ourselves to dealing with a small arc of a circle, we can treat it as a straight line, which we call a line of position, or LOP. Our position is at the intersection of the lines. This is called a fix.

This is another important concept you should remember: when we draw a line of position on a chart, our location is somewhere on that line. We don't know exactly where. The intersection of multiple LOPs gives our position, or fix.

In practice, measurements have a finite accuracy, so each line of position may have a small error associated with it. In other words, we may not be exactly on the LOP, but we should be close. Although two lines of position are required to get a fix, in reality at least three LOPs are generally used. In this way, a navigator can estimate how well the fix has been made, and can also see if a major mistake was made.

You draw two lines of position on a plot. What is the best guess for your current position?

The correct answer is b.

The best guess for your current position will be the intersection of the LOPs.

How many LOPs are generally required to provide an accurate fix?

The correct answer is c.

The correct answer is c. Three or more LOPs provide a great detail of confidence in the fix, and can reveal if a major mistake was made in plotting one of the lines of position.

There is one major point we have not addressed. We know that we need to measure the zenith distance to objects, but how does a person actually do this? This is where the sextant is used. A sextant is actually a large-angle measuring instrument. That is, it is used to measure the angle between two objects. Generally the two "objects" we use are the horizon and a star.

We use the horizon instead of the zenith because there is no way to obviously tell precisely where the zenith is. But we can see the horizon (well, when it is clear and not too dark, anyway), and we know that the horizon is about 90 degrees away from the zenith, so the horizon makes an excellent proxy for the zenith.

We'll talk more about the sextant in a few moments.

What we actually measure with a sextant is the angle between the horizon and a star; this is called the star’s **altitude**. It is simply how high in the sky a star appears from the horizon. In other words, altitude is angular **height**. In fact, navigational "shorthand" for altitude is "H". But note that this isn't height as we normally think about it in terms of feet or meters, it is height in terms of an angle from the horizon.

This is an important point, so let's spend a few moments on this. If we were to ask, "how high in the sky is that star?", how would you answer? You may think in terms of feet, or meters, or miles. But these really don't make sense for something in the sky that is unimaginably far away. Instead, to measure the height of something in the sky we use angles.

To illustrate, the distance between our horizon and our zenith is about 90 degrees. So an object about halfway between our horizon and our zenith would have an angular height of about 45 degrees. We say its altitude is 45 degrees.

Roughly speaking, if you hold out your arm, the width of your little finger is about 1 degree

and your closed fist is about 10 degrees. So remember that altitude is height, but it is measured by an angle. It is abbreviated H. We will use "H", meaning altitude, later in this module.

There is a very simple relationship between zenith distance and altitude. Zenith distance = 90 degrees minus altitude.

For example, if a star has an altitude of 89 degrees, its zenith distance is 90 - 89, or 1 degree. That is, it is very close to being directly overhead.

A star near the horizon might have an altitude of 5 degrees; its zenith distance would be 85 degrees; a very big angle between the zenith and the star.

Up to this point we have been talking about the zenith distance of objects. But because what we actually measure is altitude, the navigational publications tend to use altitude, not zenith distance. This does not cause a problem, and actually keeps us from having to convert between the two.

From the list below, select the correct values for each object shown in the figure.

Recall that zenith distance = 90 degrees minus altitude. Star A has an altitude of 60 degrees and zenith distance of 30 degrees. Star B has an altitude of 75 degrees and zenith distance of 15 degrees.

A marine sextant is used to measure the angle between the horizon and a celestial object. To use the sextant, we look through a small telescope at the horizon mirror.

The horizon mirror allows some light directly through so that on one side we see the horizon. It also has a reflective surface so on the other side we see an image from the index mirror. The index mirror is adjusted, using the index bar, until the image of the object is visible. A fine adjustment is made with the micrometer drum to bring the image of the object to line up with the horizon. The graduated arc and vernier is read to get our sextant altitude. In the animation, you see the image of the lower limb of the Sun is brought to the horizon.

**Directions:** Drag-and-Drop the labels to the boxes in red corresponding to the elements they represent on the graphic, then click Done.

Your answers should match those shown in the figure.

What is the altitude of Star A? Enter your answer in degrees.

The altitude of Star A is 60 degrees. Now that we’ve learned how a sextant provides information about altitude and therefore zenith distance, let’s examine how these measurements relate to our position, or fix.

The graphical technique used worldwide to plot our "fix" is called **The Intercept Method**. The Intercept Method, and the steps involved in obtaining a fix, are described in the next section of this module.

At this point we have learned the basic ideas behind celestial navigation. We know a star's geographic position at any time by consulting an almanac.

We can determine how far away we are from a star's geographic position by measuring the star's zenith distance. (In reality, we measure its **altitude** above the horizon using a sextant.)

This gives us a circle of position, and our position is somewhere along that circle. Doing this for more than one star will result in intersecting circles. Our position—or **fix**—is at the intersection of these circles.

The concept is straightforward, especially when looking at a globe. But what technique is utilized by sailors? That is, how is it really done in practice? As we progress through the next sections, Navy Chief Quartermaster Tim Sheedy will demonstrate these techniques and then take us through a typical day in the life of a navigator.

The technique that is used worldwide is called **The Intercept Method.** It is also sometimes called the Altitude-Intercept Method or Marcq Saint Hillaire Method. It is a graphical technique; that is, it utilizes a chart—often paper—to show lines of position and a **fix**.

