# The Physics, Astronomy and Mathematics of the Solar System

## CONTENTS OF CURRICULUM UNIT 07.03.06

## Astronomy: The Mathematician's Perspective

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## Strategies

### Graphs

Students will learn how to graph using astronomical data.

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Elements (Gases) and Diameters of Planets
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The elements, or gases, present in the atmosphere of a planet or Sun are dependent upon its mass, gravity, and temperature. If the temperature is high enough, as with Earth, gases such as hydrogen, which is a light element, can easily escape the atmosphere. That is why Earth has no hydrogen in its atmosphere. But, on Jupiter, where it is cold and the mass and the gravity of it are smaller than Earth's, Jupiter then, will keep an abundant amount of hydrogen in its atmosphere.

Description of elements in the atmospheres of the planets:

- Hydrogen is the lightest of all elements and the most abundant in the universe.
- Nitrogen is present in all living things.
- Oxygen is critical for all life on Earth.
- Argon is a highly stable chemical element.
- Carbon dioxide is an important greenhouse gas.
- Helium is the second lightest and most abundant element in the universe.
- Methane is a good fuel because of its clean burning process.
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The diameters of the planets vary according to size with Mercury's equatorial diameter being the smallest at 4,880 km and Jupiter's equatorial diameter being the largest at 142,984 km (See Table 2). Moon data, in terms of radii, can be found in Appendix A.

Gases will be used to instruct about graphs because it presents numerous data to work with, there are intriguing combinations of gases in the atmosphere on each planet, and there are fascinating comparisons of gases between planets, including the Sun. Students will be motivated to learn how to graph and to make conjectures about planets using the data from the graphs.

Diameters of planets and moons will be utilized because it offers different measurements, interesting comparison of measurements of planets in relation to the order of planets in the solar system and of moons in relation to their planets, and after graphing the atmospheres there are notable conjectures to be made about planet size and gases found.

Students will read tables, construct a graph using the horizontal and vertical axis of the first quadrant, label and increment the horizontal and vertical axis, determine the height of the graph, construct bar and line graphs, read and record data, and compare two graphs for similarities and differences.

Mercury: essentially none

Venus: 96.5% carbon dioxide, 3.5% nitrogen, 0.003% water vapor

Earth: 78.08% nitrogen, 20.95% oxygen, 0.035% carbon dioxide, 1% water vapor

Mars: 95.3% carbon dioxide, 2.7% nitrogen, 0.03% water vapor, 2% other gases

Jupiter: 86.2% hydrogen, 13.6% helium, 0.2% methane, ammonia, water vapor, and other gases

Saturn: 96.3% hydrogen, 3.3% helium, 0.4% methane, ammonia, water vapor, and other gases

Uranus: 82.5% hydrogen, 15.2% helium, 2.3% methane

Neptune: 79% hydrogen, 18% helium, 3% methane

Sun: 92.1% hydrogen, 7.8% helium, 0.1% other elements

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Table 1
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Atmospheric composition by number of molecules. Source: Universe, Freedman and Kaufmann, seventh edition, 2005.

Mercury - 4,880 km

Venus - 12,104 km

Earth - 12,756 km

Mars - 6,794 km

Jupiter - 142,984 km

Saturn - 120,536 km

Uranus 51,118 km

Neptune - 49,528 km

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Table 2
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Listed are the equatorial diameters of the planets. Source: Universe, Freedman and Kaufmann, seventh edition, 2005

### Slope

Students will learn about positive and negative slope and how to find slope using the Earth-Moon system.

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The Earth-Moon System
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Both the Earth and Moon orbit around their center of mass
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. The Earth revolves around the Sun and the Moon revolves around Earth. Because the Moon orbits the Earth in the exact amount of time that it rotates on its axis, the Moon keeps the same side or face to us at all times. The Earth and the Moon both exert the same amount of gravitational pull on each other. The Moon's orbit is slightly elliptical and therefore wobbles as it spins in orbit. Because of the wobble, more than half the Moon, almost 60%, can be seen with careful monitoring.

