# Physics Tutorial: The Moon's Movement. Eclipses. Calendars

In this Physics tutorial, you will learn:

• How does the Moon move?
• What is tropical and synodic month? How do they differ from each other?
• What are the Moon phases? When do they occur?
• What are eclipses? How many eclipses are there?
• How calendars are designed? How many calendars are there?
• What factors are considered when designing a calendar?
• How to calculate the alignment period between two planets?

## Introduction

It is a matter of fact that the Moon is the only natural satellite of Earth and its closest celestial body. It is a rocky celestial body with many craters on its surface formed by the strike from asteroids experienced in time. Moon was formed about 4.5 billion years ago, about 30-50 million years after the origin of the Solar System, out of debris thrown into orbit by a massive collision between a smaller proto-Earth and another planetoid, about the size of Mars. Gravitational forces, like in all the other celestial bodies, are the main factors that determine the movement of Moon. However, unlike in planets where only the Sun is considered when dealing with gravitational forces, as the effect of other celestial bodies in this force is negligible, there are two different gravitational forces that determine the trajectory of Moon: the gravitational force exerted by the Sun and that exerted by Earth on the Moon. The sum of these two forces (net gravitational force) makes the Moon deflect too much from the pure circular (or elliptic if you want) motion (discussed in tutorial 8.2) as shown in the figure below. (Actually Earth makes the same kind of rotation around the centre of our galaxy when revolving around the Sun but for now, we are not interested on this). ## Main Features of the Moon's Movement

The Moon also makes one complete cycle around the Earth in an elliptic-shape trajectory in a similar fashion to planetary motion around the Sun, where Earth is located in one of the ellipse foci. This cycle lasts for 27.32 days (a tropical month) when the (unmoveable) stars are taken as a reference frame and 29.53 days (a synodic month) when the Sun is taken as a reference frame. The successive new month begins when the Moon shows the same face it had in the previous month when viewed from Earth. This method for counting lunar months (i.e. to observe the same periodic view of the Moon) is used only when dealing with synodic month.

This difference in month duration is due to the Earth displacement around the Sun during the given period, similar to day duration difference in sidereal and tropical discussed in the previous tutorial (Earth day and solar day).

Aphelion (the longest distance) of Moon's orbit around the Earth is 407 000 km while perihelion (the shortest distance) is 356 000 km. This large difference indicates a long (and pressed) elliptic trajectory, more or less like this: Now, let's explain why there is a difference of about two days between tropical month (27.32 days) and synodic month (29.53 days). The following figure is very helpful in explaining this point. The Moon revolves around the Sun in the same direction as Earth does. Let's consider the initial position of Moon at M1 (Earth meanwhile is at E1). When the Moon completes one cycle in respect to remote (unmoveable) stars, it is in the position M2 (Earth is at E2). This process occurs in 27.32 days. During this time, the Earth rotates by 360°/12 = 30° (more accurate measurements give a value of 27°) around the Sun. However, in order to show the same view to the Earth, the Moon must be at M3. It takes about 2 days to the Moon to move from M2 to M3 and complete therefore the cycle of synodic month.

## Moon Phases

Like all the other celestial bodies (except stars), the Moon does not produce light by itself. It only reflects the light incident from the Sun, which always illuminates only half of the Moon. This illuminated part looks different in various periods of the month depending on the position of the Moon. This different view of the the Moon from the Earth at different times of the lunar month determines the Moon's phases. The figure below clarifies this point. Aside each Moon position there is another figure, which shows how the Moon looks at that stage when viewed from Earth. The names of the eight phases of Moon are, in order, new Moon (M1), waxing crescent (M2), first quarter (M3), waxing gibbous (M4), full Moon (M5), waning gibbous (M6), third quarter (M7) and waning crescent (M8).

When the Moon is at position M1, it is not visible from Earth. This stage is known as "new moon".

When the Moon is at position M2, only a small portion of the Moon is visible from Earth.

When the Moon is at position M3, there is half of it visible from Earth. This stage is known as "half-moon".

When the Moon is at position M4, there is more than half of it visible from Earth.

When the Moon is at position M5, the entire bright side of it is visible from Earth.

When the Moon is at position M6, the image viewed from earth is similar to M4.

When the Moon is at position M7, the image viewed from earth is similar to M3 (half-moon).

When the Moon is at position M8, the image viewed from earth is similar to M2.

When the Moon is at its initial position (M1), the new lunar month starts.

Thus, depending on the position of the Moon, its appearance changes when viewed from Earth. In the position 2 and 8 it looks like a sickle, while in positions 4 and 6 it looks like a bitten orange and so on.