Its use is so extensive largely because the steps involved are straightforward, and a navigator does not need to perform complicated math. It can only be used if we have some idea about where we are, to within a hundred miles or so. This is almost never a problem because we usually have some idea about our location.

The process of taking a sextant measurement, called a **sight**, and making it into a line of position is called a **sight reduction**.

Before going into detail of the sight reduction steps used in the Intercept Method, let's look at how the method works in a broad sense. For this process, we’ll assume that we have access to a celestial navigation calculator, which will provide some of the output values needed for mapping a line of position.

The main steps of the Intercept Method are:

1. Determine the "**Observed Altitude**", also called "**Height observed**" or **Ho**, of the star. This is our sextant measurement plus some corrections.

2. Select a position close to where we think we are. This is called our ``**Assumed Position”**, or **AP**.

3. Get the star's **"Computed Altitude"**, also called **"Height computed"** or **Hc,** and the star's **azimuth**, **Zn**. These are the altitude of the star and direction to its geographic position if we were exactly at the Assumed Position.

4. On a Universal Plotting Sheet, plot our AP and draw a line through it in the direction of the azimuth, Zn. This **azimuth line** is the direction towards the star's geographic position. Our LOP will be perpendicular to the azimuth line.

5.Determine where our LOP **intercepts **the azimuth line. This uses the Observed Altitude, Ho and the Computed Altitude, Hc. Mark this intercept position on the azimuth line.

6. Draw our **line of position**. It is simply a line perpendicular to the azimuth line at the intercept point.

7. Observe more than one star to get crossing LOPs; our** fix **is at the intersection.

Broadly speaking, these steps form the basis of the Intercept Method. In the next sections, let's look at each step in a little more detail, demonstrating the Intercept Method using a sextant observation of the star Capella as an example.

Before we move on, try the exercise below and match each definition to its proper term.

Match each term below to its appropriate definition.

If you struggled with any of these definitions, you might wish to review the previous pages. You'll learn more about how these values are used in celestial navigation as you continue through the module.

The strip form acts as a worksheet to step you through the sight reduction process. A blank strip form is available for printing here (PDF).

The first step in using the Intercept Method is to determine the "Observed Altitude" of the star. This is also called "Height observed," with a notation of Ho. Ho is important to remember.

By its name, you may expect that the observed altitude is what we read from a sextant. The sextant reading is the major portion, but Ho is actually what we would measure with a sextant if:

- the sextant were perfect;
- and if our eye were at sea level;
- and Earth's atmosphere didn't bend light;
- and if we observed the exact center of the object.

These corrections to our sextant reading sound like they are difficult, but they are not.

They are simple additions or subtractions, and they have been pre-computed in tables in **The Nautical Almanac**, which is an annual publication by the U.S. Naval Observatory and Her Majesty's Nautical Almanac Office. Also, each step is laid out in a page, called a **strip form**, used by navigators. You can think of a strip form as a paper spreadsheet that details each step in the sight reduction process.

The strip form acts as a worksheet to step you through the sight reduction process. A blank strip form is available for printing here (PDF).

Ho is the observed altitude of the star once corrections have been applied. For our example, the star is Capella. The sextant reading is 29° 46.0' at time 08:00:00 UT (or GMT) on 17 July 2018. Height of eye = 10 ft. Index correction = -0.4'. Chief Sheedy uses the information to determine observed altitude, Ho.

The sight reduction applies each of the corrections to the sextant altitude (Hs) to obtain the observed altitude (Ho). The first correction is the index correction, unique to the sextant being used.

The second correction depends on the observer’s elevation above sea level, and is called the **dip correction** because the higher we are, the lower—or more dipped—the distant horizon appears. The dip correction accounts for this visual difference. It is in a pre-computed table in the inside front cover of The Nautical Almanac. Lastly, we apply **altitude corrections**, which are essentially corrections due to Earth's atmosphere bending light rays. These are in pre-computed tables in The Nautical Almanac. For the Moon, Venus, Mars, or if the temperature is extreme, we’ll apply additional corrections.

Watch and listen as Chief Sheedy demonstrates the sight reduction process for an 0800 UT sight of the star Capella.

Let’s review. Select the appropriate label for each definition below:

The next step in using the Intercept Method is to determine an estimate of our position. This is called an **Assumed Position, or AP**. The exact location chosen for an AP isn't very important as long as it is within a hundred miles or so of our actual location. Often an AP is selected to make calculations easier (see the Supplemental section for further discussion on AP).

For our demonstration, we will use our **Dead Reckoning** position, noted **DR**, as our assumed position. Dead Reckoning is the process of determining a ship's approximate position by advancing a previously known position for course and speed. Determining a DR position is outside the scope of this module. The U.S. Navy has specific rules to obtain a DR position; only the ordered courses steered and ordered speeds are used. Drift due to current is not factored in.

Continue to the next page to view as Chief Sheedy discusses dead reckoning and assumed position and how to set up a plotting form for the current position.

For our example, the DR position is 31° 25' N, 54° 12' W. Chief Sheedy sets up a Universal Plotting Sheet centered at 32° N, 055° W and plots the DR position. The Universal Plotting Sheet contains a compass rose and lines for marking latitude and a mid-longitude.

USNO Celestial Navigation Calculator: **http://aa.usno.navy.mil/data/docs/celnavtable.php**

The next step in the Intercept method is to get the **Computed Altitude** and **Azimuth** of the star. This is done for our Dead Reckoning position at the time we took the sextant observation.

