# Angles in Moon, Earth and Sun system

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Consider the vector positions of the center of mass of Moon, Sun and Earth, $vec{r}_{ m M}$, $vec{r}_{ m S}$, $vec{r}_{ m E}$, respectively, in a given reference frame.

Consider:

• the angle between the vector $vec{r}_{ m S} - vec{r}_{ m E}$ and $vec{r}_{ m M} - vec{r}_{ m E}$,
• the angle between the vector $vec{r}_{ m E} - vec{r}_{ m S}$ and $vec{r}_{ m M} - vec{r}_{ m S}$,
• the angle between the vector $vec{r}_{ m E} - vec{r}_{ m M}$ and $vec{r}_{ m S} - vec{r}_{ m M}$.

Does any of these angles have a name in astronomy, or can be important for some specific reasons? If so, may you please give me some general indications of why this is the case, and point to some references?

## Angles in Moon, Earth and Sun system - Astronomy

#### Moon Phase

The universe is 13.7 years old.

The theory about how the universe was created is called the Big Bang Theory.

Leftover energy from the big bang: Cosmic Background Radiation.

A huge group of stars bound together by gravity: a galaxy.

A galaxy with a bulge in the middle, and arms that spiral outward is called a SPIRAL GALAXY (The Milky Way (our galaxy) is a spiral galaxy)

A galaxy that looks like a round or flattened ball elliptical galaxy.

A galaxy that does not have a regular shape is called a irregular galaxy.

## Angles in Moon, Earth and Sun system - Astronomy

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%A Kristen Miller
%A Scott Miller
%T Introductory Astronomy: Earth - Moon - Sun System
%D 2000
%U http://www.astro.umd.edu/resources/introastro/moon.html
%O text/html

%0 Electronic Source
%A Miller, Kristen
%A Miller, Scott
%D 2000
%T Introductory Astronomy: Earth - Moon - Sun System
%V 2021
%N 25 June 2021
%9 text/html
%U http://www.astro.umd.edu/resources/introastro/moon.html

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## Angles in Moon, Earth and Sun system - Astronomy

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%A Kara Lehman, (ed)
%T Sun, Earth and Moon Activity
%D 1997
%U https://web.pa.msu.edu/courses/1997spring/ISP205/sec-3/moon.act.ans.html
%O text/html

%0 Electronic Source
%D 1997
%T Sun, Earth and Moon Activity
%E Lehman, Kara
%V 2021
%N 25 June 2021
%9 text/html
%U https://web.pa.msu.edu/courses/1997spring/ISP205/sec-3/moon.act.ans.html

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## Learning Astronomy

How do seasons work? What is an eclipse? How do tides work? These questions all relate to the Earth, Moon, and Sun system. Use the links on the right to look at the topics such as: Seasons, Moon, Eclipses, and Cool Photographs relating to the Earth and Moon.

Remember to have fun learning how the universe works! This section contains astronomy animations, photos, videos, activities, questions, and lots of information!

Never Stop Exploring.

Each week, along with the lastest parts for your model, you will receive a 24-page full colour magazine containing assembly instructions and in-depth articles about space.

Detailed, step-by-step instructions for each assembly stage of the model.

Travel to the farthest reaches of the solar system. Starting from the Sun, we take you on a tour of the solar system’s marvels and mysteries. Find out all the most up-to-date information on the planets, dwarf planets, moons, asteroids and much more.

Turn to the Image Gallery for jaw-dropping space photography and photorealistic artworks. Future issues include spectacular images from the Hubble Space Telescope as well as the biggest and best of the Earth-based observatories.

Trace the history and development of astronomy from its birth in prehistoric monuments to the high-tech science of space probes and satellites. Find out about the life and works of the greatest astronomers from Copernicus to Hawking.

A new constellation with every issue, including information on all the most fascinating celestial objects found within it and practical tips on how best to observe it. Explore the night skies with specially commissioned star maps.

