Astronomy Midterm #3

Exoplanets in eccentric orbits experience large temperature swings during their orbits. Suppose you had to plan for a mission to such a planet. Based on Kepler's second law, does the planet spend more time closer or farther from the star? Explain.

A complete orbit is required in order to model or detect the presence of a planet with the radial velocity technique. From Kepler's laws, we know that the closest planets have short periods. In the case of the close-in giant planets (so-called "hot Jupiters"), these orbits can be as short as a few days. It does not take long to observe over all phases of these short orbits. However, Saturn has a 30-year orbit. Thus, we would have to observe Saturn for at least 30 years in order to detect that planet. A further complication is that when the planet is far away, the gravitational tug on the star is weaker and harder to detect. Saturn only induces a velocity in the sun of 2.7 m/s. Doppler surveys with sufficient precision to measure this signal only began around 2003. It is also worth noting here that even though small planets may be close to their parent stars and have short periods, their small masses make it much more difficult to observe either gravitational tugs or transits; thus, it is much easier to detect massive planets in short-period orbits around their host stars than it is to detect any low-mass planets or to detect high-mass planets at great distance from the star.

Describe the evolution of a massive star (say, 20 times the mass of the Sun) up to the point at which it becomes a supernova. How does the evolution of a massive star differ from that of the Sun? Why?

A massive star will leave the main sequence once its core hydrogen is depleted. After that time, it becomes a supergiant star, and it will begin fusing a variety of heavier elements as fusion thresholds are overcome. (For example, it takes about 100 million degrees for helium to fuse into carbon.) For example, the star will first do helium fusion (where helium is the fuel), then carbon fusion, neon fusion, oxygen fusion, and silicon fusion. After the inner core becomes iron, which requires energy to fuse, the star will experience a core collapse and create a neutron star from its core. The rest of the star explodes into a supernova. The evolution differs from a solar-mass star in several ways. First, the fusion history is different: A massive star can fuse more elements and create more fusion products. Second, the lifespan for each star is quite different, with a solar-mass star taking a much longer time for each stage of its evolution compared to the massive star. Third, the method by which each star loses mass is different, with a solar-mass star losing mass via the planetary nebula phase, while a massive star undergoes a supernova event. Finally, the final state of each star is different, with a low-mass star becoming a white dwarf, and a massive star becoming a neutron star. The main factor that determines the fate of each star and the path they will take to get to the fate is their initial mass on the main sequence.

How does a white dwarf differ from a neutron star? How does each form? What keeps each from collapsing under its own weight?

A white dwarf is an electron degenerate object, while a neutron star is a neutron degenerate object. A white dwarf has a larger radius and is much less dense than a neutron star. All white dwarfs are less than 1.4 MSun while neutron stars are between 1.4 and 3 MSun. A white dwarf forms after a star with an initial mass less than 10 MSun ends its core energy generation, while a neutron star forms from a star with an initial mass between 10 and 40 MSun. A white dwarf would form after the planetary nebula phase of a star, while a neutron star forms just before the rest of the star experiences a type II supernova detonation.

Describe the spectrum of each of the following: A. starlight reflected by dust, B. a star behind invisible interstellar gas, and C. an emission nebula.

A. The overall shape of the spectrum is the similar to that of the star: a continuum with some emission or absorption lines superimposed, depending on the star. The spectrum will be slightly altered in that it will contain more blue and less red light than the original stellar spectrum, due to the preferential scattering of blue light. B. The spectrum will look the same as for an unobscured star, except that there will be narrow absorption lines at certain wavelengths, corresponding to colors that can be absorbed by atoms such as calcium and sodium in the interstellar gas. C. The spectrum of an emission nebula is dominated by emission lines, most prominently the Balmer line of hydrogen, along with helium and other light elements. The continuum will be very faint or entirely undetected.

Describe the evolution of a star with a mass like that of the Sun, from the main-sequence phase of its evolution until it becomes a white dwarf.

After the main-sequence phase ends, the star will evolve toward the upper-right area of the H-R diagram as its core contracts and the outer layers expand. It becomes a red giant and will continue to expand its outer layers. This causes its luminosity to increase; the star's size can become more than 100 times its main-sequence radius. Eventually, the contracting core reaches a temperature of 100 million K, which leads to the explosive ignition of helium. When the star stabilizes, it will lose some of its outer layers as it becomes smaller, and moves back toward the main sequence region of the H-R diagram. The star will continue to fuse helium into carbon and oxygen, but for a time much shorter than the time on the main sequence. Eventually, the core will be depleted of helium, and the star once more evolves toward the upper-right area of the H-R diagram as the core contracts and the outer layers expand to an even greater extent. The outer layers of the star will be gradually blown out into space by the strong winds during this period. This leads to the formation of a planetary nebula out of the outer layers of the star. The remaining mass of the star in the core continues to contract and will eventually become a white dwarf.

