ASTR1105_lab_manual

Table of Contents Introduction 5 Essential Skills 7 Scientific Notation 7 Significant Figures 9 Graphing 11 Using Google Sheets 13 The (Virtual) Night Sky 15 Scale of the Solar System 17 Graphing and Kepler’s 3 rd Law 21 Cycles of the Sky 27 Mass of Jupiter 33 Luminosity and the Inverse Square Law 39 Telescopes and Image Formation 45 Sunspots and The Solar Cycle 55 Dark Matter 63 Analysis of Explosions and Hubble’s Law 67 White Light as a Mixture Of Colors 75 Inverse Square Law (Penny Simulation) 77 Spectral Analysis 81 3

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Significant Figures When two or more numbers with different significant figures are combined, the result of the calculation cannot show an increase in precision over the data. The use of an appropriate number of significant figures in recorded data is a way of indicating precision. The greater the precision of the measurement, the larger the number of significant figures one is justified in using. For example, in length measurements the value 25.55 has four significant figures whereas 25.5 and 25.6 have only three significant figures. In each of these values, the last digit reflects estimation on the part of the experimenter. Generally, digits that are known with certainty, and the first estimated digit, are called significant figures. To find the number of significant figures in a number, count from the left ignoring all leading zeros (i.e., zeros that appear in the number before any number that is not zero). The number of significant figures is not related to the location of the decimal point. Furthermore, zeros that are merely used to indicate the location of the decimal point are not significant. For example, 1706, 170.6, 17.06 and 0.001706 all have four significant figures. When a value is expressed as 1540, it is not clear as to whether or not the zero is significant. Scientific notation can be used to make this clear. To clearly indicate that it is significant, the value should be expressed as 1 . 540 × 10 3 . Expressing it as 1 . 54 × 10 3 shows that it is not significant. When doing calculations, the result of your calculation should be written to the correct number of significant figures. What is correct depends on the type of operation you are doing. The rules you should follow are outlined below. Addition or Subtraction The number of decimal places in the result should be the same as the quantity in the calculation that has the fewest number of decimal places. Examples: a) 145 . 23 + 22 . 6 = 167 . 8 b) 145 . 23 − 22 . 6 = 122 . 6 c) 3 . 421 × 10 2 + 4 . 2 × 10 2 = 7 . 6 × 10 2 d) 3 . 4211 × 10 3 + 4 . 2 × 10 2 = (3 . 4211 + 0 . 42) × 10 3 = 3 . 84 × 10 3 Notice that in the last example, you must compare the number of decimal places when both numbers have the same scaling factor (power of ten). 9

0 1 2 3 4 5 6 7 8 9 10 Time t (sec) 0 5 10 15 20 25 30 Velocity v (cm/sec) Will Gunton (own work) CC-BY 4.0 slope = Δ v Δ t = (25 . 0 - 7 . 0) cm / sec (9 . 5 - 2 . 5) sec = 2 . 6 cm / sec 2 = acceleration (2.5,7) (9.5,25) Quantity Symbol Units Quantity Symbol Units Title Instantaneous Velocity as a Function of Time Figure 2.1: An example of an ideal graph. See the Graphing section for a summary of the features of a properly drawn graph. 12

The (Virtual) Night Sky Using a computer, you will visit a number of popular and useful astronomy web sites. On each of these sites, you will practice finding some astronomical information. By the end of this lab, you should be able to... find the current position of the Sun, Moon and planets create a star chart for a particular date and time look up information about current and future observing conditions determine when the International Space Station will be visible use Stellarium to see the “Virtual” Night Sky. This lab has four parts: 1. Sun, Moon, and the Planets 2. Star Charts, Constellations, and your “Sun Sign” 3. Weather and Forecasts 4. The International Space Station In each of these parts, you will use the following websites: www.heavens-above.com A wonderful resource that will help you visualize what is in the night sky. Includes everything from the location of satellites (including the current position of the International Space Station) to the current position of planets and the Moon to an interactive sky chart. www.cleardarksky.com An astronomers weather forecast, with observing conditions for over 5300 sites in North America (including Vancouver and Burnaby). www.clearoutside.com Another source for an astronomers weather forcast, with 7-day hourly cloud cover and weather forcasts and other useful information like if there is a visible ISS passover. More details on each part can be found in the worksheet you will complete during the lab. This can be found in the same folder on Blackboard as this document. 15

Part 2 - Size In this part, you will calculate the relative diameter of the Sun, planets and Pluto, and put them in the correct order “to scale” (except for the Sun as it’s too big to do this reasonably). At the front of the lab room, there are an assortment of Styrofoam and plastic spheres that you can use to represent the planets. You will assume that Jupiter is the size of the largest available Styrofoam sphere. 1. Find the largest Styrofoam sphere, and measure it’s diameter. Enter this in the table (provided in the report sheet) as the diameter of Jupiter. 2. Calculate the diameter of all the other planets (and the Sun and Pluto) in the table “to scale” relative to the size of the Styrofoam Jupiter. 3. For each planet (and Pluto), find the sphere that is closest to the diameter that you calculated, and assemble your “to scale” solar system (in the correct order). Once your solar system is assembled, get one of the lab instructors to check it. As an example, if you measured the diameter of the largest sphere to be 21 cm, then the diameter of the sphere that would represent Saturn is D Saturn D Jupiter = x 21 cm −→ x = 21 cm × D Saturn D Jupiter = 21 cm × 120 , 000 km 142 , 000 km = 18 cm (3.2) Please be careful not to lose or drop any of the spheres, and return each of spheres to the correct bin when you are done. Do not write on any of the Styrofoam spheres. 18

Graphing and Kepler’s 3 rd Law By the end of this lab, you should be able to... draw a line of best fit through a set of data. determine if a straight line is a good choice to model a set of data. verify Kepler’s 3 rd Law. Theory and Background In Astronomy (and many sciences) you will look for a mathematical relationship between two variables. A very useful tool we can use to examine relationships between variables is graphing. If we have a direct relationship, then the graph of these two quantities will be straight line. For example, consider the area (A) and side length (l) of five different squares given in Fig. 4.1 . Figure 4.1: The measured area and side length for five different sqaures. A reasonable question to ask is: “What is the relationship between the area of the square and the side length?” Two possible answers to this question are plotted in Fig. 4.2 . On the left is the relationship A = l , which states that the area is equal to the side length of the square. On the right is the relationship is A = l 2 , which states that the area is equal to the side length of the square squared. The correct relationship is the one which best fits to a straight line. The one that fits best will have all the points lie on (or close to) a straight line. From these two figures, it is clear that the data in the figure on the right best fits a straight line. You should check this for yourself (and will see an attempt at a straight line fit to both plots in the tutorial video for using Google Sheets ). 21

Figure 4.2: Left: A plot of the relationship A = l . Right: A plot of the relationship A = l 2 . The correct relationship is the one which best fits to a straight line fit. Kepler’s 3 rd Law For this lab, you are going to use graphs to check or verify Kepler’s 3 rd Law. It took Kepler over a decade to identify this law! Kepler determined the orbital period of each planet (in years) and the average distance of the planets from the Sun (in AU 1 ). Kepler’s 3 rd Law relates the orbital period ( P ) to the semi-major axis of the planets orbit ( a ). In general, Kepler’s 3 rd Law states that the square of a planet’s period ( P 2 ) is proportional to the cube of the semi-major axis of planet’s orbit ( a 3 ). That is, P 2 ∝ a 3 (4.1) The symbol ∝ is the proportional symbol. What this means is that there is a direct (or linear) relationship between the period square and the semi-major axis cubed, and therefore the plot of the period squared as a function of the semi-major axis cubed will be a straight line. For objects in our solar system (meaning, objects that orbit the Sun) Kepler’s 3 rd Law states that P 2 = a 3 (4.2) where the period ( P ) is measured in years, and the semi-major axis ( a ) is measured in AU. In this case, the slope of the straight line to the plot of the period squared as a function of the semi-major axis cubed would be one. 1 1 AU is defined as the average distance between the Earth and Sun, where 1 AU = 1 . 50 × 10 11 m 22

In the first part of this lab, you will make measurements of the period and semi-major axis for several “new” planets in our solar system, and use them to confirm Kepler’s 3 rd . You will do this by making a plot of P versus a and P 2 versus a 3 to see which plot best fits to a straight line and verify Kepler’s 3 rd Law. In the second part of the lab, you will be given data for planets orbiting another star (with a different mass from our Sun) to see if the general form of Kepler’s 3 rd Law (given in Eq. 4.1 ) holds for different solar systems where the mass of the star at the center of the solar system is different from the mass of our Sun. Procedure For this lab, you will use Google Sheets to make your plots and add a straight line to your data. There are instructions for using Google Sheets in the “Lab Introduction” document in the Labs area on Blackboard. There is also a tutorial video in the folder for this lab. You must make you plots using Google Sheets and ensure they have all the features of a good graph outline in the “Lab Introduction” document. You will get your completed plots checked by the lab instructors at the relevant part of the lab. Part 1 - Kepler’s 3 rd Law In Our Solar System In order to verify Kepler’s 3 rd Law you need to make some measurements of the period and semi-major axis for several “new” (read: made up) planets in our solar system. To do this, we will use this Planetary Orbit Simulator that is part of the Planetary Orbits lab in the NAAP Labs program. We will run this simulation and take the data together at the start of the lab 2 . In this simulator, you can set the semi-major axis of a planet (in AU), the eccentricity, and the animation rate. The animation rate should be set to 1.0 yrs/s such that one second of real time when the simulation is running corresponds to one year. That is, if you measure the time for one complete orbit (the orbital period) to be 3.6 s, this would correspond to an actual orbital period of 3.6 years. You have a colleague who has already measured the semi-major axis and eccentricity for five new planets in the solar system. They have also named all the planets. These measurements are listed in Table 4.1 . For each of the planets, we will set the correct given values for the semi-major axis and eccentricity and then measure the period (the time it takes to orbit around the Sun once) using the simulation. To do this, use any stopwatch you have available (for example, on your phone). You may want to take several measurements and average your result (and we will discuss details of taking “good” measurements before the lab). 2 This simulator is part of the NAAP Labs program. If you want to access this simulation yourself, details are provided on Blackboard and can also be found in the lab folder for this lab. 23