### Build a Precision Mechanical Solar System

Issue 1
Date-Stamped Base Plate, Zodiac Calibration Ring and Feet

Issue 2
Magazine + Sun | Screws | To hat bush | Column support | Column | Drive shaft

Issue 3
Magazine + Venus gear support arm | Tooth gear | Tooth driving gear | Gear axels 1 & 2 | Screws

Issue 4
Magazine + Tooth Gear | Mercury planet arm | Screws | Planet Mercury

Issue 5
Magazine + Tooth Gear x 2 (Venus gear set) | Planet arm collar | Gear spindle | Screws | Plastic washers

Issue 6
Magazine + Moon gear arm | Planet Venus | Tooth gear x 2 (Moon gear set) | Outsize gear axles | Venus planet arm | Screws

Issue 8
Magazine + Vertical support arm | Earth’s Moon | Tooth gear | Planet Earth

Issue 10
Magazine + Earth support arm | Tooth gear | Earth Spindle bush | Earth spindle | Screws | Plastic washers

Issue 11
Magazine + Planet gears and arm to complete stage 2 phase 2

Issue 12
Magazine + Planet mars, its two moons and vertical support arm

Issue 13
Magazine + 106-tooth gear for the planet mars gear train

Issue 14
Magazine + 43-tooth gear, gear collar and planet spindle

Issue 15
Magazine + 85-tooth gear for the planet mars gear train

Issue 16
Magazine + 22-tooth gear and gear arm to complete mars phase

Issue 17
Magazine + Ceres, its 106-tooth gear and arm begin stage 2 phase 4

Issue 18
Magazine + 45-tooth gear, gear collar and planet spindle

Issue 19
Magazine + 83-tooth gear for the ceres gear train

Issue 20
Magazine + 22-tooth gear and gear arm to complete ceres phase

Issue 21
Magazine + Planet jupiter, its four major moons and support arm

Issue 22
Magazine + 106-tooth gear for the jupiter gear train

Issue 23
Magazine + 43-tooth gear, gear collar and planet spindle

Issue 25
Planet Gear Arm, Self-Tapping Screws, Grub Screws, 22-Tooth Gear, Gear Axle 1 & 2

Issue 28
48-Tooth Gear, Gear Collar, Plastic Washers, Self Tapping Screws, Grub Screws & Planet Spindle

Issue 30
Planet Gear Arm, Self-Tapping Screws, Grub Screws, 22-Tooth Gear, Gear Axle 1 & 2, 3 Base Plate Feet

