the birth and evolution of planetary systems

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objects in the Solar System. It is important for the students to realize that you can determine many of the properties of a planet by knowing its mass, size, and distance from the Sun.
Although planetary scientists are confident they have the big picture of Solar System formation correct, the details are still in question. Questions remain concerning Jupiter’s formation. It might not have been able to form via accretion but may have formed similarly to the Sun. To complicate matters further, Uranus and Neptune appear to be too large to have formed at their current posi- tions in the Solar System. Additionally, more than one hundred large extrasolar planets have been discovered in odd orbits about other stars, hinting that perhaps our Solar System is not representative of solar systems throughout the galaxy. As you teach this chapter, be sure to remind your students that the story being told is still incomplete and probably even partially incorrect. It is, however, a great demonstration of science in action, illus- trating how our understanding of the universe changes and grows as technology provides us with better observations and more data. It also is very exciting to be able to teach your students about other worlds and the fact that even more are being found as you read this and speak to them.
DiScuSSion PointS • Based on the amount of material near the Sun, how
likely is it that the Sun has siblings that formed from the same parental molecular cloud? Discuss with students the possibility of stellar siblings of any kind.
• Have students use the equation for spin angular momentum to calculate the values for the Sun, Jupiter, Saturn, Neptune, Earth, Pluto, and a typical molecular cloud. Discuss these in the context of conservation of angular momentum.
• Ask students, if they had to choose just one technique among the leading choices for finding extrasolar planets that might be as habitable as Earth, which one they would pick. What makes that method advantageous over the others?
Ideas about the origins of the Sun, the Moon, and Earth are older than written history. Greek and Roman mythol- ogy, as well as creation myths of the Bible, represent some of humanity’s earliest attempts to explain how the heav- ens and Earth were created. Thousands of years and the scientific revolution have ultimately debunked many ancient creation stories, but our new theories of solar system formation are still relatively in their infancy.
The nebular model of solar system formation was first proposed by Immanuel Kant in 1755. It has undergone significant revision in the last 250 years, but the details of the pro cess remain elusive. In broad strokes, our current understanding suggests that the Solar System formed from a collapsing cloud of interstellar gas and dust. Most of the infalling material fell to the center of the cloud, where it became the proto- Sun, but a significant fraction of the cloud’s mass was flattened into a disk- shaped nebula surrounding the proto- Sun. Because of the higher temperatures in the inner nebula, only relatively rare refractory materials (rock and metal) condensed from the nebula. However, in the outer nebula, abundant volatile materials also condensed as ices. Rock and metal particles in the inner nebula accreted into small protoplanetary seeds that gave birth to terrestrial planets. In the outer nebula, ices dominated the larger protoplanetary seeds that became the Jovian planets. All of the protoplanets captured hydrogen and helium atmospheres from the nebula; but when nuclear fusion ignited the Sun, powerful solar winds cleared the nebula of loose material and stripped the small terrestrial planets of their atmospheres. Meanwhile, because they were larger and farther from the Sun, the Jovian planets retained their atmospheres despite buffeting from the solar winds. After the Sun cleared the nebula, protoplanets on crossing orbits swept up any remaining material and collided with each other, ultimately growing to the full- sized planets we see today. Eventually the volcanically active terrestrial planets (aided by comet impacts that delivered volatile material from the outer Solar System) outgassed secondary atmospheres. As implied from the above statements, the envi- ronment created by the Sun dictated a great deal of the properties for each of the planets, as well as the other
C h a p te r 7
The Birth and Evolution of Planetary Systems
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2 Chapter 7 The Birth and Evolution of Planetary Systems
DEmonStrationS anD activitiES Demonstration: Ranking Task for Formation of Planetary Systems
Note to Instructors Ranking tasks are excellent means of helping students think about the progression of events, whether it is in space, as in sizes of objects, or in time, as in which came first, second, and so on. This ranking task asks students to think about the pro cess of the formation of a star and its planetary system. Rather than memorizing the steps, students should visualize the birth of a star and planets in steps that begin from a giant, cool, rotating molecular cloud and end with fully formed planets orbiting a genuine star. This is listed as a demonstration rather than an activity because students should be able to discuss, explain, and explore what they think is the correct order.
Learning Goals • Chapter learning goal(s) addressed: Summarize the
role that gravity, energy, and angular momentum play in the formation of stars and planets. Describe the modern theory of planetary system formation.
Required Materials • Cards or strips of paper containing the steps • Same steps attached to strips of magnetic sheeting
Instructions Make enough copies of the list of the stages in star /planet formation (included with the worksheets for this chapter) to hand out to teams of 3– 5 students. Cut the sheet into strips, shuffle the strips, and place sets of strips into envelopes. Students should work in teams to put the stages into the correct order, according to the current theory of planet formation. If events occur nearly simultaneously, then those stages should be put into the same pile. After students have had enough time to discuss the order of the stages and placed the strips in order from first to last, call on a team to read off their order, or have a representative of a team come to the board and place the magnetic strips in the order his or her team found.
Correct order according to the text (*coeval events):
1. Cloud of interstellar gas starts to collapse under the force of its own self- gravity.
2. *Gravitational potential energy of collapsing interstellar gas cloud is converted into heat and radiative energy.
• Discuss with students the significance of finding as many extrasolar planets as we have so far. Consider the discovery of Earth- like planets in habitable zones. What cultural impact does that have? What questions does it provoke? If there were life, would we be able to communicate with it?
ExPloration Exploration 1: Using the Transit Method to Detect Exoplanets
Note to Instructors This alternate Exploration is about searching for extrasolar planets (exoplanets) using transits. It pairs nicely with the Exploration in the textbook since Doppler shift is the way astronomers have found most of the known extrasolar planets to date, but transits— specifically with the Kepler mission— are how astronomers are currently searching for new extrasolar planets. Transits hold the best potential for finding Earth- like extrasolar planets.
Learning Goals • Explain thoroughly how transits can be used to dis-
cover extrasolar planets. • State what mea sur able pa ram e ters transit observations
can yield.
Required Materials • Computer with Internet access
Pre- Post- Assessment Question Assuming the mass of the parent star is known, what information about an extrasolar planet can be inferred from a mea sured light curve?
a. Orbital period b. Planet radius c. Orbital radius d. All of the above can be inferred. Answer: d
Post- Exploration Debriefing After completing this activity, have students visit NASA’s Kepler website to research the latest discoveries at http:// kepler.nasa.gov/Mission/discoveries.
Instructions Complete instructions are included on the student worksheet.
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Chapter 7 The Birth and Evolution of Planetary Systems 3
• Investigate the claim that a Jupiter- like planet could not exist at the location of 51 Pegasi b.
• Chapter learning goal(s) addressed: Describe how astronomers find planets around other stars and what those discoveries tell us about our own and other solar systems.
Required Materials • Scientific calculator
Instructions Detailed instructions for this activity are found on the student worksheet.
Activity 2: Surveying the Known Extrasolar Planets
SKill lEvEl: MEDiUM
Note to Instructors This activity has students compare the characteristics of known extrasolar planets to those of the planets in our Solar System by using an interactive table and histograms of all known extrasolar planets.
When introducing this activity, it is worth emphasizing that students are using real, cutting- edge astronomical data to answer scientific questions; students are imple- menting the scientific method in much the same way a professional scientist would.
Learning Goals • Survey the characteristics of known extrasolar
planets. • Compare the extrasolar planets to those in our Solar
System. • Use histograms and data tables to analyze the
discoveries. • Chapter learning goal(s) addressed: Describe how
astronomers find planets around other stars and what those discoveries tell us about our own and other solar systems.
Instructions Detailed instructions for performing this activity are in - cluded on the student worksheet. You may wish to pro- vide students with the scatterplot and two histograms shown on page 4 for the more difficult comparisons.
3. *Cloud of interstellar gas rotates faster and faster as it collapses because of conservation of angular momentum.
4. *Inner parts of flattening cloud begin to fall freely inward, raining down on growing object at the center.