When you do this, you should think about the precision you can reasonably measure. The precision corresponds to the number of decimal places on your time. Do you think you can reasonably measure the precision to within a second (i.e., a time like 3 seconds)? What about a precision of a tenth of a second (i.e., a time like 3.1 seconds)? Or what about a precision of a hundredth of a second (i.e., a time like 3.16 seconds)? What do you think would influence the precision of your measurement? You should be consistent with your precision for all your time measurements. Planet Semi-major Axis [AU] Eccentricity Caliban 0.91 0.13 Ziltoid 2.3 0.044 Neara 3.5 0.35 Clayman 4.1 0 Unearth 5.2 0.65 Table 4.1: Data for “new” planets in our solar system. 1. Using the Planetary Orbit Simulator, measure the period for each of the five planets. This will be done at the very start of the lab. Record your data in the table provided in the worksheet. 2. Calculate the values of the period squared ( P 2 ) and the semi-major axis cubed ( a 3 ) and add them to your table. Make sure you write your answers to the correct number of significant figures. Show a sample calculation for each calculation. 3. Using Google Sheets, make a plot of the period (on the y-axis) as a function of the semi-major axis (on the x-axis) and add a straight-line trendline. 4. Using Google Sheets, make a plot of P 2 (on the y-axis) as a function of a 3 (on the x-axis) and add a straight line trendline. Once you have made both plots, ask a lab instructor to check them and sign your lab report sheet. 5. Based on only the two plots, which one do you think is the correct relationship between the period and semi-major axis? Justify your answer based on what you see in your plots. The information in the “Theory and Background” section of this lab manual may be helpful for this. 6. What is the slope of the trendline of the plot that shows the correct relationship? Is this what you expected? Why or why not? 24

Part 2 - Kepler’s 3 rd Law In A Different Solar System You have also taken very careful data in a solar system where four planets orbit a star that has twice the mass of our Sun. The data for the period and semi-major axis, along with P 2 and a 3 are given in the table below. You have not thought of good names for the planets yet so you have just labeled them with a letter. Planet Period [Years] Semi-major Axis [AU] P 2 a 3 Planet A 0.850 1.13 0.723 1.44 Planet B 1.20 1.42 1.44 2.86 Planet C 2.10 2.07 4.41 8.87 Planet D 3.50 2.90 12.3 24.4 Table 4.2: Data for planets in a different solar system. 1. Using Google Sheets and the data in Table 4.2 , make a plot of P 2 (on the y-axis) as a function of a 3 (on the x-axis) and add a straight line trendline. Once you have made this plot, ask a lab instructor to check them and sign your lab report sheet. 2. Does this data still fit a straight line? If yes, what is the slope of the trendline? 3. Based on this plot, do you think the general version of Kepler’s Third Law given in Eq. 4.1 is still valid for objects that orbit a star that is not our Sun? Explain why or why not. If yes, would the version of Kepler’s Third Law given in Eq. 4.2 for our solar system still be the correct version to use or would any modifications be required? 4. For this question you should use your plot from the first step to answer this question. In the lab report sheet, you should sketch the plot in order to explain your work. You discover a new object that is orbiting this star with a semi-major axis of 2.5 AU. What is the orbital period of this object? Hint: Since you have a plot of P 2 vs a 3 , find the correct value of a 3 on the horizontal axis of the plot and use your straight line trendline to estimate the corresponding value of P 2 (on the vertical axis). Then use this number to determine the period. 5. Based on your results from Part 1 and Part 2 of this lab, if you had data for the period and semi-major axis for a solar system where the star at the center had a mass that was four times larger than our Sun, do you think the slope of the trendline of a plot of P 2 vs a 3 would be larger or smaller than in the plot your made in this part of the lab? Briefly explain how you decided on your answer. 25

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Cycles of the Sky In this lab, You will use models of the Sun, Earth and Moon to discover what causes the seasons, why the Moon changes its phase and what causes lunar and solar eclipses. Learning Objectives By the end of this lab, you should be able to... state the relative position of the Earth and Sun at the equinoxes and solstices. explain the reason for the seasons on Earth. state the order of the Moon’s phases. demonstrate how the Moon’s position relative to the Earth and Sun creates the phases. state and demonstrate your position at Earth at various times of the day, sunrise, noon, sunset and midnight. determine the phase of Moon given the time and position of the Moon. Theory Part 1 - Reasons for the Seasons The rotational axis of the Earth is tilted at about 23.5 degrees relative to the orbital plane of the Earth around the Sun. As it orbits around the Sun, the Northern Hemisphere receives more daylight in the June at the summer solstice and less in December at the winter solstice, while Australia receives less daylight in June and more in December. The increase in daylight during the summer is a result of Sun appearing higher in the sky at its maximum altitude (angle above the horizon). This also leads to more direct sunlight. Contrary to what you might think, the distance between the Sun and the Earth plays no role in the seasons on Earth. In fact, the Earth is actually slightly farther from the Sun during the Northern Hemisphere summer than the Norther Hemisphere winter! See Section 4.2 of the textbook for more information . 27

Part 2 - Sun and Earth at Different Times of the Day The Earth rotates around it’s rotational axis (which runs through the South and North poles of the Earth) once a day, and the Earth orbits around the Sun once a year. To specify the direction of the rotation and orbit, we say that the Earth rotates counterclockwise, and the Earth orbits around the Sun counterclockwise. However, this is only true if you are looking “down” on the North Pole of the Earth. Similarly, the Moon orbits the Earth in a counterclockwise direction from this perspective. Due to the daily rotation of the Earth about its rotation axis, the Sun appears to rise in the East and set in the West. It might be helpful to remember that the Earth rotates “towards the East”. In the lab, you will model the Earth using your head (where the top of your head is the North Pole), and you will put a dot on a diagram of the Earth to represent the location of your nose (a person) at different times of day. This means if you hold out your arms to represent the horizon your left arm represents East and your right arm represents West. For example, Figure 5.1 shows the Earth at on arbitrary time of the day. It is not noon, midnight, sunrise, or sunset. Can you determine approximately what time it is? 1 . Figure 5.1: A sketch of the Earth/Sun system looking down on the North Pole of the Earth. From this perspective, the Earth rotates counterclockwise. The dot represents the location of a person at an arbitrary time of day. It is not noon, midnight, sunrise, or sunset. Part 3 - Phases of the Moon The eight major phases of the Moon are shown in Figure 5.2 . Each phase corresponds to a different relative position of the Sun and Moon. You will investigate these relative positions in the lab. The phase and position of the Moon can also be used to determine the time. Or, given the time and the position of the Moon you can determine the phase of the Moon you would see. To keep things simple, we will always assume the the Sun rises at 6 AM in the East and the Sun sets at 6 PM in the West. At Noon, the sun will be at it highest point in the sky, directly above the Southern horizon. See Section 4.5 of the textbook for more information . 1 Hint: It is after sunset, but before midnight. 28

Figure 5.2: The eight major phases of the Moon. Image Credit: Jennifer Kirkey CC0 2016. 29

Part 4 - Eclipses Eclipses occur when the Sun, Moon, and Earth “line up” during their orbits. A solar eclipse occurs when the Moon moves between the Sun and the Earth, and a shadow of the Moon is cast on the Earth. A lunar eclipse occurs when the Earth moves between the Sun and the Moon, and a shadow of the Earth is cast on the Moon. See Section 4.7 of the textbook for more information . Procedure Part 1 - Reasons for the Seasons Using a small ball to represent the Earth and the light at the center of the room as the Sun, tilt the Earth at about 23 degrees and demonstrate how the Northern Hemisphere receives more daylight (and more direct sunlight) in June at the summer solstice and less daylight (and less direct sunlight) in December at the winter solstice, while Australia receives less daylight in June and more in December. Once you think you have your model correct, ask one of the lab instructors to verify that you are correct. Part 2 - Sun and Earth at Different Times of the Day Use the light coming from the light bulb in the classroom to represent the Sun. You head will represent the Earth, and your nose is the location of a person on the Earth (let’s assume that is the location of Douglas College). Stand in front of the lamp as if it were one of the times of the day (as asked on the report sheet) here at Douglas College and sketch the relative position of your head (Earth) and the lamp (Sun), assuming you are looking from above at the plane of the solar system down on the North Pole of the Earth. Part 3 - Phases of the Moon 1. Use the Styrofoam ball and a skewer. The Styrofoam ball is the Moon. Put it on the skewer and hold it about an arm’s length away from you. Using the light bulb as the Sun, make the Moon (ball) orbit around the Earth (your head). Make sure everything is rotating (or orbiting) in the correct direction (see the Theory section if you are not sure about this). Check with the drawing to make sure you can model the eight (8) major phases. 30

Once you think you can demonstrate all eight phases, ask one of the lab instructors to verify that you are correct. 2. Draw the eight major phases as the Moon goes through in one month. Make sure they are in the right order and you have the correct names for them. Make sure that as you draw the phases, you keep the bright regions of the Moon “bright” (i.e., white). In other words, you should fill in the areas that are not bright so they appear dark. 3. For several different times of the day and positions of the Moon (see the lab report sheet) name the phase of the Moon. You may find it helpful to model these situations using the light bulb (Sun) and Styrofoam ball (Moon). 4. Sketch the relative position of the Moon and Sun for four of the phases, as asked. This sketch is not to scale! You will do this from the side and then from above as if you were looking down onto the solar system. If the Moon would appear “behind” the Earth, draw a visible dot as if you could see through the Earth, and indicate that the Moon is on the far side of the Earth. Part 4 - Eclipses Using the lamp (Sun) and the Earth and Moon spheres, model eclipses. Demonstrate to the instructor the position of the Sun, Moon and Earth for a complete lunar and solar eclipse. Discussion and Conclusions Answer the following questions using a few brief (but complete) sentences. It may also be helpful to draw a picture, but make sure you explain what you draw. 1. Why do we experience seasons on Earth? In your answer, you must give the two main implications that follow from this reason. Why when it is summer in Northern Hemisphere it is winter in Southern Hemisphere? 2. What would you see in the sky during a solar eclipse? You must clearly explain the sequence (and direction) of events. This means you indicate how you see the eclipse began and end. For example, does the shadow start on the “East” or “West” side of the object. 31