Why have we learned a lot about star formation since the invention of detectors sensitive to infrared radiation?

As stars form (during the proto-star stage) they are collapsing and heating up. For much of the formation process they give off a considerable part of their emission in the infra-red. Star formation happens best in great (molecular) clouds of gas and dust, which have a lot of dust particles. Infrared radiation can penetrate the dust but visible light cannot. Thus, the ability to use infrared detectors lets us see through the dust deep into the clouds where the star formation is happening.

Suppose that, instead of being inside the Local Bubble, the Sun were deep inside a giant molecular cloud. What would the night sky look like as seen from Earth at various wavelengths?

At visible, ultraviolet, and X-ray wavelengths, the night sky would be completely black. The dust in the cloud would block out all starlight. At infrared wavelengths, we would be able to see other stars, and farther into the infrared, we would see a glow from the dust all around us. At radio wavelengths, the sky would be filled with very bright molecular line emission.

Why is it difficult to determine where cosmic rays come from?

Cosmic rays are charged particles, which means that they can be deflected by magnetic fields. As a result, they do not travel in straight lines. They curve due to the magnetic fields of Earth and those found in interstellar space. Because they travel in curved paths, we cannot easily trace them back to their point of origin.

Why do molecules, including H2 and more complex organic molecules, only form inside dark clouds? Why don't they fill all interstellar space?

Dark clouds are vulnerable to dissociation by ultraviolet light. Most of interstellar space is filled with ultraviolet light from stars, so any molecules that form there are quickly disrupted. Significant numbers of molecules can build up only in places where the ultraviolet light is blocked, and the only places in interstellar space that meet that requirement are inside dark clouds

Why do molecules, including H2 and more complex organic molecules, only form inside dark clouds? Why don't they fill all interstellar space?

Dark clouds are vulnerable to dissociation by ultraviolet light. Most of interstellar space is filled with ultraviolet light from stars, so any molecules that form there are quickly disrupted. Significant numbers of molecules can build up only in places where the ultraviolet light is blocked, and the only places in interstellar space that meet that requirement are inside dark clouds.

Suppose someone told you that she had discovered H II around the star Aldebaran. Would you believe her? Why or why not?

Don't believe it. Aldebaran is a red supergiant star with a surface temperature of about 4000 K. That is not hot enough to emit much light at the extreme ultraviolet wavelengths capable of ionizing hydrogen atoms and producing H II. Only much hotter stars can make H II regions.

Where does interstellar dust come from? How does it form?

Dust forms in the winds coming off stars as they lose mass at the end of their lives, or from condensing in the material from supernova explosions. Dust grains can also grow when they find themselves in dense environments like molecular clouds.

New stars form in regions where the density of gas and dust is relatively high. Suppose you wanted to search for some recently formed stars. Would you more likely be successful if you observed at visible wavelengths or at infrared wavelengths? Why?

Dust is an efficient absorber of visible radiation, and so star-forming regions often cannot be observed at visible light wavelengths. Infrared radiation does penetrate the dust, and so infrared observations are an essential tool for studying regions of star formation.

One way to calculate the size and shape of the Galaxy is to estimate the distances to faint stars just from their observed apparent brightnesses and to note the distance at which stars are no longer observable. The first astronomers to try this experiment did not know that starlight is dimmed by interstellar dust. Their estimates of the size of the Galaxy were much too small. Explain why.

Extinction by interstellar dust makes stars appear fainter than they would if there were no dust. Suppose an RR Lyrae variable could just barely be seen at a distance of 10,000 light-years in the absence of interstellar extinction. Now consider the case that there is enough interstellar extinction to dim the light from distant RR Lyrae stars by a factor of 10. The RR Lyrae stars at a distance of 10,000 light-years will become too faint to be detected. Only those RR Lyrae stars closer than (10)0.5 x 10, light-years = 3200 light-years will still be bright enough to be detected. Therefore, if no allowance is made for dimming by interstellar dust, astronomers will conclude that the Galaxy is only about one-third its true size. These arguments can be generalized to apply to any type of star and any amount of extinction.

Describe what happens when a star forms. Begin with a dense core of material in a molecular cloud and trace the evolution up to the time the newly formed star reaches the main sequence.