Computed altitude, also called **Height computed** and noted **Hc**, is what the observed altitude, **Ho**, would be if we were exactly at the DR position. Likewise, the azimuth is the direction to the star's geographic position if we were exactly at the DR position. Remember our DR position is only an approximate position. We may be there, but we most likely are not.

Up until now we have not defined what **azimuth** is. Azimuth is a measure along the horizon to the location of an object. We measure it from true north toward the east. Like altitude, it is an angular measure. Because the horizon appears like a full circle, azimuths run from 0 to 360 degrees; both 0 and 360 degrees are due north. An azimuth of 90 degrees is due east. Due south is 180 degrees, and due west has an azimuth of 270 degrees. In this module, the azimuth is the direction to the star's geographic position.

For the purpose of this module, we will assume that we have access to the internet. The U.S. Naval Observatory has a free celestial navigation calculator online.

USNO Celestial Navigation Calculator: **http://aa.usno.navy.mil/data/docs/celnavtable.php**.)

Let's use it to determine Hc and Zn.

For our example, we put in the date and time in the appropriate box. 2018, July 17. Universal time of the observation is 8 hours 0 minutes 0 seconds. Our dead reckoning position is latitude 31 degrees 25.0 minutes North and 054 degrees 12 minutes West.

The output is a table with visible objects for that date, time, and location.

In our example, we observed Capella. From the table, we can see the computed altitude is 29 degrees 28.8 minutes. And the azimuth is 51.5 degrees.

You will notice other data on this table as well, such as the GHA and Declination of each object. You may remember that this corresponds to the geographic position of the body. Other information is explained in the "Notes on the Data" section of the input page.

Although not necessary for learning the fundamentals of how celestial navigation works, in order for celestial navigation to be a true emergency navigation system, navigators should know how to determine Hc and Zn without the use of electronics. Strip forms will lead us through the steps, and special publications such as **The Nautical Almanac** and **"Publication No. 229, Sight Reduction Tables for Marine Navigation"** are published so complex math is unnecessary.

For more on determining Hc and Zn by hand, see the Supplemental Material section.

Now that we’ve determined the computed altitude (Hc) and the azimuth (Zn), we’ll continue through the intercept method and watch and listen as Chief Sheedy leads us through plotting the line of position.

For our example, Capella's computed altitude (Hc) and azimuth (Zn) at DR (AP) and time of observation are: Hc = 29° 28.8' and Zn = 051.5°. Chief Sheedy marks azimuth line, uses Ho-Mo-To to determine the intercept, and plots the LOP.

To apply the intercept method, we first plot the **azimuth line** through the DR position. The azimuth is the direction toward (or away) from the star's geographic position.

To determine if our LOP is toward the geographic position or away from it, we use the moniker "**Ho-Mo-To**" (pronounced "Hoe - Moe - Toe"). This means observed altitude (Ho) More (Mo) Toward (To). If the Ho is more than the Hc, then we plot the intercept toward the GP, that is, toward the direction of the star. **Ho-Mo-To** is an important mnemonic to memorize.

Conversely, if Ho is less than Hc, our intercept is away from the GP, that is, away from the star.

In our example, Ho is **more** than Hc, so our intercept is 12.0 arcminutes **toward** the GP. Chief Sheedy demonstrates this process to plot a line of position for Capella.

Remember, our position is somewhere on the line of position we’ve marked on the plot.

In order to find out where our position actually is, we need to take another sextant measurement of a different star in a different direction of the sky. Performing the sight reduction in the exact same way, we will get another LOP that crosses the first one. Our location is at the intersection. In reality, we want at least three LOPs, which are parts of larger circles of position, that should all cross at about the same spot. How closely they intersect gives us an indication of the accuracy of our fix.

Before moving on, try the following exercises:

1. A star's observed altitude (Ho) is 19 degrees 17.1 minutes and the computed altitude (Hc) is 19 degrees 30.2 minutes. Where along the azimuth line do you mark your intercept?

The correct answer is b.

Because Ho < Hc, your intercept will be away from the GP.

2. On a plot, the line of position is perpendicular to the azimuth line. (Select True or False.)

The correct answer is a.

This statement is True. A line of position (LOP) is always perpendicular to the azimuth line.

3. Your position is located at the intercept of the line of position and the azimuth line. (Select True or False.)

The correct answer is b.

This statement is False. Your position is located somewhere along the line of position (LOP), but not necessarily at the intercept. Determining a location requires more than one LOP. Your location will be the intersection of the LOPs. This is why a minimum of three LOPs are often needed to obtain an accurate position.

At this point we have learned the basic ideas behind how celestial navigation works and how to use the Intercept Method to plot our lines of position on a chart.

Briefly, we know a star’s geographic position at any time by consulting an almanac or a trusted web site or app. We determine how far away we are from a star’s geographic position by measuring the star’s altitude using a sextant. When we correct the sextant observation for such things as dip and index correction, we have an observed altitude, or Height observed, Ho. We select a position close to our present location, called an **Assumed Position**. Any will do, but we have chosen to use our **Dead Reckoning** position (DR) at the time of the sextant observation. We determine the computed altitude, or Height computed, Hc, and azimuth of the star at the time of the sextant observation and the DR position. (In our example, we used the U.S. Naval Observatory’s online calculator, but this could also be done using special publications produced by the U.S. and United Kingdom Navies.)

