Overview
In this unit, we will review Mathematical skills that they have previously studied. We will study scientific notation. While they have learned this, they have not used it very often and are not particularly comfortable with using it. Because most numbers used in Astronomy are quite large or quite small, this is a skill that must be reviewed. Scientific notation rewrites these numbers as a number between 1 and 10 multiplied by a power of ten. The exponent of ten comes from how many places the decimal must be moved to get the number between 1 and 10. When there is no decimal written in the number, it is understood to be at the end of the number. For very large numbers, the decimal is moved to the left and the exponent is positive. With very small numbers, the decimal is moved to the right and the exponent is negative. For example, 345,000,000,000 would be written 3.45 x 10 1 1 because the decimal at the end of the number was moved 11 places to the left. The small number .00000000000007468 would be written 7.468 x 10 - 1 4 because the decimal at the beginning of the number was moved 14 places to the right.
When multiplying numbers written in scientific notation, the decimal numbers are multiplied together and the powers of ten are multiplied. When the powers of 10 are multiplied, the exponents are added. For example, 2.45 x 10 1 2 x 3.67 x 10 - 1 8 is 8.9915 x 10 - 6. It would be acceptable to round this answer to 8.99 x 10 - 6. This particular example was not hard to multiply by hand. I will also show students how to enter numbers of this type on the TI-83+ or TI-84 Graphing Calculators. The numbers between one and ten are entered from the keypad. To enter the powers of ten, the 10 is entered from the keypad. To enter the exponent, the ˆ key located in the right column directly above the ÷ key is pushed and the exponent is entered from the keypad. If the exponent is negative, the negative sign (-) is the key on the bottom row to the left of the "enter" key. Numbers are entered on the graphing calculator in the same order that they are written on the paper. When the answer is written, the multiplication symbol and the 10ˆ will be replaced with the letter E.
When dividing numbers written in scientific notation, the exponents will be subtracted. Remember that division can
be written as a fraction. For example, 2.45 x 10 1 2 / 3.67 x 10 - 1
8 can be rewritten 
. Divide the decimals by hand or by calculator. The result is .67 when rounded to two
decimal places. When dividing the powers of 10, 12 – (-18) is 30. This gives .67 x 10 3
0. Rewrite .67 as 6.7 x 10 - 1. Multiplying by 10 3 0 gives a
final answer of 6.7 x 10 2 9. Depending on the numbers, it is usually better to do division
with a calculator.
Formulas with multiple variables will be reviewed. In the past, they have rewritten formulas for a different
variable than the one originally solved for. Again, this is something that they have done before but are not
particularly comfortable doing. I will use formulas that the students are familiar with. To solve a formula for a
different variable requires the same equation solving skills that are used in solving an equation in one variable.
The formula for the area of a rectangle is A = bh. To solve for b, students will divide both sides by h. This yields
. The
formula for the area of a trapezoid is 
. To solve
for b 1 involves more steps but works in the same manner. Multiply both sides by 2. Then divide both
sides by h. This intermediate result is 
.
Subtracting b 2 from both sides gives the final result 
.
The Pythagorean Theorem will be reviewed. Of all the skills, this is the one that will be the most comfortable for them. When given a right triangle and the lengths for two of the sides, the 3 rd side can be found by using a 2 + b 2 = c 2. Remember that c must be the side opposite the right angle. For the given right triangle, a = 5 m and c = 12 m. Using the Pythagorean Theorem, 5 2 + b 2 = 12 2. Solving involves the same good equation solving skills that were used earlier and two more skills. First, 5 2 and 12 2 must be simplified to 25 and 144. Subtracting 25 from both sides results in b 2 = 119. To find b requires taking the square root of both sides. Normally, I teach students to simplify radicals if possible. However, in this unit, it will be important to use a calculator and round to the appropriate number of decimal places for the particular problem. Because lengths of sides cannot be negative, we will write only the positive answer that b = 10.91 m. It is important to remind students that there are two answers found when taking a square root and that there are problems that will need both of these results. However, we are using the result that makes sense for our problem. It is important to review taking a square root on the graphing calculator. The square root symbol must be entered and then the 119. The square root symbol is not on a key. It is in yellow to the left above the x 2 key. This is a second function for the calculator. To utilize the square root, push the yellow 2 nd key and then the x 2 key. This places the √¯ symbol on the screen. Then enter the 119. One of the advantages of using a graphing calculator is that you see the problem you entered and the result. This is very helpful when you get a result that does not make sense with your problem. You can look to see if you entered it correctly.
