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Making the most out of our graphical telemetry computers.

Like with many complex instruments, there are multiple ways to solve certain problems and to measure the required inputs. Combining the various functions leads to a multitude of advanced uses.

## The Quadrant Telemeter: A Star Compass

When we talk about emergency navigation at night, we usually refer to the method of finding north with the North Star in the northern hemisphere or with the Southern Cross constellation in the southern hemisphere.[1]

How can we find the cardinal direction when the North Star or the Southern Cross is obscured, be it by clouds or obstacles such as mountains, forests or the like? If the sky is not completely covered, we can use all visible stars whose declination we know, as long as we can see and identify them (not the least through the constellations within which they are located).

The declination of a star compared to the declination of the Sun (almost[2]) does not change over the course of a year and remains constant regardless of the time of year and time of day. This is because the distance to the star is practically infinite compared to the diameter of the earth and also because certain stars are only visible at certain times of the year. E.g. the winter stars and constellations in autumn in the east are only visible towards the morning (shortly before sunrise), during winter they are visible all night long -- and at midnight in the south. In spring, the stars are already visible in the west (shortly after sunset), right after dark. In summer they would be visible near the Sun -- if the sunlight would not obscure them (also due to our atmosphere).

How can we use the constant declination for navigation and finding the cardinal points?

The methods described in this tutorial can be performed with any Quadrant Telemeter, not just the Horary Quadrant or Latitude Quadrant.

## Method 1

We use the rising or setting direction of the star to find east or west -- as a star compass. This method was used by Polynesian navigators centuries before the compass was developed by the Chinese to navigate between the many islands of Polynesia.

Like the Sun, the stars rise in the east and set in the west. Unfortunately, not all stars rise precisely in the east and set in the west, but the stars with a positive declination in the range between 0° and +(90 - ABS (latitude))° rise between north and east and set between west and north. The stars with negative declination between 0° and -(90 - ABS(latitude))° rise between east and south and set between south and west (Figure 1). Only the stars with a declination of 0° rise precisely in the east and set in the west.

Figure 1: Rising and setting azimuths of stars.

Figure 2: Inclination of the constellation Orion from autumn through winter to spring.

Stars with a declination greater than (90 - ABS(latitude))° and with the same sign as the latitude are circumpolar: they are visible all night and neither rise nor set. The stars with a declination greater than (90 - ABS(latitude))° and with the opposite sign as the latitude are not visible.

How can we tell where the stars rise or set when their movement is so slow that they appear almost static? Here we can use the method from the tutorial Measuring Star Time, except that we measure vertically against the horizon. Another method is to determine, by tracking known constellations, how their inclination changes on the horizon in the east, in elevation in the south and again on the horizon in the west (Figure 2, using the example of the constellation Orion).

To accurately determine east and west, we have to use the declination δ of the star to calculate its rising azimuth αrise or setting azimuth αset for the latitude φ of the observation location. To do this, we either use the formula for the sunrise azimuth from the manual or simplified:

``` αrise = cos-1⁡(sin δ / cos ⁡φ) ```

``` αset = 360° - cos-1⁡(sin δ / cos ⁡φ) ```

These formulas provide the rising azimuth from 0° for north over 90° for east to 180° for south and the setting azimuth from 180° for south over 270° for west to 360° for north.

We can also calculate the corresponding azimuth with the method described in the tutorial Direction of Sunrise with the Sine Quadrant.

After we have calculated the azimuth, we sight the star on the horizon above the vertex of the quadrant and the calculated azimuth (Figure 3, using the example of the star Betelgeuse (rising azimuth for latitude 50° north is 78.42°)). We correct the azimuth for bearing based on the quadrants in Figure 1, after which the two sides of the quadrant show the true directions.

Figure 3: Sighting a star to find the cardinal direction.

Figure 4: To make sighting easier, a small wooden bar with two small nails can be used. One nail is positioned at the vertex of the quadrant and another one outside of the quadrant. The string aligned between the vertex and the outer nail helps to set the azimuth you are looking for.