Plotting our LOP involved drawing the azimuth line through our DR position, determining where the LOP intercepts the azimuth line, and drawing the LOP perpendicular to the azimuth line. A second and third observation gives us a fix.

The examples used throughout this module up to this point use stars to demonstrate the concepts and techniques. Stars make very good navigational objects, but they are not the only ones. The bright planets (Venus, Mars, Jupiter, and Saturn) are also used, as are the Sun and the Moon.

The Sun is especially useful because it is easy to find and can be observed during the day.

In the following segments discussing a typical day in Navy navigation, we will see how the Sun plays a very important role. Additionally, the examples shown up until now did not consider the motion of the ship. In the next segments, we will see how this is taken into account, leading to **running fixes**.

To demonstrate the celestial navigation techniques used in the U.S. Navy, we will go through many of the major duties used to determine a ship’s position throughout a typical day. Each of the following duties will be described in more detail later in the module.

We will demonstrate five procedures used during a typical day:

- We begin with a known position shortly after midnight. From that, we make a DR plot until we obtain a three-star fix in morning twilight.
- We take a Sun line a few hours after sunrise, then a Local Apparent Noon Sun line at noon.
- We obtain an
**estimated position**at noon. - A few hours later, we take another Sun line. This will be combined with the morning and noon Sun lines to obtain a
**running fix**. - Finally, we obtain another three star fix in evening twilight.

For our “Day in the Life of a Navy Navigator”, we will use the date 17 July, 2018, although any date could be chosen. Time will be given in Universal Time, often called Greenwich Mean Time.

A DR plot is started from a previous established position. For our example, the last known electronic fix was at 04:00:00 UT, at location 38° 08.0' N, 053° 52.2' W. Ordered course and speed are 315°, 6 knots. Here, Chief Sheedy plots the last known electronic fix and the DR position up to 0800 UT.

A Navy navigator’s major concern is the accurate determination of the ship’s position. But it is not enough to know only the present position; equally important is calculating the ship’s position for a desired time in the future. For this, the navigator obtains the ship’s approximate present or future position by using a Dead Reckoning Plot.

In the U.S. Navy, a DR plot is always started from an established position; that is, an estimated position or a fix. In the U.S. Navy, only ordered courses and ordered speed are used to determine a DR plot. The effect of current is not considered in determining a DR position.

Although some of the details of dead reckoning are beyond the scope of this module, let’s look at our navigator making a DR plot.

Before navigators can start a series of observations, they need to decide which objects will be visible, and which will give the best lines of position. To take a sight using stars, it must be light enough to see the horizon, but dark enough to see the stars. The period when the sky brightness allows for this is referred to as "nautical twilight." The times of twilight and sunrise are determined by using The Nautical Almanac; strip forms can be used to help.

The decision of which objects to observe must be made well before morning twilight. Two methods are often used to identify the best objects. We could use a 2102-D Star Finder, also known as a Rude Star Finder, which is a plastic celestial map that shows the positions of stars used for navigational purposes.

Or we could use the book, "Publication 249 Sight Reduction Tables for Air Navigation (Selected Stars)," also known as "Pub 249." Both the Star Finder and Pub 249 indicate which objects are best to observe, and also give us their rough positions in the sky so we know where to look for them.

*Pub 249 is also known as AP 3270 outside of the U.S.*

Using the Rude Star Finder and Pub 249 are not difficult, but are outside the scope of this module. For our example location and date, the three best stars are Capella, Deneb, and Fomalhaut.

The first star in our three-star fix is Capella, with sextant altitude 29 degrees 46 minutes and the same data used earlier to demonstrate the Intercept Method. Here, the sight reduction is performed and Chief Sheedy plots the LOP for Capella.

The three stars observed are Capella, Deneb, and Fomalhaut. Let’s say each were observed at the same time, at 8:00:00 UT. Let’s reduce Capella first. Keen viewers will notice that we already reduced Capella in the Intercept Method segment of this module, so we will go through the steps quickly. If you need further explanation of a step, go back to that segment.

For our example, at time 08:00:00 UT on 17 July 2018, Hs for Deneb = 49° 02.1' and Hs for Fomalhaut = 27° 00.4'. Height of eye = 10 ft. Index correction = -0.4' and we are ready to perform the sight reduction.

Here is the strip form for our three-star fix. For Capella, the sextant altitude (Hs) is noted and the sight reduction has been completed. The sextant altitudes and index and dip corrections for Deneb and Fomalhault are also shown on the form.

From the Nautical Almanac, the appropriate Altitude Correction for Deneb is -0.8 arcminutes.

Based on the values, select the correct Ho value for Deneb:

To find the observed altitude Ho, add the index correction and dip correction, and apply them to the sextant altitude, Hs. Then find the altitude correction and apply that value to obtain Ho. Subtracting degrees and minutes might appear somewhat complex, but it is not. In the case of Deneb, the index correction and dip correction sum to -3.5'. We apply that value to the sextant altitude, Hs. Remember that 1 degree = 60 minutes, so we can always make the subtraction easier by adding 60 minutes to the minutes value and taking 1 degree away from the degrees values.

For Deneb, the corrections applied to Hs can be shown as 49° 02.1' - 3.5' = 48° 62.1' - 3.5' = 48° 58.6'

Then, apply the altitude correction: 48° 58.6' - 0.8' = 48° 57.8'

Here is what the your results for Deneb look like on the partially completed strip form.