Generally, collapse leads to central heating, which eventually leads to hydrogen ignition temperatures. Specifically, when gravity exceeds pressure in the molecular cloud, material from the outer reaches of the cloud transfers gravitational potential energy into kinetic energy as it falls in towards the center of the cloud. When it reaches the center of the cloud, the kinetic energy is converted to thermal energy as the material is either added to the central Protostar or spun out into the protoplanetary disk. The protostar increases in temperature and luminosity as more material is added. Its central pressure and temperature also increase until temperatures of 10-15 million K are achieved and nuclear fusion begins. In the meantime, other processes related to conservation of angular momentum shape the material surrounding the collapsing core and limit the amount of material involved in the collapse.

The terms H II and H2 are both pronounced "H two." What is the difference in meaning of those two terms? Can there be such a thing as H III?

H II means hydrogen that has been stripped of its electron (ionized). The Roman numeral I indicates an atom with all its electrons, II means one electron removed, III means two electrons removed, and so on. There is no such thing as H IIII because that would mean hydrogen with two electrons removed. However, hydrogen only has one electron to start with. In contrast, H2 means a hydrogen molecule: two hydrogen atoms held together by a bond.

Even though neutral hydrogen is the most abundant element in interstellar matter, it was detected first with a radio telescope, not a visible light telescope. Explain why. (The explanation given in the chapter on Analyzing Starlight for the fact that hydrogen lines are not strong in stars of all temperatures may be helpful.)

Hydrogen in interstellar space outside H II regions is so cool that all of its atoms are in the lowest-energy state. At such low temperatures, the most likely transition for a neutral hydrogen atom is merely to change the orientation of the spin of its electron relative to the spin of the proton, and this does not produce any spectral lines in the visible region of the spectrum. This could not be detected until astronomers began to build radio telescopes that can detect such lower-energy radiation.

List three ways in which the exoplanets we have detected have been found to be different from planets in our solar system.

In other planetary systems, gas giant planets can reside close to the star (hot Jupiters) or in highly elliptical orbits around the star. Super-Earths and mini-Neptunes have been found around other stars, but we don't have any in our solar system. In other solar systems, small rocky planets are commonly found inside of Mercury's orbit. Some exoplanets have been discovered to orbit binary stars.

Describe the properties of the dust grains found in the space between stars.

Interstellar dust grains are typically hundredths to tenths of a micron size. They are composed of either carbon-rich (sooty) or silicate (sandy) material. When found inside dark clouds, they also have mantles of ice, composed of water, ammonia, methane, and similar materials around them. Dust grains can have temperatures from around 10-500 K.

Why can't we use visible light telescopes to study molecular clouds where stars and planets form? Why do infrared or radio telescopes work better?

Molecules can only exist in dark clouds where ultraviolet starlight is blocked out. However, these same regions are also very cold because the same photons that destroy molecules are also responsible for heating up the gas. Regions that are dark enough to have molecules are therefore also too cold to emit much at visible light wavelengths. Furthermore, the dust that is responsible for protecting the molecules would also block any visible light from escaping to our telescopes. Both of these problems are diminished or go absent at infrared and radio wavelengths. Colder material does emit infrared and radio light, and the wavelengths are much less subject to extinction by dust.

List any similarities between discovered exoplanets and planets in our solar system.

Over 500 planetary systems are similar to our solar system in having more than one planet orbiting the same star; we find a range of planetary masses among the planets in the same system. Kepler is sensitive to co-planar planetary systems, so many systems that have been discovered appear to be co-planar like our solar system.

Why is it so hard to see planets around other stars and so easy to see them around our own?

Planets only reflect light and so are much fainter than their host stars when viewed at large distances. Therefore, glare from the host stars often completely washes out the planetary light.

What causes reddening of starlight? Explain how the reddish color of the Sun's disk at sunset is caused by the same process.

Small particles, either dust grains in interstellar space or molecules in Earth's atmosphere, can absorb and scatter light. Both absorption and scattering are more efficient for blue light than for red light. As a result, they cause more extinction of blue light than of red light, so starlight that passes through a region filled with small particles will emerge redder than it was when it entered. In the case of Earth's atmosphere, the sun appears redder at sunset than midday. Since it is lower in the sky, its light must travel a longer path through the atmosphere, with a greater chance that sunlight will be scattered. Since red light is more likely to be scattered than blue light, the Sun appears increasingly red as it approaches the horizon.

Thinking about the topics in this chapter, here is an Earth analogy. In big cities, you can see much farther on days without smog. Why?