In theory, it is surmised that the Moon was formed when an object impacted the Earth and the debris from the collision coalesced into the Moon and the Earth reshaped itself after impact.

The Earth-Moon system will be utilized because it is a fascinating idea that the Moon revolves around the Earth, in the first place, and that that revolution does not play a role in the formation of the solar system. Also, the Moon wobbles as it travels around the Earth allowing for varied distances from the Earth, positive and negative slope can be positioned by its revolution around the Earth, and it can be related to a coordinate plane.

Students will determine the imaginary lines from the Earth to the Moon as it orbits. The average distance of the Moon from the Earth is 384,400 km.
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The range of wobble is 357,000 km to 407,000 km. Students will transfer their drawn lines to a coordinate plane and find the rise and run which determines slope.

### Pythagorean Theorem and Trigonometric Ratios

Students will find distances between two points and measures of angles using the Pythagorean Theorem and trigonometric ratios. This will be demonstrated by the Earth-Moon parallax and the Sun parallax.

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Parallax
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Parallax is how the position of an object appears to change depending on the vantage point taken. For example, if one is standing before a television set below two paintings, standing to the right of the set will yield the left painting as being the background to the set. However, when standing to the left of the television set the right painting will become the background. Now, if the television set is positioned away from the wall of paintings to the center of the room each respective painting will still be the background but appear smaller. Therefore, the closer the object is to the background, the larger the parallax. Parallax is used to find measurements of angles.

The Earth-Moon parallax is based on the position of the observer on the Earth with relation to the center of the Earth and the path of the Moon. The angle of the parallax is created by the distance to the Moon from the observer and back to the center of the Earth. The Moon just above the horizon will create a larger angle parallax than the Moon at a 45
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angle from the horizon. A zero parallax will be created when the Moon is directly overhead. The Sun parallax is similar but the distances from the Earth to the Sun will be greater and the parallax angles will be smaller. (See Figure 4)

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It is imperative to discern the distances of celestial bodies such as the Sun, the planets, and the stars. For example, the distance from the Earth to the Moon allows for calculation of space travel to the Moon, and, information about light and heat is provided by understanding our distance to the Sun. Also, stars, which include our Sun, emit energy to varying degrees. By knowing the distance of stars to Earth astronomers can calculate the luminosity of a star and compare it to the Sun for intensity. Knowledge obtained by this comparison reveal how hot the star is. The best way to measure distances to celestial objects is the parallax.

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Figure 4 The diagram shows the Moon traveling around the Earth and the various angles the Moon creates with the Earth as a result of the shifting positions of the Moon. Source: author.

Parallax will be used to instruct the Pythagorean Theorem and trigonometric ratios because it has an engaging diagram, and, it also has the appealing notion that the changing of position will change the perspective of an object. A minimum and maximum of angle measurement can be determined, it can be related to a coordinate plane, it allows for simple calculation, and grasping of concept.

Students will determine the right angles that can be formed using the Earth-Moon parallax. Students will apply the Pythagorean Theorem; the square of the hypotenuse is equal to the sum of the squares of the legs, and the trigonometric ratios, sine, cosine, and tangent, to find distances and angle measurements. (The same can be applied to the Sun parallax.)

### Arc Lengths and Area of Sectors

Students will determine measures of arc lengths and areas of sectors using Kepler's Laws and the equatorial circumference of the Earth.

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Kepler's Laws
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Johannes Kepler could not ignore small discrepancies in the orbits of the planets when predicting planetary positions. He tried unsuccessfully to prove that the orbits of the planets were circular and almost succeeded with Mars, however, the small discrepancies led him to conclude that the orbits must be elongated. He abandoned the circular theory and based his work on ellipses. He was now able to predict planetary positions with more accuracy.