Besides the orbital revolution, the Moon rotates around its own axis as well. This rotation occurs at the same period as the period of revolution around the Earth (in one synodic month). This is the reason why the Moon always shows the same face to Earth. As we have explained in the previous tutorial, all natural satellites manifest the same behavior except Hyperion (one of natural satellites of Saturn). This coincidence in periods derives from the fact that the Earth and the Moon are both massive objects that don't have a very large distance between them - a fact that prevents us from considering the Moon as a point object. The gravitational force exerted by Earth on the Moon produces a moment that makes the two abovementioned periods equal. This equalization has been achieved after millions of years of Earth-Moon coexistence. Tides represent a similar effect but this time exerted by the Moon on Earth.

## Eclipses

An eclipse is an obscuring of the light from one celestial body by the passage of another between it and the observer or between it and its source of illumination. The orbital plane of the Moon forms an angle of 5° 9' on average to that of Earth. This small angle is the reason why eclipses occur.

There are two types of eclipses occurring on Earth surface. They are the solar and lunar eclipses. Let's get a closer look at both of them.

### 1. Solar eclipse

This kind of eclipse occurs when the Moon is between the Sun and the Earth. During this process, less sunlight than usual falls on a certain region of the Earth and the environment darkens, this event may occur during the daytime. At this moment the Moon is at perihelion (at the shortest distance from the Earth), view of the Sun may be blocked completely, despite the Suns' dimensions being much larger than those of the Moon. This phenomenon is known as the total solar eclipse. It may only be visible in certain regions of the Earth. In other regions where not all sunlight is blocked by the Moon, we have a partial solar eclipse, as shown in the figure below. The dark region shown on the figure by black colour represents a total solar eclipse. Thus, where no sunlight is incident in that region while the rest of the Earth experiences a partial solar eclipse as sunlight is blocked only partially.

The duration of a solar eclipse lasts from a few minutes (for total solar eclipse) to a few hours (partial solar eclipse) as both the Earth and Moon shift from the actual position as both of them transition in their respective orbits.

The maximum diameter of shadow formed by total solar eclipse is 250 km. It shifts across various location of the Earth, providing a spectacular event, where the stars can be observed in the sky during daytime. The solar corona, protuberances and chromosphere surrounding the Moon disc are only visible from the Sun.

In a partial solar eclipse the Sun is not completely blocked but it looks like a bitten orange. This view is visible in regions extending up to a few thousands km. When a solar eclipse ends, this means the shadow is no longer formed on the Earth surface but in other regions of space.

A solar eclipse is a rare phenomenon; it occurs when the Moon is precisely between the Sun and the Earth. Theoretically, a solar eclipse may occur once a month. However, in most cases, this phenomenon cannot occur because as explained earlier, there is a shift of about 5° in orbital plane of the Moon in respect to that of the Earth. This makes the Moon shadow miss the Earth surface even if the Moon is between the Sun and the Earth.

### 2. Lunar eclipse

This phenomenon occurs when the Earth is between the Sun and the Moon. During this event, the Moon take a reddish colour because the red radiation emitted by the Sun is the most malleable type of radiation pertaining to visible light. Thus, we see a dark-red Moon instead of a black one. Like a solar eclipse, lunar eclipses may be total or partial as well. Thus, when the red colour covers the entire Moon, we have a total lunar eclipse. When the red colour only appears in a certain part of the Moon, we have a partial lunar eclipse.

## Calendars

Humans invented calendars for the purpose of orientation in time. All calendars are based on the three major cycles that are visible from the Earth:

1. Solar day (24.00 h) related to the self-rotation of the Earth around its axis. This cycle involves the continuous alteration of day and night.
2. Synodic month (29.53 days) related to the orbital revolution of Moon. This cycle lasts between two identical phases of Moon.
3. Solar year (365.2422 days) - otherwise known as the tropical year - which is related to the orbital revolution of the Earth around the Sun. This cycle appears through the continuous alteration of seasons.

For many centuries, it has been a very challenging task to find a satisfactory compliance between these three cycles and to form a unique calendar. Thus, the calendar used by the majority of countries (and officially by all countries) in the world is the "Gregorian calendar". It has 365 days except the years divisible by 4, which have 366 days. However, not all years divisible by 4 have 366 days; those years ending with a double zero and with the first two digits not divisible by 4 (for example 1900, 2100, etc.) have 365 days. This is done for compensation purposes, as one solar year does not last exactly 365.25 days but slightly less (365.2422 days).

Years that have 366 days are known as leap years. The additional days corresponds to February 29 (regular years have 28 days in February).

### Example 1

How many days have elapsed between January 1st, 1700 to January 1st, 2021?

### Solution 1

From 1700 to 2021 there are 2021 - 1700 = 321 years elapsed. If this number is divided by 4, we obtain 321/4 = 80.25(1). But 1800 and 1900 have not been leap years, as 18 and 19 are not divisible by 4. Thus, the number of leap years during the given period is 80.25 - 2 = 78.25.