Issue 31
Planet Uranus, Uranus' Support Arm, Uranus' Moon

Issue 32
106-Tooth Gear for the Planet Uranus Gear Train

Issue 33
37-Tooth Gear, Gear Collar and Planet Spindle

Issue 34
91-Tooth Gear To Drive Your Planet Uranus Gear Train

Issue 35
22-Tooth Gear and Uranus Planet Gear Arm

Issue 36
Planet Neptune, Support Arm and Neptune's Moon

Issue 37
106-Tooth Gear for the Planet Neptune Gear Train

Issue 38
30-Tooth Gear, Gear Collar and Planet Spindle

Issue 39
98-Tooth Gear for the Planet Neptune Gear Train

Issue 40
22-Tooth Gear and Gear Arm to Complete Neptune Phase

Issue 41
Dwarf Planet Pluto, it's Moon Charon and Support Arm

Issue 43
41-tooth gear, gear collar and planet spindle

Issue 44
87-Tooth Gear for Pluto's Gear Train

Issue 45
22-Tooth Gear for the Dwarf Planet Pluto Gear Train

Issue 46
Dwarf Planet Eris, it's 106-Tooth Gear and Support Arm

Issue 47
Bank Gear, Gear Collar and Planet Spindle

Issue 49
Legs, Rubber Leg Pads and Allen Key Screws

Issue 50
The Base Plate, Knob and Speed Control System

Issue 51
Motor with Gearbox and Noise-Eliminating Foam

Issue 52

Sorry, that issue isn't available

### Build a Model Earth, Moon & Sun Orbiter

Issue 54
Earth Axis, Support Arm and 18-Tooth Gear

Issue 55
54-Tooth Chamfered Gear an Steel Bearings

Issue 56
14-Tooth Sprocket, Spacer Tubes and Locking Collar

Issue 57
91-Tooth Gear for the Earth-Moon System

Issue 58
Brass Spindle ollar, Feet, Large Washer and Screws

Issue 60
Moon Pillars, Steel Bearings and Screws

Issue 65
91-Tooth Gear for the Central Shaft

Issue 66
Central Shaft, Brass Collar abdb

Issue 67
2-Tooth Gear, 22-Tooth Chain Sprocket, 23-Tooth Gear

Issue 68
18-Tooth Chain Sprocket, Brass Spacers and Washer

Issue 72
19mm Steel Bearings, Washers and 6mm Screws

Issue 73
102-Tooth Gear to Rotate the Main Earth Arm

Issue 74
Model Sun for your Earth, Moon & Sun Orbiter

Issue 75
Central Column and Central Shaft

Issue 76
Central Column Support and Temporary Plastic Feet

Issue 77
91-Tooth Gear for the Sun's Central Shaft

Issue 78
Brass Collars for the Sun's Central Shaft

Issue 79
160-Tooth Gear for the Sun's Central Shaft

Issue 80
Engraved drum Plate and Year Counter Cover Plate

Issue 82
Engraved Drum Plate and Year Counter Cover Plate

Issue 83
Year Counter Mounting Block, Mounts and Gear Train

Issue 85
Center Shaft and Gear Components

Issue 88
Components for your Anomalistic Month Indicator

Issue 90
Components for the Drive Shaft and Gears

Issue 93
The Model Orbiter's Three Brass-Plated Legs

Issue 94
Drive Motor and Pre-Assembled Gearbox

Issue 95
On-Off-Reversing Switch and Speed Control System

Issue 96
Three 76mm Base Bolts

Issue 97
14-Tooth Chain Sprocket, 20-Tooth Gear, Brass Spacers

Issue 99
14-tooth chain sprocket and locking spacer

Issue 102
Sun-mounting spindle and date pointer

Issue 103
LED Ill uminating sun with Battery box

Issue 104
Power adaptor and eclipse fine-tuning stickers

Sorry, that issue isn't available

## Teaching Notes and Tips

Areas of Confusion/Misconceptions:
- The Sun does not have a shadow.
- The Moon's shadow is smaller than the Earth.
- The Earth's shadow is larger than the Moon.
- The New Moon is visible during the day.
- The Sun's light is blocked by the moon during the middle of the day.
- The Earth's shadow does not create the phases of the Moon.
- The Moon is always half lit by the Sun.
- The amount of the lit portion of the Moon remains the same but the amount that is seen from Earth changes.

Tips
- Shutting off the classroom lights will help students see shadows.
- Having the lamps (or overhead projectors) spaced apart will help prevent shadows from other directions.
- Allow students to explore the shadow concepts, but reinforce and guide struggling to being looking for shadows as they cast on other objects.

This activity is different from what I've had my students do in the past. Usually I have just done this activity with the same materials to explore Moon Phases. However, when I get to teaching about eclipses, the students usually look at diagrams and video clips, rather than do any 3-D modeling themselves. This will hopefully engage them more and help them conceptually understand the phenomena before I show them real footage of eclipses from Earth, diagrams of the eclipses from space, views of the Moon phases from Earth, and diagrams of the Moon as it revolves around the Earth from space vantage point.

## Overview

Before you begin this course, make sure you have completed the Course Orientation.

Most of us do not notice the daily and yearly changes in the sky because, frankly, we no longer have to bother! Clocks and watches have been around so long that they hardly seem to be advanced technology. Anyone can look at a calendar to tell you what the date is, and, if we are lost on an unfamiliar road, most phones have GPS functions that can help us find our way home. Even though calendars, clocks, and navigational tools have insulated us from the need to understand the motions of the sky, most of us do appreciate that our system of time and our location on the surface of the Earth can be tied directly to observations of the sky.

In this lesson, we will study the observable changes in the positions of objects in the sky and show how these data allow us to understand the relative positions and motions of the Sun, Moon, Earth, and the stars. My hope is that from this point forward, whenever you look at the sky you'll be aware of what it can tell you about time, date, and your location.