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Mass of Jupiter By the end of this lab, you should be able to... use a (virtual) telescope to measure the radius and period of an orbit. calculate the mass of a planet using the orbit of a moon. calculate the percentage error of the result Theory and Background One way to measure the mass of an object is to observe the influence that object has on something else. For example, the acceleration of an object dropped near the surface of the Earth depends on the mass of the Earth (and, it turns out, is independent of the mass of the object being dropped). If you dropped objects on different planets, you could use the acceleration of the object to determine the mass of that planet. Unfortunately we can’t travel to other planets to perform this experiment, so astronomers need another method of determining the mass of planets in our solar system. In this lab, you will use a “virtual telescope” to observe Jupiter’s four largest moons: Io, Europa, Ganymede, and Callisto. From these observations, you will determine the orbital period and orbital radius of one of the moons, and from this data calculate the mass of Jupiter. The “virtual telescope” can be found here: Galilean Moons of Jupiter . This “virtual telescope” shows you what you would see if you looked at the Jupiter through a telescope at the date and time shown. You can change the time in fixed increments, or you can use the Animate button to let time run forward at a fixed speed. Make sure you leave the Orientation in the default setting, which is “N up, W right (normal)”. Determining the Mass of a Planet To determine the mass of a planet in our solar system, one possible method is to observe the moons that orbit that planet. From class, we know that the orbital speed ( v ) of a moon depends on the radius of the orbit ( r ) and the mass of the planet ( M ), and is given by v = r GM r (6.1) 33

This also means that if we know the orbital speed ( v ) and the radius of the orbit ( r ) we could solve Eq. 6.1 for the mass ( M ), which gives 1 M = v 2 r G (6.2) Therefore, to determine the mass of a planet an astronomer would need to know the radius of the orbit and the orbital speed. While this equation is only true for circular orbits (and no orbit is perfectly circular) it is a good approximation in the case of the four moons you will observe in this lab. Required Observations and Calculations As Eq. 6.2 shows, to determine the mass of a planet, we need to know two things about one of its moons: orbital radius and orbital speed. Orbital Radius If we assume that the distance to the planet is known 2 the physical size (i.e., the diameter) of the planet can be determined from the angular size. This is a topic that will be covered in more details later in the course. For the purpose of this lab, you can assume the diameter of the planet is a known value (and it will be provided to you). Once the diameter is known, the ratio of the planet-to-moon distance and the diameter of the planet as seen through the telescope can be used to determine the radius of the moon’s orbit (which is the true planet-to-moon distance). For example, consider the two observations shown in Figure 8.4 below. This shows a (sim- ulated) observation of Jupiter’s four largest moons. The top image shows a measurement of the size of Jupiter as seen through the telescope (1.88 cm), and the bottom image shows the distance between the center of Jupiter and the moon Ganymede as seen through the telescope (14.42 cm). The ratio of the distance between Jupiter and Ganymede ( d , which is the radius of Ganymede’s orbit) to the diameter of Jupiter ( D ) in the observation image can be used to find the true radius of Ganymede’s orbit. Given that the diameter of Jupiter is 140,000 km, the radius of Ganymede’s orbit ( r ) is d D = r 140 , 000 km −→ r = 14 . 42 cm 1 . 88 cm × 140 , 000 km = 1 . 1 × 10 6 km . (6.3) 1 You should check this yourself 2 The distance to the planets can be measured using several different techniques, including measuring the time it takes a radio signal to travel to (and from) the planet. We will talk about measuring distances later in the course. 34

Figure 6.1: A simulated observation of of Jupiter’s four largest Moons. The top image shows a measurement of the size of Jupiter as seen through the telescope (1.88 cm), and the bottom image shows the distance between the center of Jupiter and the moon Ganymede (14.42 cm). Note that the radius of the orbit was written to two significant figures because the true diameter of Jupiter was only given to two significant figures. Orbital Speed Once the radius of the orbit is known, the orbital speed ( v ) can be calculated by measuring the orbital period ( P ), where v = 2 πr P (6.4) The orbital period of the moon can be measured by observing the moon over several nights, and recording the time it takes to complete one orbit around the planet. Procedure For this lab, you will use Jupiter’s moon Io for your measurements. In the “virtual telescope” this is the moon labeled I. You should refer to the information provided in the Theory and Background section for details on the calculation. Make sure you read this carefully so you have a good “big picture” view of what you are doing. Remember that the units on the quantities used in the equations must be m (meters), s (seconds), and kg (kilograms). This means you may need to do some unit conversions during this lab. Before you begin, open the “virtual telescope” and watch the orbits of the moons for several (simulated) days. Then, write a short summary of what you need to measure and what calculations you will need to perform. You will submit this summary as part of the pre-lab quiz. 35

Data Collection First, you need to make the observations and measurements (that is, collect the data) re- quired to calculate the mass of Jupiter. 1. Stop the simulation when Io is at its furthest distance from Jupiter. Since we are assuming the orbit is circular, this can be when Io is either on the right or left side of Jupiter. You may want to make some small adjustments to the time to ensure the location is correct. Record this date and time as the “start time” of the orbit. Although you do not need to hand it in, you may want to take a screenshot of this observation so you can refer to it later. 2. Move time forward until Io has completed one orbit around Jupiter (and returned to the same location). Record this date and time as the “stop time” of the orbit. Although you do not need to hand it in, you may want to take a screenshot of this observation so you can refer to it later. 3. Using either of the observation from the previous two parts 3 , measure the size of Jupiter and the distance between Jupiter and Io. In the work you hand in, you should either make a sketch of the observations and label the two distances you measured. To do this, you can use a physical ruler (and hold it up to your screen) or you can use a virtual ruler. For example, you can download a Screen Ruler here . Analysis Now that you have data on the orbit of Io, you need to do some data analysis in order to calculate the mass of Jupiter 4. Use your data to calculate the radius of Io’s orbit in meters. The true diameter of Jupiter is 140,000 km. 5. Use the date and time for the start and finish of one complete orbit (the “start time” and “end time” from Steps 1 and 2) to determine the period of Io’s orbit in seconds. Note: The time in the telescope simulation is given in 24-hour time. Make sure you take into account the date of each observation as well as the time, as the period may be longer than one day. 6. Calculate the orbital speed of Io in m/s. 3 You can use either since Io should be at the same place in both images since the second image is exactly one orbit after the first 36

7. Calculate the mass of Jupiter in kg. For reference, the actual mass of Jupiter is 1 . 90 × 10 27 kg. If your answer for the mass of Jupiter is not with a factor of 2 (i.e, within twice the mass or half the mass) of the true mass of Jupiter, then there is likely an error in your calculation. You will need to double check your calculations and your observations for the orbital radius. Compare your calculated value for Jupiter’s mass to the true value by finding the percent error. The percentage error is percent error = | true mass − calculated mass | true mass × 100% (6.5) The | | symbol in the numerator means “absolute value”. This means that if the difference between the true mass and the calculated mass is a negative number, you would simply use the non-negative (i.e., positive) value of the number. The percentage error is a quick way to estimate the error on your determination of the mass of Jupiter. If you are careful taking data in this lab (and you should be) you can achieve a percentage difference of less than 10%. Not bad for an hour or two of work! 37

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Luminosity and the Inverse Square Law By the end of this lab, you should be able to... state the relationship between intensity and distance from a light source. calculate the luminosity of a light source. explain how to determine which model best fits measured data. Theory This lab will investigate the relationship between the luminosity of a star and how bright it appears to us on Earth. Of course, we can’t use a real star in the lab. Instead, we will use a light bulb as our light source. Assuming a light source (star) radiates spherically, the apparent brightness ( b ) of the light source is proportional to the luminosity ( L ) of the source and inversely proportional to the square of the distance from the source ( d ). This relationship is called the Inverse Square Law b ∝ L d 2 (7.1) Recall that ∝ means proportional to and is a way of indicating how the brightness depends on the luminosity and the distance. Since apparent brightness is dependent on distance, two light sources of different brightness may appear to be the same if the brighter object is further away. Thus, apparent brightness is not very useful for comparing the properties of two different light sources. What is needed is a quantity that is intrinsic to the source. That quantity is luminosity and is a measure of the total power output by the source. The apparent brightness of a light source is related to the intensity ( I ) of the light at the point it is observed, and is related to the luminosity ( L ) of the light source and the the distance ( d ) from the source by I = L 4 πd 2 W m 2 (7.2) However, there are several different ways to measure these quantities. We will briefly discuss two different ways here, since you will encounter both of them in this lab 1 . 1 Note that this introduction is a simplification of some of these terms. If you would like more details, please ask you lab instructor. 39

Photometric Units Photometric units are weight by a function that represents the human eye’s sensitivity to light at a given wavelength. For example, the human eye is very good at seeing green light (at a wavelength around 550 nm), but cannot see light in the infrared part of spectrum (at wavelengths great than 800 nm). In Photometric units, the quantity that represents power is called the Luminous flux and is measured in units of lumens (lm). The quantity that represents the power per unit area (intensity) is called the Illuminance and is measured in units of lux (lx). Radiometric Units Radiometric units are not weighted by the sensitivity of the human eye, and therefore give information about the absolute luminosity of the source. In other words, these units tell you how bright a source would appear if the human eye could see all wavelengths equally. In radiometric units, power us measured in watts (where 1 W = 1 J/s) and intensity is measured in W/m 2 . These are the units that are used in class. For example, imagine you have a laser that emits light at 1064 nm at a power of 100 W. If you measured the illuminance of this laser you would find that it is zero! This is because the human eye cannot see any light at this wavelength. Measurements in this Lab In this lab, you will use a light meter to measure the intensity of light falling on a small area (the area of the light meter) in one direction from the source. This light meter measures illuminance, but uses an “old” imperial unit: the foot-candle . This unit comes from how much light a candle gives when you are one foot or 12 inches away from it. If you are outside in bright sunlight, you would measure an illuminance of about 10,000 foot-candles. After taking these measurements, you will convert from photometric units (foot-candles) to radiometric units (W/m 2 ), and work in radiometric units for the rest of the lab. This conversion depends on the type of light source used (since different light sources will emit light at different wavelengths). For the incandescent light bulbs used in this lab 2 1 foot-candle = 0 . 89 W / m 2 (7.3) 2 This conversion also takes into account the electrical efficiency of the bulb. For example, at 60 W bulb uses 60 J of electrical energy per second, but does not produced 60 J of light energy every second. Much of the energy is “lost” to other sources (like heat). 40