Smog is composed of tiny particles of pollutants. These particles absorb radiation very efficiently and keep the light emitted or reflected by distant objects from reaching us.

Why is star formation more likely to occur in cold molecular clouds than in regions where the temperature of the interstellar medium is several hundred thousand degrees?

Stars can form when gravity (which pulls things together) exceeds the local pressure (which tends to push atoms apart.) Pressure is higher in hot material and lower in cold material, so colder clouds put up less resistance to gravity and can collapse more readily. Also, at lower temperatures, molecules form. Because molecules are heavier than individual atoms, they move around more slowly and can congregate in a small volume of space, increasing the local matter density and thereby the local gravity.

Describe how the T Tauri star stage in the life of a low-mass star can lead to the formation of a Herbig-Haro (H-H) object?

Tauri stars show a stellar wind, a flow of atomic particles away from the star. While the star is still surrounded by an accretion disk around its equator, the wind is stopped in that direction, and it emerges far more effectively in two cones or jets, perpendicular to the accretion disk. As these jets plow into the surrounding material, they can occasionally collide with a somewhat-denser lump of gas nearby, transferring energy and exciting its atoms, causing them to emit light. These glowing regions are called Herbig-Haro (HH) objects, after the two astronomers who discovered them. There are often two such objects, one on each side of the star where the jets emerge. 6. Look at the four stages shown in Figure 21.8 Formation of a Star. In which stage(s) can we see the star in visible light? In infrared radiation? Visible light would be the later (last) stage once much of the dust has been driven away and the star is on the main sequence. Infrared radiation gives insight primarily in the final three stages. 7. The evolutionary track for a star of 1 solar mass remains nearly vertical in the H-R diagram for a while (see Figure 21.12 Evolutionary Tracks for Contracting Protostars). How is its luminosity changing during this time? Its temperature? Its radius? In this vertical region on the diagram, luminosity is dropping while the surface temperature remains constant. This is a stage where the material of the star is falling inward without any hindrance and since the star has less and less surface area with which to give off radiation, its luminosity is decreasing together with its radius. 8. Two protostars, one 10 times the mass of the Sun and one half the mass of the Sun are born at the same time in a molecular cloud. Which one will reach the main sequence stage, where it is stable and getting energy from fusion, first? As you can see in Figure 21.12 Evolutionary Tracks for Contracting Protostars, the more massive a star is, the more quickly it goes through each stage of being a protostar. Thus, the 10 solar mass star will become a real star (one that supplies its energy through fusion) first. 9. Compare the scale (size) of a typical dusty disk around a forming star with the scale of our solar system. The dusty disks range in size from 10 to 1000 AU. Jupiter's orbit is about 10 AU wide. The orbit of Pluto has a diameter of 80 AU. The outer diameter of the Kuiper belt of smaller icy bodies is about 100 AU. Eris' orbit has an average diameter of about 136 AU. 10. Why is it so hard to see planets around other stars and so easy to see them around our own? Planets only reflect light and so are much fainter than their host stars when viewed at large distances. Therefore, glare from the host stars often completely washes out the planetary light. 11. Why did it take astronomers until 1995 to discover the first exoplanet orbiting another star like the Sun? The first discovery of an exoplanet took place by finding the changing Doppler shift in the spectrum of the star as the planet moved in orbit around it. To measure the really small variations in the spectra of stars in a stable way over long periods of time, astronomers had to build very precise spectrometers. 12. Which types of planets are most easily detected by Doppler measurements? By transits? The Doppler technique measures the motion of the star caused by the pull of one or more planets around it. The gravitational force is proportional to the mass of the planet and inversely proportional to the square of the separation. So the easiest planets to detect with this method are massive and close to the star. That's why the hot Jupiters were found first. The transit probability (the chance that the orbit will bring the planet in front of the star for a transit) is greatest for close-in planets. The size of the planet must be big enough to give a measureable decrease in the brightness of the star. Since astronomers need to wait for three transits before they feel comfortable confirming their observation, that means that the longer the planet takes to orbit its star, the longer it will take to confirm the existence of a transiting planet. So this method works best for planets of larger size, orbiting close to their stars. 