You can now use the USNO website at http://aa.usno.navy.mil/data/docs/celnavtable.php to find the computed altitude (Hc) for Deneb at 0800 UT 17 July 2018. Remember, your assumed position is the DR and equals 31° 25.0 minutes N, 054° 12.0 minutes W. Once you have located the computed altitude (Hc), enter that value here.

Enter the azimuth value (Zn) for Deneb here:

The computed altitude (Hc) for Deneb should be 48 degrees 52.8 minutes and azimuth is 304.6 degrees. If you did not get the correct values, or if you still have questions about observed altitude (Ho) versus computed altitude (Hc), you may wish to review the previous content and then work through the exercise again.

How do Ho and Hc compare for the star Deneb?

The correct answer is a.

In this case, Ho is 5.0 arcminutes greater than Hc.

Based on your answer to the question above, where should the intercept for the line of position for Deneb be marked? (Choose the best answer.)

The correct answer is c.

The intercept should be marked along the azimuth line, 5.0 arcminutes toward the GP.

Let’s repeat the process for the Fomalhaut observation.

From the Nautical Almanac, the altitude correction for Fomalhaut is -1.9 arcminutes. What is the resulting Ho?

The Ho for Fomalhaut is 26 degrees 55.0 minutes. If you did not get this result, you may wish to work through the calculations again to locate your error. Remember that the index and dip corrections will the same as those used for Capella and Deneb, since we took the observations from the same location at the same time.

What is the computed altitude (Hc) and azimuth (Zn) for Fomalhaut at 0800 UT, 17 July 2018, based on the current assumed position (31° 25.0' N, 054° 12' W)? Recall that the U.S. Naval Observatory has a Celestial Navigation calculator available at http://aa.usno.navy.mil/data/docs/celnavtable.php.

Computed altitude (Hc):

Azimuth (Zn):

Using the USNO Celestial Navigation calculator, you find that the computed altitude (Hc) for Fomalhaut is 27 degrees 08.4 minutes. The azimuth (Zn) is 195.9.

For the star Fomalhaut, you found an observed altitude (Ho) value and a computed altitude (Hc). Based on the numbers, where would you mark the intercept on your plot?

The correct answer is d.

The Ho for Fomalhaut is 26 degrees 55.0 minutes and the Hc is 27 degrees 08.4 minutes. The azimuth (Zn) is 195.9. The intercept should be marked along the azimuth line for Fomalhaut, 13.4 arcminutes away from the GP. Next, Chief Sheedy will demonstrate plotting the three-star fix on the Universal Plotting Sheet.

For our example, Deneb's intercept is 5.0 arcminutes toward, and the azimuth (Zn) = 304.6°. For Fomalhaut, the intercept is 13.4 arcminutes away, and azimuth (Zn) = 196°. Chief Sheedy plots the LOP for each star.

During daylight hours, the Sun is used for determining our position. It makes
a good navigational object because it is easy to find in the sky and easy to
measure with a sextant. However, the Sun will only provide us with one line
of position; at least until it moves significantly across the sky, which takes
a few hours. Therefore we cannot get a fix with one Sun observation,
only a line of position. This is why we call the result a **Sun line**. Later we
will demonstrate how Sun lines taken throughout the day can be combined to
obtain a fix, but for now let’s show how a Sun line is taken.

Whenever looking or observing the Sun we must use the proper filters so we do not damage our eyes! Sextants have filters that are swung into place when making a Sun observation.

A morning Sun observation is routinely taken a few hours after sunrise. The reduction procedure is very similar to that of stars, except we must note f we observed the upper or lower limb. A strip form includes all of the steps in the reductions.

For our example, the morning Sun sight was taken at 11:46:15 UT for a sextant reading of Hs = 35° 59.1' at DR position 31° 55.0' N, 054° 25.0' W. Chief Sheedy applies the corrections and fills out the strip form. Ho = 36° 10.3'; Hc = 36° 25.6', Zn = 86.1°.

Based on the values from the strip form, how does Ho compare to Hc, and how will that relate to the intercept you mark on the plot?

The correct answer is b.

The intercept should be located 15.3 arcminutes away from the GP.

From the strip form for our example, the morning Sun sight's intercept is 15.3 arcminutes away, and the azimuth is 86.1°. Here, Chief Sheedy plots the morning Sun line and plots DR for the next few hours.

The Sun crosses our meridian at noon; more precisely, at Local Apparent Noon, also called LAN. At this moment, the Sun appears to us to be at its highest point in the sky for the day. In other words, its altitude is at its largest. Also at this moment, the Sun is either due North, or due South of us.

(Well, it’s possible that it is directly overhead, but this is very, very unlikely). This means that our line of position observed at this moment will run exactly East-West.

A LAN observation not only provides us with another Sun line, but this Sun line is a measurement of our latitude. Because our local apparent noon LOP runs directly East-West, this is equivalent to measuring our latitude. You could go through all of the steps for a full sight reduction. However, it is actually easier than that, and a different strip form is often used specifically for a LAN measurement. Our latitude is simply the Sun's zenith distance, plus or minus the Sun's Declination. Because a strip form is used, you do not need to memorize this formula: Latitude = Zenith distance(Sun) +/- Declination(Sun).