We will look at taking two similar formulas and writing them as a ratio to create a simpler expression. This is a
skill that will be useful in Calculus. For example, we have expressions for the areas of two circles. A 1
= Πr 1 2 and A 2 = Πr 2 2. Rewriting these as a ratio
yields 
. Using good Algebra, the Π that is common to the top and bottom is eliminated giving
an equation that is much easier to work with because the value of Π is no longer an issue. The new ratio is 
.
An extremely important part of this unit will be the review of units of measurement that they have previously used, and the introduction of new units of measurement. Metric units, especially meters and grams, will be reviewed. Students need to review the order from largest to smallest. I use the mnemonic King Henry Died __ drinking chocolate milk to help them remember that the order is Kilo_, Hecto_, Deka_, meter or gram, deci_, centi_, milli_. When moving from a smaller unit to a larger one, they are multiplying by a power of 10. When moving from a larger unit to a smaller one, they are dividing by a power of 10. Angle measurements in both degrees and radians will be used. It is important to remember that there are 360 degrees or 2Π radians in a complete circle. Conversions between customary units and metrics will need to be reviewed. These include:
- n = 2.54 cm
- t = .3048 m
- i = 1.609 km
- m/s = 2237mi/hr
- i/hr = .447m/s
To work with new units unique to this study, we begin with 1 degree = 60 arcminutes
(60′ ) and 1 arcminute = 60 arcseconds (60″ ). The Astronomical Unit (AU) is a unit of length used primarily in measuring across our Solar System. It represents the average distance between Earth and the Sun.
1 AU = 1.496 x 10 8 km or 8.317 light minutes or 92.96 million miles
When measuring units of length to the stars, two units of measurement are used. A light year is the distance light travels in one year.
1 ly = 9.46 x 10 1 2 km or 63,240 AU or 6 trillion miles
Another unit of measure, the parsec, is the distance at which 1 AU subtends an angle of 1 arcsec. Sometimes, kiloparsecs and megaparsecs are used.
- c = 3.09 x 10 1 3 km = 3.26 ly
- pc = 10 3 pc
- pc = 10 6 pc
When working with very small distances, three units are important to know. They are the micrometer (mm), the nanometer (nm) and the angstrom (Å).
- m = 10 3 nm
- m = 10 - 9 m
The speed of light in a vacuum is represented by the letter c.
c = 3.00 x 10 5 km/s = 3.00 x 10 8 m/s = 1.86 x 10 5 mi/s
Grams and solar masses are units used to express the mass of objects. For our study, those objects will be stars.
- g = 1000 g
- olar Mass = 1.99 x 10 3 0 kg
To measure luminosity, we will use watts.
1 watt = 1 joule per second
A joule is the amount of energy in the motion of a 2 kilogram mass moving at a speed of 1 meter/second.
To convert from one unit of measurement to the other, we multiply by a fraction equivalent of 1. For example, to
convert 524 miles into kilometers, 1 would be 
. This
comes from a conversion formula given previously. When this is multiplied, the miles units cancel and the final
answer is 843.12 km. Remember that the unit you are converting to should be on top and the one you are converting
from should be on the bottom.
Another very important subject for review is using correct units of measurement in an answer. Units of measurement should be defined at the beginning of a problem. When two identical units of measurement are multiplied, the answer is always squared. When three identical units of measurement are multiplied, the answer is cubed. For example, 2 m x 3 m = 6m 2 and 12 cm x 5 cm x 3 cm = 180 cm 3.
The Universe has always fascinated man from the earliest times. As early as 280 B.C., Aristarchus developed a plan for determining the distance from the Earth to the Sun and to the Moon. (Freedman and Kaufmann, 2004. Unless otherwise noted, all Astronomical information comes from this source.) It was first believed that the Earth was the center of the Universe and that everything revolved around it. Then, the Sun was considered the center of the Universe. It is amazing how far Astronomy advanced without the sophisticated equipment available today. As mankind developed better techniques and more sophisticated equipment, we have come to understand that our Solar System is a very small part of a vast Universe. With all the talented Astronomers and all the high tech equipment, there are still so many things we do not know and other understandings that are no longer valid. As the Star Trek television series and movies always say "Space – The Final Frontier", space truly still is a frontier.
Stars have always held a sense of wonder for the smallest child to the oldest adult. There are about 6,000 stars that can be seen without any equipment. On any given night, there may be 3,000 visible stars in the dark part of the sky. Our location will determine how many of those are visible above the horizon. While they seem to be very close, they are really very far away. One ongoing work for some Astronomers is the measurement of the distance to these stars. In 1577, Tycho Brahe developed Astronomical instruments that were very sophisticated for his time, and took very accurate measurements of star and planet positions. In the last 100 years, advancements in technology have greatly enhanced our knowledge of the Universe.
