Dark Matter Teacher Guide

8/19/2019 Dark Matter Teacher Guide

http://slidepdf.com/reader/full/dark-matter-teacher-guide 1/56

Teacher’s Guide

The Mystery ofDark Matter e r

y

r k

a t t e rT h e M y s t e r

y

D a r k a t t



© 2008 Perimeter Institute for Theoretical Physics

Contents

How to Access Supplementary Materials from the DVD-ROM

Instructions for Microsoft Windows XP

1. Insert the Dark Matter DVD in your computer.2. If your usual DVD software automatically plays the

movie, close the DVD software.3. Click Start.4. Click My Computer.5. Right-click on the DVD icon called DARK .

6. Select Explore from the drop-down menu.7. When the Explore window opens, double-click

the TeacherGuide folder to access the les.

Instructions for Mac OS X

1. Insert the Dark Matter DVD into your computer.2. If your computer automatically plays the DVD, quit the DVD player.3. Double click the DARK DVD icon on the Desktop.4. Double click the TeacherGuide folder to open it and access the les.



Contents

A. Background Information p. 2

Perimeter Institute for Theoretical PhysicsIntroductionSuggested Ways to Use this PackageCurriculum Links

B. Dark Matter in a Nutshell p. 6

A one-page summary of the video’s contents.

C. Student Activities p. 8

All the student activities in this booklet are also included on the DVD-ROM in aneditable electronic form so that you can modify them if you wish.

Hands-on Demonstrations

These activities demonstrate the concept that we can use the orbital speed of a starto infer the mass of a galaxy within the radius of the star’s orbit. They are suitable foruse before, during, or after the video.

Student Worksheets

1. Dark Matter Concept Questions(Series of multiple-choice questions)

2. Video Worksheet(Suitable as a follow-up to the video)

3. Dark Matter within a Galaxy(Calculations on dark matter based on actual scientic data)

4. Advanced Worksheet(Collection of more challenging questions for stronger students)

5. Dark Matter Laboratory: Measuring Mass using Uniform Circular Motion(Uses centripetal motion apparatus to reinforce key concepts)

Dark Matter Laboratory: Teacher’s Notes

D. Supplementary Information p. 22

The video has been divided into seven chapters.This section provides extra information on each.

Video Chapters

1 Introduction2 Measuring the Mass of the Sun3 Measuring the Mass of a Galaxy: Orbital Method

4 Measuring the Mass of a Galaxy: Brightness Method5 Evidence from Einstein6 Failed Ideas About Dark Matter7 Current Theories of Dark Matter

Who Are the Physicists in the Video?

E. Solutions to Worksheets p. 38

F. Appendix p. 50

Raw Data for Orbital Speeds of Stars in Triangulum (M33)

1



2

Perimeter Institute for Theoretical Physics

Canada’s Perimeter Institute for Theoretical Physics is an independent, non-prot, scienticresearch and educational outreach organization where international scientists gather to push thelimits of our understanding of physical laws and explore new ideas about the very essence ofspace, time, matter, and information. The award-winning research centre provides amulti-disciplinary environment to foster research in the areas of Cosmology, Particle Physics,Quantum Foundations, Quantum Gravity, Quantum Information, Superstring Theory, and relatedareas. The Institute, located in Waterloo, Ontario, also provides a wide array of educationaloutreach activities for students, teachers, and members of the general public in order to sharethe joy of scientic research, discovery, and innovation. Additional information can be foundonline at www.perimeterinstitute.ca

Dr. Damian Pope

Perimeter Explorations

This series of in-class educational resources is designedto help teachers explain a range of important topics inphysics. Perimeter Explorations is the product of extensivecollaboration between international researchers, PerimeterInstitute’s outreach staff and experienced teachers. Eachmodule has been designed with both the expert and novice

teacher in mind and has been thoroughly testedin classrooms.

About your host

Dr. Damian Pope is the Senior Manager of ScienticOutreach at Perimeter Institute for Theoretical Physics in Waterloo, Ontario, Canada. He holds a PhD in theoreticalphysics from the University of Queensland, Australia,and his area of speciality is quantum physics. He also hasextensive experience in explaining the wonders of physics

to people of all ages and from all walks of life.

Background Information



3

For thousands of years, astronomy has been about light. Astronomers have studied light fromdistant stars, galaxies, planets, and other objects in space. However, over the last few decades,physicists have come to realize that celestial objects that emit any type of light make up only atiny fraction of the universe.

Introduction

The video begins by explaining how even though physicistscannot see dark matter, they can see its effects on thespeeds of stars within galaxies. It then describes existingtheories about what dark matter is made of and nishesby outlining various current experiments to detect darkmatter on Earth. These experiments are so crucial to ourunderstanding of the universe that the rst team to succeed will likely win the Nobel Prize.

This resource is the culmination of a year-long processinvolving teachers, physicists, students, and lm-industryprofessionals. All of the material presented has beenclassroom tested on multiple occasions. We trust you willnd this resource a useful addition to your seniorphysics course.

The rest of the universe is made of unseen materialthat does not emit, reect, or absorb any type ofelectromagnetic radiation. This “dark matter” dominatesgalaxies, making up about 90% of the mass of every galaxyin the universe. Without it, galaxies (including our own) would be inherently unstable and would rapidly fall apart. As well as being one of the most important topics inphysics today, the subject of dark matter is also veryaccessible. This package introduces dark matter using theconcepts of universal gravitation and uniform circularmotion. It is aimed at senior high school students and theonly equations needed to understand the material presentedare



4

Background Information

Suggested Ways to Use this Package

In designing this package, we have tried toprovide you with exibility and choice. Inaddition to a classroom video and a seriesof hands-on demonstrations, the packageincludes ve student worksheets. Weencourage you to choose those that best

suit your needs and to modify any of theworksheets as you see t (editable electroniccopies can be found on the DVD-ROM).

Outline for One PeriodHands-on Demonstrations (5-10 minutes)Video (25 minutes)Dark Matter Concept Questions Worksheet (15-20 minutes)Homework: Video Worksheet

Outline for Two PeriodsFirst PeriodDark Matter Laboratory

Second PeriodHands-on Demonstrations (5-10 minutes)Video (25 minutes)Dark Matter Concept Questions Worksheet (15-20 minutes)Homework: Dark Matter within a Galaxy Worksheet



5

Topic Connection to dark matter Relevant materials

Uniform circularmotion

One of the strongest pieces of evidencefor dark matter comes from the orbitalspeeds of stars in uniformcircular motion.

Video: Chapters 1, 2, & 3Worksheet 2: Questions 1, 2, & 3Worksheets 3 & 5

Universal

gravitation

As we cannot see dark matter, we only

know of its existence via itsgravitational effects.

Video: Chapters 1, 2, & 3

Worksheet 2: Questions 1, 2, & 3Worksheet 3

Two-dimensionalcollisions

A number of experiments are underwayworldwide to try to detect dark matterparticles in rare elastic collisions withatomic nuclei.

Video: Chapter 7Worksheet 2: Question 4Worksheet 4: Question B

Nature of scienceand technology,scientic process

There are a number of competingtheories about what dark matter ismade of. Some physicists do not eventhink dark matter exists.

Video: Chapter 7Worksheet 2: Question 5

Mathematical

relationships

The relationship between the speed of

a star and the mass of the galaxy withinthe radius of its orbit is

Physicists use this relationship to helpinfer the existence of dark matter.

Video: Chapters 2 & 3

Hands-on demonstrations

Curriculum Links



6

B.DarkMatterin aNutshell



7

B. Dark Matter in a Nutshell

• In 1967, Vera Rubin observed that starswithin the Andromeda galaxy hadhigher-than-expected orbital speeds.

• Physicists have also observed the samephenomenon in the nearby Triangulum galaxy.

• By measuring the orbital speeds of stars withinTriangulum and using the formula

physicists have calculated that the mass of thisgalaxy within a radius of r = 4.0 × 1020 mis equivalent to 46 billion Suns.

• However, by measuring the brightness of

Triangulum, they have also calculated that itsmass within a radius of r = 4.0 × 1020 m isequivalent to 7 billion Suns.

• The discrepancy between these two resultsimplies that there is 39 billion Suns’ of unseenmass within Triangulum.

• This unseen mass is called “dark matter”.

• Physicists have observed many other galaxiesand most are now convinced that, on average,dark matter accounts for 90% of the mass ofevery single galaxy in the universe.

• Physicists also have independent evidence forthe existence of dark matter from observationsof distorted images of distant galaxies(gravitational lensing).

• Although no one knows what dark matter is

made of, physicists currently have a numberof theories.

• One of the earliest theories of dark matter wasthat it consists entirely of compact celestialobjects such as planets, dwarf stars, andblackholes. Careful observations have ruledout this theory.

• Most physicists today think that dark matter ismade of a type of subatomic particle that, todate, has never been detected in the laboratory

The two leading candidates are weaklyinteracting massive particles (WIMPs)and axions.

• Numerous experiments that are trying todetect one of these particles are currentlyunderway worldwide.

• As physicists do not yet know what dark matteris made of, they do not know the compositionof a large fraction of the universe.