A LAN sight is taken when the Sun is at its highest position in the sky. For our example, the time of LAN was found using UNSO Complete Sun and Moon Data for One Day: http://aa.usno.navy.mil/data/docs/RS_OneDay.php.

LAN was measured at 15:45:05 UT, with a sextant reading of 79° 50.0'. Using the strip form, Chief Sheedy finds the observed latitude is 32° 05.3' N.

There are two methods for observing LAN. We can observe the Sun with a sextant, making constant adjustments as needed, until the Sun begins to descend to a lower altitude. Or we can record sights at intervals before and after LAN, then approximate when LAN occurred. Because we don’t want to miss this once-per-day event, both methods require a prediction of the time of meridian passage of the Sun at our DR position.

For the purpose of this module, we will assume we have access to the internet. The U.S. Naval Observatory has a free “Rise/Set/Transit calculator” at http://aa.usno.navy.mil/data/docs/mrst.php. One can also use the UNSO Complete Sun and Moon Data for One Day site, at http://aa.usno.navy.mil/data/docs/RS_OneDay.php

.Again, if doing this by hand, strip forms and The Nautical Almanac are used.

At our DR position, we estimate LAN to occur at 15:45.

Note that this is only an estimate because our DR is only an estimate. Because of this, we need to begin our observations several minutes prior to this time in order to be sure we don't miss it. For our purposes, we will observe the Sun with a sextant, making constant adjustments as needed, until the Sun begins to descend. We note the largest sextant reading. This is our LAN measurement.

The observed altitude (Ho) of the Sun at LAN is 79° 2.2'. What is the zenith distance?

The correct answer is c.

Recall that zenith distance is just 90 degrees minus Ho.

In our example, the zenith distance is 10° 57.8'.

The LAN sight provided an observed latitude of 32° 05.3' N. On the plotting sheet, Chief Sheedy plots the LAN line, which runs exactly east-west.

The Sun's true Declination at time of LAN can be found on the U.S. Naval Observatory web site. Without internet access, we could use The Nautical Almanac.

In our example, True Declination is 21 degrees, 7.5 minutes north.

To get our latitude, we either add the Declination to the zenith distance, or subtract it, depending on a few rules.

In our case, we add, and we find our latitude is 32 degrees 5.3 minutes north.

Because a LAN line of position runs exactly east-west, it is easy to plot at our current DR position.

Advanced lines of position can provide position information when making simultaneous observations of multiple objects is impossible. Between the morning and LAN Sun observations, our course was approximately 318° and speed 6 knots. Using these values, Chief Sheedy demonstrates advancing the morning Sun line to the time of LAN.

Ideally, we would like three LOPs from simultaneous observations to obtain a fix. However, sometimes making three simultaneous observations is impossible. For example, during daytime, we only have one object, the Sun, that is observed. We need to wait a few hours in between observations. We can, in fact, use observations, and their resulting LOPs, that are separated by a considerable amount of time. The procedure is called **advancing a line of position**.

The details of advancing LOPs are beyond the scope of this module, but conceptually, to use an LOP from an earlier sight, we must take into account how our ship moved since the earlier sight was taken. Similar to getting our DR, we use the ship's ordered speed and course between the times of the LOPs to get the distance and direction the ship should have traveled. Next, we move the earlier LOP parallel to itself by this amount.

For our example, the morning Sun line was taken at 11:46, and the LAN sight was at 15:45. Our ordered direction and speed remained constant at 317.8 degrees and 5.7 knots. Between the two sights, we therefore should have gone 22.8 miles at a heading 317.8 degrees. On our chart, we **advance our morning LOP** this distance and direction, being sure to move the LOP parallel with itself.

Two Sun lines can be used to provide an estimated position. For our example, Chief Sheedy plots a new estimated position (EP) at 32° 05' N 055° 00.0' W, then plots future DR positions from this estimated position.

We can now get an **estimated position** using the two Sun lines and our DR position. In Navy parlance, the term estimated position is used when we have incomplete data or data of questionable accuracy. Having less than three lines of position constitute incomplete data. Once a third LOP is determined, we will then have a **fix**. A square is used to symbolize an estimated position on a chart.

Chief Sheedy demonstrates marking estimated position on the plot.

For our example, the afternoon Sun observation is taken at 19:35:58 UT from DR 32° 21.5' N, 055° 18.0' W. The intercept is 3.2' away, and Zn = 272.4°. Chief Sheedy plots the resulting afternoon Sun line.

An afternoon Sun observation is routinely taken a few hours after LAN. Observation and sight reduction procedures are exactly the same as those that were done in the morning.

Once again, a strip form includes all of the steps in the reductions.

The morning and LAN Sun lines are advanced to the time of afternoon Sun line to obtain a running fix. Chief Sheedy demonstrates and obtains a new running fix at 32° 21.3' N, 055° 16.0' W.

Similarly to preparing for morning observations, a navigator needs to decide which objects will be visible, and which will give the best lines of position during evening twilight. The times of sunset and twilight are determined by using The Nautical Almanac; strip forms can be used to help.

As with the morning three-star fix observations, we can use either the 2102-D Star Finder or Pub 249 to find which stars are the best objects to observe, and to get their rough positions in the sky so we know where to look for them.

For our location and date, the three best stars are Altair, Spica, and Dubhe.

In our example, the evening three-star fix observations are taken at 23:10:25 UT using stars Altair, Dubhe, and Spica. The DR is 32° 23.5' N, 055° 44.3' W.