5. Material makes its final inward plunge, landing on a thin, rotating accretion disk.
6. Motions push smaller grains of material back and forth past larger grains; smaller grains stick to larger grains.
7. Planetesimals form that are massive enough to have gravity that begins to attract nearby bodies.
8. Planetesimals continue to accrete material until they become large enough to be called planets.
Activity 1: 51 Pegasi: The Discovery of a New Planet
SKill lEvEl: MEDiUM
Note to Instructors This activity walks students through a review of the graphed 1995 discovery data of the first extrasolar planet orbiting a Sun- like star and interpretation of the radial velocity data for 51 Pegasi that led to that discovery. Stu- dents determine the planet’s orbital period and the radial velocity amplitude of the parent star and then use these data with equations derived from Kepler’s and Newton’s laws in order to find the radius of the orbit and a lower limit on the mass of the planet. Students then compare 51 Pegasi b to planets in our Solar System to gain an appre- ciation for how bizarre many of the known extrasolar planets are when compared to what we think of as conven- tional for planets.
The current working theory for how a gaseous planet got so close to its star is also addressed as students work through the same calculations that astronomers used immediately after the discovery. This activity gives students direct insight as to how science works when new and unexpected discoveries are made.
Learning Goals • Experience the basic steps involved in the pro cess of
discovering a planet orbiting another star. • Apply Kepler’s and Newton’s laws to find orbital and
physical characteristics of an exoplanet. • Compare an extrasolar planet to the more familiar
planets of our Solar System. • Summarize the current theory on the formation of the
planet orbiting 51 Pegasi. • State how we know the motion of a star.
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SummarY SElf- tESt 1. d, b, f, a, e, c 2. a, b, h, c, e, d, f, g 3. (c) Since angular momentum L = mvr, we see that
if the distance r is halved, the speed v must be doubled.
4. (b) Since the inner material was hotter than the outer Solar System, inner planets were not able to capture the large amounts of gas that are present in the Jovians.
5. (b) It is hard to guarantee that every star has planets, but it is also hard to argue that only a few stars have them.
6. (e) While not all methods are very efficient, they have all been used.
7. (b) The distance at which planets form from their central star controls their composition.
8. (a) Large planets close to the star tug the stars by a large amount with short periods, making them the easiest to detect.
9. (c) The disk instability theory proposes that a disk could break up into large planet- like components, which happens very rapidly, as opposed to the slow pro cess of accretion.
10. (b) Some books or instructors suggest that if a theory is wrong, it has to be thrown out. Strict adherence to that disproven theory has to be stopped, but the theory can be modified in a consistent and realistic way to include new observations.
truE/falSE anD multiPlE choicE 11. True: Aside from chemical bonds that hold together
rocks or chunks of metal, almost everything in the universe that is held together is done so by gravity.
12. True: Gravity is what causes the cloud to collapse and what allows larger planetesimals to grow. Angular momentum spins up the protostellar disk as it con- tracts, which increases the rate of collisions of small particles, allowing them to grow into planetesimals.
13. True: Volatile materials turn into liquids and vapors at moderate to high temperatures.
14. True: The same can be said for all stars, too! 15. True: The difference between microlensing and tran-
sits is that in the latter, the bright object is dimmed; while in the former, the object becomes brighter.
16. (a) This is an expression of the conservation of angular momentum.
17. (b) In the early Solar System, collisions between particles and clumps cause them to stick together and grow in size.
Te m
pe ra
tu re
Mass (MJupiter)
N um
be r
0 5 10 15 20 25 30 35
S em
i-m aj
or a
xi s
(A U
Mass (MJupiter)
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level. Another way to think about this is that conserved quantities are like a bank, because if you bor- row from the bank, you have to pay it back. Consider ice skaters, who have a fixed amount of energy in their bodies. If they skate very quickly (i.e., a lot of energy of motion), they must get that energy from somewhere, in this case, they convert some of their own biochemical energy. An object can always gain or lose mass, energy, momentum, or angular momentum, but it has to come from somewhere.
30. Spin is a conserved quantity, but it is mea sured by the product of how fast you rotate and how far away you are from your axis of rotation. Thus, when an ice skater spins slowly with her arms extended, she has a fixed amount of spin. As she brings her arms in toward her body, that distance gets smaller, so the rate of rotation has to increase to keep the product of the two constant. Thus, the skater spins much faster.
31. Just as a skater spins rapidly as she pulls her arms in toward her body, so would the Sun that formed at the center of the presolar nebula in early theories of solar system formation. These early models did not consider that angular momentum was transferred from the collapsing star to the accretion disk and so pre- dicted that the star that formed would spin so rapidly it would shred itself apart.
32. An accretion disk is the thin, rotating disk that forms as a gas cloud collapses on itself. It is out of this disk that the central star and planets form. These disks are also found in a wide variety of astrophysical environ- ments, as we shall later learn. Broadly speaking, this disk allows material to spiral into the star by convert- ing its orbital energy into heat. When material spins in a disk around a star, there is friction between the different particles, which heats up that material and allows it to give off energy. As particles give off that energy, they spiral deeper down the disk toward the star, eventually landing on the star.
33. Small grains of dust moving about the Sun in similar orbits randomly collide with each other at gentle speeds. These small grains stick to each other electro- statically to form increasingly larger seeds. As the seeds grow, gravity becomes more important. Rather than awaiting random collisions, the largest seeds begin to gravitationally attract nearby particles, and their growth rate increases exponentially. Eventually, the largest seeds reach planetesimal sizes. When planetesimals collide, the resulting accretion leaves planet- sized objects.
34. As I blow “dust bunnies” toward each other (provided they don’t fly apart), they tend to become tangled with each other and grow in size. While on Earth they
18. (d) One of the hallmarks of good science is demon- strated when the same conclusions are reached from many different avenues.
19. (a) As shown in Math Tools 7.1, spin angular momentum scales as L ∝ R2/P so if the radius is cut in half, the period must drop by a factor of 22 = 4.
20. (d) Angular momentum, whether spin or orbital, depends on all three factors listed.
21. (a) Spectroscopic radial- velocity mea sure ments pick up the Doppler shift of a star being tugged by its orbiting planet, much like a very small chihuahua tugs on its owner when being walked.
22. (b) Volatiles are easily evaporated or broken up with heat, and it was too hot in the inner Solar System for these compounds to be present.
23. (d) Remember that the “primary” atmosphere is the one the planet was formed with; the terrestrial planets all lost theirs soon after formation.
24. (d) Asteroids and comets are made almost exclusively of the pristine material out of which the Solar System formed.
25. (c) Jupiter- like planets cannot form close to their stars, according to our theories of planet formation; therefore, they must have migrated inward to be located close to their stars, as we currently find many of them to be.
thinking aBout thE concEPtS 26. The hydrogen and most helium atoms that make up
our Sun and Solar System all came from the Big Bang, while everything else was formed in previous genera- tions of stars.
27. Stellar astronomers looking at young stellar objects have noticed that many of them are located within dark, dusty disks, as shown in Figure 7.2. Planetary scientists looking at our own Solar System noticed that the planets all lie in a disk orbiting in the same direction, and by studying meteorites, they found evi- dence of larger objects being built up from smaller ones. This suggests that our Solar System formed from a disk of gas and dust. The two findings suggest a common origin for solar systems.
28. A protoplanetary disk is the spinning disk of gas and dust out of which planets form around the central star. The inner part will be hotter because it is closer to the central star and because the inner material has gained energy when moving from the outer regions in.
29. “Conservation” of a quantity means that the sum total of that quantity within a system must keep a fixed
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pler shifts of emission lines from the star due to an orbiting planet dragging the star around in space, which requires the system’s orbital plane be close to edge- on; (2) transit method: we watch a star regularly change brightness due to a planet passing in front of the star and eclipsing it; this is only possible if one star passes in front of the other, which means the orbital plane must be even closer to edge- on than the first method; (3) microlensing method: a planet passes in front of a distant star, and its gravity bends some of the star light toward Earth, making the star brighten for a short period of time; this requires an exact alignment of the two stars, which is very rare; and (4) direct imaging: we take an image that resolves the planet from its central star, which is only possible if the stars are near enough to Earth to be resolved.
39. Stars are very bright and far away, while planets appear very close to their stars. Thus, it is difficult to mask out the light of the star and still see close enough to it to see the reflected light of a planet.