8

C.Student Activities



9

Hands-on Demonstrations

Introduction

The series of demonstrations on the following

pages illustrates the principles underlying howphysicists use the orbital speeds of stars tomeasure the mass of a galaxy within acertain radius.

Although dark matter is a very abstract topic, some of the basicconcepts underlying the evidence for its existence can be explainedvia concrete, hands-on demonstrations. These demonstrations centrearound the centripetal motion apparatus illustrated in Figure 1.

Materials

rubber stopper stringglass or plastic tubepaper clipwashers

The apparatus consists of a rubber stopper tied to the end of a pieceof string. The string passes through a narrow glass or plastic tubeand a number of washers are attached to the other end of the string.

A paper clip (or other similar object) is attached part way along thestring and is used to x the radius of the stopper’s orbit, as describedbelow.

To use the apparatus, hold the tube vertically, attach the paper clip tothe desired location along the string, and swing the stopper so that itmoves in a horizontal circle. Changing the speed of the stopper willalter its orbital radius and thus the height of the paper clip. Adjust thestopper’s speed so that the paper clip sits just below the bottom ofthe tube, as in Figure 2. The orbital radius and speed of the stopperare now constant.

Figure 1 Centripetal motion apparatus.

Figure 2 Set the orbital radius by adjust-

ing the speed of the stopper so that the

paper clip lies just below the bottom of

the tube.



10

In Step 5, the stopper will orbit with a noticeably faster orbital speedthan in Step 2. This is because, for constant orbital radius, themathematical relationship between the stopper’s orbital speed v

S and

the mass of the washers, mw, is

(See Appendix A on the DVD-ROM for details). Thus, the greater massof washers in Step 5 leads to a higher orbital speed.

Surprisingly, the relationship in the previous equation is of the sameform as the one between the orbital speed v of a star in a spiral galaxy

orbiting at radius r and the mass Mgal within r :

Thus, we can use this demonstration to illustrate how physicists usethe orbital speeds of stars to infer the masses of galaxies within certain

radii.

Student Activities

Effect of Dark Matter

on the Orbital Speeds of Stars

Demonstration 1

1. Place a number of washers on the end of the string and use apaper clip to x the orbital radius. The washers represent the inner

stars in a galaxy.

2. Swing the stopper around in a horizontal circle at constant orbitalspeed. Make sure the paper clip sits just below the bottom of thetube and that students observe the stopper’s orbital speed. Thestopper represents an outer star in a galaxy.

3. Tape some more washers together with black tape to representdark matter.

4. Attach the “dark matter” to the original washers, as shown inFigure 3, thus increasing the mass.

5. Again, swing the stopper around in a horizontal circle with thesame orbital radius as before.

This demonstration illustrates how a star’s orbital speed depends on the mass around whichit orbits. Thus, measuring a star’s orbital speed can reveal the presence of unseen mass.

Figure 3 Increasing the mass of the washers



11

Concealing the Mass

Demonstration 2 Demonstration 3

This demonstration allows students to directlycompare the orbital speeds for two differentmasses of washers.

1. Make two centripetal motion apparatuses with identical orbitalradii. One should have three times the mass of washers asthe other.

2. Have two students each swing one of the stoppers around in ahorizontal circle at the same time so that the rest of the class cancompare the orbital speeds in the two apparatuses. The studentsshould see that the speed depends on the mass of the washers.

3. Use plastic cups (or other similar objects such as envelopes) tocover both sets of washers so they are hidden.

4. Have the two students leave the classroom. While they are outside,have them either swap their apparatuses or not. Tell the rest of theclass about this, so that they will not know who has which mass.

5. Have the two students return and get them to swing the stoppersaround in horizontal circles with identical orbital radii. Ask the classto guess who has the larger mass of washers. The stopper thatmoves with the fastest orbital speed should be attached to thegreater mass.

For a more vivid demonstration, we can modifythe centripetal motion apparatus to make itresemble the real-life situation (i.e., visible starsand dark matter) more closely.

1. Tape a small light-emitting diode (LED) connected to a 1.5 Vcalculator or watch battery to the top of the stopper so that the LEDshines ( Figure 4 ). The LED represents an outer star in a galaxy.

2. Attach some washers to the other end of the string, turn out thelights, and get the LED to move in a horizontal circle at constantorbital speed. The LED represents a star orbiting in a hypotheticalgalaxy without dark matter.

3. Turn the lights back on, and increase the central mass by attachingextra washers taped together with black tape. Turn out the lights andagain get the LED to move in a horizontal circle at constant orbital

speed with the same orbital radius as before. The LED’s orbitalspeed will be noticeably higher. The LED now represents an outerstar orbiting at a much higher orbital speed due to the presence ofdark matter.

Demonstration in the Dark

Figure 4 A light-emitting diode taped to the end of the

rubber stopper represents an outer star in a galaxy.



12

Student Activities

Student Worksheets

Editable electronic copies of all worksheets arelocated on the DVD-ROM so that you can modifythem if you wish.

Worksheet 1 Dark Matter Concept Questions

Worksheet 2 Video Worksheet

Worksheet 3 Dark Matter within a Galaxy

Worksheet 4 Advanced Worksheet

Worksheet 5 Dark Matter Lab



13

Worksheet 1

Dark Matter Concept Questions

Name:

1. Which of the above graphs best shows the v against r relationship for planets orbiting the Sun?

a) A b) B c) C d) D

2. Which of the above graphs best shows the v against r relationship observed for stars orbiting the centre ofa galaxy?

a) A b) B c) C d) D

3.When astronomers measure the mass of the galaxy Triangulum using the Brightness Method the resultthey get is much less than when they measure the mass using the Orbital Method. This fact can best beexplained by the fact that

a) the stars in Triangulum are made of lighter elements than those in the Sun. b) the Orbital Method underestimates the mass of Triangulum. c) Triangulum contains a large amount of unseen mass. d) Triangulum contains only mass that emits visible light.

4. Observations that indicate the presence of dark matter have been made in

a) only the Andromeda and Triangulum galaxies. b) only the Andromeda, Triangulum, and Milky Way galaxies. c) every galaxy that has been examined for dark matter. d) many, but not all, galaxies that have been examined for dark matter.

5. Evidence for dark matter comes from observations of

a) the orbit of the Moon around Earth and gravitational lensing.b) the orbits of stars and gravitational lensing.

c) the orbit of the Moon around Earth and the orbits of stars. d) the orbits of stars, the orbit of the Moon around Earth, and gravitational lensing.

6. Dark matter is called “dark” because it

a) only emits high-energy radiation such as X-rays and gamma rays. b) only emits low-energy radiation such as microwaves and radio waves. c) reects light but does not emit other radiation like stars do. d) does not emit or reect any type of radiation or light.

7. Most physicists think that most dark matter is made of

a) WIMPs or axions. b) brown dwarf stars. c) black holes or planets. d) stars like the Sun.

8. Which of the following is true?

a) Physicists know exactly what dark matter is made of. b) Physicists have no idea what dark matter is made of. c) Only some physicists know what dark matter is made of. d) Physicists have some ideas about dark matter, which they are currently testing by experiments.

v v v v

r r r r



14

4. In rural Minnesota, U.S.A., there is a dark matter detector knownas the Cryogenic Dark Matter Search (CDMS) located 700 munderground in an abandoned mine. It involves a number of 250 gcrystals of germanium (Ge) that are cooled down to just aboveabsolute zero (–273o C).

According to the weakly interacting massive particle (WIMP) theoryof dark matter, billions of WIMPs from outer space are raining downon Earth each second. Although they typically pass through solidobjects as if they are not there, there is a very small chance that aWIMP will collide with a nucleus of an atom within any material ithappens to pass through.

As a result, at CDMS there is a very small probability that a WIMPwill collide with the nucleus of a germanium atom within thedetector, as illustrated below:

a) In the gure above, a WIMP with a mass of 1.07 × 10–25 kg andan initial speed of 230 km/s collides with a stationary germaniumnucleus with a mass of 1.19 × 10–25 kg. If the WIMP is deectedand its speed is reduced to 75 km/s, use conservation of energyto determine how much energy is transferred to the nucleus. (It isthis energy that scientists must somehow detect.) In calculatingyour answer, assume that the collision is elastic.

b) How many times smaller is this energy than the energy requiredto lift a grain of sand by one millimetre (1 x 10–7 J

)?

5. A friend sends you an email that expresses skepticism about theexistence of dark matter. It says: “I thought science was aboutobservation, and objects you can see? How can you say that darkmatter exists when no one can see it?” Write a ve to ten sentencereply describing the evidence for dark matter and defending thestance that something does not have to be visible in order tobe understood by science. In your reply, give an example fromeveryday life of something that exists but is not visible.

Student Activities

Worksheet 2

Video WorksheetUseful formulas

1. Mars orbits the Sun in uniform circular motion. The radius ofMars’ orbit is 2.28 × 1011 m and its orbital speed is 2.41 × 104 m/s.

a) Draw the free-body diagram for Mars and use it to derive anexpression for the mass of the Sun in terms of Mars’ orbitalspeed, the radius of its orbit, and the universalgravitational constant.

b) Use the expression derived in part a) to determine the mass ofthe Sun.