### What will we learn in Lesson 1?

By the end of Lesson 1, you should be able to:

• identify the objects visible in the night sky to the unaided eye
• describe the three dimensional geometry of the Earth / Moon / Sun system
• describe the various motions in the sky that result from Earth’s rotation and orbit
• explain the reason that the Earth experiences seasons
• describe the process and appearance of eclipses and the phases of the Moon.

### What is due for Lesson 1?

Lesson 1 will take us one week to complete. Please refer to the Calendar in Canvas for specific time frames and due dates.

There are a number of required activities in this lesson. The following table provides an overview of those activities that must be submitted for Lesson 1.

Lesson 1 Activities
Lesson 1 Quiz Your score on this Canvas quiz will count towards your overall quiz average.
Discussion: Teaching and Learning about the Moon Participate in the Canvas discussion forum "Teaching and Learning About the Moon."

### Questions?

If you have any questions, please post them to the General Questions and Discussion forum (not email). I will check that discussion forum daily to respond. While you are there, feel free to post your own responses if you, too, are able to help out a classmate.

## Including Students in a Model of the Earth, Moon, and Sun System

Including Students presents a single, kinesthetic model that can be used to explain such varied concepts as rotation, revolution, phases of the Moon and seasons. The outlined activities are meant to be used throughout an entire Astronomy unit. In fact, the authors caution that this collection of activities cannot be completed in one classroom period. Because the activities increase in complexity, the authors recommend that teachers take their time in introducing the model. Then, students will be comfortable in modifying it to explore the more complex concepts. The required materials are easily obtainable and relatively inexpensive. The authors have taken great pains to address misconceptions and to provide questions that teachers can use to probe students' prior knowledge.

#### Performance Expectations

MS-ESS1-1 Develop and use a model of the Earth-sun-moon system to describe the cyclic patterns of lunar phases, eclipses of the sun and moon, and seasons.

Clarification Statement: Examples of models can be physical, graphical, or conceptual.

Assessment Boundary: none

This resource is explicitly designed to build towards this performance expectation.

Including Students presents a kinesthetic model that uses a lamp, students or globes, and Styrofoam balls to represent the Sun, Earth and Moon, respectively. The outlined activities cover lunar phases and seasons however, the authors note that, with modifications designed by teachers and students, the model could be used to represent other scientific phenomena such as eclipses and rising/setting times of lunar phases. Students do not develop the initial model from scratch rather, they learn to use the described model in order to make predictions and construct explanations. Throughout the activities, students are encouraged to discuss the model's limitations with an eye to improvement.

#### Science and Engineering Practices

This resource is explicitly designed to build towards this science and engineering practice.

Throughout the activities, the authors present multiple opportunities for students to use a model to construct explanations about such scientific phenomena as lunar phases and seasons. The authors lean towards classroom discussion to check understanding to monitor individual understanding, teachers may wish to design prompts for students to discuss in their journals/notebooks. For example, students might be asked to describe the necessary geometry for viewing the various phases of the Moon from Earth or they might be tasked with explaining how Earth’s tilt affects the duration of sunlight received at different geographic locations. This will give students the added advantage of recalling their explanations when returning to the model at a later date.

#### Disciplinary Core Ideas

This resource is explicitly designed to build towards this disciplinary core idea.

Part 3: Moon phases uses a model to explain how the Moon's revolution around Earth results in our observance of phases. In this activity, students use their own heads and Styrofoam balls to represent the Earth and Moon, respectively. By revolving the ball around their heads, students will be able to note illuminated differences in its appearance. Using this information, students can then construct an explanation for lunar phases.

This resource is explicitly designed to build towards this disciplinary core idea.

Part 4: Seasons presents an elegant model to investigate how the tilt of the Earth leads to seasons. Using a globe and two nails, students will see how the duration of sunlight received by the Earth on any given day may differ between the two hemispheres. In particular, students will be able to qualitatively measure the hours of sunlight received on both the equinoxes and solstices. This information will assist students in understanding why seasons occur.

#### Crosscutting Concepts

This resource is explicitly designed to build towards this crosscutting concept.