Finding Luminosity To find the luminosity of the light bulb, we will assume that the source radiates equally in all directions, and thus the total power of the source is spread over the surface area of a sphere with a radius equal to that of the distance from the source that the measurement is made ( A = 4 πd 2 ). The luminosity can be found by solving Eq. 7.2 for the luminosity ( L ) and using you measurement for intensity ( I ) at a given distance ( d ). Predicting Relationships Although you already know how the intensity of light scales with the distance from the source of the light 3 you will use your data to determine the correct relationship. Answering the question “What is the correct relationship between two quantities” is the type of question that scientist encounter quite often. One way to answer this question is to suggest several different models, and see which one best matches your data. To do this, you will predict the intensity of light at several distances based on one of your measurements. You will plot the prediction for two different models, and then plot your measured data on the same plot. From this, you can choose which model best matches your data. The two possible models you are going to check are: Model A: Intensity ∝ 1 distance (7.4) Model B: Intensity ∝ 1 distance 2 (7.5) Again, recall that ∝ means proportional to . In this context, these two equations are a way of indicating how the intensity depends on the distance from the source. There are two different methods you can use the predict the intensity at a different distance based off of a known intensity. Make sure you read through both methods carefully. They achieve the same result, but show you two different approaches you can take. 3 You know this because you have read the preceding part of this theory section, and you have will have covered it in class 41

For example, assume you are trying to predict the intensity at a distance of 30.0 cm from the source based on your measured intensity of 516 W/m 2 at a distance of 10.0 cm from the source. 4 The first method is based on using the factor by which the distance changes (that is, the ratio of the distances) to find how much the intensity will have changed (which depends on the model). Then, based off of this change in intensity, the intensity at the new location can be found. To use this approach, first determine how the distance has changed by using a ratio of the new distance to the original distance 30 . 0 cm 10 . 0 cm = 3 . 00 . (7.6) This means the distance has increased by a factor of 3. The change in intensity that this results in depends on how the intensity scales with the distance. In other words, it depends on which model is used. Next, we can use how each model depends on the distance to determine the new intensity. Using Model A (see Eq. 7.4 ) intensity at 30 cm = 1 3 . 00 × intensity at 10 cm = 516 3 = 172 W / m 2 (7.7) Using Model B (see Eq. 7.5 ) intensity at 30 cm = 1 3 . 00 2 × intensity at 10 cm = 516 9 = 57 . 3 W / m 2 (7.8) In this case, the factor of 3.00 is squared because this model predicts that the intensity scales like the distance squared. The above examples work because we know that the intensity should decrease as we move away from the source, and each model provided information about how the intensity should vary with distance. 4 If you measured a different value, that’s OK! This number is not necessarily the same as what you will measure. 42

Another approach is to use the ratio of the intensity at the two distances. For example, consider Model B. Let I 1 be the intensity at d 1 = 10 . 0 cm and I 3 be the intensity at d 3 = 30 . 0 cm (which we are trying to find). The ratio of I 3 to I 1 gives I 3 I 1 = 1 d 2 3 1 d 2 1 = 1 d 2 3 × d 2 1 1 = d 1 d 3 2 (7.9) The third step follows from the math “trick” that dividing by a fraction is equivalent to multiplying by the inverse of the same fraction. From Eq. 7.9 the intensity at a distance of 30 cm is I 3 = d 1 d 3 2 × I 1 = 10 . 0 30 . 0 2 × I 1 = 1 3 . 00 2 × 516 = 57 . 3 W / m 2 (7.10) This gives the same result as the example shown in Eq. 7.8 above (which illustrates an alternative approach to solving the same question). You can apply the same approach for model A, except that the intensity has a different scaling with the distance. Procedure Part 1 - Measurements 1. Use the light meter to measure the illuminance (apparent brightness) of your “star” at the distances given on the worksheet. The light meter measures in units of foot-candles and has 2 scale settings. It is recommended that you start on the more sensitive scale at the largest distance and approach the light source. If the meter overloads, switch to the less sensitive scale. Make sure you place your light meter at the correct distance from the source (the source may not be located at a position of 0), and ensure that you hold the light meter in a consistent position for each measurement. It must be at the same height as the light source, and the detector must be pointed directly at the source. 2. Convert your illuminance reading at each distance to the intensity, using the conversion given in Eq. 7.3 . 43

Part 2 - Model Testing To build your two possible models, you will start by assuming you only have data at the 10 cm position, and use this intensity to predict the intensity at other positions. After you have made predictions, you will compare with the rest of the data you have taken. In an ideal situation, you would take your data after making predictions, but doing so in this lab can introduce too much error. 1. In a table (see the worksheet) fill in your measured intensity at 10 cm. 2. Based on the two models (see Eqs. 7.4 and 7.5 ) and your measured intensity at 10 cm (which you wrote in the table), calculate the predicted intensity of the light at a dis- tance of 20 cm, 50 cm and 100 cm. Record your answers in the table on the worksheet. In the next few steps, you will create a plot with your predictions and measured data to check which model is correct. There is a template already created for you in Google Sheets where you can enter your predictions and measured data to create this plot. The link can be found in lab folder on Blackboard. Make sure you follow these instructions carefully. 3. Enter your data for the predicted intensity for Model A and Model B in the corre- sponding columns in Google Sheets . The plot will show the predicted data points and a trendline showing the prediction for how the intensity should change a function of distance. 4. Add your data for the measured intensity versus distance to the corresponding column in Google Sheets . Your measured data will appear on the same plot as your predictions. 5. Download your plot as a PDF or PNG image and upload it to Blackboard (using the “Lab Hand-In” test that will be available during the lab). Only one person in your group needs to do this. 6. Based on the plot, which model best matches your data? Justify your answer based on what you see on your plot. Does this agree or disagree with the expected theoretical relationship? Make sure you state what the expected relationship is. Part 3 - Luminosity Calculation In this part, you will calculate the luminosity of your source (which corresponds to the power of light bulb) based on your intensity measurement at 40 cm. Astronomers can use the same process to determine the luminosity of stars. Does your calculated luminosity in watts agree with the luminosity marked on the light bulb? Suggest reasons for any discrepancies (if there is one). 44

Telescopes and Image Formation By the end of this lab, you should be able to... classify and measure the focal length of lenses and mirrors calculate the magnification of a lens system construct a simple telescope calculate the angular resolution of several optical instruments (including your own eye) calculate angular size from an object’s physical size and distance from the viewer. calculate the minimum size object an imaging device can see at a certain distance given the device’s angular resolution. Theory Image Formation A lens is a piece of transparent material (usually glass) used for forming images by the refraction (bending) of light rays. To refract the incoming light, at least one of the surfaces must be curved. In general, lens where the glass curves outwards is called a convex lens, and a lens where the glass curves inwards is called a concave lens. However, the two sides of the lens can have different curvatures. A selection of several different types of lenses and their names are given in Fig. 8.1 . When light from a distant object, such as a star, is incident on a lens, an image is formed at a distance, called the focal length, from the lens. Figure 8.2 shows image formation for a biconvex lens. In this case, the light rays from the object enter the lens parallel to each other, and come to a focus at the focal point of the lens. We can assume the light rays from the object are parallel only if the object is very far away from the lens. A curved surface mirror can also be used to form images, as shown in Fig. 8.3 . However, the deflection of the light rays is accomplished by reflection rather than refraction. 45

Figure 8.1: A selection of several different types of lenses and their names. You can identify the type of lens by examining the curvature of both sides of the lens. Figure 8.2: Image formation for a biconvex converging lens. The image is located at the focal point of the lens. Telescope Construction A simple telescope can be constructed using two converging lenses as shown in Figure 8.4 . The image formed by the first lens, called the objective (or primary) lens, becomes the object for the second lens which is called the eyepiece. When you look at the light emerging from the eyepiece lens, distant objects will appear magnified. The amount of magnification ( M ) is simply the ratio of the objective lens focal length ( f o ) to the eyepiece lens focal length ( f e ) M = f o f e . (8.1) 46

Figure 8.3: Image formation for a concave converging mirror. The magnification of a telescope is easily changed by changing they eyepiece. Most telescopes come with a range of eyepieces that allow you to choose the magnification you want to use. Figure 8.4: A simple refracting telescope, constructed using two converging lenses. 47

A telescope that uses a lens to gather light is called a refracting telescope or a refractor. Telescopes that gather light with a mirror are called reflecting telescopes or reflectors. Re- flecting telescopes are common because they are less expensive than refracting telescopes of equivalent size. The difficulty with reflecting telescopes is that the image is formed on the same side of the mirror as the object and attempting to view it will block the incoming light. The most common solution to this problem, named after Isaac Newton, is to use a small secondary mirror to reflect the light from the large primary mirror out the side of the telescope. As with a refractor, the magnification is the ratio of the objective and eyepiece focal lengths (see Eq. 8.1 ). The schematic for a Newtonian reflector is shown in Figure 8.5 . Figure 8.5: The schematic for a Newtonian reflector. This type of telescope uses a small secondary mirror to reflect the light from the large primary mirror out the side of the telescope. A catadioptric telescope uses both lenses and mirrors to gather light. The most common type of catadioptric is the Schmidt-Cassegrain where light enters the telescope through a thin lens at the front of the telescope, reflects from a large concave mirror at the back of the telescope, reflects again from a small convex mirror on the back of the lens, and finally exits through a hole in the large mirror (see Figure 8.6 ).The combination of the two curved mirrors and the lens allows Schmidt-Cassegrains to have short tubes but long focal lengths. Most Schmidt-Cassegrains have a focal ratio of 10, meaning the effective objective focal length of the lens and mirror system ( f o ) is 10 times the diameter of the objective lens ( D o ). That is, focal ratio = f o D o = 10 −→ f o = 10 × D o (8.2) 48