2. The plot below relates the orbital speed of the planets to the radiusof their orbits.

a) What is the force that keeps the planets in their orbits?

b) Why do the “outer” planets travel slower than the “inner” planets?

c) Rearrange your answer to 1a) to nd the equation for thegraph above.

3. Astronomers have studied galaxy UGC 128 for many years. Theyhave measured its brightness and calculated that the mass of starswithin a radius of 1.30 × 1021 m is 3.34 × 1040 kg. Stars orbiting atthis radius has been measured travelling at a speed of 1.30 × 10 5 m/s.What percentage of the mass within this radius is dark matter?

2.28 x 1011 m

Sun Mars

WIMP

WIMP Ge

Ge

230 km/s

7 5 k m

/ s

v

r



15

S p e e d ( × 1 0 5 m

/ s )

Orbital Radius (×1020 m)

1.00

2.00

1 2 3 4 5 6

Name:

Worksheet 3

Dark Matter within a Galaxy Useful formulas

Astronomers have analysedthe stars in the galaxy UGC11748. They found that mostof the stars lie within a radius

r = 1.64 × 1020 m and that thetotal mass within this radius is1.54 × 1041 kg, or 77.4 billiontimes the mass of the Sun.

It is expected that the stars thatlie outside this radius will orbitin the same way that planetsorbit the Sun. In this activity youwill analyse the motion of starslocated in the outer regions ofUGC 11748.

Orbital radiusof star(× 1020 m)

Measuredspeed(× 105 m/s)

Calculatedspeed(× 105 m/s)

Gravitationalmass(× 1041 kg)

MissingMass(%)

1.85 2.47 2.36 1.69 8.99

2.75 2.40 1.93

3.18 2.37 1.80

4.26 2.25 1.55

6.48 2.47 1.26

1. Use the values from the table above to plot measured speed againstorbital radius on the graph provided. Label this line “measured”.

2. a) For each orbital radius, calculate the speed expected if the only mass is the luminous mass of 1.54 × 1041 kg. Record your

answers in the “Calculated speed” column.

b) Show a sample calculation. c) Plot calculated speed against orbital radius on the graphprovided. Label the line “calculated”.

3. Compare the “measured” and “calculated” plots.Discuss a possible explanation for any differences.

4. a) Use the measured speeds to calculate the mass of the galaxycontained within each orbital radius. Record your answers in the“Gravitational mass” column.

b) Show a sample calculation.

5. For each orbital radius, calculate the difference between thegravitational mass within this radius and the total mass of the stars(1.54 x 1041 kg). Represent this difference as a percentage of thegravitational mass within the orbital radius. Record your answers inthe “Missing Mass” column.

6. Do your results support the following statement? “It is reasonable to expect that stars orbit around the gravitational

mass contained within the radius of their orbit in the same way thatplanets orbit around the Sun.” Discuss.

7.Explain the shape of your plot for measured speed against

orbital radius.



16

The angle (measured in radians) by which light from a distant star orgalaxy is bent by a mass M is given by the following formula

where G = 6.67 x 10-11 Nm2 /kg2, d is the closest the light comes to thecentre of the object, and c is the speed of light.

Student Activities

Worksheet 4 Advanced Worksheet

Name:

A. Gravitational Lensing

Some of the most convincing evidence for dark matter comes from a phenomenon known asgravitational lensing. This was rst predicted by Einstein in his theory of relativity. The theorypredicts that large masses in outer space, such as clusters of galaxies, bend light that travelsnear them. So as the light from a distant star passes by a large mass its path is distorted bygravity. Gravitational lensing was rst observed experimentally in 1919 when physicist Arthur

Eddington observed light from a distant star being bent by the Sun.

1. Given that the mass of the Sun is 1.99 × 1030 kg and its radius is6.96 × 108 m, calculate the angle of deection for light from a distantstar that passes very close to the Sun’s surface.

2. A ray of light that passes within a distance of 16 million lightyears from the centre of a cluster of galaxies is bent by an angle of2.0 x 10-5 radians. Use gravitational lensing to calculate the massof the cluster.

3. Order the following three scenarios according to the angle ofdeviation (from highest to lowest) for light that just passes by theedges of the clusters.

a) A cluster of galaxies with a mass of 1014

times the mass of theSun and a radius of 107 light years.b) A cluster of galaxies with a mass of 5 x 1014 times the mass of the

Sun and a radius of 3 x 106 light years.c) A cluster of galaxies with a mass of 2 x 1014 times the mass of the

Sun and a radius of 4 x 106 light years.

M

d

Where starappears to be

Actualposition of star



17

4. If dark matter is made of WIMPs then billions of these particlesfrom space are raining down on Earth each second. Although theytypically pass through solid objects as if they are not there, thereis a very small chance that a WIMP will collide with a nucleus ofan atom within any material it happens to pass through.

So, at CDMS there is a very small probability that a WIMP willcollide with the nucleus of a germanium atom in the detector.This collision would be elastic, and is illustrated at right.

You have been hired as a consultant by CDMS and some of thephysicists ask you for help with the following problem:

Suppose a WIMP has a mass of 1.07 × 10–25 kg and an initialspeed of 230 km/s. It collides with the nucleus of a stationarygermanium atom with a mass of 1.19 × 10–25 kg. The germaniumatom is deected with an energy of 10 keV (1 eV = 1.60 × 1019J).The physicists would like to know in which direction thegermanium atom travels after the collision.

Find the answer to this problem and write a clear, detailedexplanation of how you arrived at it so that you can send it tothe CDMS physicists.

C. Density of Dark Matter (Challenging)

5. The total mass of dark matter, Mdark, within a galaxy increaseslinearly with distance r from the centre of the galaxy, i.e.,

Assuming that dark matter is distributed in a sphericallysymmetric fashion, use this fact about the mass of dark matter towrite a proportionality statement (e.g., ) for the relationshipbetween the density of dark matter and the distance from thecentre of a galaxy.

B. “Seeing” Dark Matter on Earth: WIMP Collisions

One of the many experiments currently underway on Earth in the search for dark matter is locatedin rural Minnesota, U.S.A. It is 700 m underground in an abandoned mine and is called theCryogenic Dark Matter Search (CDMS). The experiment involves a number of 250 g crystals ofgermanium cooled down to just above absolute zero (–273o C) and is designed to detect darkmatter if it is made of weakly interacting massive particles (WIMPs). To date, the experiment has

not detected any WIMPs.

WIMP

WIMP Ge

Ge

230 km/s

Worksheet 4 Advanced Worksheet Cont’d

Name:



18

Student Activities

Worksheet 5

Dark Matter Lab

Measuring Mass using Uniform Circular Motion

An object moving at a constant speed in a circular path is accelerating (i.e., the direction ofthe velocity vector is constantly changing). This acceleration is caused by an unbalanced forceacting towards the centre of the circle (centripetal force). Any change in the unbalanced force willproduce a change in the orbital motion of the object.

PredictHow will the speed of an orbiting body change as the applied forceincreases, if we keep the orbital radius constant?

Materials

rubber stopper stringglass or plastic tube

Procedure

1. Measure and record the mass of (i) the stopper and (ii) all of thewashers combined.

2. Your teacher will show you how to construct the apparatus.

3. Set the radius of revolution of the stopper between 40 and 80 cm

by keeping the paper clip just below the bottom of the tube. Recordthe distance from the top of the tube to the middle of the stopper.

4. Attach eight washers to a second paper clip tied to the free endof the string. Spin the stopper in the horizontal plane, keeping thepaper clip suspended just below the bottom of the tube. Once youhave the stopper orbiting at a constant rate, record the time takenfor 10 cycles.

5. Increase the number of washers by two, keeping the radiusconstant. Record the time for another 10 cycles. Repeat until youhave results for at least ve different masses.

Application

You are given an object of unknown mass. Follow the proceduredescribed above and record the time taken for 10 cycles.

Analysis1. Use the geometry of a circular path to convert the period of motion

to linear speed for the stopper.

2. Plot the speed v of the stopper against the mass mW

of the washers.What relationship between speed and mass is suggested by theshape of the plot?

3. Replot the data using v 2 against mW. Calculate the slope of the line

(remembering to include the correct units).

4. Draw free-body diagrams for the washers and the stopper.

5. Use these free-body diagrams to derive an expression that relatesv 2 to m

W. The angle between the string and the horizontal should

be relatively small for all your results. Given this, let this angle equalzero in your calculation.

6. Use the expression derived in Step 5 to give a physical interpretationfor the slope of the plot of v 2 against m

W

. Compare the slope withthe value you would expect to get from the expression derived inStep 5.

7. Use the results to calculate the unknown mass. Compare youranswer to the value obtained using a balance.

Questions

1. Two students are spinning identical stoppers at equal orbital radii.One of the stoppers is moving noticeably faster than the other.What can you infer about the number of washers attached to thefaster stopper?

2. Earth orbits the Sun because of gravitational attraction. How couldyou use Earth’s orbital data to measure the mass of the Sun? Findthe relevant data and calculate the Sun’s mass.

3. The Sun orbits the centre of the Milky Way galaxy at a radius of7.6 kpc (1 parsec = 3.26 light years) and at a speed of 220 km/s.

Determine the mass of the Milky Way contained within theSun’s orbit.