40. We have only detected planets around a few hundred stars, which is a very small sample of the stars in the Milky Way, and not all of them look like our Solar System. In par tic u lar, our technology for detecting extrasolar planets is biased toward detecting giant planets in close orbits about their parent stars. We are only now finding Earth- like planets in Earth- like orbits. Based on this sample and our current techno- logical limitations, it is too early to make any general conclusions about whether our Solar System is unusual.
aPPlYing thE concEPtS 41. Setup: The conversion from m/s to mph is
× =1 m s
2.3mph
Solve: In Figure 7.20, the maximum radial velocity is about 35 m/s, or 35 × 2.3 = 80.5 mph, which is a fac-
tor of =67,000 80.5
832 smaller than the Earth’s orbit
around the Sun. Review: As expected, a star moves much less than the planet orbiting it. We see here that astronomers are indeed detecting very small radial motions in a star to detect the planet orbiting it.
42. Setup: The table gives us values of planetary masses in terms of Earth’s mass, so for this question we must sum up the planet masses and then compare them to Jupiter and Earth.
will never grow enough in size to pull on each other in a meaningful way, this could certainly happen in outer space. The Earth’s gravity and friction with the ground keep the bunnies from forming self- gravitating collections.
35. Rocky materials are refractory and condense at very high temperatures. As such, we expect that rocky materials will solidify in all regions of the nebula, including near the proto- Sun. Volatiles condense only at very low temperatures. Only the regions of the solar nebula far from the proto- Sun are cool enough for these materials to solidify. This picture is consistent with modern observations of our Solar System. Furthermore, barring other issues (such as migration of giant planets inward owing to complex gravitational interactions), we expect the same to be true of other planetary systems.
36. The giant planets had several advantages over the terrestrial planets to enhance the growth of their atmospheres. (1) Giant planet seeds formed from more abundant volatile materials, whereas terrestrial planet seeds formed from the much less abundant refractory materials. (2) Giant planets had a much larger area from which to accumulate raw materials, whereas terrestrial planets could draw material only from a narrow zone in the immediate vicinity of the forming proto- Sun. (3) Accreting in the outer nebula left giant planets less vulnerable to the powerful solar winds emanating from the proto- Sun; the terrestrial planets were scoured clean due to their proximity to the proto- Sun. Because of their ability to quickly grow very large and the lower temperature of the gas, the Jovian planets captured and retained significant amounts of the primordial hydrogen and helium from the solar nebula. Meanwhile, the terrestrial planets were too small and too exposed to the young Sun to capture or retain significant atmospheres from the solar nebula where the gases were hotter.
37. Debris from the formation of the Solar System exists in the form of asteroids and comets (some of which still occasionally cross planetary orbits). Jupiter keeps the outer regions of the terrestrial planet zone gravitationally stirred, thereby preventing the asteroids in the asteroid belt from accreting into another terrestrial planet. The Oort Cloud was formed when the giant planets ejected comets to the fringes of the Solar System. Finally, the Kuiper Belt is composed of icy planetesimals that were too sparsely distributed for accretion into large worlds.
38. The four methods that astronomers currently use to search for extrasolar planets are (1) spectroscopic radial- velocity method: we watch the periodic Dop-
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about its axis, which is why it has so little spin angular momentum.
45. Setup: For this problem, we will use the equation of orbital angular momentum Lorb = mvr. However, instead of computing L for each planet, let’s directly compare the values of both by constructing a ratio. This has two added advantages: (1) it gets rid of the constants in our formula, and (2) we can use what ever units we want as long as we are consistent, that is, masses in terms of Earth mass, speeds in km/s, and distance in astronomical units.
Solve : L

=

=

× =
Review: Given Jupiter’s huge mass and greater distance from the Sun, it is no wonder that it has so much more angular momentum around the Sun than the Earth.
46. Setup: Conservation of angular momentum states that the cloud’s angular momentum just before it collapses must equal its angular momentum after it has become the Sun. Expressing this as an equation, we
have L L m R
P
2 2π π .
Solve: Since a large percentage of the cloud becomes the Sun, we will assume the mass does not change. Solving above, we have
P R
R P
P R
R P
10 . 66 81 96 10 0 6yr yr s= × =−. . .
Review: Recall that angular momentum depends on how fast an object is spinning and how far that object is away from its spin axis. Thus it stands to reason that a huge cloud that collapses into a star would spin at an almost mind- numbing rate, if there were no means by which that cloud could transfer its angular momentum away from the central object. Thank goodness for accretion disks.
47. Setup: To compute the density of Vesta, we will assume that Vesta is a sphere. The volume of a sphere is
π π=r d 4 3 6
.3 3 Density is ρ = mass volume
.
ρ π
3,464 kg/m . 20
5 3 3
Solve: (a) The sum of the eight planets in our Solar Sys- tem is Mp = (0.055 + 0.815 + 1.00 + 0.107 + 317.83 + 95.16 + 14.54 + 17.15) M⊕= 446.66 M⊕. (b) To find Jupi- ter’s percentage of that mass, divide its mass by the total: M
M J
.
. . . %.

M
M
M
MP
Review: It is no surprise that the Solar System’s planetary mass is dominated by Jupiter, given how large and massive it is compared to everything else. We can simply see from inspection that almost all the mass is in Jupiter and Saturn.
43. Setup: For this problem, we will use the equation of
spin angular momentum L mR Pspin =
4 5
2π and orbital
angular momentum Lorb = mvr. Remember to use consistent units (i.e., time in seconds, distance in meters). Solve: For Earth’s spin angular momentum, P = 24 h = 86,400 s, so
Lspin = ⋅ × ⋅ ×
⋅ =
4 5.97 10π 24 6 26 378 10 5 86 400
7 kg m
Lspin = ⋅ × ⋅ ×
⋅ =
4 5.97 10π 24 6 26 378 10 5 86 400
7 kg m
, .00 1033 2× kg m s/
By comparison, using 1 AU = 1.5 × 1011 m Lorb = mvr = 5.97 × 1024 kg ⋅ 29.8 × 103 m/s ⋅ 1.5 × 1011 = 2.7 × 1040 kg m2/s which is about 3.8 million times more than the spin angular momentum. Review: Recall that angular momentum depends on how fast an object is spinning and how far away that object is from its spin axis. Thus, it stands to reason that all the angular momentum of a planet is in its orbit, since the planet’s distance from the Sun is so much greater than its size.
44. Setup: For this problem, we will use the equation
of spin angular momentum L mR Pspin =
4 5
2π , however
=

Review: Although Venus is practically Earth’s twin in terms of mass and size, Venus rotates very slowly
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51. Setup: This question asks for distance from a star knowing the orbital period, so we must use Kepler’s third law. Since the star’s mass is that of the Sun, this is simply P2 = A3; however, we are given the period in days, so we must convert to years.
Solve: (a) = × = × −3.525 day yr
365.25 day 9.65 10 yr,3P
so solving for distance we find the planet is A = P2⁄3 = (9.65 × 10−3)2⁄3 = 0.045 AU from HD 209458. (b) Mercury is located at 0.387 AU from the Sun. That is 8.5 times farther from the Sun than Osiris is from HD 209458! Osiris is very likely to be tidally locked to its parent star, with one side of the planet experiencing temperatures much higher than any planet in our Solar System. Because Osiris is a gas giant, the thick atmosphere of the planet probably transports the heat efficiently to the dark side of the planet. Review: Do we expect Osiris to be very close to its star? Compare its period to that of Mercury (about one- half year), and you find that it must indeed have a very small distance to have such a short period.
52. Setup: The dimming of the light from HD 209458 is proportional to the cross- sectional area of Osiris. Expressed as an equation, we can write this as
= =Dimming Osiris
where A represents the
cross- sectional area of the object and D represents the object’s diameter. Solve: (a) Rewriting our dimming equation to solve for the diameter of Osiris and substituting in appropriate numbers yields = = × = ×Dimming 1.7 10 km 0.017 2.2 10 km.Osiris star
6 5D D
= = × = ×Dimming 1.7 10 km 0.017 2.2 10 km.Osiris star 6 5D D (b) Taking the ratio
= × ×
1.585.Osiris
J
5
5
Osiris is approximately
59 percent larger than Jupiter. Review: From Problem 51, we know that Osiris is extremely close to its host star, and from this problem we know it is larger than Jupiter, which shows us that a very large planet is orbiting extremely close to its star. Certainly this begs the question, how can a giant planet be right next to its host star if our planet formation scenario requires that giant planets only form far from their stars?
