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Frank L. Preuss

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How can I find the angle by making a drawing?
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I now want to find the angle between Capella and Aldebaran by making a drawing.

Astronomers describe the positions of objects in the sky by imagining that they are all at the same distance from the Earth, on the surface of an imaginary globe called the celestial sphere.

As information I use the coordinates given by the astronomers. For Capella it is 5.28 hours, which is 79^{o}, and +46^{o}, and for Aldebaran it is 4.60 hours, which is 69^{o}, and +16.5^{o}.

I now take this globe and from this globe I only take the upper half and from this upper half I only take the right hand quarter and from this I only take the back eighth, because that is where these two stars are.

And I do not have to draw my drawing to scale, because it is only about angles and the radius in this imaginary globe can in any case be of any dimension.

So of this eighth of my globe I draw a plan and an elevation and a side elevation.

And then I draw the positions of my two stars in all three pictures and connect the lines from the centre to Capella and then to Aldebaran and then back to the centre in all three pictures.

So all three pictures show the same triangle, but the triangle looks different in all three pictures, because it is not in the same plane as my drawing is. And to get it into the plane of my drawing I have to tilt the triangle, twice, to get the angle I am after.

All seizes of the angles of these three triangles are not the real seizes, because none of these triangles is in the same plane as the picture is.

So I now want to find the real seizes of the angles.

I now tilt the triangle in the "Elevation" so that the line between the centre and Capella is in the vertical plane. The axis around which this tilting is done is the axis going from south to the centre and to north.

I therefore have now at least one side of the triangle in a plane shown on my drawing. This line is now in one of the planes shown on my drawing, but it is not the right length. The right length is from the centre to the surface of the imaginary globe.

I now project this line on to the "Side Elevation" and there I now have the right length of the line. It is the length from the centre of the globe to the surface of the globe, therefore the radius of the globe. Both stars, and also all stars, have this distance from the centre, because they are all on this surface of this imaginary globe.

With this line, the blue one, from the centre to Capella I now have the real length of this line, but the rest of the triangle, the two dotted grey lines, are still not the real length, because this triangle is not in the plane of the "Side view."

So in the "Elevation" this turning of the triangle is shown as parts of circles. And in the "Side view" it is shown as vertical lines going up. And in the "Plan view" it is shown as horizontal lines going to the left. But in all three cases it is a turning about the south north axis.

This was the first tilting and now comes the second, and I do this in the "Side view."

Now I tilt the rest of this triangle around the blue line between the centre and Capella and I do this by erecting a perpendicular to the blue line. And this perpedicular goes through Aldebaran, the little circle shown with a dotted line in green ink.

And this perpendicular goes through the surface of the globe. And that is that part of the surface of the globe that is shown on the "Side view" as a quarter of a circle.

And where the perpendicular intersects with the circle, there is the final position of Aldebaran.

And now I have the triangle in its right size. The three sides of the triangle are the blue line from the centre to Capella and the green side from the centre to Aldebaran and the red line from Capella to Aldebaran. And the angle at the centre, therefore opposite of the red line, is the angle sought.

So this triangle is now the triangle that shows the correct angles and therefore the lengths of the three sides of the triangle are in right relationship to each other.

So with this imaginable Globe we have a situation, where all stars are on the surface of this globe. And this makes dealing with this system quite simple. Also on our drawing all stars are always on the surface of this globe, except after the first tilt, when Aldebaran is not on the surface and is therefore shown as little circle with a dotted line in green ink. But only when both stars are both on this quarter circle, only then can we really measure the angle we are looking for.

And when we measure the angle then we get 30^{o}.

And as the triangle is a three dimensional triangle, we have to do this tilting twice. If the triangle would be a two dimensional triangle, one tilting only would do the trick.

We have a system with three axes: one from Zenith to Nadir, one from north to south and one from west to east. If the plane of the triangle is not parallel to at least one of these three axes, then we have to tilt the triangle twice. If the plane of the triangle is parallel to one of these three axes, then one tilting would be enough. If the plane of the triangle is parallel to two axes, then no tilting would be necessary.

So this is an alternative method to determine the distance between two objects in the sky. It has the advantage that one really gets a feeling for the location of the stars in the celestial sphere.

In the "Plan view" for instance one can see the orbits on which the two stars move. The smaller the orbit gets the closer the star is to the axis between Zenith and Nadir. Looking at a time-exposure photograph one can nicely see these star trails. And as the celestial sphere is similar to the terrestrial sphere, these circles correspond to the degrees of latitude or parallels, which mark positions on the surface of the earth.

In the "Plan view" the angles go from 0^{o} in the north to 90^{o} in the east to 180^{o} in the south and to 270^{o} in the west and to 360^{o} in the north, which is also 0^{o}.

Now this system for the celestial sphere makes more sense than that for the terrestrial sphere, because the system for the terrestrial sphere should be the same as that of the celestial sphere. Parting the terrestrial sphere in two halves, in east and west, should have been avoided, because now a second detail is necessary to describe a position, the sign "plus" or "minus". And that is not the only problem, the other is that their system becomes confusing when the position of a place is discussed that is on the other side of the globe, far away from London, then a place nearby might be to the east, but is in the far west of their system. So they should have London, or any other place they want to choose, as 0^{o} and also as 360^{o}. So the system chosen for the celestial sphere very nicely shows how to do something like this. To divide the world into east and west has some advantages, but also disadvantages, for the designation east or west should relate to the present location only. The old system was to speak of morning and evening, and that could be applied to any place wherever one was.

If they had taken 180^{o} for London, then the Date Line could also have been there, where it is today. And then one could still devide the world in east and west, because that does not really depend on the system.

For the position of 0^{o} or 360^{o} here some quotes:

The first point of Aries

In ancient Greek times, over 2,000 years ago, the vernal equinox - the point at which the ecliptic crosses the celestial equator - lay near the border of Aries and Piscies.Right ascension is the angle of the object measured from the celestial meridian, which intersects the celestial equator at the first point of Aries.

First point of Aries (vernal equinox point) is the origin for right-ascension measurements.

Celestial meridian - the line of 0

^{o}right ascension.

So that System was chosen for a good reason. That was for the celestial sphere. Now for the terrestrial sphere London was chosen. They picked London, because that was the capital of their country and that was where they were. Others did the same. The French for example picked Paris. So the English system became the generally accepted one.

Now the need to do this determining of the angle between two objects in the sky arises, when one for example sees two stars in the sky and wants to check if they are really the ones, one thinks they are.

And the purpose for this is to search may be for the star Regulus in the constellation Leo and to look at this star and be reminded of the greatness of the universe and of the greatness of God.

One advantage of this graphic solution is that I find the angle of the triangle so, that I can picture the matter to myself, while with a determining by calculation I first have to believe in the correctness of the formula.

This is the end of "Astronomical question and answer 281"

To the German version of this chapter:
Astronomische Frage und Antwort 281

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