Figure 8.6: The schematic for a Schmidt-Cassegrain catadioptric telescope. In this type of telescope light enters the telescope through a thin lens at the front of the telescope, reflects from a large concave mirror at the back of the telescope, reflects again from a small convex mirror on the back of the lens, and finally exits through a hole in the large mirror. Image Credit: Celestron Owners Manual 2015. Angular Resolution The ability of an imaging device (such as our eyes or a telescope) to see an object depends on the angular size of the object, not its physical size. For example, we can happily read fine print at close distance, but when the text is far away it needs to be larger in order to be able to read it. We can determine the angular resolution of an imaging device by finding the smallest angular size that it is able to resolve clearly. When we reduce the angular size of an object, either by reducing its physical size or moving it further away, until our imaging device can only just see it clearly, the angular size of the object is equal to the angular resolution of the imaging device. The angular size of an object (measured in radians) is the ratio of the object size to the object distance θ size [in radians] = object size object distance = d r (8.3) If you are able to determine the minimum angular size you can observe with your imaging device (be that you eyes or a telescope), then the minimum angular size is equal to the angular resolution of your imaging device. 49

For example, in this lab you will stand 7 to 10 m from a chart that has vertical lines of varying spacing. You will determine the minimum separation of lines (the object size) that you can resolve. Let’s say you stand 9.5 m ( r ) from the chart and can distinguish the separation of the lines spaced 6.0 mm apart ( d ). The angular resolution of your eyes is θ res = d r = 6 . 0 mm 9 . 5 m = 6 . 0 × 10 − 3 m 9 . 5 m = 6 . 3 × 10 − 4 radians (8.4) In astronomy, we often work with angles in units of arcseconds instead of radians, where 1 radian = 206265 arcseconds . (8.5) In arcseconds, you angular resolution (from Eq. 8.4 ) would be θ res = 6 . 3 × 10 − 4 radians × 206265 arcseconds 1 radian = 130 arcseconds (8.6) The equation for the angular size of an object (see Eq. 8.3 ) can be written in terms of arcseconds directly as θ size [in arcsec] = 206265 d r (8.7) which follows from the unit conversion between radians and arcseconds in Eq. 8.5 . You can also use Eq. 8.3 or 8.7 to determine the smallest size of an object ( d ) that you could see given you know the distance to the object ( r ) and the angular resolution (or resolving power) of your telescope ( θ size , which would correspond to the smallest angular size you can resolve). You can also determine the resolving power of an imaging device (like your eyes or a telescope) based on the diameter of the objective lens (for your eyes, this is the size of your pupil). In addition to the diameter of the telescope, angular resolution depends on the wavelength (colour) of the light. Measured in radians or arcseconds, the angular resolution is θ res [in radians] = 1 . 22 λ D or θ res [in arcsec] = 252000 λ D (8.8) where λ is the wavelength of the light and D is the diameter of the objective lens. 50

Telescope Mounts Have you ever tried to hold a 25 kg telescope up to your eye and keep it steady at 200 times magnification? It’s not possible. Thus, telescopes are usually attached to a mount. The simplest type of telescope mount is called an alt-azimuth mount because it has an axis that rotates the telescope vertically (altitude) and an axis that rotates the telescope horizontally (azimuth), as shown in the left image in Fig. 8.7 . An alt-azimuth mount is simple to construct, however it requires adjustment of both axes to track a star unless you are at the North Pole, the south pole, or the equator. Because the stars appear to rotate about the north star, Pointing one axis of a mount at the north star allows tracking of a star while adjusting only one axis. This is equivalent to moving an alt-azimuth mount to the equator of the Earth, hence the name, equatorial mount. A schematic of an equatorial mount is shown in the right image in Fig. 8.7 . Figure 8.7: Left: An alt-aximuth mount. Right: An equitorial mount. 51

Procedure Part 1 - Building A Telescope You will be provided with the parts you will use to build your own telescope (along with a set of instructions). Before you assemble the telescope, you will need to: 1. Determine the type of lenses you are using (see Fig. 8.1 ). 2. Measure the focal length of the lenses. To do this, you will need to produce an image of a distant light source uses the lens and then measure the distance from the lens to this image (see Fig. 8.2 ). Details of how to do this will be explained at the start of the lab. Assemble your telescope following the instructions provided in the lab. Then, 3. Calculate the magnification of the telescope 4. Measure the objective diameter of your assembled telescope. 5. Calculate the angular resolution of the telescope at a wavelength of 550 nm in arcsec- onds (using Eq. 8.8 ). Once you have assembled your telescope, get on of the lab instructors to check it. Bring it out into the hallway where you will use your telescope to look at a distant object at the far end of the hallway. Part 2 - Your Eyes versus Your Telescope In this part of the lab, you will compare the angular resolution of your eyes to the angular resolution of your telescope, and predict the size of a crater on the Moon you could see with either your eyes or the telescope. To determine the angular resolution of your eyes, follow these steps: 1. Find the resolution chart on the wall. 2. Stand at least seven (7) meters from the chart. You will have to measure the distance with a meter stick (or you lab instructor may indicate where you should stand). 3. Have your partner cover the chart with the blank paper. Tell your partner to move the paper very slowly up the chart, keeping the paper horizontal. When you start to see the chart lines separate (i.e, you can resolve the separation between the vertical lines) just below the paper, tell your partner to hold the paper in place. 52

4. Your partner will read the line spacing printed on the chart nearest to the bottom of the paper. Record this value. This value is your object size. 5. Calculate the angular resolution of your eye in arcseconds using Eq. 8.7 . Now you have the angular resolution of the small telescope you built (from Part 1) and your eyes (from the previous steps). Given this information, you will determine (and compare) the smallest size of a crater on the Moon you could see with your eyes or the telescope. You should give your answer in km. It will be helpful to know that the distance from the Earth to the Moon is 384,000 km. Part 3 - A Large Commercial Telescope In the lab room will be one of the large telescopes that Douglas College owns, which is a Schmidt-Cassegrain telescope made by Celestron. For this telescope: 1. State the type of mount the telescope is attached to. 2. Determine the angular resolution of the telescope in arcseconds. To do this, you will need to make a measurement of something on the telescope. You will need to decide what to measure during the lab. Please be careful around this telescope and do not touch any of the optics! 3. Calculate the size of the smallest crater you could see on the Moon with this telescope. Give your answer in km. Note that these last two calculations are the same as what you did for the telescope you built. The only difference is that this commercial telescope is larger! 53

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Sunspots and The Solar Cycle By the end of this lab, you should be able to... describe and define a sunspot. calculate the average rotation speed of the Sun. predict when the next maximum number of sun spots will occur. Theory and Background This information was compiled primarily from “The Galileo Project”. You can find more about this project and Sun spots by visiting: http://galileo.rice.edu/sci/observations/sunspots.html Sunspots are dark areas of irregular shape on the surface of the Sun. Their short-term and long-term cyclical nature has been established in the past century. Spots are often big enough to be seen with the naked eye. While direct observation of the Sun in a clear sky is painful and dangerous, it is feasible when the Sun is close to the horizon or when it is covered by a thin veil of clouds or mist. Records of naked-eye sunspot observations in China go back to at least 28 BCE. In the West, the record is much more problematical. It is possible that the Greek philosopher Anaxagoras observed a spot in 467 BCE, and it appears that there are a few scattered mentions in the ancient literature as well. However, in the dominant Aristotelian cosmology, the heavens were thought to be perfect and unchanging. A spot that comes and goes on the Sun would mean that there is change in the heavens. Given this theoretical predisposition, the difficulty of observing the Sun, and the cyclic nature of spots, it is little wonder that records of sunspots are almost non-existent in Europe before the seventeenth century. A very large spot seen for no less than eight days in 807 was simply interpreted as a passage of Mercury in front of the Sun. Other mentions of spots seen on the Sun were ignored by the astronomers and philosophers. In 1607 Johannes Kepler wished to observe a predicted transit of Mercury across the Sun’s disk, and on the appointed day he projected the Sun’s image through a small hole in the roof of his house (a camera obscura) and did indeed observe a black spot that he interpreted to be Mercury. Had he been able to follow up on his observation the next day, he would still have seen the spot. Since he knew that Mercury takes only a few hours to cross the Sun’s disk during one of its infrequent transits, he would have known that what he observed could not have been Mercury. 55

The scientific study of sunspots in the West began after the telescope had been brought into astronomy in 1609. Although there is still some controversy about when and by whom sunspots were first observed through the telescope, we can say that Galileo and Thomas Harriot were the first, around the end of 1610; that Johannes and David Fabricius and Christoph Scheiner first observed them in March 1611, and that Johannes Fabricius was the first to publish on them. His book, De Maculis in Sole Observatis (”On the Spots Observed in the Sun”) appeared in the autumn of 1611, but it remained unknown to the other observers for some time. In 1610, shortly after viewing the sun with his new telescope, Galileo Galilei made the first European observations of sunspots. Daily observations were started at the Zurich Obser- vatory in 1749 and with the addition of other observatories continuous observations were obtained starting in 1849. The sunspot number is calculated by first counting the number of sunspot groups and then the number of individual sunspots. The ”sunspot number” is then given by the sum of the number of individual sunspots and ten times the number of groups. Since most sunspot groups have, on average, about ten spots, this formula for counting sunspots gives reliable numbers even when the observing conditions are less than ideal and small spots are hard to see. Monthly averages (updated monthly) of the sunspot numbers show that the number of sunspots visible on the sun waxes and wanes with an approximate 11-year cycle. Note: There are actually at least two ”official” sunspot numbers reported. The International Sunspot Number is compiled by the Solar Influences Data Analysis Center in Belgium. The NOAA sunspot number is compiled by the US National Oceanic and Atmospheric Administration. Figure 9.1: Left: An image of the sun with several sunspots (the dark spots) visible. Right: A sunspot drawing by Galileo Galilei from June 23, 1612. 56