4. Physicists estimate the mass of luminous matter in a galaxy bymeasuring the galaxy’s brightness. They have observed that starswithin many galaxies orbit around their galactic centres at speedshigher than expected. Using ideas from this lab, give an explanationfor these observations.

paper clip16 washersstopwatch

electronic balanceunknown mass



19

Worksheet 5

Dark Matter Lab

Teacher’s Notes

Roughly speaking, dark matter is matter that we can only detect via its gravitational interactionwith other objects. Some of the most compelling evidence for it comes from the analysis of starsthat are moving much faster than expected, based on the surrounding visible mass.

By introducing students to the mass–speed relationship in this activity,you give them the physics they need to explore one of the greatestunsolved problem in physics today.

Safety

This laboratory activity involves swinging objects around on a string,which is an inherently dangerous thing to do. If the string breaks or theknots come undone, the rubber stopper will be projected outwards.The rubber stopper can also hit the student holding it if sufcient careis not taken. Therefore

1. Always inspect the equipment before performing the lab.Replace any worn strings.

2. Have students wear safety goggles during the lab to prevent

eye injuries.3. Ensure students have sufcient room to do the lab without

endangering other students or themselves.

Introduction

In this activity, students will explore the relationship between the massof a number of washers and the orbital speed of a rubber stopper thatrotates around the washers. The activity demonstrates that we canuse the speed of an orbiting body to infer the magnitude of the massresponsible for the rotation. A series of questions then leads studentsto see how they can use this relationship to analyse astronomicalobjects and, ultimately, nd evidence for the existence of dark matter.

When the stopper is in orbital motion, it orbits below the top of thetube because of its mass. So the length of string between the stopperand the top of the tube deviates from the horizontal plane, as shown in Figure 5. The angle measures the extent of this deviation. If students correctly incorporate when analysing this laboratory, theywill end up with a somewhat complicated expression for v (the orbitalspeed of the stopper) in terms of m

W

(the mass of the washers) thatdoes not readily lend itself to being analysed (details can be found in

Appendix B on the DVD-ROM). This fact and the fact that is alwaysrelatively small are good reasons why students should let = 0 inSteps 5 and 6. This will allow them to ignore extraneous, complicatingdetails and help them to focus on the most important concepts. Background

Before they perform this activity, students will need to be familiar withthe following concepts:

• The circumference of a circle is given by 2π r , where r is the radius.• The term “period” describes the time taken for one cycle.• Centripetal motion is caused by an unbalanced force acting towards

the centre of a circle.• The unbalanced force that acts on planets is the force of gravity

and its magnitude and direction are given by Newton’s Universal Lawof Gravity.

Modifications

Depending on the particular emphasis in your course you may wish totry some of these modications:

• Do not weigh the washers. Have your students just discover aqualitative relationship between mass and speed.

• Have students use graphical techniques to nd the relationshipbetween v 2 and m

W, rather than telling them to plot v 2 against m

W.

• Have students compare results for different values of orbital radius toexplore how this quantity effects the orbital speed.

Figure 5 The stopper orbits below the

top of the tube when in orbital motion.



20

3.

To calculate the slope of the line:

4. mw represents the mass of the washers. Let m s represent the mass

of the stopper, T the magnitude of the tension in the string, g theacceleration due to gravity, and the angle at which the stringbetween the top of the tube and thestopper deviates from horizontal.

The free-body diagrams are as follows:

Analysis

Sample Data

Mass of rubber stopper = 12.4 gMass of 16 washers = 86.3 gLength of string from top of tube to middle of stopper = 62 cm

Unknown MassMass as measured on a balance = 54.0 gTime taken for 10 cycles = 7.68 sSpeed of stopper = 5.07 m/s

1. Sample Calculation

Let r represent the orbital radius of the stopper:

where P is the period of the stopper’s orbit.

2. The shape of the plot suggests a “square root” relationshipbetween v and m

W, i.e.,

Student Activities

Number of washers 8 10 12 14 16

Mass of washers (g) 43.2 53.9 64.7 75.5 86.3

Time for 10 cycles (s) 8.51 7.58 6.92 6.40 6.00

Speed of stopper (m/s) 4.60 5.14 5.63 6.08 6.50

Speed of stopper against mass of washers

8

6

4

2

0

0 20 40 60 80 100

mw (g)

v

( m / s )

60

40

20

0

0 20 40 60 80 100

mw (g )

Speed of stopper squared

against mass of washers

v 2

( m 2 / s 2 )

Dark Matter Lab

Teacher’s Notes Cont’d

T

F G =m

W g

washer

F G =m

S g

T

stopper



21

As is relatively small for all values of mW

, letting it equal zero in thecalculation above gives an accurate approximation. If the value of θ is incorporated the relationship between v 2 and m

W is considerably

more complicated and does not readily lend itself to analysis bystudents. More details on this can be found in Appendix B on theDVD-ROM.

6. From Equation 4, the slope of the plot of v 2 against mW

depends on

the mass of the stopper, the orbital radius, and the acceleration dueto gravity. Using the following data: m

S = 12.4 g , R = 62 cm, and

g = 9.8 m/s2 yields

Hence, the measured slope of 5.0 × 10 2 m 2 / (kgs 2 ) is 2% greaterthan the value predicted by the free-body diagrams.

7. Using the sample data of R=62 cm and v =5.07 m/s and the formula

Note: Final answer was obtained by using all of the digits present in intermediate

answers and not rounding off.

Thus, the mass of the unknown mass, as calculated from the stopper’s

orbit, is 53 g. This differs from the balance mass of 54.0 g by just 2%.

5. As the stopper orbits in the horizontal plane, the magnitude of thenet force acting on it equals the horizontal component of the tensionon the string. From the free-body diagram for the stopper, we cansee that this component equals T cos . As we let = 0, it reducesto T .

As the stopper is in uniform circular motion, its acceleration isgiven by

(1)

where R is the distance between the stopper and the top of the tube.

Substituting T and the expression for aC from Equation 1 into New-

ton’s second law of motion for the stopper yields

(2)

From the free-body diagram for the washers, we can seethat T = m

w

g and thus

(3)

Solving this equation for v 2 yields

(4)

Thus, the square of the speed of the stopper is proportional to themass of the washers, with the constant of proportionality beingequal to

(5)

Dark Matter Lab

Teacher’s Notes Cont’d



22

D.SupplementaryInformation



23

D. Supplementary Information The video has been divided into sevenchapters, each of which can be accessed viathe DVD menu. In this section, we provide anoverview of the video and outline the contentof each chapter.

Video Chapters

1 Introduction2 Measuring the Mass of the Sun3 Measuring the Mass of a Galaxy: Orbital Method4 Measuring the Mass of a Galaxy: Brightness Method5 Evidence from Einstein6 Failed Ideas About Dark Matter7 Current Theories of Dark Matter



24

This chapter of the video

introduces Vera Rubin and her measurements of the orbital speedsof stars in Andromeda.

explains that Rubin expected the orbital speeds of the outer starsto decrease the farther the stars were from the centre of Andromeda

(like the orbital speeds of planets in the Solar System).

Supplementary Information

Chapter 1

Introduction

Vera Rubin

shows that, instead, the orbital speeds remained constant withdistance and were higher than expected.

explains that, over time, Rubin’s observations led physicists to rethinkthe composition of the entire universe.

Historical Background

Vera Rubin is an astronomer from Washington, D.C. Between 1967 and 1969 she, withassistance from her colleague Kent Ford, used two telescopes (at Kitt Peak and LowellObservatory, both in the U.S.A.) to observe the orbital speeds of stars within the

Andromeda galaxy.

Although other astronomers had measured the orbitalspeeds of stars in various galaxies before, what madeRubin’s observations unique was the technology she used.Ford had recently built a highly sensitive spectrometer ableto collect data from the faint outer regions of galaxies. This allowed Rubin to observe phenomena previously

inaccessible to astronomers.

The orbital speeds of the planets in the Solar Systemdecrease the farther they are from the Sun and so Rubinexpected to see a similar decrease in the orbital speeds ofstars in Andromeda. Instead, the observed orbital speeds were as shown in Figure 6. In the outer region of the galaxy, the orbital speeds wereconstant at around 225 km/s as far out as Rubin couldmeasure. She also made observations in the inner region, which we do not include in the video. Here, she saw theorbital speeds increase linearly. We did not include thisdata as the pattern it exhibits is what we would expect inthe absence of dark matter. Thus, the inner data does notprovide compelling evidence for dark matter.



25

Fritz Zwicky

100

200

Observed speeds

Expected speeds

Distance from centre of galaxy

S p e e d ( k m / s )

Figure 6 Expected and observed orbital speeds for stars in Andromeda.

The expected speeds are based on the assumption that the vast majority of

Andromeda’s mass lies within the galaxy’s core where most of the stars are found.

Figure 7 The Andromeda galaxy. The stars within it orbit in uniform

circular motion.

Earlier Evidence for Dark Matter

Although Rubin’s observations in the 1960s led to darkmatter entering mainstream physics, tentative evidencefor dark matter existed much earlier. In 1933, Swiss-bornastronomer Fritz Zwicky examined the speeds of individual

galaxies within the Coma Cluster of galaxies and foundthat they were so high they exceeded the cluster’s escape velocity. This meant that the cluster should have beenunstable and falling apart, when clearly it was not.