The simplest way of measuring the distance to stars is stellar parallax. This was first successfully done in the 1800s. To understand the idea of parallax, use your hand extended in front of you. The background should have obvious differences. Cover one eye and then cover the other eye. Did your hand appear to stay in the same location? No, it did not. Our eyes work together to determine where our hand really is. To measure stars using stellar parallax requires two different observations of a star. These need to be done at two times when the Earth's orbits are the farthest apart. (See illustration 1). These measurements would be taken six months apart. The line of sight from the first observation forms one side of a skinny triangle. The line of sight from the second observation forms the second side of the triangle. The third side is the diameter of the Earth from one observation to the other. The height of this triangle is the distance to the star. Half of the diameter is the distance from the Earth to the Sun (1 Astronomical Unit). The parallax angle (p) that we need is half of the angle bisected by the height of the triangle. This triangle gives the equation 1 AU = dp where d is the distance to the star in parsecs and p is the measure of the parallax angle in arcseconds. Solving this equation for d gives the equation:
All stars have a parallax angle of less that 1 arcsecond. This is less than 
of 1
degree.
When calculating the parallax angle, Astronomers must take into account the proper motion (Μ) of the star. Proper motion is how many arcseconds a star appears to move in a year on the celestial sphere. This motion is always in one direction. To determine proper motion requires three observations of the star. The third would be six months after the 2 nd when the Earth had returned to its original observation position. The distance between the original and final position of the star would be the proper motion for an entire year. Removing half this measure would give an accurate parallax measurement. (See illustration 2) (Seminar, 2005)
We cannot measure the parallax for stars that are very far from the Earth. To measure some of these, the parallax can be measured by satellites in space. For others, there are other methods to find the distance. However, it is still important for stars that are relatively close to have their distances measured using stellar parallax. This is essential for the other methods used in measuring distances to stars that are the farthest away. The result in parsecs can then be converted to light years or to kilometers using conversions given earlier. For calculating luminosity, we will need to convert it to meters.
When looking at the sky, one might think that the stars that are the brightest are the closest. This is not true.
The brightest stars are often farther from us than some dimmer stars. Brightness has to do with thermonuclear
reactions taking place within the star. Luminosity is how much energy from these reactions is emitted into space
each second. If we know the brightness of the star and its distance, we can determine its luminosity. To do this, we
use the Inverse-Square Law: 
where b is the brightness given in watts per meter squared, L is the luminosity in
watts and d is the distance in meters. We may not know the exact value of the brightness but may know its brightest
in relation to another star or to the Sun. To find luminosity, it is helpful to solve the equation for L.
Multiplying both sides by 4Πd 2 gives L = 4Πd 2b. Develop this equation for the star
whose luminosity you want to find using the subscript 1. Develop the same equation for another star using the
subscript 2. Create a ratio using the two equations.
The 4Πs cancel leaving a simpler equation.
You will need to know the ratio of the brightness of the two stars. It is not necessary to know the actual value of the brightness for either star. You will also need the distance to each star or the ratio of their distances. Again, the result may not be the actual luminosity of either star but a ratio of the luminosity of one star to the other. In my class, we will be finding the luminosity ratio but, if 2 of the ratios are known, the third can always be found. For two stars with the same brightness, the one that is the farthest away must have a greater luminosity for us to be able to see it. A dimmer star that is closer would have less luminosity than a star that is brighter and farther away.
To the average person, stars appear to be stationary in the sky but this is not the case. They have space velocity. Space velocity tells the direction they are moving and the speed with which they are moving. To calculate space velocity, there are two components – tangential velocity and radial velocity. Tangential velocity is the movement across the sky or perpendicular to our line of sight. Radial velocity is movement in our direction or away from us or parallel to our line of sight.
Calculating tangential velocity requires us to know the proper motion and the distance to the star. To find
tangential velocity (v t), use the formula 
where
Μ is the proper motion measured in arcseconds and d is the distance measured in parsecs. The resulting velocity
will be measured in kilometers per second.
Radial velocity (v r) involves the measurement of wavelengths of spectral lines coming from the star. If the star is coming toward us, the shift is decreasing wavelength or blue-shifted. If the star is going away from us, the shift is increasing wavelength or red-shifted. The Doppler shift equation used to find radial velocity is:
where Λ is the star's wavelength in nanometers, Λ 0 is the wavelength of the line at rest in the laboratory in nanometers and c is the speed of light in kilometers per second. Multiplying both sides by c gives the following equation:
This velocity will also be measured in kilometers per second.