13. List three ways in which the exoplanets we have detected have been found to be different from planets in our solar system. In other planetary systems, gas giant planets can reside close to the star (hot Jupiters) or in highly elliptical orbits around the star. Super-Earths and mini-Neptunes have been found around other stars, but we don't have any in our solar system. In other solar systems, small rocky planets are commonly found inside of Mercury's orbit. Some exoplanets have been discovered to orbit binary stars. 14. List any similarities between discovered exoplanets and planets in our solar system. Over 500 planetary systems are similar to our solar system in having more than one planet orbiting the same star; we find a range of planetary masses among the planets in the same system. Kepler is sensitive to co-planar planetary systems, so many systems that have been discovered appear to be co-planar like our solar system. 15. What revisions to the theory of planet formation have astronomers had to make as a result of the discovery of exoplanets? We now understand that planets can migrate in the protoplanetary disk through gravitational friction or drag. For example, Jupiters can migrate inward and be quite close to their stars (hot Jupiters). We now understand that planet formation is more chaotic and less orderly than we imagined. Instead of all the planets orbiting in one plane and in the same direction, we now see some planets orbiting at right angles to the plane of the other planets or even moving backward. We also learned that it is possible to have stable planets orbiting a system of two stars. 16. Why are young Jupiters easier to see with direct imaging than old Jupiters? Young Jupiters have more internal heat from the process of accretion and formation. This energy is radiated as infrared, and so young Jupiters will be brighter infrared sources. Since the planet cools with time, younger Jupiters are more luminous and easier to see with direct imaging techniques. Thought Questions 17. A friend of yours who did not do well in her astronomy class tells you that she believes all stars are old and none could possibly be born today. What arguments would you use to persuade her that stars are being born somewhere in the Galaxy during your lifetime? There are many different ways to answer this question. For example, we see some very luminous stars that are radiating energy at such a prodigious rate that they will use all their fuel in a few million years. These stars must have formed in the last few million years, which is very recent, given that the universe itself is about 14 billion years old. Unless we live in a unique time that just happens to be within a few million years of when this type of star formed (and astronomers never like to think we live in such a unique time), then new stars must be forming now, so that in a few million years, highly luminous stars will still be present in the Galaxy. Also, we observe regions of gas and dust where it appears that star formation is taking place right now, such as the Orion molecular cloud region. According to our models, some of these stars are less than 1 million years old. While we cannot actually see stars forming inside giant molecular clouds, observations with the Hubble Space Telescope have shown us a number of protostars and disks at the edges or in cleared-out regions of dusty clouds that show evidence of being extremely young. We also see stars in all of their various stages of evolution, including many that are just beginning to form and many that are nearing the ends of their lifetimes. In much the same way, when you see human beings of all ages, you assume the birth-growth-death cycle continues and that you are not looking at the last generation that will ever be born. 18. Observations suggest that it takes more than 3 million years for the dust to begin clearing out of the inner regions of the disks surrounding protostars. Suppose this is the minimum time required to form a planet. Would you expect to find a planet around a 10-MSun star? (Refer to Figure 21.12 Evolutionary Tracks for Contracting Protostars.) According to Figure 21.12 Evolutionary Tracks for Contracting Protostars, it takes only about 100,000 years for a 10-MSun star to reach the main sequence. If planets take 3 million years to form, which seems to be the case from observations of disks around lower mass stars, then it is highly unlikely that any planets could form during the pre-main-sequence phase of evolution of a 10-MSun star. Since main-sequence stars do not have disks around them with enough material to form planets, it also seems impossible for planets to form during the main-sequence phase of evolution. Therefore, we do not expect to find planets around massive stars. 19. Suppose you wanted to observe a planet around another star with direct imaging. Would you try to observe in visible light or in the infrared? Why? Would the planet be easier to see if it were at 1 AU or 5 AU from its star? The planet will be easier to see if it is farther away from its parent star; closer planets would be even more likely to be lost in the glare of the parent star. The ratio of the brightness of the planet to the brightness of the star is very small at all wavelengths, but will be larger in the infrared than in visible light so observations should be made in the infrared. 