Procedure This lab has two distinct sections. Make sure you read the procedure for both parts before you start the lab. 1. In Part 1, you are going to look at images of the Sun taken over about two weeks which contain several sunspots. You will use an average of the observations for these sunspots to calculate the period of the Sun’s rotation. The data for this part will be provided at the start of the lab. The images come from the Solar and Heliospheric Observatory (SOHO). An example of one of these images is shown in Fig. 9.2 . 2. In Part 2, you are going use data for the sunspot number (which is a measure of the number of sunspots on the sun over the course of a year) to make a plot of the sunspot number as a function of year, and use this data to make predictions about the number of sunspots in future years. the annual sunspot numbers are from the National Oceanic and Atmospheric Administration of the United States Department of Commerce, National Geophysical Data Centre (you can find the data here 1 ). Part 1 - Rotation Rate of Sun We are pretending that for each recording image we are stepping outside at about the same time each day, projecting the sunspots onto paper, and tracing where they appear. You will use these observations to calculate the period of the Sun’s rotation. The data provided is from June 22 to July 3 in 2002 when there were lots of sunspots. If this lab were done using images from (for example) 2009 it would be a very boring lab as there were more than 200 days without sunspots in that year. An example of the data (on June 26) is provided in Fig. 9.2 . Reminder: Latitude measure the angle above or below the equator. This correspond to the horizontal lines in the image of the Sun. Longitude measure the angle East or West of a fixed point on the equator. This corresponds to the vertical lines lines on the image of the Sun. Since the Sunspots move horizontally across the surface of the Sun, we are interested in the change of the longitude coordinate. When you record the position of the sunspots, you should be able to estimate the location of each sunspot to with 1 ◦ of longitude. For example, using the image provided in Fig. 9.2 , sunspot B appears at a longitude of − 27 ◦ on June 26. 1 https://www.ngdc.noaa.gov/stp/space-weather/solar-data/solar-indices/sunspot-numbers/ american/lists/list_aavso-arssn_yearly.txt 57

Procedure For each labeled sunspot (which are labeled A, B, and C in the provided images): 1. Record the date and longitude (i.e., the position) of the sunspot when it is first visible and when it is last visible. You should consider the sunspot visible if it is labeled in the images. 2. Using the data you recorded in Step 1, determine the change in longitude of the sunspot and the corresponding time (number of days) this change in longitude occurred over. 3. Using the data you collected in Step 1 and Step 2, calculate the average degrees of longitude moved each day by the sunspot. At this point, you should have the average degrees of longitude moved each day by each of the three labeled sunspots. For the next part, we will take an average of these three results, and work with this value for the remainder of the analysis. The reason that the sunspots are moving is because the Sun is rotating. This means you can use the data to calculate the rotation rate (degrees/day) or rotational period (days) of the Sun. You will do this in the following steps: 4. Calculate the average degrees of longitude moved each day by a sunspot on the sun. This is the average of your result for each sunspot from Step 3. 5. Over this time period, the Earth has also been moving around the Sun. This movement is in the same direction, and is at a rate of about 1 . 0 ◦ per day (since a circle is 360 ◦ and it takes about 365 days for the Earth to orbit the Sun). To account for this, add 1 . 0 ◦ / day to the result from Step 4. This is now your best estimate for the actual degrees of longitude a sunspot moves in one day due to the rotation at the Sun. 6. Using your answer to Step 5, calculate how long it would a sunspot to circle around the Sun once. Moving around the Sun once means the sunspot would need to move 360 ◦ . Express your answer in days. This is the rotational period of the Sun! 58

Part 2 - Sunspots Through the Years Astronomers have found that the number of sunspots increases and decreases on a regular cycle called the solar cycle or sunspot cycle . In this lab, you will make a plot of the number of sunspots from 1945 to 2016 and then answer a series of questions. The annual sunspot numbers are from the National Oceanic and Atmospheric Administration of the United States Department of Commerce, National Geophysical Data Centre (you can find the data here 2 ). 1. Transfer the sunspot data into a Google Sheets document. You can either type the data in year by year or follow these instructions to save a bunch of time: Open the link to the data, and select and copy the data from 1945 to 2016 (including the two column headers “Year” and “SSN”). Paste this data into Google Sheets . All this data will be in one column. Select the column with the data, and then click “Data” (on the menu bar at the top of the page) then “Split text to columns”. Now the data will be arranged in two columns, as it should be. 2. Add additional years up to 2030 to your columns for “Years”. Do not enter anything in the sunspot number (“SSN”) column. 3. Create a plot of the data and select Column Chart as the chart type. The year should be on the x-axis (horizontal), and the sunspot number should be on the y-axis (vertical). Print a copy of your plot (the lab instructors will explain how to do this during the lab). 4. Like all data, the measured sunspot number will not follow a perfect pattern. However, you can use the data to visualize the average pattern that the sunspot number follows. One way to do this is to draw a smooth line through or near the tops of each column (which represent the number of sunspots in a given year) to help average out the data. Draw this line on your printed plot 2 https://www.ngdc.noaa.gov/stp/space-weather/solar-data/solar-indices/sunspot-numbers/ american/lists/list_aavso-arssn_yearly.txt 59

Use your plot to answer the following questions: 4. Which years correspond to solar maximums? Since there are seven solar maximums in this time range, you should answer with seven years in chronological order. Do your best to estimate these years based on the trends shown in your plot. 5. Which years correspond to solar minimums? Since there are six solar minimums in this time range, you should answer with six years in chronological order. Do your best to estimate these years based on the trends shown in your plot. 6. What is the average number of years between solar maximums? 7. What is the average number of years between solar minimums? 8. If the number of sunspots follows the same pattern past 2016, when would we expect the next solar maximum and solar minimum? It’s possible the “next” date will be in past (i.e., occur before this year). As part of your answer, explain how you determined these two dates. 9. Predict the sunspot number from 2016-2030 by extending your smooth “average” line out to the year 2030 on your plot. Briefly justify your predictions by making specific reference to features, patterns, or trends you see on your plot. 60

June 26 A B C Figure 9.2: Sunspots on June 26 2002. 61

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Dark Matter By the end of this lab, you should be able to... differentiate between the Brightness Method and Orbital Method for determining the mass of a galaxy. explain the work done by astronomer Vera Rubin that lead to the discovery of dark matter calculate the amount of dark matter present in a galaxy. Theory and Background In 1967, Vera Rubin observed that stars within the Andromeda galaxy had higher-than- expected orbital speeds. Physicists have also observed the same phenomenon in the nearby Triangulum galaxy. Figure 10.1: Left: Andromeda Galaxy (M31). Right: Triangulum Galaxy. Image Credits: NASA CC0. The mass of a galaxy can be determined by measuring the orbital speeds of stars within the Triangulum Galaxy and using the following formula gravity force = circular motion force G Mass star Mass galaxy radius 2 orbit of star = Mass star velocity 2 star radius orbit of star (10.1) which simplifies to Mass galaxy = velocity 2 star radius orbit of star G or M = v 2 r G (10.2) 63

Physicists have calculated that the mass of the Triangulum Galaxy within a radius of r = 4 . 0 × 10 20 meters is equivalent to 46 billion Suns. This method of determining the mass of the galaxy is called the Orbital Method because it uses information about the orbital speed of stars and gas in the galaxy. However, the mass of the galaxy can also be measured using the brightness of galaxy, and determining how mass is required to produce the observed brightness. This method of determining the mass of the galaxy is called the Brightness Method (no explanation required). By measuring the brightness of Triangulum, physicists have also calculated that its mass within a radius of r = 4 . 0 × 10 20 is equivalent to 7 billion Suns. The discrepancy between these two results implies that there is 39 billion Suns’ of unseen mass within Triangulum. This unseen mass is called “dark matter”. “Dark” because we cannot see it, but “matter” as it has a gravitational effect on other objects. Physicists have observed thousands of other galaxies and most (but not all) are now convinced that, on average, dark matter accounts for 90% of the mass of every single galaxy in the universe. Physicists also have independent evidence for the existence of dark matter from observations of distorted images of distant galaxies (gravitational lensing). Although no one knows what dark matter is made of, physicists currently have a number of theories. One of the earliest theories of dark matter was that it consists entirely of compact celestial objects such as planets, dwarf stars, and black holes. Careful observations have ruled out this theory. Most physicists today think that dark matter is made of a type of subatomic particle that (to date) has never been detected in the laboratory. The two leading candidates are weakly interacting massive particles (WIMPs) and axions. Numerous experiments that are trying to detect one of these particles are currently underway worldwide. As physicists do not yet know what dark matter is made of, they do not know the composition of a large fraction of the universe. This is reproduced and adapted with permission from the Perimeter Institute . 64

Procedure and Analysis Note: You will be given the data that is referenced in the following two parts at the start at the lab time. Part 1 - Galaxy Rotation Curves In this part, you will make a plot of the observed orbital speed and the expected orbital speed that you calculated based on the observed mass of the galaxy from the brightness method as a function of distance from the centre of the galaxy. You will use the comparison between the observed and calculated orbital speed to determine if there is any “missing” matter in this galaxy. The expected orbital speed is the orbital speed of the star you would expect given the observed (that is, visible) mass. This mass has been measured using the brightness method and will be given to you as part of the lab data. As a reminder, the orbital speed of a star can be found using v star = r Gm galaxy r (10.3) where m galaxy is the mass of the galaxy contained within a radius of r from the center of the galaxy. The data provided are for objects that are very far away from the center of the galaxy, such that the mass of the galaxy determined using the brightness method can be assumed to be the same for every orbital radius. 1. For each orbital radius, calculate the expected orbital speed using the mass determined by the brightness method (which is the mass given to you). Record your answers in the “Expected Speed” column of a table like the one provided to you with the lab data. 2. Plot the expected speed (which is the speed you calculated) and the measured speed as a function of the orbital radius on the same plot using Google Sheets. In this plot, use the Smooth line chart option. Make sure you label each line, label each axis, and adjust the scaling of the plot to match your data. 3. Compare the plots of the “expected” and “measured” speed and explain what differ- ences you observe. Provide a clear explanation for what could explain the differences you observe, and why/how your explanation leads to the observed difference. You should use the information provided in the lab manual and the video to write your explanation. 65

Part 2 - Determination of Missing Mass In this part, you will determine how much mass appears to be “missing”, and how this amount of “missing” mass depends on the distance from the centre of the galaxy. In this case you will use Eq. 10.2 , which tells you the amount of mass contained within a radius r based on the measure orbital speed of the stars. 1. Use the measured speeds to calculate the mass of the galaxy that must be contained within each orbital radius. Record your answers in the “Calculated Mass” column of a table like the one provided with the lab data. 2. For each orbital radius, calculate the difference between the calculated mass within this radius and the total mass of the stars that can be seen with the brightness method. Represent this difference as a percentage of the calculated mass within the orbital radius. Record your answers in the “Percentage of Missing Mass” column of your table. The percentage difference is percentage of missing mass = m calc − m brightness m calc × 100% (10.4) where m calc is the amount of gravitational mass (the number you calculated) and m brightness is the amount of mass that can be seen with the brightness method. This number represents the percentage of the total mass of the galaxy that is “missing mass” (that is, mass we cannot see). 3. If you assume that all of the “missing mass” is dark matter, do you think most of the dark matter in a galaxy is found close to the center of galaxy, far away from the center of the galaxy, or evenly distributed within the galaxy? Clearly justify your answer based on the data and calculations you did in this part of the lab. 66