Zwicky concluded that there must be a vast amountof unseen mass within the cluster holding it together via gravity. However, Zwicky’s data contained largeuncertainties and other physicists were skeptical.



26

Difference between Circular and Elliptical Orbits

As planets orbit the Sun in elliptical orbits, as in Figure 9, we can derive the Sun’s mass from Kepler’s third lawof motion:

(2.1)

where T is the planet’s period, a is the semi-major axisof the orbit, and M

S is the mass of the Sun. Rearranging

Equation 2.1 to solve for M S, we obtain

(2.2)

Jupiter’s orbit is very close to circular. It is so close thatthe difference in the result for M S we obtain by modelling Jupiter’s orbit as being circular instead of elliptical is just0.0064%. This corresponds to the difference obtained byusing the equation

(2.3)

instead of

(2.4)

Thus, using a circular orbit for Jupiter leads to a highlyaccurate result for the mass of the Sun. Furthermore, usingsuch an orbit for any other planet also yields an accurateresult for this mass.


shows how we can use Newton’s theory of universal gravitation tocalculate the mass of the Sun from the orbit of any planet.


Chapter 2

Measuring the Mass of the Sun

Elliptical Orbits

In calculating the mass of the Sun, we model Jupiter’s orbitaround the Sun as being circular, as in Figure 8. Althoughits orbit is actually elliptical, this fact makes little differenceto the result. Jupiter’s orbit is only slightly elliptical and thedifference between the masses calculated assuming circularand elliptical orbits is less than 0.01%.

The method for measuring mass discussed in this chapter of the video is commonly known as theDynamical Method. However, we use the term Orbital Method to emphasize its connection to theorbital speeds of stars.

Figure 9 Planet orbiting the Sun in an elliptical orbit. a is the

semi-major axis of the orbit.

a

Figure 8 We can calculate the mass of the Sun from the orbit of

any planet.



27

A second issue related to the Orbital Method and massdistribution is why the Orbital Method only measures the

mass of a galaxy within a certain radius? That is, why doesn’it also measure the mass of the galaxy farther out? Ananswer to this question can be found in Appendix C on theDVD-ROM.

Doppler Effect

The basic principle underlying how physicists measurethe speed of a star using the Doppler effect is the same asthat for measuring the speed of an ambulance from theDoppler shift of its siren, as in Figure 10. In the latter case, we measure the siren’s apparent frequency f and calculate itspeed v (towards or away from us) from knowing its actualfrequency f 0 and using the formula

(3.1)

where v m is the speed of sound in air and the ± sign isminus when the ambulance is moving towards us and plus when it is moving away.


explains how to calculate the mass of a galaxy within a given radiusfrom the orbital speeds and orbital radii of stars (the Orbital Method).

Distribution of Mass

An interesting feature of the Orbital Method is that weemployed the same equation used to calculate the mass of

the Sun

to calculate the mass of a galaxy within a certain radius. This leads to the following question: By using this equationfor a galaxy, aren’t we implicitly assuming that the mass ofa galaxy is concentrated at its centre, like the mass of theSolar System?

The answer to this question is that we do not make thisassumption. The equation

applies to any spherically symmetric distribution of mass,including both spread out and localized distributions. Thus,it applies to a galaxy in which mass is spread over alarge volume.

The reason the equation applies to any sphericallysymmetric distribution of mass is that such a distributionproduces a gravitational eld outside of its radius that isidentical to one that would have been produced if all of themass had been located at the centre of the distribution. Thus, the outside gravitational eld for such a mass is

independent of the distribution of mass and only dependson the total mass M contained within the distribution.

We implicitly use this fact when calculating the gravitationalattraction between any two extended bodies (e.g., the Sunand Earth) using the centre-to-centre distance as thedistance r in Newton’s law of universal gravitation (e.g.,between the Sun and Earth).

Chapter 3

Measuring the Mass of a Galaxy: Orbital Method

applies the Orbital Method to the Triangulum galaxy and calculates a mass of 46 billion Suns within a radius of 4.0 × 10 20 m.

Figure 10 We can use the Doppler effect to measure the speed of

an ambulance.



28

Physicists use a similar equation to nd the speeds of

stars within galaxies, but there are two differences. First,physicists do not directly measure the frequency shift oflight waves emitted by a star. Instead, they measure thefrequency shift of radio waves emitted by hydrogen gasorbiting at the same speed as the star. This allows them tocalculate the speed of the gas and thus the speed ofthe star.

The second difference is that physicists use a slightlydifferent equation from Equation 3.1 becauseelectromagnetic radiation travels at the speed of light andmust be handled using Einstein’s theory of relativity, asdetailed in Appendix D on the DVD-ROM.

Note that in the animation of Doppler shifted stars inthe video we have greatly exaggerated the colour changesassociated with the Doppler effect in order tohighlight them.

Doppler Effect and Orientations of Galaxies

It is important to note that the Doppler effect onlymeasures the speed of a star towards or away from us. Itdoes not measure any sideways or transverse motion withrespect to Earth.

If a galaxy is oriented so that we can only see the edge

of its orbital plane (an edge-on galaxy) some of its stars aremoving directly towards us and some are moving directlyaway from us, as in Figure 11. We can measure the orbitalspeeds of these stars using the Doppler effect.

However, if a galaxy’s orbital plane faces Earth directly(a face-on galaxy) then all of its stars move with purelytransverse velocities relative to Earth, as in Figure 12.In this case, both the stars and hydrogen gas do not exhibitany frequency shift and we cannot measure their speedsusing the Doppler effect. Only a small fraction of galaxiesare face-on.

Most galaxies lie somewhere in between the two extremesof being face-on or edge-on. They have orbital planestilted at some angle towards us. So, physicists can use theDoppler effect to measure a component of the velocitiesof their stars. Physicists then nd the total velocities bydetermining the angle of the tilt and adjusting the measuredDoppler speeds accordingly.

Where did the data for Triangulum come from?

The speed ( v = 123 km/s) and radius ( r = 4.0 ×1020

m) values for Triangulum used in the video came from recentmeasurements of this galaxy. The complete set of data is inthe Appendix in Section F.

Measuring the Orbital Radius of a Star

Another question that arises from this chapter of the videois how physicists measure the radius of the orbit of a starin a distant galaxy. Appendix E on the DVD-ROM answersthis question.

Figure 11 Edge-on galaxy

Figure 12 Face-on galaxy. For an edge-on galaxy, we can measure the

orbital speeds of stars via the Doppler effect. For a face-on galaxy, we

cannot.




29


outlines how to measure the mass of a galaxy from its brightness(the Brightness Method).

applies the Brightness Method to the Triangulum galaxy and obtains a mass 39 billion Suns less than the value obtained from theOrbital Method.

A more detailed breakdown of the steps involved in theBrightness Method is as follows:

1. First, physicists measure the distribution of light withina galaxy by looking at an image of it.

2. Next, they use this distribution to calculate the galaxy’stotal apparent brightness within a radius r by adding up

all the light within r .3. Then, they use knowledge of how far away the galaxyis to determine its total actual brightness (i.e. luminosity) within r .

4. Next, they estimate how much mass within the galaxy,on average, produces one unit of brightness. Thisquantity is a conversion factor from brightness tomass and can be estimated by a variety of means. Forexample, physicists sometimes use knowledge of therelative abundances of different types of stars within agalaxy, along with the brightness and mass of each type,to estimate the conversion factor.

5. Finally, they multiply the total actual brightness within

r by the conversion factor to obtain a result for mass within the radius r .

Chapter 4

Measuring the Mass of a Galaxy: Brightness Method

Seven Billion Suns

The Brightness Method result of 7 billion Suns is thecombined mass of all of the following within a radius of4.0 × 1020 m: (i) all of the stars, (ii) the hydrogen gas, and(iii) the helium gas. Helium is the second most abundantelement in the universe after hydrogen and there is asignicant amount of it within Triangulum.

How Accurate is the Brightness Method?

Like many calculations in astronomy, the BrightnessMethod contains an appreciable amount of uncertainty.However, this fact is not overly signicant because thediscrepancy between the Brightness and Orbital Methods isso large. Even if the actual mass of the stars and gas withina radius of 4.0 × 1020 m was double the value of 7 billionSuns (a 50% error), there would still be a discrepancy of 32billion Suns.

In addition to the numerical discrepancy between theOrbital and Brightness methods, the overall pattern of the

orbital speeds of the stars within galaxies (constant orbitalspeed with increasing distance) is fundamentally differentfrom the expected pattern (orbital speed declining withincreasing distance). Thus, even if the actual masses ofstars in distant galaxies were higher than current estimates,this would only have the effect of moving the plot forexpected orbital speed in Figure 6 upwards. It wouldnot alter the plot’s overall pattern so that it matched the

As might be expected, the method that physicists actually use to calculate a galaxy’s mass from itsbrightness is considerably more complicated than the approach presented in the video. However, thecore principle underlying the method used is that galaxies with greater mass tend to contain morestars and thus tend to be brighter.

shows how this discrepancy can be explained by the existence of avast amount of unseen mass called dark matter.

explains that all galaxies examined to date for dark matter have beenfound to contain vast quantities of dark matter.