Hence, the number of days elapsed during the given period is

N = 321 × 365 + 78
= 117243 days

Prior to Gregorian calendar introduction in 1582, a similar calendar (Julian calendar) was used in most of the world. The main difference between Julian and Gregorian calendars is that an average year in Julian calendar is 365.25 days while an average year in Gregorian calendar is 365.2422 days. The Gregorian calendar is the normal calendar we currently use to determine the date. The Julian calendar was used from 46 B.C to 1582. Then, it was replaced by the Gregorian calendar. In Christian Orthodox countries of Eastern Europe (except Greece and Cyprus) the Julian Calendar is still in use when performing religious practices. Thus, in these countries Christmas is not celebrated on December 25 but on January 7 due to the shift in dates caused by using different calendars. When this slight difference in time (365.25 - 365.2422 = 0.0078 days is multiplied by 2068 (2021 + 46 = 2067 is the number of years elapsed from the first introduction of Julian calendar to this date), we obtain:

Shift = 2067 × 0.0078 = 16.1226 days

This gap is somehow reduced by corrections made during the last centuries. However, there is still a 13 days gap between these two solar calendars.

Muslim countries on the other hand, use lunar calendars in their religious practices. This calendar, based on the Moon's phases, is about 11 days shorter than solar ones because one lunar month lasts for 29.53 days and when this number is multiplied by 12, it gives 354.36 days. Therefore, the duration of lunar months is 29 or 30 days, unlike solar calendars where a month is 30 or 31 days long (except February).

The common feature all calendars in use share is the number of months (12).

### Example 2

The current year (2021) corresponds to the years 1442 - 1443 in Muslim countries. When did this (lunar) calendar start?

### Solution 2

The number of days elapsed since the beginning of Muslim (lunar) calendar is

N(d) = 1442 y × 354.36 d/y
= 510987 d

When converted to solar (tropical) years we obtain

N(s.y.) = 510987 d/365.2422 d/y
= 1399 years

Since we are in 2021, the lunar calendar used in Muslim countries did start in 2021 - 1399 = 622. This is the year according the Gregorian calendar when the Muslim lunar calendar started.

## Calculating the Time Needed for the Earth and Another Planet to Realign Again

Let's select an outer planet for this task (one from Mars to Neptune). We take as initial value of time the instant when the Sun, Earth and the planet are collinear. The Earth revolves around the Sun at angular velocity

ωE = /TE

where TE is the stellar (sidereal) period of Earth revolution.

On the other hand, the other planet revolves around the Sun at angular velocity

ωp = /Tp

where Tp is the stellar (sidereal) period of planet's revolution. Given that, in this instance, we are considering an outer planet (its distance from the Sun is longer than that of the Earth), the stellar period of planet is larger than stellar period of the Earth because the angular velocity of planet is smaller than that of the Earth (the planet rotates slower than the Earth around the Sun). This means the Earth has made more revolutions around the Sun than the given planet when they align again in the same direction. Obviously, during this time the Earth has made more than one revolution around the Sun (assuming that the planet has made at exactly one revolution around the Sun).

We denote by θ the time required for this process (i.e. the synodic period of the planet). We have

2π + /Tp ∙ θ = /TE ∙ θ

where is the complete angle in radians.

Simplifying both sides by , we obtain

1 + θ/Tp = θ/TE
1 = θ/TE -θ/Tp

Dividing both sides by θ, we obtain

1/θ = 1/TE -1/Tp

This approach can be used to calculate the realignment time of the two inner planets (Mercury and Venus) as well. The only difference is that the Earth rotates slower this time. Hence, the above formula becomes

1/θ = 1/Tp -1/TE

### Example 3

Assuming that today the Sun, Earth and Mars are perfectly aligned, calculate the next time when this phenomenon will occur. Take the following values for periods of revolution around the Sun: TE = 365.24 days and TM = 686.97 days.

### Solution 3

Mars is an outer planet, so we must use the first formula. We have

1/θ = 1/TE -1/Tp
= 1/365.24 + 1/686.97
= 0.0027379 + 0.0014557
= 0.0041936

Thus, the new alignment of planets will occur in

θ = 1/0.0041936
= 238.46 days

## Summary

The Moon is the only natural satellite of the Earth and its closest celestial body. It is a rocky celestial body with many craters on its surface formed by asteroid strikes. The Moon was formed about 4.5 billion years ago.

Gravitational forces, like in all the other celestial bodies, are the main factors that determine the movement of the Moon. There are two different gravitational forces that determine the trajectory of the Moon: gravitational force exerted by the Sun and that exerted by the Earth on the Moon. As a result, the Moon revolves around the Earth showing always the same face to it.