53. Setup: Following Math Tools 7.3, we can estimate the percentage reduction of light during a transit as
. 2
2
R
In this problem, we are given units in terms
of a solar radius (7 × 108 m) and Earth’s radius (6.4 × 106 m).
(b) Vesta’s density is significantly higher than that of water or rock. This implies that Vesta’s composition must include a significant component of metal in addition to rock. Review: This confirms what we have found in mete- ors that have landed on Earth: they are mostly composed of metals.
48. Setup: The Sun’s brightness will drop in propor- tion to how much area of the Sun is blocked. This ignores limb darkening, which will be discussed in chapter 14. Solve: The Sun’s radius is 7 × 108 m, and Jupiter’s radius is 7.2 × 107 m. Since area is proportional to radius
squared, Jupiter’s area is ×
×

as much
as the Sun. So the Sun’s light would drop by this amount, or about 1 percent. Review: Aliens should easily be able to detect this drop with even crude instruments. However, this drop only happens once every 11.8 years, so they will have to watch the Sun very carefully and for a long time to detect it.
49. Setup: Doppler shifting tells us that the relative shift of a line away from its center equals the relative speed of the moving object compared to the speed of the wave. In this case, the wave is light, so our formula
can be written λ λ
= . 0
λ λ= = ×
= × − 0 8
6575 1
v c
/ /
. .
Review: This shows us that to observe Doppler shifts in visible light, large velocities are required.
50. Setup: Doppler shifting tells us that the relative shift of a line away from its center equals the relative speed of the moving object compared to the speed of the wave. In this case, the wave is light, so our formula
can be written λ λ
= . 0
and in this problem we are
looking for the shift Δl away from a central wavelength l0 = 500 nm. Solve: Solve our Doppler- shift equation for
λ λ= = ×
1.5 10 nm.0 8 7v
c
Review: This is much smaller than the size of sub- atomic particles (10− 15 m) and thus shows how nearly hopeless it is to ever be able to detect such a gravitational signal with radial velocities.
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mum mass, the semimajor axis, period, and eccentricity of its orbit, its density, and radius if available, and whether it is Jovian or terrestrial.
58. (a) At the time of this writing, Kepler has confirmed 105 planets and has 2,740 candidates. Follow- up observations consist of direct imaging of the star to rule out eclipsing binaries and spectroscopy to determine the spectral type of the star, to rule out eclipsing binaries, and to try to determine the planet’s orbital details through radial- velocity mea sure ments. (b) Answers will vary.
59. Answers will vary. The answer will include why it is better to have many people looking at data, rather than one person, and a copy of any stars observed for the citizen- science project.
60. Gaia will detect the wobble of stars being tugged by their planets, as well as transits. The science goals include confirmation of suspected planets, a more complete census of nearby faint stars, mea sure ment of the orbital planes of planets, mass estimates, and creation of a database of targets for upcoming planet- finding mis- sions. The launch, on a Soyuz- STB/Fregat rocket from Kourou, French Guiana, is delayed until October 2013.
ExPloration 1. Earth’s orbital eccentricity is very small, meaning the
orbit is close to circular. 2. Mercury has the most eccentric orbit, differing from a
circle by about 20 percent. 3. In general, our planets have mostly round orbits. 4. Mercury has the largest orbital inclination. 5. The inclination of orbits is defined relative to ours. 6. The Solar System is a flat disk, like a pancake. 7. Venus rotates clockwise, i.e., opposite all other planets. 8. The rotations of the Solar System bodies suggest that
they all formed together at the same time from the same body.
9. The Solar System is revolving counterclockwise and mostly rotating in that same direction.
10. Our Solar System today is ordered the same as a flat spinning disk of gas, which, we believe, happens when gas clouds collapse.
11. If there were insufficient gravity in the cloud, only a star, and no planets, might form.
12. If angular momentum is not conserved, the system might not flatten to a disk, and only a star, and no planets, might form.
13. A giant impact by a planet- sized body might have affected Venus such that its original rotation (in a CCW sense) was halted and slowly reversed (CW).
⋅ × ⋅ ×
1.4 10 6 2
8 2 3 or 0.14%.
Review: Transits of planets should produce very small drops in brightness given how tiny the planet is compared to the star, as we have found here. Note that in Math Tools 7.3, we see that Kepler- 11C produces a drop of only 0.0008 or 0.08%, which confirms that this planet is larger than Kepler- 11C.
54. Setup: Volume scales as size V ∝ R3 and density ρ = M / V. Solve: (a) Since volume scales with size cubed, the ratio of the volumes will be the cube of the ratios of
the sizes, i.e.,
or the new planet has
4.9 times more volume. If the density is the same, then M ∝ V so the new planet has 4.9 times more mass as well. Review: Note that part (b) is based on an assumption that may not be true. However, if we can use the orbital details of the planet to find its mass, we can combine these data to find the density.
55. Setup: We are comparing to Jupiter, so note that MJ = 1.9 × 1027 kg and Rj = 7.15 × 107 m Solve: (a) M = 2.33 MJ = 2.33 ⋅ 1.9 × 1027 kg = 4.43 × 1027 kg. (b) R = 1.43 RJ = 1.43 ⋅ 7.14 × 107 m = 1.02 × 108 m.
(c)
4 3
(1.02 10 m) 4.44 10 m .3 8 3 24 3
(d) ρ = = × ×
0.997kg/m 27
24 3 3M
V , which is
just slightly less than the density of water. This planet must be gaseous. Review: We expect that planets this large and massive will all be gas giants, and the density we found confirms this expectation.
uSing thE WEB 56. Answers will vary. The answer will include what meth-
ods are used to detect planets from the ground and from space, whether planets have been found in each of two projects, and if any were, what types of planets they are. A future project will also be presented, including when it will begin and what method(s) it will use.
57. Answers will vary. Part (a) will show a graph showing the distances of exoplanets from its central star, noting the masses and sizes of each planet, and comparing them to our solar system. (b) will give a recently discovered and published planet, including its mini-
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Name Date Section
ExPloration: using the transit method to Detect Exoplanets Go to the StudySpace at wwnorton .com/studyspace and open the “Exoplanet Transit Simulator” from the list of Inter- active Simulations for Chapter 7.
On the light curve graph, move the red line back and forth to see the correlation between the graph and a planet’s motion.
1. What is happening when the planet’s brightness drops?
2. Why doesn’t the brightness drop instantly?
Vary the radius of the planet (in the “Planet Properties” box) and observe the effect this has on the light curve.
3. Can a planet’s radius be inferred from the light curve data? Explain why or why not, supporting your answer.
4. Can a planet’s orbital period be inferred from a light curve? Explain why or why not, supporting your answer.
5. Assume the parent star’s mass is known. How can Kepler’s laws be used to find the planet’s orbital radius?
Planetary systems are oriented randomly with respect to our point of view, so they aren’t always viewed edge- on. Play with the “Inclination” slider (in the “System Orientation and Phase” box) and see what happens to the light curve when a planet is not edge- on.
6. Can transits be used to find all extrasolar planets? Explain why or why not, supporting your answer.
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DEmonStration: ranking task for formation of Planetary Systems
Cloud of interstellar gas starts to collapse under the force of its own self- gravity.
Gravitational potential energy of collapsing interstellar gas cloud is converted into heat and radiative energy.
Cloud of interstellar gas rotates faster and faster as it collapses because of conservation of angular momentum.
Inner parts of flattening cloud begin to fall freely inward, raining down on growing object at the center.
Material makes its final inward plunge, landing on a thin, rotating accretion disk.
Motions push smaller grains of material back and forth past larger grains; smaller grains stick to larger grains.
Planetesimals form that are massive enough to have gravity that begins to attract nearby bodies.