30

Is Dark Matter the Same as Dark Energy?

Dark matter is distinct from dark energy, a recentlydiscovered unseen energy that many physicists also thinkmakes up a large fraction of the universe. Dark energy isanti-gravitational and is thought to be making the universeexpand at an ever-increasing rate.

Dark Matter within Triangulum

As stated in the video, there are 7 billion Suns of luminousmass in Triangulum within a radius of 4.0 × 1020 m and39 billion Suns of dark matter within the same radius. There are very few stars beyond this point, although smallquantities of hydrogen can be found farther out. Darkmatter extends far beyond 4.0 × 1020 m and so, overall,

there is much more than 39 billion Suns of dark matter within Triangulum.

observed plot. Thus, the stars and gas alone cannot explain

the observed speeds of stars, no matter how large theircombined mass is.

Mass-Luminosity Relationship

The graph shown in Figure 13 plots brightness(i.e., luminosity) against mass for individual stars. It showsthe well known mass–luminosity relationship. Althoughthe relationship shown is linear, we have used logarithmicscales on both axes for the purposes of simplication.

The actual relationship is

(4.1)

where M S and L

S are, respectively, the Sun’s mass and

luminosity. M and L are the mass and luminosity ofthe star in question. (Note that the exponent 4 is onlyapproximate and sometimes a different one is used,e.g., 3.5 or 3.9)

Density of Dark Matter

Measurements of orbital speed can be made at distancesmuch farther out than the outermost stars by lookingat faint concentrations of hydrogen gas. Physicists havefound that the speeds measured remain constant with

distance and are much higher than expected far beyond where the stars end.

From the shape of the resulting graph of orbital speedagainst orbital radius, physicists have determined that thetotal mass of dark matter, M

dark , within an orbital radius of

r increases linearly with r ,

(4.2)

As dark matter gravitationally attracts other dark matter,it tends to be found clumped together. As a result, in theimage of dark matter ( Figure 14 ) that appears near the end

of this chapter of the video, the density of dark matter isgreatest at the centre and gradually decreases as we movefarther out.

Figure 13 Mass-Luminosity Relationship

Figure 14 Representation of a cloud of dark matter surrounding a galaxy.




31


explains that large masses in outer space bend nearby rays of light(gravitational lensing).

The idea that mass bends light that travels near it comes from Einstein’s theory of generalrelativity. In fact, Einstein rst achieved worldwide fame in 1919 because another physicist, ArthurEddington, observed light being bent by the Sun, conrming the existence of this phenomenon.

The amount by which even a large cluster of galaxiesbends light is typically only very small, much less than onedegree. However, in the video, we have exaggerated theeffect in order to highlight it.

Clusters of Galaxies

Gravitational lensing allows physicists to study dark matteron the scale of clusters of galaxies, that involve largenumbers of individual galaxies, as in Figure 15. They have

found that the ratio of dark matter to stars and hydrogengas on this scale is signicantly greater than inindividual galaxies.

Some gravitational lensing observations are difcult, ifnot impossible, to explain without dark matter and manyphysicists think that gravitational lensing provides thestrongest evidence for the existence of thiselusive material. Furthermore, as gravitational lensing is a feature ofEinstein’s theory of general relativity and not Newton’stheory of universal gravitation, it provides evidence for thepresence of dark matter that is independent of evidencefrom the orbital speeds of stars within galaxies.

Chapter 5

Evidence from Einstein

Albert Einstein

Figure 15 Gravitational lensing in a cluster of galaxies.

explains how we can use the amount of distortion we see in imagesof distant galaxies to infer the presence of dark matter within clustersof galaxies.



32


shows that the bulk of dark matter is not made of planets, browndwarf stars, or black holes.

Planets

The main problem with the idea that planets make upthe bulk of dark matter is that there would need to be somany of them. For example, there is 39 billion Suns ofdark matter in Triangulum within a radius of 4.0 × 1020m. Jupiter’s mass is one thousandth of the mass of the Sunand so we would need more than ten trillion Jupiter-like planets to account for all of this dark matter. Thiscorresponds to thousands of planets for every star within aradius of 4.0 × 1020 m, as in Figure 16. Given that our SolarSystem has only eight planets, this seems highly unlikely.

Brown Dwarfs and Black Holes

Two other dark matter candidates are brown dwarf stars(also known as brown dwarfs) and black holes, as inFigures 17 and 18. Both have mass but give off so littlelight that we cannot readily see them using telescopes. Inspite of this, physicists can detect their presence via small-scale gravitational lensing experiments. Instead of tryingto observe distortions in the images of entire galaxies, they

look for distortions in the images of individual stars. Thesedistortions are the temporary brightening of stars causedby dwarf stars, black holes, or other “dark” objects bendingnearby light and acting like a converging lens. Physicistshave found some unseen mass this way, but not anywherenear enough to account for all dark matter.

Chapter 6

Failed Ideas About Dark Matter

A second reason why it is unlikely that the bulk of darkmatter is made of black holes relates to the explosionscalled “supernovas” that accompany their creation.Supernovas occur when a very massive star reaches the endof its lifetime and collapses under gravity. This is followedby an incredibly bright explosion (a supernova) that spewsout vast quantities of a wide range of chemical elements,as in Figure 19. If the star’s mass prior to the supernovais greater than around 25 times the mass of the Sun, theforce of gravity in the collapse is so strong that it results ina black hole.

One of the earliest theories of dark matter (from the 1970s) was that it is made of known celestialobjects such as Jupiter-sized planets, brown dwarf stars, and black holes. These, and otherrelated objects, are known collectively as massive astrophysical compact halo objects(MACHOs) and so the theory that dark matter is made of them is called the MACHO theory ofdark matter.

Figure 16 If dark matter is made entirely of planets, there would need

to be thousands of planets for every star within a galaxy.




33

The elements created in a supernova emit characteristic

emission spectra and can be readily detected by physicists.So, the creation of a black hole leaves a highly visibletrace. If dark matter were made entirely of black holes,there would be vast quantities of the elements created ina supernova spread throughout the universe. However,current observations indicate that there are nowhere nearenough to support the idea that dark matter is composedentirely of black holes.

Visualizing a Black Hole

At various points in this chapter of the video, we showan image of a black hole surrounded by brightly colouredtrails, as in Figure 18. They represent electromagnetic

radiation emitted by nearby matter as it falls into the blackhole, a common occurrence. Hydrogen Gas

Another dark matter candidate is sparsely distributedhydrogen gas. Hydrogen is the universe’s most abundantelement and there are vast quantities of it within galaxies,as well as among them. When it is sparsely distributed, itcan be challenging to detect.

However, there is strong evidence that hydrogen gas(or anything else made of atoms) does not make up thebulk of dark matter. This evidence comes from the highly

successful theory of Big Bang Nucleosynthesis, whichallows physicists to estimate the total amount of mass inthe universe made up of any type of atom (the baryonic mass). The universe’s total baryonic mass is only one fthof the total mass of dark matter and so it seems that,at best, only a small fraction of dark matter is made ofhydrogen gas.

Neutrinos

Yet another theory about dark matter is that it is made ofneutrinos. These are tiny, very light subatomic particles thatpass through solid objects as if they were not there, makingthem very hard to detect. There are vast quantities of

neutrinos throughout the universe and so some researchersin the 1980s thought they might make up the bulk of darkmatter. Recently, however, physicists have been able toestimate the mass of the neutrino and have found that it istoo small to account for the majority of dark matter.

Figure 19 Supernova: an explosion that occurs at the end of the life

of a massive star.

Figure 17 Brown dwarf star

Figure 18 Black hole.



34


discusses the possibility that there is no dark matter and that, instead,we need to modify our existing laws of gravity.

describes how most physicists think that dark matter consists of an as-yet-undetected type of subatomic particle.

Planets

All of the dark matter theories mentioned in Chapter6 of the video involve objects that have been detectedexperimentally. Their failure suggests one of the twofollowing possibilities:

• dark matter is made of objects that have never beendetected in experiments.

• dark matter does not exist. Much of the evidence forit comes from effects related to gravity (e.g., the orbitalspeeds of stars, gravitational lensing). Thus, if our currentlaws of gravity do not apply on a galactic scale, thisevidence is undermined.

WIMPs

Having ruled out all experimentally detected types ofmatter as making up the bulk of dark matter, manyphysicists turned to undetected varieties. One of the mostpopular ideas is that dark matter is made of hypotheticalsubatomic particles called “weakly interacting massiveparticles” or WIMPs, as in Figure 20.

WIMPs are many times more massive than a protonand have no electric charge. Electromagnetic radiationis produced by charged particles, so since WIMPs are notcharged, they do not emit electromagnetic radiation ofany frequency and thus appear dark. Many physicistsare condent that dark matter is made of vast clouds of WIMPs travelling rapidly in all directions.