20. Why were giant planets close to their stars the first ones to be discovered? Why has the same technique not been used yet to discover giant planets at the distance of Saturn? A complete orbit is required in order to model or detect the presence of a planet with the radial velocity technique. From Kepler's laws, we know that the closest planets have short periods. In the case of the close-in giant planets (so-called "hot Jupiters"), these orbits can be as short as a few days. It does not take long to observe over all phases of these short orbits. However, Saturn has a 30-year orbit. Thus, we would have to observe Saturn for at least 30 years in order to detect that planet. A further complication is that when the planet is far away, the gravitational tug on the star is weaker and harder to detect. Saturn only induces a velocity in the sun of 2.7 m/s. Doppler surveys with sufficient precision to measure this signal only began around 2003. It is also worth noting here that even though small planets may be close to their parent stars and have short periods, their small masses make it much more difficult to observe either gravitational tugs or transits; thus, it is much easier to detect massive planets in short-period orbits around their host stars than it is to detect any low-mass planets or to detect high-mass planets at great distance from the star. 21. Exoplanets in eccentric orbits experience large temperature swings during their orbits. Suppose you had to plan for a mission to such a planet. Based on Kepler's second law, does the planet spend more time closer or farther from the star? Explain. The planet spends more time farther from the star. Recall from the chapter on Orbits and Gravity that Kepler's second law says that in equal intervals of time, a line between the Sun and a planet sweeps out equal areas. To do this, the planet has to move faster when it is close to the star and slower when it is farther away. So, unless the planet has an extreme tilt in the hemisphere you are visiting, or you are planning to stay through a periastron, it makes much more sense to plan for longer periods of relatively cooler temperatures. ​Figuring for Yourself ​22. When astronomers found the first giant planets with orbits of only a few days, they did not know whether those planets were gaseous and liquid like Jupiter or rocky like Mercury. The observations of HD 209458 settled this question because observations of the transit of the star by this planet made it possible to determine the radius of the planet. Use the data given in the text to estimate the density of this planet, and then use that information to explain why it must be a gas giant. Density is defined as mass divided by volume. The mass of HD 209458 is about 70% the mass of Jupiter and the radius is about 35% larger than Jupiter. For Jupiter, the radius = 70,000 km, the mass of HD 209458 = 1.898 × 1027 kg, and the density = 1326 kg/m3. Mass of HD 209458 = 0.7 × mass of Jupiter = 1.33 × 1027 kg. Radius of HD 209458 = 1.35 × radius of Jupiter = 9.45 × 107 m. Volume of HD 209458 = 4/3πr3 = 4/3 π(9.45 × 107 m)3 = 3.53 × 1024 m3. Density of HD 209458 = mass/volume = 1.33 × 1027 kg/3.53 × 1024 m3 = 377 kg/m3. This is substantially lower than the density of water! Compare this to the mean density of Jupiter: 1326 kg/m3. 23. An exoplanetary system has two known planets. Planet X orbits in 290 days and Planet Y orbits in 145 days. Which planet is closest to its host star? If the star has the same mass as the Sun, what is the semi-major axis of the orbits for Planets X and Y? Planet Y is in the shorter period orbit and therefore resides closest to the star. For Planet X, the period is 0.79 years, so (using Kepler's laws) the semi-major axis is p2 = a3. So p = 290 d × (1 y/365.25 d) = 0.79 y, p2 = (0.79 y)2 = 0.624 y2, 1 y2 = 1 AU3, 0.624 AU3 = a3, a = 0.85 AU. ; for Planet Y, the period is 0.40 years so the semi-major axis is p2 = a3. So p = 145 d × (1 y/365.25 d) = 0.40 y, p2 = (0.40 y)2 = 0.16 y2, 1 y2 = 1 AU3, 0.16 AU3 = a3, a = 0.54 AU. 24. ​Kepler's third law says that the orbital period (in years) is proportional to the square root of the cube of the mean distance (in AU) from the Sun (Pµ a1.5). For mean distances from 0.1 to 32 AU, calculate and plot a curve showing the expected Keplerian period. For each planet in our solar system, look up the mean distance from the Sun in AU and the orbital period in years and overplot these data on the theoretical Keplerian curve. 25.​Calculate the transit depth for an M dwarf star that is 0.3 times the radius of the Sun with a gas giant planet the size of Jupiter.​ This is easily detected with Kepler or even with ground-based observations. 26.​If a transit depth of 0.00001 can be detected with the Kepler spacecraft, what is the smallest planet that could be detected around a 0.3 Rsun M dwarf star?​ Put all radii in units of the Sun, then:​ Earth is about 0.001 times the radius of the Sun. 27. What fraction of gas giant planets seems to have inflated radii? Based on Figure 21.25 Exoplanets with Known Densities., there are 3-4 times more gas giant planets with inflated radii that are bigger than Jupiter.