For our example, let's say there was an ordered speed and course change after the afternoon Sun line was taken. It gives us a DR position of 32 degrees 23.5 minutes north, 55 degrees 44.3 minutes west at time 23:10:25 UT, when we take our evening sights.

We will maintain the same index correction (-0.4 arcminutes), and the same dip correction (-3.1 arcminutes). The three stars we observed are Altair, Spica, and Dubhe.

The sextant readings are 21 degrees 29.3 minutes; 40 degrees,1.1 minutes; and 41 degrees 33.4 minutes, respectively. Using these, we obtain Ho for each star.

A partially completed strip form is available for printing here. Fill in the values on the form, then enter your results in the boxes below. If you do not have a Nautical Almanac, you can view an altitude correction table here.

Enter the Ho for each star here:

Altair

Spica

Dubhe

Altair 21° 23.3'

Spica 39° 56.4'

Dubhe 41° 28.8'

Using the U.S. Naval Observatory web site, find Hc and Zn for each star. Go to **http://aa.usno.navy.mil/data/docs/celnavtable.php** to get this information. You’ll need to enter the date and time (17 July 2018, 23:10:25 UT) and the assumed position (32° 23.5' N, 55° 44.3' W).

Enter the computed altitude (Hc) and azimuth (Zn) for each star:

Altair Hc Zn

Spica Hc Zn

Dubhe Hc Zn

Altair Hc 21° 19.8' Zn 92.9°

Spica Hc 39° 45.2' Zn 214.1°

Dubhe Hc 41° 33.9' Zn 326.1°

Next, find the intercept for the line of position of each star based on difference between Ho and Hc. Enter your answers in the boxes below.

For each star, enter the distance and direction to intercept:

Altair

Spica

Dubhe

Altair 3.5 minutes toward

Spica 11.2 minutes toward

Dubhe 5.1 minutes away

If you did not get these values, you may wish to recheck your work and make sure you understand each step before proceeding. You can compare your completed strip form with the filled-in version (available as pdf).

The differences, and Ho-Mo-To, are used to plot the three lines of position. Print the pdf of the plotting form to work on drawing the corresponding lines of position. When you have finished, continue to the next page to view Chief Sheedy correctly plotting the three-star evening fix.

Print the completed Universal Plotting Sheet to compare the solution with the plot you created for the three-star evening fix. If your values do not match the correct answer, you may wish to go back and check your work. You can work through this exercise or any of the sight reductions and position plots in the module to practice and hone your celestial navigation skills.

The resulting fix is 32 degrees 13.6 minutes north, 55 degrees 44.6 west. Now that we have our fix, a new DR is started from this position.

Print the completed Universal Plotting Sheet to compare the solution with the plot you created for the three-star evening fix. If your values do not match the correct answer, you may wish to go back and check your work. You can work through this exercise or any of the sight reductions and position plots in the module to practice and hone your celestial navigation skills.

AstroNavigation, Vanderbilt University: https://my.vanderbilt.edu/astronav/

This module introduced you to the basic principles of celestial navigation and provided some examples of how this practice is used by the U.S. Navy. The previous section demonstrated many of the major duties performed during a typical day of celestial navigation.

These include the morning three-star fix; morning, LAN, and afternoon Sun lines; and the evening three-star fix. It is also possible to make observations throughout the night if the horizon is illuminated, such as on a cloudless, moonlit night.

The supplemental segments following provide more details about certain aspects of celestial navigation and will be useful for understanding some finer points of the process.

Remember that celestial navigation requires practice. Now that you have an idea of some of the fundamentals, taking routine observations and using them to plot fixes will help you gain familiarity with the process and allow you to refine your skills. For more information about using celestial navigation, also see AstroNavigation, an online course hosted by the Vanderbilt University BOLD program.

Thank you for completing this training on Principles of Celestial Navigation. To test your knowledge of what you’ve learned in this module, please complete the quiz.

The core purposes of this module are to give viewers a concept of how celestial navigation works, to give an idea of the **Intercept Method,** and to show how celestial navigation is used in a typical day at sea. We assumed we had access to the internet, and specifically to the U.S. Naval Observatory's celestial navigation online calculator. We used this web site to determine objects' **Computed altitude, Hc**, and **azimuth, Zn**. This is appropriate for this module, but in order for celestial navigation to be a true backup system, we should know how to obtain these values without the use of electronics. The steps are simple and **strip forms** will lead us along. However, explaining and demonstrating the steps are beyond this module's time limit. In order to give the viewers an idea of the steps, we will broadly outline them here.

To obtain Hc and Zn without electronics, two publications are important, "The Nautical Almanac" for the given year, and "Publication 229." The Nautical Almanac contains, among other data, all of the information needed to compute the **Greenwich Hour Angle** and **declination** of the observed object at the precise moment the sextant measurement is taken. The Nautical Almanac contains these data for all navigational objects, be it one of the navigational stars, the Sun, Moon, Venus, Mars, Jupiter, or Saturn. The Nautical Almanac is produced jointly by the U.S. Naval Observatory (USNO) in the United States and Her Majesty's Nautical Almanac Office (HMNAO) in the UK. It is available in the U.S. via the Government Printing Office, and elsewhere via the United Kingdom Hydrographic Office distributors.