By working through the model, students will soon understand that the Moon's repetitive orbit around Earth, coupled with our changing viewpoint, results in the different phases of the Moon that we view over the course of a month. Students may initially need help in determining the location of the terminator, but once they accomplish this, they will easily identify the different phases. In Part 4: Seasons, students utilize Earth's tilt and the cyclical nature of its revolution to explain how differences in sunlight duration and intensity result in seasons.

In this very practical workshop you can make the model either as a painted picture onto the wall in a 2-dimensional form or use pupils presenting "constellations" in a 3-dimensional form in the middle of the classroom or on the yard of the school. We will use now the latter method.

In the middle of the construction there will be the Sun and the Earth with its orbit. The zodiacal constellations are around this system in the right positions due to the seasons. If you imagine yourself being on the surface of the Earth and looking at the direction of the Sun, you know that there is a constellation behind the Sun. The constellations through which the Sun "goes" during the year form the Zodiac.

When we construct the model, we have to think of the positions of the zodiacal constellations compared with the positions of the Earth in its orbit. We have to take into account the summer and the winter, i.e. the positions where the Earth is at the nearest or farthest away, in perihelion or aphelion.

We will use the overhead projector for drawing the constellations (only the brightest stars) on the cardboard in an as so enlarged size as we want. For making this model more understandable we could paint the mystic figures onto the constellations freehand or using transparencies. The model could work as a two-dimensional version on the wall or it could be more illustrative as a three-dimensional version in the middle of the class. Also we can demonstrate these things by letting the pupils be the parts of this model, each carrying a cardboard of one constellation. So we need twelve "constellations", one "Earth" and one "Sun". There are many astronomical subjects you can teach with this model.

The main theme in our summer school this year is "Astronomy and Navigation". In our workshop we will have some discussions on how to use using stars in the navigation like in the "old" times and also how to use sky photographs in teaching constellations. Our main subject is still the construction of the zodiacal model.

#### Using the stars in the navigation

In the college of navigation the students have to learn to navigate also like in the "old" times. It means that they have to learn to know how to find the necessary information for navigation on the Sun, the Moon, the Zodiac and the biggest stars as well as the eclipses of the Sun and the Moon. Usually only 20-30 brightest stars are used in the astronomical observations of the navigation.

As an example we can learn to find some stars by using constellations we know, e.g. Ursa Major and Ursa Minor. (Fig. 1 and 2)

#### The path of the Sun during one year

The Earth is circulating the Sun in the plane of the Ecliptic. We mean by the Ecliptic the big circle, which forms as a section line of the plane of the orbit of the Earth and the celestial sphere. If you imagine yourself looking at the direction of the Sun, it seems to be in some constellation behind the Sun (fig. 3). These constellations are located near the Ecliptic and this combination is called the Zodiac. The angle between the Ecliptic and Equator is 23º 27' and the intersectional points are the equinox and autumnal equinox. The time which the Sun needs to move from the equinox to the equinox again is called the tropical year. Its lenght is 365 ¼ days, which means that the Sun moves on the celestial sphere about 1 º = 4 m during one day.

The Zodiac's breadth is a 16 degree zone on both sides of the Ecliptic. The signs of the Zodiac devide the ecliptic into 12 part, each one 30 degrees. Because of the motion of the equinox (Aries earlier, now Pisces, one circle in 26000 years) the places of the signs of the Zodiac deviate by 30 degree from the zodiacal constellation with the same name. Only 30 stars of those listed in the Nautical Almanac (NA) is usually used in the navigation. In the almanac you can find very important information on the stars, the planets, the Moon and the Sun, e.g. SHA= Sideric Hour Angle, Dec= declination, conversions of arcs to time units and different corrections.

The names of the zodiacal constellations in order from the equinox are: Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricornus, Aquarius and Pisces. Inside this zodiacal zone are the Moon and the planets also moving because the angles between the planes of their orbits and the Ecliptic are less than 8 º.

In the navigation only four planets are significant. They are Mars, Jupiter, Venus and Saturn. Information on these planets, which is important for the navigation, is available in the Nautical Almanac. You can recognize the planets by their even light, which differs from the twinkling light of the stars. Venus and Jupiter are much brighter than any star. The light of Mars is twinkling red. In the ancient times the stars were classified according to their brightness in the classes of the magnitudes The faintest stars, that could be observed with naked eyes, belonged to the sixth class. Nowadays the brightness is determined by photometric measures.