Kepler's first law concerns the orbits of planets around the Sun. Until this time it was thought that the planets' orbits were circular. But, Kepler's first law states that the planets orbit in an ellipse (see Figure 5a and Figure 5b). The eccentricity of the ellipse is quite small but not small enough to be circular. In fact, the use of their near circular path will allow the assumption of circular orbits to determine measurements of arc lengths and areas of sectors that will be used in the student lessons. The eccentricity (the deviation of an ellipse from a circle) for Venus is .007, Earth .017, Neptune .010 and so on in ascending order. The smaller the eccentricity the nearer to a circle is the orbit.

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Figure 5a
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An ellipse is an elongation that has two foci (plural for focus) that creates its shape. An eccentricity of zero, e=0, will yield a circle. The solar system has only one focus, the Sun, as illustrated above. Source: author.

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Figure 5b
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Take another more familiar example of an ellipse: the circular cone. A plane intersects the top cone at an angle creating a conic section called an ellipse as opposed the plane intersecting the bottom cone creating a conic section that is a circle. Source of figure: www.Wikipedia.org.

Kepler's second law deals with how the planet moves around its orbit. Each planet cuts out the same area in the same amount of time whether the planet is further away from the Sun or on the opposite side of the Sun and nearer to it. (See Figure 6 below)

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Figure 6
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In the same time interval, when a planet is furthest away from the Sun, called the aphelion, it carves out an area that is equal to the area carved out when the planet is nearest the Sun, the perihelion. Source of shape and shaded areas: www.Wikipedia.org. Descriptive enhancements and removal of one focus were done by the author.

Kepler's third law states that the more distant the planet is from the Sun the slower its orbital speed. Refer to Figure 6. (Note that speed differs from velocity in that speed is the rate at which an object moves, speed = distance/time, and velocity is both speed and direction, for example, the velocity is 50 mph due east.)

Planet: Distance from the Sun (106km)

Mercury: 57.1

Venus: 108.2

Earth: 149.6

Mars: 227.9

Jupiter: 778.3

Saturn: 1,427.0

Uranus: 2,870.0

Neptune: 4,497.0

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Table 3
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Scale for lesson plan 3: 1 inch=57.1, 1.89 inches=108.2, 2.62 inches=149.6, 3.99 inches=227.9, 13.63 inches=778.3, 24.99 inches=1,427, 50.26 inches=2,870, 78.76 inches=4,497 Source: The Cosmic Perspective, Bennett, Donahue, Schneider, Voit, fourth edition, 2007.

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Kepler's Laws are utilized because the planet's sweep carves out an arc and sector that is measurable. There is a notable relationship between what happens on both sides of the Sun as the planet orbits the Sun, there are curious comparisons between arcs and sectors of planets regarding the measurement of the arc and the measurement of the area of the sector based on orbital speed of the planet and when a planet speeds up or slows down, and there is intriguing seasonal data regarding measurements of arcs and areas of sectors. Students will be motivated to apply formulas to get results.

Students will create a model of one planet's orbit using astronomical data scaled to workable numbers, map out a seasonal sweep, calculate the arc length, calculate the area of the sector, and compare it to the other planets. Conjectures will be made regarding measurements of arc lengths and areas of sectors between planets.

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Circumference of the Earth, Great Circle
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The circumference of the Earth was first estimated by Eratosthenes by the formula 7/360*circumference of Earth=5000 stadia in 240 B.C. Applying modern day measurements it is amazing how accurate he was. He estimated the circumference of the Earth to be 250,000 stadia based on the Greek stadium. Today, we figure a stadium to be 1/6 of a kilometer and Eratosthenes' estimate to be 42,000 km. Current measurements estimate the circumference to be 40,075 km at the equator and 40,008 km from North to South Pole.
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(The Earth is not a completely circular.)

The equatorial circumference of the Earth is used to reinforce using the formulas to find measures of arc lengths and areas of sectors.

Students will determine the angles formed between two locations and the center of the Earth and apply the formulas. The equatorial circumference of the Earth will be used as the Great Circle in all examples in this exercise.