The clinometer

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Since it is difficult to estimate the height of an object in the sky, we use an inclinometer (clinometer) to determine this quantity. To make a sufficiently accurate instrument for our purposes, we need nothing more than a semicircular 180° protractor and a 20 cm long ruler, the center of which we mark with a fine line along the long side. We fix the ruler in the center of the protractor with a small screw and slightly weight it at the lower edge with, for example, two magnets, allowing it to rotate freely and align itself vertically on its own. In this way, it allows us to determine the angle of elevation without the aid of the horizon (the zenith, the apex of the sky, is always exactly 90° above the horizon). And so we save ourselves a lot of headaches, because already the concept of the horizon causes misunderstandings again and again, and also purely physically, its plane is often difficult to determine because it is regularly covered by mountains, hills or high buildings.

In everyday language, we usually mean by the horizon the outermost limit of our range of vision at which heaven and earth merge. And this is quite correct when transferred to the somewhat confused language of geometry, where the visible, apparent or local horizon is defined as the circle around any observer whose plane is perpendicular to the respective location and which appears to delimit the visible earth from the sky. If we neglect the refraction of light in the atmosphere and idealize the earth as a perfect sphere, we can calculate the distance between observer and horizon approximately as follows:

Formula 4

D = Distance between observer and horizon
H =  Height above the earth's surface in meters

The following viewing distances may serve as a guide: 27 km at 50 m altitude, 38 km at 100 m altitude, and 120 km at 1000 m altitude. The true horizon, on the other hand, is a great circle parallel to the visible horizon, which intersects the earth at its center.

The software we will use in the following generates earth-based coordinates, which we then transfer to a local system using the longitudes and latitudes we have entered. Our observation altitude above sea level only matters in this context if we are aiming for accuracy in the range of angular minutes or seconds. – This is a goal that we cannot aim for with the hand compass and the self-made clinometer we probably use, and for our purposes, we do not need to aim at all. We measure approximations, which provide us with a significant advantage.

Now, to determine the angle of elevation, take a bearing over the straight side of the protractor and fix the ruler with two fingers when it has leveled out. The value on the scale below the center mark is the elevation angle you are aiming for. When reading, remember that the clinometer shows the bearing’s angle below the zenith. The exact vertical position of the ruler at 90° thus reflects the plane of the horizon and corresponds to an elevation angle of 0°; the parallel position of the ruler and protractor at 0°, on the other hand, points to the zenith and corresponds analogously to an elevation angle of 90°. Therefore, a measured position of 60° is equivalent to 90° minus 60°, which equals 30° above the horizon.

As mentioned at the beginning, we can also determine the elevation angle „over the thumb“ with our fist. When the fist is held on the horizontally extended arm in front of us, it corresponds to an elevation angle of approximately 10°. However, to ensure accuracy, it’s advisable to compare this value with the clinometer, given that individual differences in body composition exist.

But no matter how you proceed, if you want to measure the elevation angle of the sun directly, do not look straight into the sun! Rather, protect your eyes with one of the inexpensive glass inserts that welders also use in their shields. The darkest glass with the largest index is just good enough.

Now we have all the information about the position we would like the star to be in and can, thanks to the computer, leave all further calculations to the programs that clever people have devised for this purpose. Let’s go through it point by point with two practical examples.

Next Doing it in practice

Main Natural light

Previous The compass

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