Analysis of Explosions and Hubble’s Law By the end of this lab, you should be able to... explain how objects created in an explosion move as a function of time use plots of object speed versus object distance to determine if the objects originated in an explosion describe a Hubble Plot and determine the Hubble Constant from a Hubble Plot use the Hubble Constant to estimate the age of the Universe. Background and Theory This section outlines the information you need to complete this lab, including a brief intro- duction to a new math tool that is used in this this lab: the idea of the “inverse” of a number or an expression. The sections that follow this cover the analysis of explosion (in general) and then a specific example related to our Universe (Hubble Plots). Make sure you carefully read over the entire lab manual before the lab. You will likely find it helpful to refer back to this Background and Theory section as you work through the lab. A New Math Tool: The Inverse In the context of the this lab, the inverse of a number or an expression means to divide 1 by that number or expression. For example, the inverse of the 2.0 is 1 / 2 . 0 = 0 . 5 and the inverse of x is 1 /x . This idea can also be extended to fractions and even to units. For example, the inverse of 4 / 5 is 1 4 / 5 = 5 4 The inverse of a speed (measured in m/s) is sometimes called “pace” (that is, the time it takes to cover one meter). If an object is moving at a speed of 24 m/s, then the inverse of the speed is 1 24 m / s = 0 . 42 s/m which means it takes the object 0.42 s to cover one meter. Notice that in the above expression the units are also inverted (or “flipped”), because 1 m / s = s m 67

Analysis of Explosions Imagine that a collection of moving particles originated from an initial “explosion”. If this was the case, how could we determine this based off of information from the particles? It turns out that if the particles all started (before the explosion) at the same place at the same time there should be two characteristic properties: 1. All the particles should be moving “away” from the same point. 2. Particles which are farther away must be moving faster, since all particles have been traveling for the same time. This means a plot of the particles velocity as a function of distance should be an straight line with a positive slope. For now, we will limit ourselves to consider an explosion in two-dimensions. To examine a collection of moving particles, we could make a diagram where the position of the particle is represented by a dot on the plot (where the x-axis and y-axis are the two dimensions that the particle can move in), and the velocity of the particle is represented by an arrow starting at the dot and pointing in the direction of motion. The length of the arrow represents the speed of the object (i.e., how fast it is moving). The larger the arrow, the higher the speed. An example of this type of plot is shown in Fig. 11.1 . You can use this type of plot to check the first characteristic of an explosion (see above). You could also examine a collection of moving particles by plotting the speed of each particle as a function of the distance of each particle from the “centre”. You can use this type of plot to check the second characteristic of an explosion (see above). You can also use this type of plot to determine how long ago the explosion occurred. Recall that the distance that an object travels ( d ) depends on the speed of the object ( v ) and the time it has been traveling for ( t ), and is given by the equation is d = vt . This also means that if you know the speed of the object and the distance it has traveled, you can find the time it has been traveling for d = vt −→ t = d v (11.1) In the case of an explosion, this is the time since the explosion occurred (that is, how long ago the explosion occurred). If the particles really did originate from an explosion, then every particle (regardless of position) should have been traveling for the same time. This also explains why particles found further from the “center” must be have a higher speed! You will use this idea in part two of the lab. 68

Figure 11.1: A plot showing the position of a collection of particles and the velocity of each particle. The position of the particle is represented by a dot on the plot (let’s assume that our particles can only move in two-dimensions), and the velocity of the particle is represented by an arrow starting at the dot and pointing in the direction of motion. The length of the arrow represents the speed of the object. The table on the right gives the distance from the “centre” (r) and the speed (v) of each particle. 69

Figure 11.2: A Hubble Diagram, simply a plot of galaxy velocity versus galaxy distance. This diagram is based upon distances found using Supernovae. The slope of the line is the value of the Hubble Constant ( H 0 ) and is given directly on the plot. Our Universe and Hubble Plots Astronomers are able to measure the speed of and distance to many objects in the universe. If the universe is the result of an explosion that happened many (many) years ago, a graph of speed (on the vertical axis) versus distance (on the horizontal axis) should display the properties outline in the previous section. These plots are called Hubble Plots , an example of which is shown in Fig. 11.2 . The slope of the straight line on the plot is called the Hubble Constant ( H 0 ), and relates the velocity of a distant object to the distance from our solar system by v = H 0 d (11.2) As you can see from Fig. 11.2 , the units on a Hubble Plot are not the standard units of velocity and distance that you are used to. On these plots, velocity is measured in km/s and distance is measured in Mpc (megaparsecs). As a reminder, 1 Mpc = 10 6 pc 1 pc = 3 . 09 × 10 16 m 1 km = 1000 m (11.3) 70

This means that the units of the Hubble constant are (km/s)/Mpc. However, with the appropriate unit conversions, this can be written in our “normal” units of (m/s)/m. You will need to do this during the lab. To more easily visualize this conversion, it is helpful to write the units in the following way: km/s Mpc = km s Mpc 1 = km s × 1 Mpc (11.4) The last step follows from the fact that dividing by a fraction is the same as multiplying by the inverse of the fraction. At this point, if a unit conversion is done to convert both km and Mpc into meters (or either km or Mpc into the other, if you are so inclined) then the unit for length will cancel, and the units that remain are 1/s. This is what is expected given that the slope of the line to a plot of speed as a function of distance is v/d = 1 /t which has units of 1/s. This follows from d = vt −→ t = d v −→ 1 t = v d (11.5) This also implies that the inverse of the slope of a velocity versus position plot (like a Hubble Plot ) has units of time (that is, seconds)! This is a very important idea that will come up several times during this lab, when you are looking for a quantity that has units of time (for example, the time since the explosion occured). 71

Procedure Part 1 - Explosion? In this part of the lab, you will look at three different data sets that show the trajectory and position of particles moving in 2D. You will determine whether each set of data is a result of an explosion by checking for both characteristics of an explosion discussed in the Background and Theory section. The three data sets are available in the lab blackboard folder on Blackboard. Note that while these data sets look similar to the example shown in Fig. 11.1 , the example data shown in Fig. 11.1 is not used in this lab. 1. For each set of data, graph the speed (on the y-axis) as a function of distance (on the x-axis) using Google Sheets . This should be three different graphs, not one that contains both data sets. Use a “Scatter chart” as the chart type, and adjust the scale of each axis to ensure the origin (zero) is visible for each axis. On each plot, add a line of best fit and ensure that the equation for the line of best fit is displayed on the plot. Once you have made your plots, get them checked by one of the lab instructors. Note: The position (r) and speed (v) of each particle is given in the table to the right of the plots 2. For each data set, use the provided trajectory plots (on Blackboard) and the graphs you made in Step 1, to determine if the particles trajectory is the result of an explosion. For each set of data, you must clearly justify your answer referencing both characteristics of an explosion discussed in the Background and Theory section. Part 2 - How Long Ago Was The Explosion? For this part, use the data set that you determined was the result of an explosion in Part 1. If two data sets were the result of explosion, you can use either one. In this part of the lab, you will use the data from Part 1 to determine how long ago the explosion that created the observed trajectory of particles occurred. Then, you will find a connection between the time since the explosion occurred and the slope of the graph you made in Part 1. 1. Choose any three particles and calculate the time that it took each particle to travel from the “center” to its current location. This distance the particle has traveled is the value in the “r” column to the right of plot in Fig. 11.1 . Make sure you express each time to the correct number of significant figures. 72

2. Are the three times you calculated in the previous step the same? Is this what you expected? Briefly justify your answer. 3. Based on your data, how long ago did the explosion occur? Briefly explain how you determined this. 4. What is the slope of the graph of speed versus distance for this data set from Step 1 of Part 1? Remember that you can find this from the equation for the line of best fit. Your answer for the slope must include the correct units. 5. What is the relationship between the time since the explosion (your answer to Step 3) and the slope of the graph of speed versus distance (your answer to Step 4)? You must “check” your relationship by showing that it gives the same answer you found in Step 3. Hint: What are the units of these two numbers? An an example of a relationship between two vales, imagine have two numbers a and b where a = 2 m and b = 4 m 2 . In this case, the relationship between these numbers is b = a 2 . You can check this relationship by using the numbers given. In this case, b = a 2 = (2 m) 2 = 4 m 2 . Obviously, these numbers, units, and the relationship are not the correct answer to this part of the lab! Part 3 - Our Universe For this part, you will use the Hubble Plot shown in Fig. 11.2 . Note that the slope of the Hubble Plot is already given to you on in the figure. You do not need to calculate the slope. This is not exactly the same as the current accepted value that we will use for other questions in the course. This was done on purpose. However, the result you find for the age of universe will be similar to the current accepted age of the universe (about 13.8 billion years). 1. Does the Hubble Plot have the correct pattern for a universe which started with an “explosion”? Clearly explain why or why not. 2. Convert the slope of the Hubble Plot (i.e., the Hubble constant) into units of 1/s. The “Our Universe and Hubble Plots” section of the lab manual for details on how to do this. 3. Use your answer from the previous step to calculate the time since the “explosion” which started these particles moving. The explosion was the “Big Bang” and your answer is the age of universe! Give the time in seconds and years. If you unsure of how to do this, look back over your answers from Part 2 of the lab. 73

