Trajectory of the Moon around the Sun

Assigning students the seemingly simple task of drawing the Moon’s trajectory in the heliocentric system can ignite a profound discussion about the intricacies of the trajectory. This article presents a diverse set of plots depicting suggested trajectories, alongside a detailed discussion of their differences and the implications of various details. The provided reasoning exemplifies analytical thinking, showcasing how even a small dataset enables us to dismiss certain suggested trajectories (hypotheses).


Introduction: draw a trajectory
Have you ever challenged your students to depict the trajectory of the Earth and the Moon around the Sun in the heliocentric system?Encourage them to consider the fundamental facts of Earth and Moon motion: the Earth orbits the Sun, and the Moon orbits the Earth.Prompt them to illustrate these trajectories while imagining the view from far above the Sun, looking at the plane of the orbits from the north.From the begining we assume negligible tilt between the Moon's and the Earth's orbital planes.
I have done this activity many times, almost every year, particularly when engaging with students, prospective physics teachers, in discussions about basic astronomy.Recently, I have given this task to a group of 16 international students participating in the Erasmus exchange program at our faculty (of education).They come from different European countries and are enrolled for one semester in the course Fundamentals of Astronomy with Didactics.It is important to note that these students have different academic backgrounds and none of them are majoring in physics.However, their common goal is to become primary school teachers.
The Moon's trajectory has been discussed in papers [1][2][3][4].At a certain point, every student learns about the Moon's orbit around the Earth and its orbital period.Drawing the Moon's trajectory around the Sun, however, challenges them to shift between two frames of reference: from geocentric (Moon orbiting the Earth) to heliocentric (Moon orbiting the Sun).This transition is definitely challenging for primary school students [5], and can still be challenging for high school students [6,7] and even for many K-8 science teachers [8].Utilizing different frames of reference (geocentric, heliocentric, topocentric) can also serve as a model for organising observable astronomical phenomena while demonstrating the application of a scientifc approach [9,10].

B Rovšek
The task assigned to the students has resulted in the creation of a series of figures, that provide a valuable opportunity to discuss the nuances in their representations-highlighting both differences and similarities.Using these drawings, we can illustrate how a simple task can greatly enhance the overall understanding of interconnected facts about the Sun, Earth and Moon.These basic concepts are familiar to every child at some point, and analysing the different representations improves students' understanding of these celestial relationships.
In figure 1, a 'screenshot' of the whiteboard shows four proposed drawings illustrating the trajectories of the Earth and the Moon.The students made these detailed drawings themselves.As they grappled with the task, I walked around the room and observed their efforts.I selected four students whose drawings differed significantly in important details and asked them to redo their illustrations on the board at the front of the classroom.

Observed differences in plots
After presenting these different ideas on the board, I gave the students a second task: to identify the differences between the proposed trajectories.Below is a summarised list of the observed differences in the characteristics of the trajectories.

A Trajectory of the Moon:
1 has loops (P1, P2 and P3) or has no loops (P4), 2 if there are loops, they are directed inwards (P1 and P3) or outwards (P2), 3 the trajectories have a different number of loops (P1: many, P2: fewer, P3: about 13), (there are also about 13 'waves' in the trajectory of the Moon in P4), 4 it may have a different average deviation from the Earth's trajectory (different distance of the Moon from the Earth compared to the distance between the Sun and the Earth; P1, P2 and P4: larger, P3: smaller).B The Earth's trajectory: 1 is circular (P3 and P4) or obviously elliptical (P1 and P2), 2 the Sun is in the centre (P1, P3 and P4) or outside the centre (P2), 3 seasons are noted (P2), 4 there are details of the orbit (exaggerated; not very conspicuous, but wavy, P3).C The direction of motion is marked (P1, P2 and P4) or not (P3).
Each of the identified peculiarities can be discussed and analysed, which can be an extremely illustrative and didactic example of how the scientific mind works.Let us therefore proceed with the discussion of the observations listed above.We will follow the order of the list, but from the back and some points will be discussed together.

Discussion on the observed differences
C If you look at the orbits of the Moon and the Earth from a distant perspective above the plane of their orbits and from the north side, both revolutions and rotations occur in the same-positivedirection, as shown in figure 2. All students who have taken the trouble to indicate these directions have done so correctly.It seems that this particular feature of motion is firmly embedded in students' minds, possibly due to the fact that they have seen numerous illustrations of the system of the Sun, Earth and Moon in which this characteristic of motion is clearly emphasised.
B4 The author of P3 demonstrates a more solid understanding of basic physics concepts compared to the other students in the group.In particular, he correctly recognised that the centre of mass of the Earth-Moon system traces (almost) a circular orbit.In his analysis, the Moon, being less massive, was incorrectly shown to have a looping orbit, whereas the Earth, being more massive, exhibits only a slight deviation from the orbit of the centre of mass, which is correct.Furthermore, the author of P3 offered a valuable insight by noting that the Earth's deviation in his diagram is even less pronounced than depicted.He also pointed out that the Moon is on the opposite side of the Earth with respect to the orbit of the center of mass of the Earth-Moon system.
B3 The author of P2 holds a widespread misconception about the origin of the seasons.In his plot he claims that summer occurs when the Earth is closer to the Sun, and winter when it is furthest from the Sun.
B1 and B2 We have dealt elsewhere [11] with the misconceptions about the position of the Sun in relation to the Earth's orbit and also about the shape of the orbit.For the sake of completeness, we repeat once again that the basis of the two diagrams (P1 and P2) was the idea of the ellipticity of the Earth's orbit, which is, however, extremely exaggerated in these diagrams; the eccentricity of the Earth's orbit is far too small to distinguish the ellipse from the circle with the naked eye.The authors of P3 and P4 are aware of this fact, while the authors of P1 and P2 are not.Another distinguishing feature is the position of the Sun, which is in the centre or outside the centre (at the focal point of the ellipse).This detail can be attributed to knowing more about ellipses and Kepler's laws (P2) or being confused by the many illustrations in textbooks and on various websites that are used to explain the origin of the seasons and use a side view perspective on the ecliptic plane to also show the tilt of the Earth's axis of rotation on the same illustration.
A4 Only the author of P3 thought about distances and was able to remember the fact that the distance between the Moon and the Earth is much less than the distance between the Earth and the Sun.He was able to show this explicitly on his diagram and at the same time draw some details of the trajectory (even if these details are only partially correct).He additionally commented verbally that the deviation of the Earth from the trajectory of the center of mass (of the Earth-Moon system) is much less than he drew.The authors of P1, P2 and P4 have not given any thought to the distances and scale they need to use to draw these trajectories.The general fact that should be known to most, and which is related to these distances, is the time it takes for light from the Sun or Moon to reach the Earth.When we discussed this feature of the diagrams with the students afterwards, almost all of them knew that when we observe the Sun at sunset, at the moment when it seems to disappear below the horizon, in reality it has already disappeared below the horizon for 8 min.And some of them later remembered that it takes just over 1 s for the light from the Moon to reach the Earth.