Planetesimals continue to accrete material until they become large enough to be called planets.
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Name Date Section
activitY: 51 Pegasi: The Discovery of a new Planet In just the past few years, astronomers have announced discoveries of hundreds of planets orbiting nearby stars. These discoveries seem to finally answer the question of whether or not our solar system is unique. We should note, however, that when astronomers state that they have discovered a new planet, what they are really saying is that their data can best be interpreted as a planet orbiting a star. One cannot “prove” that these other planets exist; one can only state that, until the hypothesis is disproved, a planet orbiting the star best explains the observations. We can mea sure only indi- rectly the influence each one has on its parent star as the star and planet orbit their common center of mass. The planet makes the star “wobble,” so we can use the Doppler method to detect it. That is the method we explore here. We enter this realm of discovery by working with actual discovery data from observations of the star 51 Pegasi made at the Lick Observatory in California. These data are the mea sure ments of the Doppler shift of the wavelengths of the absorption lines seen in the spectra of 51 Peg.
Observations
Take a look at the graph shown here. It shows the mea sured radial velocities as a function of time recorded in days. As you can see, the radial velocities are sometimes positive (the light is redshifted) and sometimes negative (the light is blueshifted), indicating that sometimes the star is receding (redshifted) from us and sometimes approaching us (blueshifted) when viewed from our frame of reference, Earth. This wobble of the star was the first indication that the star 51 Pegasi had an invisible companion.
V el
oc ity
50
0
–50
–100
Time (days)
Figure 1 Discovery data for the planet orbiting the star 51 Pegasi.
The above plot represents the observed radial velocities over a period of about 33 days. The data were obtained by mea- sur ing the Doppler shift for the star using the formula
λ λ λ
rest (Eqn. 1)
( ) . .
3 10 18 8× × −
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Here, c is the speed of light and v is the velocity of the object being observed. Note that the difference in the shifted wavelength and the rest wavelength is extremely small for these planetary detections and that the ratio of the change in the wavelength to the rest wavelength means the units come out to be what ever units in which the speed of light is expressed.
Procedure and Questions
1. A period is defined as one complete cycle, that is, where the radial velocities return to the same position on the curve but at a later time. How many cycles did the star go through during the 33 or so days of observations?
Number of cycles = ___________
2. What is the period, P, in days of one complete cycle? (Number of days for these observations divided by number of cycles.)
Period = ___________ days
3. What is P in years? (Hint: divide the period in days by the number of days in a year; the answer will be a decimal number smaller than 1.)
P = ___________ years
4. What is the uncertainty in your determination of the period? That is, by how many days or fractions of a day could your value be wrong? (This is a number that you decide. There is no set “rule” for this, but the uncertainty has to be less than a day here.)
Uncertainty = ___________ days
5. What is the amplitude, K? To find this, take 12 of the value of the full range of the velocities. For example, if the velocities went from +12.4 m/s to – 13.8 m/s, 12 of the full range (26.2 m/s) would be 12 of 26.2 m/s = 13.1 m/s.
K = ___________ m/s
Uncertainty = ___________ m/s
We make some assumptions in order to simplify the equations we have to use for determining the mass of the planet. In this case, we use Jupiter as our reference planet. The equation we use is
M
(Eqn. 2)
P should be expressed as a fraction of a year, and K in m/s. Twelve years is the approximate orbital period for Jupiter and 13 m/s is the magnitude of the “wobble” of the Sun caused by Jupiter’s gravitational pull. The answer we get is the ratio of the mass of the planet (Mplanet) to the mass of Jupiter (MJupiter), since that is so much easier to envision than a number times 10 raised to the power of 27.
7. Put in your values for P and K and calculate the mass of this new planet in terms of the mass of Jupiter. That is, your calculations will give the mass of the planet as some factor times the mass of Jupiter. Astronomers like doing these kinds of direct ratios so that the large numbers basically cancel each other, as do all units like m/s, and seconds, and years.
Mplanet = ___________ MJupiter
We assume that the parent star is 1 solar mass and that the planet is much, much less massive than the star (the case with our solar system). We can then calculate the distance this planet is away from its star, in astronomical units, using Kepler’s third law:
=1 3
or a3 = P2 or =( )2 1 3a P (Eqn. 3)
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8. Again, P is expressed as a fraction of a year, and a represents the AU’s. Solve for a: a = ___________ AU
9. Do a search for the actual values for the following quantities and list them in column 2, giving your references in the space provided. List your results in column 3.
Characteristic Published Value My Value
Mass
References:
How close were you for all of these values? (Be sure to list all quantities.)
10. Compare the orbit of this planet to those in our Solar System by referring to this rough scale model. Mercury is 0.4 AU from the Sun; Venus, 0.7 AU; Earth, 1.0 AU; Mars, 1.5 AU; Jupiter, about 5 AU. Where would the new planet fit in if it were in our Solar System?
MSun V E M J
11. Science is based on the ability to predict outcomes. However, nothing prepared astronomers for the characteristics of this “new” Solar System. Consider the mass of this planet as well as its distance from its star. Why was the discovery such a surprise when compared to our Solar System?
12. If this actually is a planet, is it possibly hospitable to life? Comment on what it would be like on this planet.
13. Summarize the current working theory on how the planet orbiting 51 Pegasi came to rest so close to its planet. Use what ever information sources are available to you.
14. In your own words, summarize how astronomers determine the velocity of a star and how they know when the star is coming toward us and when it is going away from us based on the spectrum of the star.
One of the original pessimistic views held by many astronomers after this discovery was that this could not be a gaseous planet so close to its star because the planet would not have been able to hold onto its atmosphere for billions of years. We can test that hypothesis with a few simple equations. First, let’s just give this planet the mass and radius of Jupiter. We know that Jupiter held onto its atmosphere from when and where it formed. The question becomes: would Jupiter have been able to hold onto its atmosphere if it migrated to the inner part of our solar system to ~0.05 Au?
15. If we graph the approximate temperature versus distance from the Sun, we can extrapolate to the temperature at the distance that this planet is away from its star. The fit is definitely a power function (see Eqn. 4); the temperature is proportional to the inverse distance squared. What would be the approximate temperature in the inner part of the planetary system, 0.05 AU, where this planet is located?
T = _________ K
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=T K D( ) 389 (AU)
(Eqn. 4)
Te m
pe ra
tu re
700
600
500
400
300
200
100
0 0 3 6 9 12 15 18 21 24 27 30
Distance from Sun (AU)
Temperature of the Solar System vs distance from the Sun
Figure 2. Temperature of the Solar System vs. distance from the Sun. Temperatures were approximated using the Stefan- Boltzmann law, a surface temperature of the Sun of 5700 K, solar radius = 7.0 × 108, 1 AU = 1.5 × 1011.
We need to find out if this temperature is great enough to evaporate the planet’s atmosphere, and that involves determining the velocity of the gas in its atmosphere versus the escape velocity of the planet.
Escape velocity of Jupiter: v GM rescape m s= ≈
2 60 000, / (Eqn. 5)
The condition for a planet to hold onto its atmosphere for billions of years is that the gas velocity must be less than
one- sixth of the escape velocity of the planet: v vgas escape< 1 6
. (Eqn. 6)
The velocity of the gas can be calculated by v m sgas temperature
molecule s mass ( / ) ,=157 (Eqn. 7)
16. Assume that the planet’s atmosphere is primarily molecular hydrogen with a molecular mass of 2. Then answer this question: If the planet orbiting 51 Pegasi formed at a distance where it had a substantial molecular hydrogen atmosphere, would the planet retain that atmosphere at its current distance of 0.05 AU.? Show all calculations here.
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Name Date Section
activitY: Surveying the known Extrasolar Planets Go to the Extrasolar Planets Encyclopaedia website at http:// exoplanet .eu /catalog
Part 1: The Data Table
The planets at the Extrasolar Planets Encyclopaedia can be sorted by pa ram e ters by clicking on a column heading. The mass and radius of a planet are expressed as comparisons to Jupiter. The period is expressed in Earth days, and the distance from its star (a) is expressed in astronomical units. Sort the data appropriately to answer the questions below.