The second important book is "Publication No. 229, Sight Reduction Tables for Marine Navigation," usually just called Pub 229. It is also known as "NP 401" in the United Kingdom. Pub 229 is six volumes, and it is static; that is, it does not change from year to year. It contains the data to get Hc, Zn, given an object's **local hour angle (LHA)**, Dec, and the observer's latitude. (Note, we have not discussed local hour angle in this module; just know it is easily calculated from the object's GHA and our longitude.) Pub 229 was developed collaboratively by USNO and HMNAO, and distributed by National Geospatial-Intelligence Agency (NGA). Copies can be found for free download on the NGA site http://msi.nga.mil/NGAPortal/MSI.portal?_nfpb=true&_pageLabel=msi_portal_page_62&pubCode=0013.

A major point to remember is that the two publications are used together. The Nautical Almanac, along with an estimate of your position, yields the object's LHA and Dec. Using these, we obtain the object's Hc and Zn from Pub 229.

This sounds easy, and it is. However, both books involve tables, and therefore we must interpolate between table entries. For example, The Nautical Almanac gives the position of the Sun every hour throughout the year.

Because we normally do not observe an object exactly on the hour, we must interpolate between tabulated hourly entries. The interpolations are not difficult; strip forms and other tables in the book can be used to guide you.

Similarly, Pub 229 provides a very large table of Hc and Zn for given LHAs and latitudes. In order to keep Pub 229 to only six volumes, the tables must be entered using only whole degrees of LHA and whole degrees of latitude.

We therefore select our **Assumed Position, AP**, to make this true. Remember that our AP just needs to be close to our actual position, maybe within a hundred miles or so. In this module we chose our Dead Reckoning position (DR) as our AP. However, if using Pub 229, you will need to select a different AP, because it is very, very rare that at the time of your sextant reading, your DR longitude and latitude would be at one of Pub 229's tabulated entry points. Therefore, we carefully select an AP to make this true. That is, we select a new longitude—near our DR longitude—so the LHA of the object is a whole degree. And we select a new latitude—near our DR latitude—so it is a whole degree. The resulting position is our **Assumed Position**, and it will differ from our DR. With this carefully selected AP, we can then use Pub 229 to look up the object's Hc and Zn.

Two tricky points. First, the Hc and Zn from Pub 229 are valid at the Assumed Position, not the Dead Reckoning position. So the azimuth line, intercept, and LOP are drawn from the AP, not the DR.

The second tricky point is that the AP is valid only for one object observed. If you observe three objects as you would do in a three-star fix, you will have three different APs. The reason is that our AP longitude is computed to make the LHA of the object a whole degree. Each object will have its own LHA, therefore we compute a new AP for each object.

One point of emphasis. The only reason you might use an Assumed Position that is different from your DR position is if you are using a book like Pub 229 to look up tabular values of Hc, Zn. In the examples in this module, we used the USNO web site, therefore everything was determined and plotted at our DR position.

The use of an Assumed Position that differs from a DR position is one of the trickier concepts in celestial navigation. If it seems unclear right now, do not worry about it. However, you will want to become familiar with it should you need to navigate using the publications such as Pub 229.

You will notice on certain illustrations, we have the starlight hitting the Earth in parallel rays; that is, the light appears to come from the same direction in the sky. You don’t really need to understand why this is true to know how celestial navigation works, but the illustrations will make a more sense if you do.

So, why does starlight hit the Earth in parallel light rays? Simply, it is because stars are so very far away. To illustrate, let’s look at light sources at three locations.

First, let’s have the light source be located directly over Location 1, but pretty close to it. We see that the light rays arriving all the locations are at very different angles. In this illustration, the rays hitting Location 2 and Location 3 are about 90 degrees different. Let’s now move the light source farther, but still keeping it above Location 1. We see that the light rays arriving at all the locations are still at varying angles, but the differences are smaller. Simply by moving the light source farther, the light rays arriving at different positions on Earth are become more parallel.

Now imagine that we move the light source very, very far off the diagram, say many miles away, but still keeping it above Location 1. The light rays arriving at Locations 1, 2, or 3—actually any location on Earth—will appear parallel. (In reality there is a very, very slight difference in the angle, but it is way too small to measure).

Bowditch, N., The American Practical Navigator, 2002 Bicentennial Edition, National Imagery and Mapping Agency, http://msi.nga.mil/NGAPortal/MSI.portal?_nfpb=true&_pageLabel=msi_portal_page_62&pubCode=0002

Sight Reduction Tables for Air Navigation, Pub. No. 249, National Imagery and Mapping Agency, http://msi.nga.mil/NGAPortal/MSI.portal?_nfpb=true&_st=&_pageLabel=msi_portal_page_62&pubCode=0012

Sight Reduction Tables for Marine Navigation, Pub. No. 229, National Imagery and Mapping Agency, http://msi.nga.mil/NGAPortal/MSI.portal?_nfpb=true&_st=&_pageLabel=msi_portal_page_62&pubCode=0013

USNO Celestial Navigation Data, http://msi.nga.mil/NGAPortal/MSI.portal?_nfpb=true&_pageLabel=msi_portal_page_62&pubCode=0002

USNO Complete Sun and Moon Data for One Day, http://aa.usno.navy.mil/data/docs/RS_OneDay.php

USNO Rise/Set/Transit Times for Major Solar System Bodies and Bright Stars, http://aa.usno.navy.mil/data/docs/mrst.php

USNO/UK Hydrographic Office, The Nautical Almanac (published yearly), http://aa.usno.navy.mil/publications/docs/na.php

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