A3 The authors of P3 and P4 took into account the well-known fact that the Earth makes one revolution around the Sun in a year, while the Moon makes about 13 revolutions around the Earth in the same period.The authors of P1 and P2 did not take this fact into account when they drew the trajectory of the Moon.
A2 The direction of the loops is linked to the relative direction of the motions.If we look at the actual sense of revolutions, as shown in figure 2, the loops (if any) are directed inwards, as shown in P1 and P3.Of course, the orbits also have the same shape when viewed from the other side of the ecliptic plane (from the southern side), the loops are directed inwards and for all motions the sense should be marked in the opposite direction.The orientation of the loops in P2 does not match the marked direction of the motions in P2.
A1 The trajectory of the Moon around the Sun is a cycloid.Whether or not there are loops in the Moon's trajectory depends on the speed of the Earth and the Moon.Since the speed of the Earth (of the Earth-Moon system) around the Sun (v E ≈ 30 km s −1 ) exceeds the orbital speed of the Moon around the Earth (v M ≈ 1 km s −1 ), there are no loops in the Moon's trajectory.Proof of the non-existence of loops can be provided by analysing the motion of the Moon at certain times when it is closest to the Sun (in the new moon phase) since there the magnitude of the speed of the Moon in heliocentric system is the smallest.At these times, the Moon moves within the Earth-Moon system (geocentric) in the opposite direction to the overall motion of the system within the Solar System (heliocentric).Nevertheless, in the heliocentric system, the Moon moves forward alongside the Earth, albeit at a slightly slower speed v M,h = v E − v M ≈ 29 km s −1 .For the loops to exist, there would have to be moments (positions in the Moon's trajectory) when the Moon is moving backwards (in the opposite direction to the Earth), as seen from the heliocentric system and shown in figure 3.
Even without knowing the specific values for the speed of the Moon and the Earth-Moon system, one can arrive at the same conclusion by considering well-known facts about distances and orbital periods.The distance between the Moon and the Earth is r M , the distance between the Earth and the Sun is r E ≈ 400 r M , the orbital period of the Moon is t M (slightly less than 1 month), and the orbital period of the Earth is t E ≈ 13 t M .Using these established facts, the ratio of the two speeds can be easily calculated as v E /v M = 2π rE tE • tM 2π rM = 400/13 ≈ 31, leading to the same reasoning as described in the previous paragraph.
In discussing this issue, I could use a Wolfram's simulation [12] but instead I use my own GeoGebra application [13] that shows the approximate trajectory of the Earth's satellite around the Sun as a function of the distance between the satellite and the Earth, assuming (even if this is not physically justifiable) that the orbital periods remain unchanged.The trajectory, which corresponds to the real orbital parameters, is obtained by introducing a dimensionless factor r = r M /r E = 1/400 = 0.0025 and is shown in figure 4(a), while, for example, if r = 0.055 or r = 0.120, the trajectory changes from one with prominent deviations of the Moon away from the Earth with no loops (r = 0.055, figure 4(b)) to one where loops are present (r = 0.120, figure 4(c)).
Finally, we will examine two specific aspects that arise indirectly from the discussions of the Moon's trajectory.The first concerns the conditions under which the Moon's trajectory exhibits loops when observed in the heliocentric system.The second concerns the curvature of the Moon's orbit within the same system.Both questions are addressed within the framework of a simplified model that assumes circular orbits of the Moon around the Earth and of the Earth around the Sun, while at the same time assuming r E ≫ r M .