Axions

A second theory involving undetected particles is that darkmatter is made of hypothetical subatomic particles called“axions”. Axions are many times lighter than electrons

Chapter 7

Current Theories of Dark Matter

and have no electric charge. One of the main differencesbetween WIMPs and axions is their mass. Thus, thedifference between the two theories (WIMPs or axions) isthat dark matter is either made of a large number of lightparticles (axions) or a smaller number of heavierparticles (WIMPs). Searching for Dark Matter on Earth

Earth lies within the Milky Way galaxy which is dominatedby dark matter. This means that if dark matter is made of WIMPs or axions, billions of unseen particles are passing

through your body each second, as in Figure 21. Physicistsshould be able to detect a tiny fraction of these particles(if they exist) using highly sensitive experiments. Thus,numerous groups worldwide have set up a number ofsuch experiments, with some of the most promising onestaking place 2 km underground in a working nickel mine atSNOLAB in Sudbury, Ontario, Canada.

discusses the two leading candidates for this new particle, weakly interacting massive particles (WIMPs) and axions.

interviews a number of dark matter researchers on their opinions.

discusses some of the experiments worldwide trying to detect dark matter directly.

Figure 20 Many physicists think that dark matter is made of WIMPs.




35

PICASSO Dark Matter Experiment

One of the experiments at SNOLAB is the PICASSO(Project in Canada to Search for Supersymmetic Objects)experiment ( Figure 22 ), which is highlighted in the video.It consists of millions of tiny droplets of superheatedliquid Freon (C

4F

10 ) suspended in a gel. There is a very

small chance that a WIMP passing through the experiment will collide with a uorine nucleus within one of thedroplets. When this happens, energy is transferred to thedroplet, causing the liquid to vapourize and a tiny bubbleto form. The bubble then rapidly expands, sending out ashock wave that physicists detect using acoustic sensors.

ICE CUBE Dark Matter Experiment

Another dark matter experiment is located at the SouthPole. The ICECUBE experiment consists of a vast arrayof sensitive light detectors located in 1 km deep holesin the ice. If dark matter is made of WIMPs, then darkmatter that is gravitationally trapped within the Sun andEarth should, indirectly, cause light to hit the detectors in adistinctive pattern.

CERN and the LHC

Yet another experiment that might detect dark matter istaking place just outside Geneva, Switzerland at CERN, the world’s largest particle accelerator ( Figure 23 ). Using theLarge Hadron Collider (LHC), physicists hope to actually

create dark matter particles (WIMPs) via extremely high-energy collisions between subatomic particles. If theysucceed, this will provide evidence for the WIMP theoryof dark matter.

Modifying Newton

A small minority of physicists advocate a radical solutionto the mystery of dark matter: modifying Newton’s theoryof universal gravitation on the scale of a galaxy (or larger).One theory is called Modied Newtonian Dynamics(MOND) and it can explain the mass discrepancy betweenthe Orbital and Brightness Methods. MOND does thisby altering the relationship between the magnitude of thegravitational force F and distance r from

to

for very large distances.

Although MOND can explain the evidence supporting theexistence of dark matter within galaxies, it cannot explainthe evidence for it from gravitational lensing. In addition,changing a basic physical law is a very rare occurrence inphysics and most physicists do not believe that this is thesolution to the mystery of dark matter.

Conclusion

The race to be the rst to detect dark matter here on Earthis intense. Whoever succeeds rst will, for the rst timeever, have directly observed the particle that makes up, onaverage, 90% of the mass of every galaxy in the universe.

They are almost certain to win a Nobel Prize.

Figure 21 If dark matter is made of WIMPs or axions, then billions of

dark matter particles are passing through your body each second.

Figure 22 PICASSO dark matter experiment in Ontario, Canada.



36



37

Who are the Physicists in the Video?

What are the Four Main Pieces ofEvidence for Dark Matter?

At one point in this chapter of the video, Chris Jillings alludes to whathe views as the four primary pieces of evidence for dark matter.They are:

• the orbital speeds of the stars within galaxies• gravitational lensing• the large-scale structure of the universe• cosmic microwave background radiation

Professor Cliff Burgess is awell-known particle physicistat McMaster University andPerimeter Institute for TheoreticalPhysics. He has a broad rangeof interests that include stringtheory, cosmology, andparticle physics.

Dr. Claudia de Rham is apostdoctoral researcher at

McMaster University andPerimeter Institute for TheoreticalPhysics. She works onextra-dimensional theorieswithin cosmology.

Dr. Chris Jillings is anexperimental physicist workingon a dark matter experimentat SNOLAB called DEAP (Darkmatter Experiment with Argonand Pulse-shape-discrimination).

Dr. Justin Khoury is acosmologist at PerimeterInstitute for Theoretical Physicswho has researched, amongstmany other things, an alternativeto the standard big bang /inationary theory that saysthat the universe existed priorto the moment of the big bang(ekpyroptic universe).

Professor Joao Magueijo is aphysicist at Imperial College andis most well known for proposinga theory that says the speedof light was faster in the earlyuniverse (variable speed of light).

Professor Slava Mukhanov is aphysicist at Ludwig-MaximiliansUniversity of Munich who isextremely well-known as one ofthe pioneers of inationary theoryThis theory says that the universeexpanded by a vast amount for asmall fraction of a second duringthe early universe.

Associate Professor Ubi Wichoskis an experimental physicist

at Laurentian University. Afterworking in theoretical cosmologyand cosmic-ray physics, he isnow engaged in two undergroundexperiments: PICASSO (Projectin Canada to Search forSupersymmetric Objects)and EXO (EnrichedXenon Observatory).

Figure 23 The ATLAS detector at the Large Hadron

Collider (LHC) experiment at CERN



38

E.Solutions toWorksheets



39

E. Solutions to Worksheets Worksheet 1 Dark Matter Concept QuestionsWorksheet 2 Video WorksheetWorksheet 3 Dark Matter Within a GalaxyWorksheet 4 Advanced WorksheetWorksheet 5 Dark Matter Lab



40

Substituting this into the answer to 1a) we obtain

2.a) The Sun’s gravitational attraction keeps the planets in their orbits.

b) From the law of universal gravitation, the Sun’s gravitational elddecreases with increasing distance from the Sun. Therefore, theouter planets travel more slowly because they experience a weakergravitational eld and thus accelerate less and have lower orbitalspeeds, as dictated by the equation

c) Beginning with the answer to 1a)

rearrange to solve for v

This is the equation for the graph in the worksheet.

3. We are given the following information

Worksheet 1

1. b 2. a 3. c 4. c 5. b 6. d 7. a 8. d

Worksheet 2

1.a) Let F G,S represent the magnitude of the Sun’s gravitational force onMars. Also, let mM and v represent the mass and orbital speed ofMars, r the radius of Mars’ orbit around the Sun and MS the mass

of the Sun.

b) We are given the following information

Worksheet 1 Dark Matter Concept Questions

Worksheet 2 Video Worksheet

Solutions to Worksheets

Sun Mars

F G,S



41

The amount of dark matter Mdark is the difference between the total

mass Mgal and the luminous mass Mstars

We can calculate Mgal as follows

The amount of dark matter is

(Note: Final answer obtained by using all of the digits present in intermediate

answers and not rounding off.)

4. a) We are given the following information

As the collision is elastic, we can calculate the energy of thegermanium nucleus from the change in energy of the WIMP using theprinciple of conservation of kinetic energy.

Before the collision

After the collision

Therefore, the amount of energy transferred to the germaniumnucleus is

b) As we need 1 x 10-7 J of energy to lift a grain of sand by 1 mm

So, the amount of energy transferred to the germanium nucleus is

40 million times less than the energy needed to lift a grain of sand by1 mm.

Therefore, scientists need a highly sensitive detector if they hope toever detect a WIMP.

5. A good answer to this question might include points such as thefollowing:

• One of the main pieces of evidence for dark matter comes fromobservations of stars in galaxies. The stars are moving at higherthan expected speeds if galaxies only contain visible mass.

• There are many ways to detect objects other than by seeing them.Sight is only one of our ve senses.

• There are many examples of scientists coming to accept theexistence of things they could not see, eg. radio waves, atoms,magnetic elds.

Examples of things from everyday life we believe to existbut cannot see:

• Air: although we cannot see it, we can detect its movement(i.e., wind).

• X-rays: we cannot see them but they allow doctors and dentists toobtain images of various parts of the body.

• Microwaves: although we cannot see them, they heat our food inmicrowave ovens.



42

1. and 2.c)

2.b)

3. The “measured” plot is fairly level. The speeds do not change muchas the orbital radius increases. The “calculated” plot shows aninverse relationship. The speeds decrease as the orbitalradius increases.

To obtain the calculated speeds, we assumed that only visible mass

exerts a gravitational force on the stars. If there was unseen masswithin the galaxy, the visible mass would underestimate the totalmass. Thus, as there is a relationship between mass and speed, thiswould also mean that the calculated speeds would underestimatethe actual speeds (i.e., the measured speeds). So, the presence ofunseen mass within the galaxy can explain the difference betweenthe “measured” and “calculated” plots.

4.b)

5.b) Let Minner represent the mass within a radius of 1.64 x 1020 m

Thus, for r = 1.85 × 1020 m

Note: Final answer was obtained by using all of the digits present in intermediate

answers and not rounding off.