Describe how the 21-cm line of hydrogen is formed. Why is this line such an important tool for understanding the interstellar medium?

The 21-cm line is formed when hydrogen atoms in which the proton and electron are aligned "flip" so that the proton and electron are anti-aligned. Hydrogen atoms that are in the anti-aligned state will be excited into the aligned state by collisions, and will subsequently emit a photon with a wavelength of 21 cm, giving rise to the line. The line is important because it is produced by neutral hydrogen everywhere. Since cold hydrogen atoms make up the largest part of the interstellar medium, the 21-cm line allows us to study the most common component of interstellar gas.

Which types of planets are most easily detected by Doppler measurements? By transits?

The Doppler technique measures the motion of the star caused by the pull of one or more planets around it. The gravitational force is proportional to the mass of the planet and inversely proportional to the square of the separation. So the easiest planets to detect with this method are massive and close to the star. That's why the hot Jupiters were found first. The transit probability (the chance that the orbit will bring the planet in front of the star for a transit) is greatest for close-in planets. The size of the planet must be big enough to give a measureable decrease in the brightness of the star. Since astronomers need to wait for three transits before they feel comfortable confirming their observation, that means that the longer the planet takes to orbit its star, the longer it will take to confirm the existence of a transiting planet. So this method works best for planets of larger size, orbiting close to their stars.

Give several reasons the Orion molecular cloud is such a useful "laboratory" for studying the stages of star formation.

The Orion cloud is nearby (astronomically speaking) so we have a close-up view of star formation processes. We know the distance to Orion's star-forming regions, so we know how luminous objects are from simply measuring the brightness of the sources. We can see a number of stages in the formation process of stars and protoplanetary disks in the region. Orion is also close to the celestial equator so it can be observed from all the large observatories in the Northern and Southern Hemispheres.

We can detect 21-cm emission from other galaxies as well as from our own Galaxy. However, 21-cm emission from our own Galaxy fills most of the sky, so we usually see both at once. How can we distinguish the extragalactic 21-cm emission from that arising in our own Galaxy? (Hint: Other galaxies are generally moving relative to the Milky Way.)

The answer is by using the Doppler effect. External galaxies are moving relative to ours, and thus, their emission is shifted to a slightly different wavelength. In a spectrum, this makes the 21-cm emission from other galaxies distinguishable from the 21-cm emission from ours.

Compare the scale (size) of a typical dusty disk around a forming star with the scale of our solar system.

The dusty disks range in size from 10 to 1000 AU. Jupiter's orbit is about 10 AU wide. The orbit of Pluto has a diameter of 80 AU. The outer diameter of the Kuiper belt of smaller icy bodies is about 100 AU. Eris' orbit has an average diameter of about 136 AU.

Why did it take astronomers until 1995 to discover the first exoplanet orbiting another star like the Sun?

The first discovery of an exoplanet took place by finding the changing Doppler shift in the spectrum of the star as the planet moved in orbit around it. To measure the really small variations in the spectra of stars in a stable way over long periods of time, astronomers had to build very precise spectrometers.

Stars form in the Milky Way at a rate of about 1 solar mass per year. At this rate, how long would it take for all the interstellar gas in the Milky Way to be turned into stars if there were no fresh gas coming in from outside? How does this compare to the estimated age of the universe, 14 billion years? What do you conclude from this?

The interstellar medium contains about 7 billion solar masses of gas, so at a rate of 1 solar mass per year of consumption, it would take about 7 billion years to turn all the gas into stars if there were no fresh gas supplies. This is less than the age of the universe, which suggests that either there must be fresh gas supplies, or that we are living at a special time when the supply of gas in the Milky Way is just about to run out, or that the rate of star formation is not constant.

The 21-cm line can be used not just to find out where hydrogen is located in the sky, but also to determine how fast it is moving toward or away from us. Describe how this might work.

The key here is to use the Doppler effect. The 21-cm line can be redshifted or blueshifted just like any other radiation, and this shift can be used to measure the velocity of the hydrogen that is emitting. If the line is shifted slightly redward of 21 cm, the hydrogen is moving away from us. If it is shifted blueward, the hydrogen is moving toward us. The amount of shift indicates the speed.

Astronomers recently detected light emitted by a supernova that was originally observed in 1572, just reaching Earth now. This light was reflected off a dust cloud; astronomers call such a reflected light a "light echo" (just like reflected sound is called an echo). How would you expect the spectrum of the light echo to compare to that of the original supernova?

The light echo spectrum should look similar to that of the original supernova, with the exception that it will be bluer than the original. That is because the blue light is be scattered more efficiently than the red light in the dust cloud.

According to the text, a star must be hotter than about 25,000 K to produce an H II region. Both the hottest white dwarfs and main-sequence O stars have temperatures hotter than 25,000 K. Which type of star can ionize more hydrogen? Why?

The main-sequence O star is much more luminous than the white dwarf and produces more ultraviolet photons capable of ionizing hydrogen. Therefore, the O main-sequence star can ionize more hydrogen than the white dwarf even though both have the same temperature.