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White Light as a Mixture Of Colors By the end of this lab, you should be able to... define diffraction and dispersion demonstrate that white light is a mixture of different colors. state whether the speed of light in glass is larger for long or short wavelengths. Theory Sir Isaac Newton was the first to understand that white light consists of a mix of light rays, each with a different color. Newton also understood that white light can be separated into its components because the path of each ray of color is changed by the glass of the prism by a different amount. This effect (the changing of the path of a ray of light) is called refraction . He realized, for example, that red light is consistently less deviated than violet light. As a result, Newton understood that when white light passes through a transparent medium (like air) into another (like glass), its components are deflected the first time according to their color, and once again when they reemerge (back into air, for example). This creates a spread of colored light rays from red to violet, like the colours of the rainbow (see Fig. 12.1 ). This ordered separation of colored rays is known as dispersion of the white spectrum. The spectrum of white light consists of six basic colors arranged in a specific order: red, orange, yellow, green, blue and violet. Colored objects appear different colors because they absorb some colors (wavelengths) and reflect or transmit other colors. The colors we see are the wavelengths that are reflected or transmitted. For example, a red shirt looks red because the dye molecules in the fabric have absorbed the wavelengths of light from the violet/blue end of the spectrum. Red light is the only light that is reflected from the shirt. If only blue light is shone onto a red shirt, the shirt would appear black, because the blue would be absorbed and there would be no red light to be reflected. White objects appear white because they reflect all colors. Black objects absorb all colors so no light is reflected. The primary colors of light are red, green and blue. Mixing these colors in different proportions can make all the colors of the light we see. This is how TV and computer screens work. It is called additive mixing. 75

Figure 12.1: White light sent through a prism creates a spread of colored light rays from red to violet, like the colors of the rainbow. The deflection of the path of the light ray is called diffraction. The variation in the amount of deflection as a function of wavelength is called dispersion. Image from https://www.flickr.com/photos/121935927@N06/13580411493 and is used under the CC BY 2.0 license. Procedure You may do only one of the two parts to this lab. Your lab instructor will provide you with details. Part 1 - White Light Dispersion Using a source of white light and a prism, split the white light into different colours and obtain a dispersed polychromatic image of a rainbow on a paper screen. Make a sketch of your setup. The sketch contains the light source, the prism, and the paper screen. You have to show the distribution of two boundary basic colours: red and violet. Part 2 - Additive Mixing of Colors Observe what happens when red, green, or blue light sources shine light on an object in front of a screen. Observe what happens if two colors are combined. If all of these colors of light are shone onto a screen at the same time, you will see white. Record your observations of the colors of the screen behind the object and the color of the object’s shadow. 76

Inverse Square Law (Penny Simulation) By the end of this lab, you should be able to... state the inverse square law. graph the inverse square law, and discuss the theoretical shape. Theory Many things obey an inverse square law. When plotted, the graph would look like the one from your textbook that deals with the force of gravity as a function of distance, shown in Fig. 13.1 . This shape of a graph happens because “something” spreads evenly. Another example is the intensity of light as you move away from the source of light. The light spreads out evenly over the surface area of a sphere, as shown in Fig. 13.2 . As the radius of a sphere is doubled, its area goes up by a factor of four. If the radius is tripled, the area goes up by a factor of nine. If the radius is quadrupled (factor of four) then the area goes up by factor of 16. Calculations To help you get used to thinking about the inverse square law, and performing some cal- culations related to it, image you are spraying paint onto a large board. The paint goes outwards in straight lines. Like gravity and light, the strength or intensity of the paint obeys an inverse square law. This is because it paint spreads out evenly as it moves away from the source (the can of paint). The bigger the area, the thinner the paint. Or, the less intense the light, or the less force of gravity. Fill in the blank spaces in the following calculations (these will be provided in the lab report sheet). The first two are done as an example. 1. At 1 m from the can, you paint an area of 1 m by 1 m this has an area of 1 m 2 and the paint is 1 mm thick. 2. At 2 m from the can, you paint an area of 2 m by 2 m this has an area of 4 m 2 and the paint is 0.25 mm thick. 77

0 1 2 3 4 5 6 Distance r (arbitrary units) 1 1/4 1/9 1/25 Force F (arbitrary units) Figure 13.1: The inverse square law for the force of gravity. As the distance between the two objects increases, the force of gravity decreases because it depends on the inverse of the square of the distance between the two objects. Image Credit: Will Gunton (own work) CC-BY 4.0. Figure 13.2: The inverse square law for the intensity of light. The light spreads out evenly over the surface area of a sphere. As the radius of a sphere is doubled, its area goes up by a factor of four. If the radius is tripled, the area goes up by a factor of nine. This is because the area of the sphere depends on the square of the radius of sphere. Image From OpenStax Astronomy, and is used under the CC BY 4.0 license. 78

3. At 3 m from the can, you paint an area of by this has an area of and the paint is thick. 4. At 4 m from the can, you paint an area of by this has an area of and the paint is thick. Activity With Pennies 1. Take 36 pennies and stack them up so all 36 are in one stack. In this case, the area is “one penny” and the flux is “36”. 2. Now make a square that is two pennies by two pennies wide. What is area (in units of “pennies”). What is the flux? Record these values in a table. 3. Repeat the previous step, making the square one penny wider each time. Record your values in a table. 4. Plot the flux as a function of the square size, and see if it looks like the inverse square law you saw earlier (for gravity, light, or even paint). Make sure you put your names on the plot and staple it to you lab report sheet when you hand it in. 79

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Spectral Analysis By the end of this lab, you should be able to... differentiate between continuous, absorption, and emission spectrum. identify the composition and relative temperature of several “stars”. calculate an object’s radial velocity from its Doppler shift. Theory In 1857, Gustav Kirchhoff experimented with laboratory chemical spectra and determined that each chemical element has its own unique spectral signature – set of spectral lines. Furthermore, Kirchhoff summarized the three important observations, called Kirchhoff’s Laws. To understand the laws, it is important to understand the concept of a ”blackbody.” A Blackbody is not an object that is black or even dark, instead it is a theoretical abstract, a body that emits all light and radiation that is directed to it. If 100% of light were to illuminate a blackbody, 100% of that light will be given off by the source (or emitted or emanated). Kirchhoff’s Laws 1. A blackbody emanates a continuous spectrum, free of any spectral lines. 2. A hot, transparent gas will produce emission spectrum - a series of bright lines against a dark background. 3. A cool transparent gas in front of a blackbody will produce absorption spectrum - dark lines over a continuous spectra that would appear in the same place as the emission spectrum of a hot gas cloud comprised of the same elements. These laws are summarized in Figure 14.1 . Astronomers study the spectra of stars and galax- ies based on these three laws using an instrument known as a spectrometer. A spectrometer is a device that separates light into all the different colors its is composed off, and lets you see the relative amount of light (the intensity) at each color. You will be using hand-held spectrometers in this lab called “spectroscopes” that use a diffraction grating to break the light into the different wavelength components. 81

Figure 14.1: Kirchhoff’s Three Laws for Blackbody radiation explain the ap- pearance of continuous, absorption and emission spectra. Credit: Adapted from a diagram by James B. Kaler, in “Stars and their Spectra”, Cambridge Univer- sity Press, 1989. Doppler Effect If the star and the observer are moving toward or away from each other, the observed spectrum is shifted since the speed of light is finite. This shifting of the spectrum is known as Doppler Shift. A particular wavelength λ 0 , as measured in a stationary laboratory, will be modified by a wavelength shift ∆ λ = λ – λ 0 , where λ is the observed wavelength of light, if the source of the light is now moving with a radial (along the line of sight, toward or away) velocity v rad ∆ λ = v r c λ 0 (14.1) where c = 3 . 00 × 10 8 m/s is the speed of light in vacuum. If the observed light has a wavelength that is longer than when a stationary light, then the Doppler shift ∆ λ is positive, and the source of light is moving away from us (which corresponds to a positive velocity). We say that the light is redshifted because red light has a wavelength longer than blue light. If the observed wavelength is smaller, then the Doppler shift ∆ λ is negative, and source of light is moving towards us (which corresponds to a negative velocity), and we say that the light is blueshifted. 82

In most cases, astronomers use the observed wavelength shift (the Doppler shift) to determine the radial velocity of the star or object they are observing. The radial speed can be found by solving Eq. 14.1 for v r ∆ λ = v r c λ 0 −→ v r = c ∆ λ λ 0 (14.2) Procedure Part 1 - Spectral Analysis Around the lab room, a variety of model “stars” will be set up for you to examine using a hand-held spectroscopes. Your lab instructors will explain how to use these at the start of the lab. 1. For each “star”, determine whether the “star” exhibits a continuous spectrum or an emission spectrum. We have no sources that exhibit absorption spectra 1 . 2. If the “star” exhibits an emission spectrum, measure the wavelength of the lines (colors) and identify the composition of the “star”. You should record the overall appearance of each of these stars, and (at least) the brightest 5 spectral lines. For each line, record the color and wavelength of the line. 3. If the “star” has a continuous spectrum, compare it to the other “stars” with a con- tinuous spectrum and decide which has the highest temperature and which has the lowest. Cooler objects will appear redder and hotter objects will appear bluer since Wien’s law states that the wavelength of peak emission is inversely proportional to the temperature of the object. It may be easier to do this without the spectroscope and just use naked eye observations of apparent color. There may be two different types of spectroscopes available to you, which have slightly differ- ent scales (from which you determine the wavelength of emission lines). If both spectroscopes are available, you can use either one. Try both and see which you like best! Black Spectroscopes These spectroscopes measure in hundreds of nanometers. For example, 6 is 600 nm. Try to measure to the nearest 5 nm. For example, if a line appears to be between 6.2 and 6.3 on the scale, record it as 625 nm. If the line is at 6.2, record it as 620 nm. Assume all your wavelength measurements are written to three significant figures. 1 Can you think of a reason why? It’s not because we don’t want to... 83

Blue Spectroscopes These spectroscopes have two scales. The top scales gives the energy of the light measured in eV (which is short-form for electron-volt). Don’t use this scale. Instead, use the bottom scale which gives the wavelength in nanometers. Each tick line is a step of 5 nm, and the longer lines refer to a step of 10 nm. That is, the longer tick lines refer to a value like 520 nm or 530 nm, whereas the short tick lines refer to a value like 525 nm. Note that the short wavelengths are on the right, and the long wavelengths are on the left. Assume all your wavelength measurements are written to three significant figures. Do not rotate the disc that is inserted into the region where you look into the spectroscope. If you can’t see anything when you look at the classroom lights, or what you see isn’t clear please ask your lab instructor for help. Part 2 - Doppler Effect In this section, you will use the spectrum of the “star” that you identified the composition as hydrogen. If there are two (or more) stars that correspond to hydrogen, just choose one of the “stars” for this part. The actual wavelength of the red emission line from a cloud of hydrogen at rest is 656.285 nm. You will assume that the difference between this wavelength and your measured wavelength is because the “star” is moving either directly towards you or directly away from you, and calculate this velocity. 84