Mechanics of the loops
To determine the distance r M at which the Moon should orbit the Earth (while the Earth remains at a constant distance r E = 150 • 10 9 m from the Sun) for loops to manifest in Moon's trajectory, we apply Newton's law of gravitation and equations of motion.Assuming r M is significantly smaller than r E we can neglect the curvature of the Earth's trajectory.Loops occur when the Moon's velocity v M exceeds the Earth's velocity v E .By utilizing the expression for radial acceleration in circular motion and recognizing gravity's role, we derive v M = √ Gm E /r M and v E = √ Gm S /r E where m E and m S are the masses of the Earth and Sun, respectively.Loops in the Moon's trajectory appear if r M < r E m E /m S = 450 km and this is much smaller than the Earth's radius (which is 6400 km)! (The Moon should be inside the Earth.There are further complications arising from this result; for example, the expression used for the gravitational force between the Earth and the Moon is not valid in these circumstances.But that goes beyond the scope of this article.)

Curvature of the moon's trajectory
Another question that arises concerns the curvature of the Moon's trajectory in a heliocentric system.The local curvature of the trajectory is determined by the resultant force, which is a vector quantity, acting on the Moon, and the resulting acceleration.The change in velocity has the same direction as the resulting force.There are two gravitational forces acting on the Moon: one from the Earth and one from the Sun.These forces act in the same direction when the Moon is in position 1 (see figure 3) and in opposite directions when the Moon is in position 2. Consequently, the resultant force is at its maximum at position 1 and at its minimum at position 2.
Certainly, the centre of curvature for the segment of the trajectory with the Moon at position 1 is on the side where both the Earth and the Sun are relative to the Moon.However, if the Moon is at position 2, the situation is extreme, and it is not immediately obvious where the centre of curvature is for this segment of the trajectory.Within our simplified model of circular motion, three regimes are expected, as shown in figure 5: the trajectory can be curved towards the Sun (a), curved towards the Earth (b), or remain straight (c).Which of these possibilities is realised depends on the magnitudes of the gravitational forces: if the Sun's pull of the Moon towards itself F g,S→M = Gm S m M /r 2 E is larger than the Earth's pull of the Moon away from the Sun F g,E→M = Gm E m M /r 2 M , the Moon's trajectory in position 2 ≈ 2, confirms the curvature depicted in figure 5(a) is correct.This implies that when the Moon is in position 2 and also throughout the entire trajectory, it undergoes a change in velocity, redirecting itself towards the Sun (and not away from it).The trajectory is least curved in position 2 and most curved in position 1, but in both cases, the centre of curvature is on the same side of the trajectory as the Sun.

Conclusions
I aimed to illustrate how a seemingly simple task in basic astronomy can serve as a rich source of discussion about various concepts, stimulating a thorough re-evaluation of students' understanding and connections between key ideas.One such key idea is the change of perspective from one frame of reference to another, such as transitioning from geocentric to heliocentric perspectives.This shift is often recognised as challenging but also fruitful, as it makes the concepts of relativity of motion in space, and more broadly, more explicit.Another important idea is the sense of motion (revolution), which depends on the viewpoint of the observer (from the north or from the south).When marking the motion in schematic figures to illustrate the trajectory, the relative sense of revolutions of objects involved also influences the qualitative shape of the trajectory.This highlights the significance of perspective in understanding celestial (and also earthly) phenomena.
The observations and results of the activity described in this article were qualitatively similar to the results obtained in groups of our own students, prospective physics teachers.In the group of a similar number of our students, there are always students who draw similar versions of trajectories, only there are fewer who manage to combine several inappropriate features in one trajectory.This can be attributed to the fact that our students have a more solid scientific background.One of the compulsory subjects they take is astronomy, and this fact is reflected in their better average answers.

Figure 1 .
Figure 1.A screenshot of the whiteboard with 4 different ideas about the trajectory of the Earth and Moon around the Sun.They are labelled P1, P2, P3 and P4.

Figure 2 .
Figure 2. The Earth and the Moon orbit and rotate in the same sense.Seen from the north side, in a positive sense.Distances (and radii) are not to scale.

Figure 3 .
Figure 3.At any moment the velocity of the Moon in heliocentric system is⃗ v M,h = ⃗ v E +⃗ v M .When the Moon is in opposition to the Sun (position 1) its velocity's magnitude is v M,h = v E + v M ≈ 31 km s −1 and it aligns with the direction of Earth's velocity.When the Moon is between the Earth and the Sun (position 2) its velocity's magnitude in the heliocentric system is v M,h = v E − v M ≈ 29 km s −1 and it coincides with the direction of Earth's velocity.Distances (and radii) are not represented to scale.

Figure 4 .
Figure 4. Trajectory of the Moon around the Sun, depending on the ratio r = r M /r E , while keeping the ratio of the orbital periods n = t E /t M ≈ 13 the same.The situation without loops in (a) corresponds to the true ratio r = 0.0025, in (b) with r = 0.055 there are still no loops and in (c) with r = 0.120 there are loops.

Figure 5 .
Figure 5.The local curvature of the Moon's trajectory depends on the direction of the resultant force acting on the Moon.When the Moon is in position 2 (between the Sun and the Earth), the resultant force can be directed (a) towards the Sun, (b) away from the Sun, or (c) equal to zero.The last situation would occur only when the gravitational pulls of the Sun and the Earth on the Moon are of the same magnitude.