The Moon Trajectory on a flat Earth

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).

Summer solstice 71.5°
Winter solstice 24.7°

Having  these two values in mind, let’s now calculate the height of the sun in winter and in summer.

H=tg71.5°*2050= 6127km
H=tg24.7°*7250= 3334km

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

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