Table 1: List of navigation stars (excerpt).

Table 1 shows the 59 "navigation stars" with constellation, declination and, for the sake of completeness, with SHA (sidereal hour angle) and hour angle RAh for the epoch 2020. These stars are bright enough to be recognized in the sky with moderate light pollution.

The table also shows the calculated azimuths of the stars for the latitude 50° north and the star altitude 0° (on the horizon). The Excel spreadsheet is available for your own calculations.

What can we do when a well-known star that we want to sight at its rise is already over the horizon or has not yet reached the horizon by the time it sets? If the distance of the star to the horizon is still less than 30°, we can "project" the position of the star onto the horizon, as shown by the example of the star Mintaka (δ Orionis) (Figure 5). The star Mintaka has a declination of -0.30°, which is pretty close to the celestial equator. That is why our "projection" is so close to the cardinal points east and west.

Figure 5: Projecting the position of a star onto the horizon.

This "projection" is based on the fact that the stars move parallel to the celestial equator. The celestial equator lies at an angle of ±90 - |φ|. In the west we use the angle φ because we are using the other side of the quadrant.

Figure 6: Rising and setting azimuths of the visible and non-circumpolar stars from the table of navigation stars for the latitude 50° (for the northern and southern hemispheres) for the true horizon.

This method can only be used to a limited extent for land navigation, as the formula used only works for the true horizon as seen at sea.

To solve this problem we can do the following:

1. Measure the height hs of a star (as low as possible above the horizon) with the quadrant.[3]
1. Calculate the rising azimuth and setting azimuth with the following formulas:

``` αrise = cos-1((sin δ - sin φ · sin hs) / (cos ⁡φ · cos ⁡hs)) ```
``` αset = 360° - cos-1((sin δ - sin φ · sin hs) / (cos ⁡φ · cos ⁡hs)) ```

These extended formulas provide a rising azimuth of 0° for north, 90° for east, 180° for south and a setting azimuth of 180° for south, 270° for west, 360° for north.

1. Sight the star with the calculated azimuth as described above and determine the cardinal points.

## Method 2

We use the altitude of the culmination of a star to find south or north. At the culmination, a star reaches its highest level (the maximum angle of elevation) above the horizon and at that moment it crosses the local meridian, which runs from north across the zenith of the observation point to south. If we can determine the culmination point of a star in this way, we find the azimuth of true south or north depending on the star and its declination.