#### Construction of the zodiacal model

We are going to build a concrete model of the Zodiac, the Sun and the Earth. We will use all of you as living models to make those astronomical subjects perfectly understandable. You can verify the different situations, e.g. the summer, the winter, the periods of the year etc. "The Earth" may circulate around the Sun and "the Sun" is shining very brightly! "The zodiacal constellations" should be in the right positions but some of them might have difficulties in "hanging up" upside down.

You can work in groups of 2-3 persons. Each group will take one constellation to paint. First we will draw the zodiacal constellations to the cardboards. We'll use the overhead projectors for reflecting the pictures from the transparencies to the wall. The distance between the overhead projector and the wall should be the same all the time. (Fig. 4)

After drawing the constellation (only the brightest stars) you can change the transparency with the mystic figure and draw it onto the constellation in the right position and size. If you want, you can be an artist yourself and draw the mystic figure. You can use some colours, too.

After painting all the constellations we are going to collect all parts together to form a compact system of Zodiac-Sun-Earth. In the middle of the classroom we'll make an elliptic circle, in the middle of it there will be the Sun and the Earth. The orbit of the Earth can be marked on the floor, if you think it's necessary. Try to form the right order for the constellations and to take the seasons into account. (Fig. 5)

If you are on a mysterious mood, you can listen to the ancient stories concerning the constellations. The mysterious figures were very warlike, but some of them were also very romantic. Enjoy yourself!

Kaila, K. 1979. Tähtitaivaan opas, Tähtitieteellinen yhdistys URSA ry, Jyväskylä: Gummerus.

Lovi, G., Tirion, W. 1989. Men, monsters and the modern universe.

Löfgren, K-E. 1997. Merenkulkuopin perusteet III: Avomerinavigointi 1993, Helsinki: Suomen navigaatioliitto.

Staal, J. D. 1988. The New Patterns in the Sky, Myths and Legends of the Stars, The McDonald and Woodward Publishing Company, Blacksburg, Virginia.

## Angles in Moon, Earth and Sun system - Astronomy

Lunar Phases are relatively easy to understand but they nevertheless are a source of constant confusion to the public at large.

The moon shines by reflected sunlight - its phase as viewed form the earth depends only on the moon-earth-sun angle. In the figure above, the sun is off to the right. There are 8 positions shown.

Position 1 - New Moon - the moon-earth-sun angle is 0 degrees and the moon is in the same part of the sky as the sun and hence can't be seen from the earth because the illuminated side is facing away from the earth.

Note that if the orbital planet of the earth-moon system and that of the earth-sun system were exactly the same, then each new moon would be marked by this event . In practice, the earth-moon plane is inclined by about 5 degrees so that full solar eclipses are rare. Note the angular sizes of the moon and the sun are both about 1/2 a degree thus making it possible for total solar eclipses to occur. This similarity of size is a coincidence.

Seasonal Variations on the Earth:

The rotation axis of the Earth is inclined with respect to its orbital plane by 23.5 degrees. This is shown in the following:

Because the earth's axis is tilted with respect to the incoming sunlight, the hemispheres receive unequal distributions of solar radiation at certain times of the year. This is shown in the following:

Seasonal variation is caused by the earth's tilted rotational axis.

Position 1 = Winter in the Northern Hemisphere and Summer in the Southern Hemisphere. The Northern Hemisphere is tilted away from the sunlight.

The 23.5 degree tilt of the earth's rotational axis is manifest by the position of the sun in the sky. Over the 6 month period June 21 - Dec 21 the sun moves from +23.5 degrees N. Latitude to -23.5 degress S. Latitude.

If you lived at the equator (Latitude = 0 degrees) the sun would be within 23.5 degrees of overhead at all times - this is why its hot year around at the equator

The following figure shows the position of the Sun at noon with respect to an observer at 45 degrees N. Latitude on Dec 21 and June 21. The path of the sun (yellow dotted arcs) in the sky is much higher on June 21 than Dec 21. On Dec 21 the highest point in the sky above the horizon will be 21.5 degrees as shown: