When considering earth as stationary and flat, the sun must be moving on a conic trajectory . (Maybe you can remember my article “Our Earth Pond and the Tent of the Sun”). In the same way the moon, in its movement in the space/ time, has to follow a trajectory moving in the shape of a cone. And this will be the basic theme of the following consideration.
How to calculate the cone
How could we clarify that the sun and the moon trace trajectories that are developing on a cone shape? I’ll quote from my post “The ecliptic”.
You could do like me and download an application that gives the height of the sun for every place in the world. So, I’ve downloaded “Sun Surveyor” from Google Play.
However, you should be careful since these applications are intended for the use relatively to a globe Earth geometry and data are calculated according to a globe geometry that has never actually been measured. Anyway, when you consider values relating to a single place in the Northern hemisphere, experience shows that heights can be used for a first, very approximated calculation.
Let’s suppose to be in Rome. Rome has a latitude of 41°54’ N. You have to consider that in a globe geometry each degree of latitude corresponds to a 111Km arc.
Rome is distant 41°54’-23°27’=18°27’ (2050km) from the summer tropic (Cancer) and 41°54’+23°27’=65° 21’ (7250km) from the other tropic (Capricorn).
Sun surveyor gives these two heights angles for the solstices at Rome. (We are taking into consideration the declination angle that the sun makes over the horizon).
Having these two values in mind, let’s now calculate the height of the sun in winter and in summer.
When we consider the angles of the moon we can do the same trigonometric calculation. The only point at issue is that the motion of the moon is a little more complicated than that of the sun. For example the moon speed is slower than that of the sun. It loses, in respect of the major luminary, 12° every day.
Let’s consider from sun surveyor some data that will show us that the moon trajectory is on a cone.
For example on the 22nd of February 2017 the moon touches the lowest point of the month. As you can see in the image the angle of height (declination) of the moon is 29.2°.
On the other hand on the 6th of March 2017 the moon rises to an angle of 67.4° that is the maximum declination for that month. This is possible because on the 22nd of February the moon is on the lower orbit of the cone while on 6th of March it is on the upper orbit.
Here below you can find a table with the max and min declinations of the moon during this year, 2017.
|Date||Minimum declination||Maximum declination|
|21 march 2017||29°|
|3 april 2017||67,4°|
|18 april 2017||29°|
|1 may 2017||67,7°|
|15 may 2017||28,5°|
|28 may 2017||67,6°|
|11 giu 2017||28,5°|
|24 giu 2017||68°|
|8 july 2017||28,6°|
|21 july 2017||67,8°|
|4 aug 2017||28,5°|
|18 aug 2017||67,9°|
|1 sept 2017||28,5°|
|14 sept 2017||67.8°|
|28 sept 2017||28,4°|
|11 ott 2017||67,8°|
|25 ott 2017||28,2°|
|8 nov 2017||68,4°|
|22 nov 2017||28,1°|
|6 dic 2017||68,3°|
|19 dic 2017||27,8°|
As you can notice, the moon travels on a cone that has the lowest orbit at about 28° (considering the latitude of Rome) and the upper orbit at about 68°.
We can conclude that the moon, as the sun, also stands on a cone and this cone is completed up and down in 27,32 days, that is the sidereal period of the moon, that means the period that the moon takes to reach two times the same star on the celestial “semi sphere”.
So, in the time the sun makes a cycle on his cone, the moon repeats its cone for more than 13 times.
The cone of the moon, even if it is very similar to that of the sun, is not exactly the same. We have clear in mind the fact that the angles of the sun at solstices are 71,5° and 24,7°, values that are slightly different from the average values of the moon cone. The slope of the two cones is different: astronomers say that there is an angle of 5° of difference from the ecliptic (the trajectory of the sun) and the trajectory of the moon.
Interesting enough is the fact that this inclination changes in the course of the years. I will add now a table for the year 2004 to clarify this aspect.
|Date||Minimum declination||Maximum declination|
|6 jan 2004||75,6°|
|20 jan 2004||20,7°|
|2 feb 2004||75,7|
|16 feb 2004||20,6°|
|29 feb 2004||75,6°|
|15 mar 2004||20,5°|
|28 mar 2004||76,1°|
|11 apr 2004||20,2°|
|25 apr 2004||76,1°|
|8 may 2004||20,2°|
|21 may 2004||75,6°|
|5 june 2004||20,3°|
|18 june 2004||76,1°|
|2 july 2004||20,3°|
|26 july 2004||75,9°|
|29 july 2004||20,3°|
|12 aug 2004||76,2°|
|25 aug 2004||20°|
|8 sept 2004||76,5°|
|22 sept 2004||20,2°|
|6 opt 2004||76,5°|
|19 opt 2004||19,8°|
|2 nov 2004||76,7°|
|15 nov 2004||19,8°|
|29 nov 2004||76,5°|
|12 dec 2004||20,1°|
|27 dec 2004||76,4°|
As you can see from the picture, angles of the moon in 2004 are different from those in 2017. It seems that the cone of the moon changes angle around the cone of the sun. This happens in a cycle of 18,5 years.
In the following picture you’ll find the cones of the moon in 2004 in comparison with the cone of the sun.
In this second picture there are both cones of the moon (the two more external envelopes of trajectories that are travelled by the moon in the cycle of 18,5 years) one over the other.
Now, let’s add a summary.
The sun has a trajectory that covers a complete cone, up and down, in one year.
The moon covers a cone in the same way but it goes up and down in 27,32 days.
The cone of the moon is slightly tilted in comparison with that of the sun with an angle of about plus/ minus 5°. This angle changes year after year from (about) minus five to plus five in 18,5 years. This movement is called moon libration in latitude.
In a future post I’ll try to explain the moon phases over a flat Earth.
In the following table I’ll briefly try to explain some terms used to describe astronomic coordinates that have to be adapted to the Flat Earth model.
|Celestial sphere: astronomers refer to the sky as a sphere surrounding the globe Earth. Over a Flat Earth we should imagine not a sphere but a semi sphere, something similar to the picture you can see aside.|
|Astronomic coordinates. When we want to individuate a star or maybe the moon in the sky we need two angles. One is the Azimuth. The white angle measured from “South” (from the point of view of the observer) and the red one is the declination or Height Angle|