The culmination altitude and direction of a star with the declination δ for a latitude φ can be calculated using the formula "Sun altitude via hour angle".

``` hs = sin-1(sin ⁡φ · sin ⁡δ + cos ⁡φ · cos δ · cos τ) ```

To calculate the altitude of a star for its upper culmination, we use a value of 0° as the hour angle τ and a value of 180° (12 hours · 15 °) for the lower culmination.

This formula provides the elevation angle of a star in the range from +0° (southern horizon) to +90° (zenith) if the star is south of the observation point and from -90° (zenith) to -0° (northern horizon) if the star is north of the observation point.

If there is no pocket calculator at hand, the altitude of the culmination can be calculated with the quadrant using the declination δ.

First we calculate the altitude of the celestial equator he based on the latitude φ of the observation location:

• for the northern hemisphere: `he = 90° - φ`
• for the southern hemisphere: `he = 90° + φ`

The Latitude Quadrant, as used in this tutorial, shows at the symbols ♈ / ♎ the position of the celestial equator on the calendar scale (Figure 7). For the latitude 50° north this is 40° above the horizon.

After we find the altitude of the celestial equator, we can calculate a star's culmination altitude based on its declination.

Figure 7: Calculating the altitude of a star's culmination from the southern horizon.

Figure 8: Calculating the altitude of a star's culmination from the northern horizon.

Let us consider the star a (Figure 7), whose declination is negative, at -15°. We subtract 15° from the angle of the celestial equator and read the culmination of 25° from the angle scale. The upper culmination of the star a with the declination -15° is 25° from the southern horizon. The lower culmination for this star will not be visible to us, because it will take place 6 months later during the day at an altitude of -55° (i.e. below the horizon).

Let us consider the star b (Figure 7), whose declination is positive, at 20°. We add 20° to the angle of the celestial equator and read off the culmination of 60° from the angle scale. The upper culmination of the star b with the declination 20° is 60° from the southern horizon. The lower culmination for this star will not be visible to us because it will take place 6 months later during the day at an altitude of -20°.

A star with the declination δ equal to the latitude φ is a zenith star or overhead star. In our example these are all stars with a declination of 50°.

Know your zenith stars and you will always find your way home![4]

Let us consider the star c (Figure 8), whose declination is positive, at 70°. We add 70° to the angle of the celestial equator and get an elevation angle of 110°, which is 20° above the zenith. This means that this star is between the zenith and the northern horizon. We rotate our quadrant so that 0° of the angular scale corresponds to the northern horizon (rotated and mirrored in Figure 8). Now we subtract 20°, the angle above the zenith, from 90°, and read the culmination of 70° on the angle scale. The upper culmination of the star c with the declination of 70° is at 70° from the northern horizon. Since it is a circumpolar star (declination δ of the star is greater than (90 - latitude φ) with the same sign), we can also calculate the lower culmination for this star. To do this, we determine the angular distance between the upper culmination and the pole altitude, which is equal to the latitude φ. We now apply this angular distance from the pole altitude to the northern horizon and read the lower culmination on the angular scale. In our example this is 30°. Accordingly, the lower culmination of the star c with the declination of 70° is at 30° from the northern horizon.

Now that we have calculated the star's culmination, we can measure the altitude of the star (by taking several measurements) until the star has reached its highest point (altitude of culmination). Then the star is exactly on the meridian and shows us south or north.

If we reverse the method of culmination altitude described above, we can first measure the culmination of a star with a known declination δ from the southern horizon in the northern hemisphere or from the northern horizon in the southern hemisphere and thus calculate the latitude φ of the observation point. To do this, we subtract the star declination δ from the determined culmination altitude hs in order to get the altitude of the celestial equator he.

`he = hs - δ` (Note the sign!)

The desired latitude φ is then calculated using the following formula:

``` φ = 90° - he = 90° - (hs - δ) ```

## Time Difference between Culmination and Star Setting or Rising

With the Horary Quadrant we can also calculate the time difference between culmination and star setting or star rising:

1. We apply the declination δ of the star from the celestial equator with the string, as described in the method for determining the height of the culmination for stars a and b.
2. Then we place the cursor on the sine curve and move the cord to the 0° mark.
3. Then we read the time difference until the star's set or rise.
4. Now we can calculate the time of the star's culmination after the star has risen above the horizon.

(Thanks to Telemeter user Andrej for this tutorial!)

Resources:

Footnotes:

1. Both methods are described in the fourth edition of the Quadrant Telemeter manual in the section "Determining local latitude". Further information can be found in relevant literature and on websites on the subject of makeshift navigation.
2. Depending on the star, the coordinates change by a few arc minutes or even a few arc seconds over the course of a year or even multiple years. That is why the term epochs is used when data is given in the tables and star catalogs.
3. The advantage of the quadrant for the measurement of the star height in the field is that the quadrant carries out and converts the measurement from the zenith (the vertical) and one does not need a true horizon as with the sextant.
4. If you are on the road and want to know how far south or north you are from home, you can measure the altitude of a zenith star and subtract the measured value from 90° (zenith altitude of the current observation location). The result is the distance in degrees. We use the following definitions to convert degrees to length: 1° corresponds to 60 minutes of arc. 1 arc minute corresponds to 1 nautical mile (NM). This means that 1° corresponds to 60 NM or 69.05 miles or 111.12 km, in a south-north direction. For the east-west direction, this value must be multiplied by cos φ.