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"Himmlische Verhaeltnisse"

(Two translations would fit:
"Heavenly conditions" or "Cosmic proportions"
therefore I kept the German pun)

Beginning of astronomy - Astronomy for beginners
by Klaus Kohl

What do we see?
The moon is a sphere, lit up by the sun far away. Sometimes in front (full moon), sometimes right aside (half moon) sometimes nearly from behind (a narrow sickle to be seen shortly before sunrise or shortly after sunset. That is repeated "monthly", each day moonrise is about one hour later. Only a sphere can show us these forms (we will not believe that the moon is cut off at its backside or in any other way crazy formed) And as it shows to us always the same face we can be shure that it is not light on one half and deep black on the other.
The phases of the moon give us proof: The moon is a sphere.
The sun is very far away, far behind the moon as we see its narrow sickle. How far - Aristarch tried to find it out more than 2000 years ago, probably by regarding the angle he saw between 'exact' half moon and the sun. A little bit less than a right angle, this one will be at the moon's place in the triangle earth-moon-sun. Aristarch thus found the sun to be 20 times farther away than the obviously far away moon. [According to these observations a cosmic system orbiting the earth - sudden death! But this conclusion was nearly unbelievable, even Archimedes showed doubts on that. Aristarch nearly he would have been killed as blasphemer - his doubts to the commonly believed geocentric system with gods in its center were unbearable! It took time until Copernicus revealed this nearly forgotten idea, precised and dared to publich it cautiously]. Later measurings, -more precise- revealed, not 20 times farther away but 20 x 20 times, 400 times farther away than the moon - this is the distance of the sun! We cannot test this, have to believe it to the astronomers:
The sun is 400 times farther away than the moon.

And as the sun seems of the same size as the moon:
The sun is 400 times bigger than the moon.

To see a 1.6 cm sized cent as big as the full moon (or the sun) we have to hold it by a distance of 1.6 m, a 2.5 cm 2-Euro piece 2.5 m. Always the distance is about 100 times the diameter [1]. That means:
The distance earth-moon is equal to 100 diameters of the moon
The distance earth-sun is equal to 100 diameters of the sun.

As roughly simplified these measurings are, as rough are the numbers obtained by this way. Their advantage is: they can easily be memorized. That applies too to the following proportions. The exact numbers - only the astronomers have to work with them! - have been obtained by different methods.
Another performance is on the scene occasionally: By full moon the moon is just opposite to the sun (seen from the earth). Just opposite? No, normally it slips through that position a little above or beneath. But about twice a year it really crosses the line - and the the earth throws its shadow on the moon. And this shadow always has a circled border! We call the performance a lunar eclipse and as we know that only a sphere makes always a circle-shaped shadow we know:

Lunar eclipses give the proof: The earth is a globe.

And additionally the circle of the shadow is about four times larger than the circle of the moon:

Earth is four times bigger than the moon.

The sun is hundred times bigger than the earth.

Now we know the cosmic proportions:

Diameter Moon - Earth - Sun
as 1 : 4 : 400

Diameter : Distance
Moon: 1 : 100
Sun: 1 : 100

25 diameters of Earth fit into the distance Earth - Moon
10 000 diameters of Earth fit into the distance Earth - Sun

We got the proportions, but a measure is missing, how many yards, miles, meters, kilometers?
Eratosthenes was obviously the first to give a rather correct result (it was the time of Aristarch). He knew that by noon in summer (June 21) in Syene (today the place is named Assuan) the sun is shining from right above (by that time the northern tropic passed nearly right through Syene) but in Alexandria there it shone a bit from south, giving a small shadow to vertical poles. Erathostenes is assumed to have used a sundial formed as a hemisphere with a vertical pole in it as long as to reach the center of the sphere (a "scaphe"). And as the length of the shadow was fitting 50 times into the circumference of his sphere he concluded that the circumference of the earth globe is 50 times longer than the distance from Alexandria to the northern tropic. And as he knew the distance between Alexandria and Syene being 5000 "Stadia" he could give a circumference of the earth globe of 250000 Stadia.

An encyclopedia may give us exact numbers, found out by others for us:
Diameter of the earth 12 713 507 ± 3 m (!) from pole to pole
Diameter of the moon 3476 km
Distance earth-moon 384 403 km
Distance earth-sun 149 597 870 km (the "Astronomic Unit" AU)
Diameter of the sun 1 393 980 km

And now we try to imagine this -
Beginning by earth we take it as a ball of 13 cm (10 inches) in diameter instead of 13000 km shrunk by 1:100'000'000
Then the moon has a distance of 3.8 m and a diameter of 3.5 cm. And the sun? 14 m in size and in a distance 0f 1.5 km... no, that is not handy! We shrink it once more by 1:10 and now a 1000 km is only 1 mm... Now earth is as big as 13 mm, moon is 3.5 mm -small- in a distance of 38 cm from our earth and sun is a balloon of 1.4 m diameter in a distance of 150 m.
And then there are the other planets:

Planet Diameter comp. to earth Distance from sun comp. to earth
Mercury 4'880 km 0.38 58 Mio km 0.39 AU
Venus 12'100 km 0.95 108 Mio km 0.72 AU
Earth 12'760 km 1 150 Mio km 1 AU
Mars 6'800 km 0.53 228 Mio km 1.52 AU
Jupiter 143'800 km 11.30 778 Mio km 5.20 AU
Saturn 120'000 km 9.40 1'427 Mio km 9.54 AU
Uranus 52'300 km 4.10 2'870 Mio km 19.20 AU
Neptune 49'500 km 3.90 4'497 Mio km 30.06 AU

In our model thousand kilometers were shrunk to one millimeter, the decimal point goes nine steps left.
Oh, neptune is then 4.5 kilometers away from the sun and its diameter is nearly 5 cm.
What about a "Planet walk" in your neighborhood? It's a an hour walk to Neptune and then you are marching with a velocity five times to lightspeed... but the next fixed star to our solar system is 4 light years away (as light goes 300000km in a second a light year is a distance of about 10'000'000'000'000 km) so in our model planet walk it would be a journey to "Alpha Centauri" just around the globe... Good bye!
The German text has also been published by "Astronomie+Raumfahrt" No. 6/06, p. 8-10 (Friedrich Verlag, Seelze)

[1] It is always interesting to ask people about this. Most will guess their stretched arm to be long enough to hold a small coin so that it will show not bigger than the sun.
If you put the question by the form (healthful especially by sunlight!): "How far should that coin be held away from the ground or a wall that it has no longer that (well visible) core shadow?" the speculations are much to big. The truth is -as often- between and with astonishment they realize, that it is exactly the same situation.
And additionally this experiment shows another fact: The just rising full moon is not bigger than by its position high at sky. It is a -sometimes life-saving- illusion which shows to us the high up an deep away smaller (and thus exaggerated far away) than the things before us, we can reach without danger by walking. We are not birds!

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