The Moon makes one complete cycle around the Earth in an elliptic-shape trajectory in a similar fashion to planetary motion around the Sun, where the Earth is located in one of the ellipse foci. This cycle lasts for 27.32 days (tropical month) when the (unmoveable) stars are taken as a reference frame and 29.53 days (synodic month) when the Sun is taken as a reference frame.

Aphelion (the longest distance) of the Moon's orbit around the Earth is 407 000 km while perihelion (the shortest distance) is 356 000 km.

The different view of the Moon obtained from the Earth in different times of lunar (synodic) month determines the Moon's phases. The names of the eight phases of the Moon are, in order, new Moon (M1), waxing crescent (M2), first quarter (M3), waxing gibbous (M4), full Moon (M5), waning gibbous (M6), third quarter (M7) and waning crescent (M8).

In addition to its' orbital revolution the Moon rotates around its own axis as well. This rotation occurs at the same period as the period of revolution around the Earth (in one synodic month). This is the reason why the Moon always shows the same face to the Earth.

An eclipse is an obscuring of the light from one celestial body by the passage of another between it and the observer or between it and its source of illumination. The orbital plane of the Moon forms an angle of 5° 9' on average to that of the Earth. This small angle is the reason why eclipses occur.

There are two types of eclipses that occur on the Earths surface. They are the solar and lunar eclipses. In total solar eclipse, the Moon lays between the Sun and the Earth and therefore, it prevents almost entirely the sunlight from reaching the Earth. As a result the view is obscured. Partial solar eclipses occur when the Sun is partially blocked by the Moon, causing a partial obscuration of the view.

Lunar eclipse occur when the Earth is between the sun and the Moon. In total lunar eclipse all sunlight is blocked from reaching the Moon and as a result, it looks red when viewed from the Earth. On the other hand, in partial lunar eclipse this phenomenon occurs only partially.

Humans have invented calendars for the purpose of orientation in time. All calendars are based on the three major cycles that are visible from Earth:

1. Solar day (24.00 h) related to the self-rotation of the Earth around its axis. This cycle involves the continuous alteration of day and night.
2. Synodic month (29.53 days) related to the orbital revolution of Moon. This cycle lasts between two identical phases of Moon.
3. Solar year (365.2422 days) - otherwise known as the tropical year - which is related to the orbital revolution of the Earth around the Sun. This cycle appears through the continuous alteration of seasons.

The calendar used by majority of countries (and officially by all countries) in the world is the "Gregorian calendar". It has 365 days except the years divisible by 4, which have 366 days. This rule excludes full-century years the first two digits of which are not divisible by 4.

Years that have 366 days are known as leap years. The additional days corresponds to February 29 (regular years have 28 days on February).

Prior to Gregorian calendar introduction in 1582, a similar calendar (Julian calendar) was used in most of the world. The main difference between Julian and Gregorian calendars is that an average year in Julian calendar is 365.25 days while an average year in Gregorian calendar is 365.2422 days.

Muslim countries on the other hand, use the lunar calendar in their religious practices. This calendar based on Moon's phases is about 11 days shorter than solar ones Therefore, the duration of lunar months is 29 or 30 days, unlike in solar calendars in which a month is 30 or 31 days long (except February).

The common feature all calendars in use share is the number of months (12).

The synodic period θ (the period of two consecutive alignments with Earth) of an outer planet (from Mars to Neptune) is calculated by

1/θ = 1/TE -1/Tp

and that of an inner planet (Mercury or Venus) is

1/θ = 1/Tp -1/TE

where TE and Tp are the stellar (sidereal) periods of the Earth and planet revolution respectively.

## The Moon's Movement. Eclipses. Calendars Revision Questions

1. How many days did the 20th century have?

1. 36 500
2. 36 523
3. 36 524
4. 100 000

2. Calculate the average diameter of total solar eclipse if the distance Earth-Moon is 384 000 km and the distance Sun-Earth is 149.6 million km. Take diameter of Sun = 1.39 million km, diameter of the Earth = 12 576 km and that of Moon = 3476 km. Use the similarity formula of trapezium D/d = H/h where D is the greatest diameter, d is the smallest diameter, H is the greatest distance (height of large trapezium) and h is the smallest distance (height of smaller trapezium). 1. 200 km
2. 89.2 km
3. 8.92 km
4. 3.476 km

3. Assuming that today the Sun, Earth and Venus are perfectly aligned, find when this phenomenon will occur again for the next time. Take the following values for periods of revolution around the Sun: TE = 365.24 days and TV = 224.65 days.

1. 0.0071893 years
2. 589.89 days
3. 294.94 days
4. 139.1 days