# 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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## Overview

The astronomical data used to instruct mathematical concepts will be varied according to mathematical concept. For instance, elements such as hydrogen, helium, and carbon occur naturally in the universe. But abundance differs on each planet, Moon, and the Sun. Comparing the abundance of elements in the solar system will allow students to analyze astronomical data by using graphs.

It will be interesting for the student to note how the atmospheric gases differ from the four terrestrial planets of Mercury, Venus, Earth, and Mars to the gaseous planets of Jupiter, Saturn, Neptune, and Uranus. Of course, interest in Earth compared to the other planets always stirs conversation about habitat for life and the elements needed for it.

Other data as using diameters of planets and moons can offer interesting comparisons. Students will discover how the diameters of all the planets compare with each other and likewise the moons. They can sort the planets according to size or position in the solar system using graphs.

The Earth and Moon relationship offers a good way to teach slope. Positioning the Moon in its orbit around the Earth can demonstrate positive and negative slope. It is intriguing to note that the Moon wobbles in its orbit and is not consistently the same distance away from the Earth. This discrepancy will allow for changing data as the Moon revolves around the Earth.

A by-product of this tactic is to remove the visual and reveal the coordinate plane.
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This will allow a smooth transition from reality to the structured and sometimes weary coordinate plane that now takes on added interest. The point can be made that the removal of the picture supports a less obstructed view of the data and the calculation of it.

On the geometry side the Earth-Moon parallax and Sun parallax can be used to teach the Pythagorean Theorem and trigonometric ratios. The Earth-Moon parallax and the Sun parallax do not offer many options to use right triangles but offer two that can be considered to be positioned in all four quadrants of a coordinate plane. Two right triangles in each quadrant is enough information to demonstrate the Pythagorean Theorem and trigonometric ratios using the Earth-Moon parallax and then emphasize the mathematical concepts by having the students use the Sun parallax. This will allow for reinforcement and questions. (Other parallax options exist by using any planetary parallax or even a stellar parallax.)

Using the orbits of planets around the Sun can teach about arc lengths. It can also be utilized to find the area of sectors. Since orbital speed varies among planets, the sweep a planet makes in a season will vary between planets and comparisons on which planets' sweep covers a larger area will be explored and may produce surprising results. It can be hypothesized to a student that an outer planet traveling at a slower speed may carve out a smaller area than expected than an inner planet traveling at a faster speed. For example, the average distance of the Earth from the Sun is 1 AU (astronomical unit)
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and its orbital speed is 29.79 km/s whereas Saturn's average distance from the Sun is 9.572 AU and its orbital speed is 9.64 km/s. (See Appendix D Table 4 for a list of orbital speeds of the planets.) The student will find that Saturn carves out the larger area.

The circumference of the Earth can be used to reinforce using the formulas of finding the measure of an arc length and the area of a sector. Students can determine the distance they want to travel, for instance, London to Paris, determine the angle formed using the two locations and the center of the Earth and apply both formulas since the radius of the Earth will be known. This information can be compared to the distance indicated on a map.