1. How many planetary systems, planets, and multiple planet systems does this cata log include?
2. What was the least massive planet discovered most recently? 3. What is the mass of that planet in terms of Jupiter’s mass? 4. The distance of that planet from its star is AU, and it takes days or years for 1 orbit. 5. List the name, mass, period, orbital distance (a), and eccentricity (e) of the planet that was most recently discov-
ered (and has values for all of these characteristics) by clicking on Col. 10, Discovery.
6. Find the planet HD 180314b, which orbits star HD 180314, by clicking on the star’s name. Note the masses of the planet and its star. Also find the orbital period in years and the distance the planet is from its star. Does the relationship between the period and orbital distance of this exoplanet obey Kepler’s third law, which assumes the star is Sun- like and contains 99.99% of the mass of the planetary system? Click on the planet name to find out more about the parent star to understand any discrepancies and summarize here. Show any calculations here.
7. For the next question, you will need to set criteria for determining whether a planet is Earth- like in terms of mass and Earth- like in terms of orbital distance. What numbers are you going to use?
One of the goals of extrasolar planet hunters is to find other Earths. Since the masses of the planets in this database are expressed in terms of Jupiter, you need the fact that Earth’s mass is 0.003 that of Jupiter’s. Be sure to use the criteria you set above; you are free to change your numbers, but if you do, make sure you note that.
8. How many extrasolar planets are Earth- like in terms of mass? How many extrasolar planets are Earth- like in terms of orbital distance (Col. 5)? How many extrasolar planets are Earth- like in terms of both of these pa ram e ters?
9. How many of the known extrasolar planets orbit their parent star closer than Mercury orbits our Sun (0.39 AU)? What percent of the known extrasolar planets is this?
10. How many of the known extrasolar planets have masses that are 1 MJupiter or more? What percent of the known extrasolar planets is this? (Hint: Figure out how many are listed on each screen and then multiply by how many screens it takes for the list. A rough estimate is fine.)
Part 2: Histograms
For this part, you need to click on “Diagrams” in the top menu. Also choose “Histogram plot” shown just under the top menu. Select the category you want for the x-axis. (Note: You can get a better estimate for these answers by clicking on log scale for the x-axis and remembering that 100 equals 1.)
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11. Approximately how many of the extrasolar planets have masses that are 10 MJupiter or less? What percent of the known extrasolar planets is this?
Part 3: Scatter Plots
Change your option to “Scatter plot.”
12. Is the typical extrasolar planet similar to Earth? To answer this question, set the x-axis Planetary Mass max value to 0.01. Earth’s mass is roughly 0.003 that of Jupiter’s. Set y-axis Semi- Major Axis min to 0.4 and max to 2. Explain using your analysis of the scatter plot. It is best to use log scales for both axes and remember that 100 equals 1 and 10− 2 equals 0.01. Compare your answer here to your answer in question 7.
13. Choose x-axis “Year of discovery” and Y axis “Distance to a host star” and plot. What does this scatter plot tell you about our rate of discoveries (be sure to unclick log scale)? What is the distance at which the number of discoveries drops off significantly (state both in parsecs, pc, and light years)? Extrapolate the information into the future. Predict what would happen to our rate of discoveries if we were able to accurately detect planets past 1000 pc (3,260 light-years).
14. Pick two of the characteristics of the exoplanets from the many options listed and create a scatter plot from those data. State what you graphed and summarize what you learned from that graph.
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matter was poorly understood at the time. In 1901 Annie Jump Cannon made some progress by noting that the 22 known stellar spectral classes could be combined into 7 and reordered as a temperature sequence. With the discoveries of the electron by J. J. Thompson in 1898 and the atomic nucleus by Ernest Rutherford in 1911, the general structure of the atom was determined. Work in the bur- geoning field of quantum mechanics in the 1920s led to a more complete understanding of how light interacts with atoms, finally providing the Rosetta stone astronomers needed to fully interpret stellar spectra. Today we can quickly determine the surface temperature of a star by noting the relative strengths of par tic u lar spectral lines. We have also learned how to uncover compositional differences from stellar spectra, leading to the discovery of two stellar populations: metal- rich and metal- poor stars. Our ability to interpret stellar spectra has matured to the point that we can use subtle gradations in line strength to subdivide Annie Cannon’s seven spectral classes into nine subclasses each. We can now literally distinguish two separate subclasses of stars whose surface temperatures may vary by as little as 100 Kelvin!
In the early 1900s, the astronomical toolbox expanded dramatically. The most important tool for stellar astronomers joined the toolbox in 1910, when Ejnar Hertzsprung and Henry Norris Russell in de pen dently discovered a relationship between a star’s luminosity and its surface temperature. The relationship is best illustrated by a diagram that bears the names of its discoverers: the Hertzsprung- Russell diagram. Plotting a star’s absolute magnitude or stellar luminosity on the vertical axis against its surface temperature (in reverse), color, or spectral classification on the horizontal axis reveals that 90 percent of stars fall along a narrow band from the upper left to the lower right of the diagram. This band is called the main sequence, and stars spend approximately 90 percent of their lives in this stage. The remaining stars generally fall into smaller clusters on the diagram. Stars in their final red giant or red supergiant stages are large and luminous with cool surface temperatures, and they plot to the upper right of the main sequence. White dwarf stars are small and faint with very hot surface temperatures, and they plot to the lower left of the main sequence. The H-R diagram is so fundamental to stellar astronomy that it is used to show the relationships
The stars are so far away that we see them only as pin- points of light, even through telescopes. Apparent brightness and color are the only semiquantifiable mea sures of stars that our eyes can perceive directly. It is not at all surprising that even as recently as the turn of the 20th century, astronomers knew almost as little about stars as the ancient Greeks of 2,500 years ago. Yet with some clever human ingenuity, combined with ever- more- sophisticated technology, 21st- century astronomers routinely mea sure fundamental properties of stars like distance, size, luminosity, surface temperature, composition, and mass. Today we plot these properties against each other using the Hertzsprung- Russell diagram (also known as the H-R diagram) to develop an understanding of stars that would have astonished astronomers of just a century ago. Two important breakthroughs in the previous century set the stage for our modern understanding of stars: high- precision parallax mea sure ments and the mat- uration of the field of spectroscopy.
Mea sur ing distances across the Solar System with parallax is easy to do. Even the most distant planets show mea- sur able motion across the sky over a few weeks. However, the stars are another matter entirely. The nearest bright star, Alpha Centauri, lies 4.4 light- years from the Sun. That is nearly 7,000 times farther than Pluto! A parallax- second (or parsec) is defined to be the distance at which Earth’s maximum angular separation from the Sun is 1 arcsecond (1/3,600°). This angle is so small that we would have to travel 3.26 light- years from the Sun before the radius of Earth’s orbit would be contained in such a narrow angle of sky. Yet the nearest star resides 30 percent farther still. At Alpha Centauri’s distance of 4.4 light- years (1.3 parsecs), the entire diameter of Earth’s orbit subtends an angle of only 1.54 arcseconds (1/2,340°). Shifting to the perspec- tive of an Earthbound observer, it is Alpha Centauri that displays that maximum parallax shift of 1.54 arcseconds when viewed from opposite ends of Earth’s orbit. This shift is so tiny that it was not mea sur able until the middle of the 19th century! Modern instruments now permit us to mea- sure stellar distances out to ~50 parsecs (our immediate stellar neighborhood) using parallax.
By the late 19th century, astronomers were routinely observing stellar spectra. Unfortunately, interpreting spectra was difficult because the interaction of light with
C h a p te r 1 3
Taking the Mea sure of Stars
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22 Chapter 13 Taking the Mea sure of Stars
• Explain why each element absorbs radiation at exactly the same wavelengths as it emits.
• State the relationship between absorption line strength and temperature, and give an example.
• Chapter learning goal(s) addressed: Determine the composition of stars from their spectra. Classify stars, and or ga nize this information on an H-R diagram.
Pre- Assessment Question How many other elements have the same emission spectrum as hydrogen?
a. 0 b. All the alkali metals c. All the gases Answer: a
Instructions Students follow the guidelines given within the Spectrum Explorer pages.
Post- Exploration Questions 1. What does the observation that each element has its
own unique spectral signature tell us about the energy levels in the atoms of the different elements?