Worksheet 3

Dark Matter within a Galaxy

S p e e d ( × 1 0 5 m

/ s )

Measured Speed

Calculated Speed

Orbital Radius (×1020 m)

1.00

2.00

1 2 3 4 5 6




43

6. It is not reasonable to expect the stars to orbit like planets. As theresults in the “missing mass” column show, there is a great dealof dark matter spread throughout UGC 11748. Thus, this galaxy’smass is not concentrated near the galactic centre, in contrastto the way the mass of the Solar System is concentrated at itscentre. As a result, we would not expect the orbital speeds of the

stars to follow the same pattern as the speeds of planets in theSolar System.

For spherically symmetric distributions of mass, a spread-outdistribution produces a gravitational eld outside of the distri-bution that is identical to the eld produced by a concentrateddistribution with the same total mass. However, the gravitationaleld produced by the spread-out distribution inside its distribu-tion is different. Thus, the gravitational forces acting on objects,such as stars, inside the distribution differ from the gravitationalforces inside the same region for the concentrated distribution. Asa result, there is a difference between the speeds of objects withinthis region for the two distributions.

7. The measured speeds remain constant as the orbital radiusincreases.The simplest explanation for this is to postulate theexistence of some kind of mass we cannot see (i.e., dark matter).

The fact that the mass difference increases with increasing orbitalradius beyond where most of the stars l ie indicates that dark mat-ter extends past where the luminous mass ends.

Orbitalradius of star(× 1020 m)

Measuredspeed(× 105 m/s)

Calculatedspeed(× 105 m/s)

Gravitationalmass(× 1041 kg)

MissingMass(%)

1.85 2.47 2.36 1.69 8.99

2.75 2.40 1.93 2.37 35.2

3.18 2.37 1.80 2.68 42.5

4.26 2.25 1.55 3.23 52.4

6.48 2.47 1.26 5.93 74.0



44


Worksheet 4

Advanced Worksheet

3. From the formula

the angle of deviation is proportional to the mass of the cluster

divided by the radius. Given this, For a) is proportional to 1014 /107 = 107. For b) is proportional to (5 × 1014 /3 × 106 ) = 1.67 × 108. For c) is proportional to (2 × 1014 /4 × 106 ) = 5.0 × 107.

Thus the order in terms of highest angle of deviation to lowest angle ofdeviation is b, c, a.

A. Gravitational Lensing

1. We can solve this problem using the formula

(1)

where G = 6.67 × 10-11 Nm2 /kg2, c = 3.00 × 108 m/s, d = 6.96 × 108 m,

and M = 1.99 × 1030

kg. Substituting these values into Equation 1,we obtain

2. One light year is the distance light travels in a year and so

1 light year

Using the equation

we have

This mass is equivalent to 5.1 × 1014 or 510 trillion times the mass ofthe Sun.



45

Substituting in values for mW ,v

Wand v

Ge, we arrive at

From conservation of momentum in the x direction

(2)

From conservation of momentum in the y direction

(3)

As we wish to nd , let us eliminate by nding expressionsfor and from Equations 2 and 3 and using theidentity

.

From Equation 2

Squaring both sides, we arrive at

(4)

From Equation 3

and so

(5)

B. “Seeing” Dark Matter on Earth 4. In solving this problem, let us label the angles of deection for the

WIMP and the germanium nucleus as follows

Using the principle of conservation of energy

(1)

The nal energy of the germanium atom is 10 keV which is equal to

Thus, the germanium atom’s nal speed is

Rearranging Equation 1 to solve for v W , we obtain

WIMP

WIMP Ge

Ge

230 km/s

θ

Worksheet 4

Advanced Worksheet Cont’d



46

Adding Equations 4 and 5 together, we obtain

(6)

Let us now solve this equation for cos and then nd a numerical value for this quantity. Expanding the right-hand side of Equation

6 yields

(7)

Substituting the following values into Equation 7

we obtain

Therefore, = 41.2 degrees.

Thus, a germanium atom with an energy of 10 keV is deected by anangle of 41.2 degrees downwards from horizontal (i.e., 41.2 degreesclockwise from the positive x -axis).


Worksheet 4




47

C. Density of Dark Matter

5. If M r then

M =

kr (1)

where k is some unknown constant.

Differentiating both sides of Equation 1 with respect to r , we obtain

(2)

dM is equal to the amount by which the mass within a radius r increases when r increases by an innitesimal amount dr .

Thus, as we assume that the mass is spherically symmetricallydistributed, dM is also equal to the mass within a thin hollowspherical shell of inner radius r and outer radius r + dr , and constantdensity .

(3)

Combining Equations 2 and 3, we obtain

The associated proportionality statement is thus

Worksheet 4




48

One light year is the distance light travels in a year: 9.46 x 1015 mThus,

Therefore, r sun

= 2.3 x 1020 m.

Substituting this result into the expression above for M

galaxy

4. The simplest explanation for the higher-than-expected orbitalspeeds observed is that the mass within a galaxy that exerts agravitational force on the stars is greater than expected. Thus, itseems that physicists’ calculations of mass within galaxies based ontheir brightness are missing something (i.e., there is mass that theycannot see that has not been included in their calculations).

1. If the objects are identical and the strings are the same length, wecan infer that the stopper that is moving faster must have a greatermass (i.e., more washers) attached to its string.

2. Earth orbits the Sun due to the gravitational attraction of the Sun onEarth. The centripetal force F

C needed to keep Earth in orbit must be

equal to the force of gravity F G exerted on Earth by the Sun.

3. The Sun is held in its orbit by the gravitational force produced by themass of the Milky Way galaxy contained within its orbit. Equating thecentripetal and gravitational forces acting on the Sun

Worksheet 5

Dark Matter Lab





49



50

50

F. Appendix



51

Raw data for Orbital Speeds of

Stars in Triangulum (M33)

Reference: Corbelli, E. and Salucci, P.: “The Extended Rotation Curveand the Dark Matter Halo of M33” Monthly Notices of the Royal

Astronomical Society, Vol. 311, No. 2, January 2000, pp. 441–7.

Radius (m) Speed (km/s)

1.23 × 1019 37.0

2.47 × 1019 54.7

3.70 × 1019 67.5

4.94 × 1019 79.1

6.17 × 1019 85.3

7.41 × 1019 89.1

8.64 × 1019 93.1

9.87 × 1019 96.5

1.11 × 1020 99.6

1.23 × 1020 101

1.36 × 1020 103

1.73 × 1020 106

2.11 × 1020 109

2.49 × 1020 117

2.86 × 1020 119

3.24 × 1020 118

3.61 × 1020 120

3.98 × 1020 123

4.38 × 1020 131

4.75 × 1020 136



52

“The Mystery of Dark Matter”Teacher’s Guide Credits

Complete scientic, educational, production,and funding credits can be viewed on the DVD.

Author

Dr. Damian PopeSenior Manager of Scientic Outreach


Educational Advisors / Associate Writers

Dave FishSir John A. Macdonald Secondary School

Waterloo, Ontario

Roberta TevlinDanforth Collegiate and Technical Institute

Toronto, Ontario

Michael BurnsWaterloo Collegiate Institute

Waterloo, Ontario

Tim LangfordNewtonbrook Secondary School

Toronto, Ontario

Alan LinvilleW. P. Wagner School of Science and Technology

Edmonton, Alberta

Brent McDonoughHoly Trinity Catholic High School

Edmonton, Alberta

Producers

Peter ConradGamut Moving Pictures Inc.

Dr. Damian Pope

John MatlockDirector of External Relations and Outreach


Designer

Jeff Watkins

Special thanks to

Professor Stacy McGaughDepartment of Astronomy, University of Maryland, College Park, U.S.A.

for data in Worksheet 3 and assistance with the detailed description of the BrightnessMethod in Chapter 3 of the Supplementary Information.

Image Credits

Helix Nebula, pp. 6-7

NASA, ESA, C.R. O’Dell (Vanderbilt University),

M. Meixner and P. McCullough (STScI)

Crab Nebula, p. 22-23

NASA, ESA, J. Hester and A. Loll (Arizona State University)

Vera Rubin, p. 24

Vera Rubin and Mark Godfrey

Fritz Zwicky, p. 25

The Caltech Institute Archives

Sombrero Galaxy M104, p. 28

NASA and The Hubble Heritage Team (STScI / AURA)

Whirlpool Galaxy M51, p. 28

NASA, ESA, S. Beckwith (STScI),

and The Hubble Heritage Team (STScI / AURA)

Albert Einstein, p. 31

Bettman / CORBIS

Abell 2218, p. 31

NASA, ESA, Andrew Fruchter and the ERO Team [Sylvia Baggett STScI), Richard Hook

(ST-ECF), Zoltan Levay (STScI)] (STScI)

ATLAS detector, p. 36

The ATLAS detector at CERN

Milky Way galaxy, 38-39

NASA, JPL-Caltech, S. Stolovy (SSC/Caltech)

Made in CanadaCopyright 2008 Perimeter Institute for Theoretical Physics



For information on other educational outreach initiatives, please visit

www.perimeterinstitute.ca


31 Caroline Street North

Waterloo, Ontario, Canada N2L 2Y5Tel: (519) 569-7600 Fax: (519) 569-7611

T h e M y D

Dark Matter Teacher Guide

Documents

Transcript of Dark Matter Teacher Guide