Tangential velocity and radial velocity are perpendicular to each other and form two sides of a right triangle. The space velocity is the hypotenuse of that right triangle. (See illustration 3) To find the space velocity, use the Pythagorean Theorem.
v 2 = v t 2 + v r 2
Taking the square root of both sides gives the formula:
The results will be in kilometers per second.
Why should we be interested in distances to the stars, the luminosity of the stars or their velocity? There is so much still to learn about our Universe. The value of the Hubble Constant, the rate at which the Universe expands, is still not firmly established. It is considered to be approximately 71 km/s/megaparsec. Without an accurate value for this, we cannot get an accurate age for the Universe. We have the Hubble Telescope, the Chandra X-Ray Observatory and the Hipparcos Satellite in orbit allowing us to measure farther and with more accuracy the stars in distant galaxies. We have more powerful Earth-based telescopes and other Astronomical instruments to more accurately measure. Both Cepheids and Type Ia supernovae are used as "standard candles" to calculate distances to galaxies. (Finkbeiner, 1998)
Strategies
We are on a 4x4 block schedule. Students attend 4 classes each day for 90 minutes each. They earn 4 credits in the fall term and 4 in the spring term. I have my students 90 minutes per day for 90 days. These lessons will be taught in a student-centered classroom. The students will be working in groups of 3 or 4 students.
Our lesson plans are set up to include the standard, rationale, materials needed, the lesson layout, and the homework. Mathematics lessons are divided into three parts:
1) The Launch is where the topic for the day is introduced to the students. This can last from 15 to 30 minutes.
2) The Explore is where students work cooperatively on the activities of the day. The teacher circulates among the students to be sure that they are on task and not heading off in the wrong direction. The teacher does not tell the students how to do things but guides them by good questioning techniques back in the right direction. This section should last 45 – 60 minutes. In Mathematics classes, it is usually broken into sections. As students finish one type of activity, student presenters share their work with the class. Students can ask questions of the presenter, and the teacher can provide any needed clarifications and guide a summary of what the students have discovered. The teacher then sets the stage for the next activity, and the same procedure follows as with the first Explore activity. An explore period may have one or several Explore activities depending upon the length of time needed for each activity.
3) The Summary is when the teacher brings the class together as a large group to tie together and summarize all that has been accomplished during that period. This usually lasts 5 – 15 minutes.
I plan to begin with a KWL to find out what my students know about the stars, what misconceptions they have, and what they want (or think they want) to know. A short movie, "The Life Cycle of Stars" will be shown. We will do a review of the Mathematical skills needed prior to delving into our study of the stars.
I will introduce the study with an overview of how our knowledge of the stars has changed through the years, and what has been learned even as recently as the last few months. Each group will then be assigned a star whose distance can be determined using the parallax angle. They will be asked to calculate the distance to their star using a formula. These results will be converted from parsecs to light years to meters. Students will share their results with the rest of the class.
The distance results will be used to begin a discussion of brightness and luminosity. Students will be asked "which star do you think will be the brightest and why?" This discussion will lead into our next formula, the Inverse-Square Law. As a large group, we will solve the formula for L, and write it in terms of the star Sirius A using the subscript 2. They will write the formula for their stars using the subscript 1. Groups will be given the brightness ratio of their star and Sirius A. Using this, the distance to their star and the distance to Sirius A, they will determine the luminosity of their star as it relates to Sirius A. Students will share their results and be asked the question that started the lesson. This will lead to a summarizing discussion of luminosity and brightness.
To begin the next lesson, the question asked will be "do stars move?" This will lead into a discussion of space velocity. Each group will use the distance found for their star and the proper motion provided to find the tangential velocity of their star. Necessary information will be provided for groups to find the radial velocity. Students will use their results to find the space velocity for their star. Groups will share their results with the class. With our results, we will discuss how fast stars are moving, and in what direction.
As a concluding activity, a field trip will be taken to the Museum of Science and History Planetarium.
Each evening's assignment will be 5 to 6 problems relating to the type of problems done in class on that day. Students will also be asked to write about a topic of interest to them relating to the stars. These topics could include:
- writing about a particular star or a particular type of star
- writing about an Astronomer and his impact on Astronomy
- writing about the impact of the Hubble Telescope, the Hipparcos Satellite, or the Chandra X-Ray Observatory, etc.
- writing about a particular area of Astronomy that is of interest.
- writing about an aspect of Astronomy that is still not understood
- writing about an aspect of Astronomy that has not changed over a long period of time
- writing about a particular galaxy
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