2. Reproduce the diagram of the energy levels of the hydrogen atom and include the first four energy levels.
a. How many absorption lines are possible? b. How many emission lines are possible? c. Are the energy differences in each of the transitions
in absorption and emission the same? d. Will the wavelengths be the same? 3. The original spectral sequence went alphabetically.
Why did it have to be changed?
Exploration 2: Using Wien’s Law to Estimate the Surface Temperatures of Stars
Note to Instructors Stars are not perfect blackbody radiators by the very fact that we can see them. The fit of a blackbody curve to a star’s spectrum, then, is, at best, an estimate— sometimes good enough, but most of the time just rough. There is a lot of stellar astrophysics involved that is well beyond an
between stars, to determine the ages of star clusters, to plot the life track of a single star, and to identify all the physical properties of an individual star. In short, just about everything we know about stars can be expressed with the H-R diagram.
DiScuSSion PoinTS • Discuss the utility of the spectral classification of stars.
Have students look for information about the new spectral classes defined in the 21st century such as “L” and “T” dwarfs.
• Discuss the Sun in terms of its relation to other stars concerning size, temperature, mass, luminosity, and its spectrum.
• In thinking about binary star systems, discuss what effects could be observable and what information could be inferred from those observations if an extra- terrestrial observer could see the orbital plane of Earth around the Sun edge-on.
• Discuss the possibility of finding planets in the habitable zones of stars and the difficulties that could be encountered when considering different types of stars.
• Discuss the history of the stellar classification system, including its original incarnation and its rearrange- ment once temperature was realized to be the important quantity.
ExPloraTionS Exploration 1: Spectrum Explorer
Note to Instructors This alternate Exploration has students using the “Spec- trum Explorer” Interactive Simulation found on the StudySpace to examine emission and absorption line spectra. The primary lesson of this activity has important ramifications for understanding the science of astronomy. Students benefit from seeing firsthand that elements have unique spectra and that this makes it possible for astronomers to infer a wealth of information about a distant object using only the light it emits. Questions 6 through 9 ask students to probe the reasons for both O and M stars— stars on the opposite ends of the temperature scale— having weak hydrogen lines. Review of key parts of Chapter 5 may be necessary.
Learning Goals • Explain how we know that each element has its own
unique spectral signature.
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Chapter 13 Taking the Mea sure of Stars 23
connection between the units and the equivalent in nanometers.
We have estimated an uncertainty for the “best fit” blackbody temperature by manipulating the temperature of a star until we get one that is “too hot” and one that is “too cool” (but still not unreasonable fits). For each spectral type, we then subtract the coolest possible temperature allowed from the hottest possible temperature allowed and divide by 2 to get our uncertainty.
The Java applet also shows how the peak wavelength for O and B stars is well into the UV part of the spectrum. When working with the M5V spectrum, the broad swaths of absorption by molecules shows how inexact this method is for such cool stars. Emphasize, however, that as inexact as this science is, we still get an estimate of the star’s temperature to within a few hundred or a thousand degrees, and that’s not bad, considering our thermometer.
Exploration 3: Details of the HR Diagram
Note to Instructors This exploration is a slightly different version from the one in the textbook, asking the student more analytical questions. It uses the simulation “H-R Diagram Explorer,” found on the StudySpace.
Learning Goals • Demonstrate an understanding of the interrelationship
among a star’s surface temperature, radius, and luminosity.
• State what the classifications of stars between brightest and nearest imply about their distances and actual numbers.
Pre- Exploration Question Locate the following sections for the H-R diagram shown here:
Stars that are hot and have high luminosity _____ Stars that are hot and have low luminosity _____ Stars that are cool and have low luminosity _____ Stars that are cool and have high luminosity _____ Region of stars that have the largest radii _____ Region of stars that have the smallest radii _____ The location of the Sun on this H-R diagram _____
introductory text. Even so, with a brief introduction and practice fitting curves to the spectra of stars covering a range of spectral types, students can get a good understanding of what is involved. They then should be able to find the surface temperatures of a sample of stars to within 20 percent or so.
Learning Goals • Infer surface temperatures of stars from their peak
wavelength using Wien’s law. • State the shortcomings of relying entirely on Wien’s
law for surface temperatures of stars. • Chapter learning goal(s) addressed: Infer the tempera-
tures and sizes of stars from their colors.
Required Materials • Computer access • Projection system
Pre- Post- Assessment Questions • How close to perfect blackbodies are stars? • What are some of the difficulties in fitting a blackbody
curve to stellar spectra?
Instructions Stellar spectra contain many bumps and wiggles that correspond to absorption lines because of a variety of elements present in their atmospheres. Our efforts in infer- ring a star’s surface temperature start with the shape of its continuous spectrum and an estimate of the peak wavelength. Perhaps the hardest part of this is getting a feel for how to determine the shape of the star’s continuum. The shape is hard to figure out because of the many absorption lines that affect the spectra of stars, pulling out light at certain wavelengths. A-stars have what’s known as the Balmer jump, where light from the star short- ward of around 400 nm is almost totally absorbed. That radiative energy has to get out somehow, and the photons jostle around until their energies are equivalent to wavelengths where the opacity isn’t so great, and they escape.
It is important that the students read the instructions carefully, as they need to know how to select a region of the spectrum and thus how to zoom in on the part that they are interested in. Also emphasize that the y-axis scale will change depending on how much of the spectrum is being viewed. Have students leave the fiducial wavelength at 5000.0 angstroms (Å). Since the x-axis uses a scale that might be hard for students to understand, emphasize the
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sented by steps on the ladder. As with all quantum phe- nomena, a macroscopic visual aid is useful in conveying to students the unintuitive nature of the quantum world.
Learning Goals • State what is meant by the “discrete energy levels” of
an atom. • Explain the pro cesses by which energy is conserved
during transitions of an electron. • Chapter learning goal(s) addressed: Summarize how
the energy levels of an atom determine the wavelength of the light that the atom emits and absorbs (Chapter 5).
Required Materials • Short stepladder (four steps is sufficient), or a lecture
hall with stairs • Rainbow assortment of colored plastic balls—
VIBGYOR—like those typically used in ball pits • Bucket or mitt to catch the balls
Pre- Post- Assessment Questions • What do we mean when we talk about electron
“jumps” or transitions in an atom? • What is meant by the “discrete energy levels” of an atom? • Can electrons exist in between those energy levels? • How is energy conserved during transitions of an elec-
tron? What transpires?
Instructions 1. Stand next to the stepladder and explain to students
that electrons in atoms cannot have just any energy, just as you can’t stand in between steps. Electrons can have only certain energies, just as you must stand on specific steps. Having discrete levels of energy means that energy levels for electrons in atoms are “quantized.” Review energy conservation and the fact that because the energy spacing between quantized levels is set for each element, photons must have just the right energy to be absorbed.
2. Hand out balls of different colors. Decide for yourself beforehand what colors represent what amounts of energy. This is important later as you “transition” to different levels of the stepladder.
3. Start by standing on ground level next to the stepladder (“atom”). Explain that you will represent an electron and the stepladder represents the atom and its energy levels that you can inhabit. State that you are starting at the ground state energy level.
4. Have a few students gently toss balls to you. If a ball has the right color for you to transition to the first energy level of the “atom,” grab it and step up. Ask, “What happened?”
Instructions Complete instructions are given within the student worksheet for this exploration. If you assigned the activity that had the students calculate the distances, temperatures, brightness, luminosities, and radii of the four stars in Cyg- nus, this Exploration provides a review.
Post- Exploration Questions 1. How can a star be cool, say, ~3500 K, and have a very
high luminosity? 2. What is the characteristic of a star that is both extremely
hot and has low luminosity? 3. The Sun, of course, is both the nearest star and the
brightest star. Is it represented on the H-R diagram in this simulation?
DEMonSTraTionS anD acTiviTiES Demonstration: Atomic Spectrum
Note to Instructors This is a variation on the demonstration used for Chapter 5. It provides a different macroscopic visual before further discussion about emission and absorption spectra, most likely necessary to refresh students’ memories. Balls of various colors represent different energy photons while you represent the electron at various energy levels repre-
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Learning Goals • Apply Wien’s law to real spectra and estimate the sur-
face temperatures of 14 stars. • Classify stars by their surface temperatures.