Suppose you wanted to observe a planet around another star with direct imaging. Would you try to observe in visible light or in the infrared? Why? Would the planet be easier to see if it were at 1 AU or 5 AU from its star?

The planet will be easier to see if it is farther away from its parent star; closer planets would be even more likely to be lost in the glare of the parent star. The ratio of the brightness of the planet to the brightness of the star is very small at all wavelengths, but will be larger in the infrared than in visible light so observations should be made in the infrared.

Why do nebulae near hot stars look red? Why do dust clouds near stars usually look blue?

The red color of nebulae comes from the Balmer line produced when ionized hydrogen captures an electron and becomes neutral again. Nebulae are filled with ionized hydrogen that combine with an electron to produce this line. Dust clouds are blue because the light we see from them is scattered starlight. Blue light scatters more than red light, so the light we see coming off the clouds is bluer than the light from the stars.

Suppose that, instead of being inside the Local Bubble, the Sun were inside an H II region. What would the night sky look like at various wavelengths?

The sky would look much the same as it does now at X-ray and most other wavelengths, but it would also be filled with bright light at particular visible wavelengths, particularly the red color corresponding to the Balmer line. The sky would glow red.

A friend of yours who did not do well in her astronomy class tells you that she believes all stars are old and none could possibly be born today. What arguments would you use to persuade her that stars are being born somewhere in the Galaxy during your lifetime?

There are many different ways to answer this question. For example, we see some very luminous stars that are radiating energy at such a prodigious rate that they will use all their fuel in a few million years. These stars must have formed in the last few million years, which is very recent, given that the universe itself is about 14 billion years old. Unless we live in a unique time that just happens to be within a few million years of when this type of star formed (and astronomers never like to think we live in such a unique time), then new stars must be forming now, so that in a few million years, highly luminous stars will still be present in the Galaxy. Also, we observe regions of gas and dust where it appears that star formation is taking place right now, such as the Orion molecular cloud region. According to our models, some of these stars are less than 1 million years old. While we cannot actually see stars forming inside giant molecular clouds, observations with the Hubble Space Telescope have shown us a number of protostars and disks at the edges or in cleared-out regions of dusty clouds that show evidence of being extremely young. We also see stars in all of their various stages of evolution, including many that are just beginning to form and many that are nearing the ends of their lifetimes. In much the same way, when you see human beings of all ages, you assume the birth-growth-death cycle continues and that you are not looking at the last generation that will ever be born.

Describe the characteristics of the various kinds of interstellar gas (HII regions, neutral hydrogen clouds, ultra-hot gas clouds, and molecular clouds).

These are the characteristics of various kinds of interstellar gas. H II regions consist of ionized hydrogen and other elements. They have temperatures close to 10,000 K, and glow brightly in visible light due to fluorescence. H II regions are produced by the ultraviolet light of hot, young stars. • Neutral hydrogen clouds are regions where the hydrogen is neutral, that is, not ionized. They can have a wide range of temperatures and densities. Temperatures range from about 100-8000 K, and densities from about 0.1-100 atoms per cm3. This is the dominant phase of the interstellar medium. • Ultra-hot gas is gas at temperatures of millions of degrees. It is produced by supernova shockwaves that heat up gas as they propagate through the interstellar medium. It emits X-ray and ultraviolet light. • Molecular clouds are regions where the ultraviolet light of stars has been blocked out, allowing the hydrogen atoms to combine to form H2. They are cold, with temperatures around 10 K, and dense (in comparison to other phases of the interstellar medium), with densities of hundreds to thousands of atoms per cm3. Within them, atoms can form complex chemical compounds.

What revisions to the theory of planet formation have astronomers had to make as a result of the discovery of exoplanets?

We now understand that planets can migrate in the protoplanetary disk through gravitational friction or drag. For example, Jupiters can migrate inward and be quite close to their stars (hot Jupiters). We now understand that planet formation is more chaotic and less orderly than we imagined. Instead of all the planets orbiting in one plane and in the same direction, we now see some planets orbiting at right angles to the plane of the other planets or even moving backward. We also learned that it is possible to have stable planets orbiting a system of two stars.

Why are young Jupiters easier to see with direct imaging than old Jupiters?

Young Jupiters have more internal heat from the process of accretion and formation. This energy is radiated as infrared, and so young Jupiters will be brighter infrared sources. Since the planet cools with time, younger Jupiters are more luminous and easier to see with direct imaging techniques.

What fraction of gas giant planets seems to have inflated radii?

there are 3-4 times more gas giant planets with inflated radii that are bigger than Jupiter.