Required Materials • Computer with Internet access • Printer • Calculator
Instructions Detailed instructions for performing this activity are contained in the student worksheet, which students can fill out and turn in.
Suggestion: Print the spectra and attach to index cards, with enough sets to use with teams. Students then place the spectra in order of temperature, reinforcing the shift not only in the peak wavelength but also in the strengths of the hydrogen absorption lines. Students can then discuss the proper fit of a thermal radiation curve. The student instructions suggest copying each image into a document editor for more rapid reviewing.
Depending on the speed of the college or university’s Ethernet system, the number of students accessing the data simultaneously, and the SDSS computers, finding the data for the individual spectra can take time, especially if students need to access each spectrum multiple times.
Activity 2: Finding Distances to Stars Using Parallax Mea sure ments
SkiLL LEvEL: inTERMEDiATE
Note to Instructors As simple as the equation is for determining the distances to stars via their mea sured parallaxes, students enjoy discovering the relationship themselves through simple experimentation and mea sur ing. This activity asks students to predict what they think the relationship is, based upon their blinking alternate eyes or maybe holding their thumbs in front of their faces, and then test that predic- tion. Even students who are familiar with the concept may be surprised.
The idea of how those results were obtained are then applied to the mea sured parallaxes of stars, including the use of arc seconds and parsecs. The desirability of a longer baseline versus the feasibility of actually mea sur- ing from a greater distance, such as from Pluto, is explored. The activity ends with ranking of distances based on actual parallax values and calculation of distances in parsecs given the actual parallaxes of another set of stars.
5. Have a few more students gently toss balls to you. If a ball has the right “energy” for the next transition (or higher transitions), “absorb” it and move up to the appropriate level. Explain what has just occurred.
6. Now, toss back a ball and move down one or more energy levels, as appropriate, depending on the color of the ball. Ask students, “What happens when an electron goes (when I go) down in energy levels?”
7. At least once, absorb a single photon that transitions you up by two energy levels and then move back down two, one step at a time, emitting two different photons. Ask students, “Is energy still conserved if an electron emits (I “emit”) two photons?”
8. At some point, work your way up to the “top” of your step’s “energy levels.” When a student tosses a ball of sufficient energy to you, jump off the ladder and run away, representing ionization. Explain.
9. When you return to the “atom,” toss out a ball of the right color in order to recombine with the atom and return to a previous energy level. Explain what has just occurred.
Activity 1: Spectral Classification of Stars
SkiLL LEvEL: EASY
Note to Instructors This activity uses real astronomical data from the Sloan Digital Sky Survey (SDSS). Students look at the spectra of several stars, fit them with a curve to find the continuum, find the peak wavelength, calculate the star’s temperature using Wien’s law, and then classify the stars based on the estimated surface temperature.
This activity is a subset of a larger activity posted on the SDSS website ( http:// www .sdss .org). The full activity has students classify stars by both temperature and hydrogen absorption line strength and poses a series of questions about why the stars with the stron- gest hydrogen lines are neither the hottest nor the coolest.
If this is the first time your students have looked at real spectra data, you will want to give them a “tour” of one of the spectra. Specifically, you will need to show them what noise looks like, what absorption lines look like (both narrow and broad), and how to see the under- lying blackbody curve through the many bumps and squiggles of a spectrum. This activity has students begin on a Web page that walks them through this very tour.
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Instructions Full instructions guiding the students through the predic- tions, experimentation, consideration of their results, and extension to the stars are given on the student worksheet. The activity ends with a short writing exercise. Students can come very close to the actual y = 1/x graph, as noted in the sample shown here that came from actual student data.
Using any kind of projector, reproduce a mea sur ing grid such as the one shown here to give the students reference points for their mea sure ments.
Activity 3: Deriving Stellar Properties for Four Stars in Cygnus
SkiLL LEvEL: ADvAnCED (SCiEnTiFiC noTATion, RATioS, ESTiMATing UnCERTAinTiES)
Note to Instructors Students seem to enjoy working problems when they realize that they are using real data and references that professional astronomers use. This activity uses SIMBAD (SIMBAD database, operated at CDS, Strasbourg, France), which has the complete details on not only stars but also galaxies, nebulae, observations at different wavelengths, and more. If earlier in the course you discussed celestial coordinates, then you can draw student attention to the listed equatorial coordinates.
The demonstration of Wien’s law is graphically displayed using an applet at http:// www .jb .man .ac .uk /distance /life /sample /java /spectype /specplot .htm and given as an Exploration for this chapter is useful for seeing how a continuum blackbody spectrum should be fit to the data. It is also important to make sure that students thoroughly understand Wien’s law and how extremely hot stars have blackbody spectra that peak far into the ultraviolet, and extremely cool stars have spectra that peak far into the infrared. For these extremes, only a maximum peak wavelength (minimum surface temperature) can be estimated for the hot stars and only a minimum peak wavelength (maximum surface temperature) can be estimated for the cool stars. Some students are uncomfortable with not being able to get the “exactly right” answer, but often in astronomy we just do the best we can with the observations we have.
Some students are also unable to figure out their uncertainties in the peak wavelength. We have found it helpful to give them lots of practice work on similar stars before attempting this activity.
Here is another opportunity for students to work in teams, sharing computations, or even set up a spreadsheet to streamline their work.
Learning Goals While working in pairs with a meter stick and toothpick, students will:
• Demonstrate mea sured parallax by noting scale of “jumps” at different distances from eyes.
• Derive the relationship between the distance of the toothpick and the corresponding “jumps.”
• Approximate the limit to the detection of the “jumps.” • Apply this knowledge to the parallax of stars. • Chapter goal(s) addressed: Use the brightness of
nearby stars and their distances from Earth to determine their luminosity.
Required Materials • Meter sticks or long bamboo skewers • Toothpick (anything very thin) • Protractor • Overhead or pre sen ta tion slide with finely spaced grid
(sample provided here)
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Distance from eyes (cm)
Measuring grid for deriving the distance versus parallax relationship
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12. False: As blackbodies, hotter stars are bluer. 13. False: A star with no carbon absorption could have no
carbon, or the atoms could not be in the right energy state to absorb light.
14. True: Conservation of momentum dictates that the more massive star moves more slowly.
15. True: Once one knows the mass of a star, then its size, temperature, and luminosity are known, but provided one also knows its evolutionary state.
16. (d) Since parallax depends on the distance of the planet from the Sun, a larger orbital distance would make the pc longer and would allow us to see parallax motion of more distant stars.
17. (a) More distant stars have smaller parallax angles. 18. (d) A radian is about 57.3 degrees while an arcmin
and arcsec are fractions of a degree. 19. (a) Brightness b changes as L/d2, so the fact that star
A appears twice as bright but is twice as far away means that L ∝ b × d2 = 2 × 22 = 8 using proportional reasoning.
20. (b) The high temperatures of O stars mean that most atoms are either ionized or the electrons are in such high- energy states that they will not absorb many photons to produce absorption lines.
21. (b) Helium has four times the mass of hydrogen. 22. (a) L ∝ R2 T 4, so if one star is hotter, it must be smaller
to have the same luminosity as the cooler one. 23. (b) Capella will be made of yellow (G-type) and red
(M-type) stars, and since both pairs are close to each other, the brighter G-type stars will dominate. There- fore, the color will appear yellow.
24. (d) While the H-R diagram shows the temperatures, colors, and luminosities of stars, we combine these with the paths that stars take along the diagram to understand how they evolve.
25. (a) Most stars are main- sequence stars; since this stage accounts for about 90 percent of any star’s lifetime, we see most stars during their longest- lived phases.
Thinking abouT ThE concEPTS 26. The sky is essentially a two- dimensional sphere with
distances mea sured in angles rather than in linear units. Because astronomers mea sure parallax as the angular shift of foreground objects in the sky, they prefer to define a distance unit that can be obtained directly from parallax mea sure ments. Therefore, the parsec is the distance unit of ch

the birth and evolution of planetary systems

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