PH 104 AstronomyMidterm Exam Spring 2011 NAME: _______SOLUTION______________ Answer True or False to each statement (1 point each) 1. __F__ There is a solar eclipse of some kind every new moon. There is a new moon every 29 days. Not every new moon causes a solar eclipse. 2. __T__ A blue star has a higher surface temperature than a red star. 3. __T__ A planet (or comet) will speed up as it approaches the Sun. This is a consequence of elliptical orbits and the law of “equal areas in equal times.” 4. __F__ Constellations are close clusters of stars, all at about the same distance from the Sun. The stars appear close together as observed in our sky, but they can be quite far apart in the galaxy. 5. __F__ The larger a wave's wavelength, the greater its energy. Shorter wavelength is higher frequency is greater energy. 6. __T__ A telescope design that uses a lens and no mirrors is a refractor. 7. __T__ Stars do not twinkle; the instability of the atmosphere causes this effect. 8. __T__ The larger the parallax shift, the closer an object is to us. 9. __F__ Galaxies look the same whether viewed in visible or X-ray wavelengths. X-ray images of galaxies reveal much different structure than we can see in visible. 10. __F__ In the sky, right ascension is measured in degrees north or south of the celestial equator. Right ascension is measured in hours and minutes, and corresponds to “East – West” directions. 11. __T__ In the Doppler effect, a red shift of spectral lines means the source is receding from us. 12. __T__ As viewed from Earth, the Sun and Moon have about the same angular diameter. 13. __T__ According to Newton's second law, if you double the force acting on an object then the acceleration will double. 14. __T__ An emission line results from an electron falling from a higher to lower energy orbital around its atomic nucleus. 15. __T__ Spectroscopy of a star can reveal its temperature, composition, and line-of-sight motion. Select the one best answer (1 point each) 16. A wave’s velocity is equal to the product of a. The frequency of the wave times the period of the wave. b. The period of the wave times the energy of the wave. c. The amplitude of the wave times the frequency of the wave. d. The frequency of the wave times the wavelength of the wave. e. The amplitude of the wave times the wavelength of the wave. As has been stated many times in class, in equation form, speed c = λf. 17. In an annular eclipse a. The Sun is totally blocked by the Moon. b. The Moon is totally blocked by the Earth. c. The Sun appears as a thin, bright ring. d. The Moon appears as a thin, bright ring. e. The Sun is partially blocked by the Earth. Annular eclipses of the Sun happen when Moon is slightly farther from Earth and its shadow has a slightly smaller angular diameter than that of the Sun. 18. Which type of telescope did Galileo turn skyward in 1610? a. A refractor. b. A Newtonian reflector. c. A Cassegrain reflector. d. A Coudé reflector. e. A prime focus reflector. Galileo’s telescope used lenses to create a focused image by refraction (bending of light through glass). 19. The Sun’s observed spectrum is a. A continuum with no lines, as shown by the rainbow. b. A continuum with emission lines. c. Only absorption lines on a black background. d. A continuum with absorption lines. e. Only emission lines on a black background. Due to low density gas in the Sun’s surface and in our atmosphere, absorption lines are observed. 20. The light-gathering power of a telescope varies a. Directly with the diameter of the lens or mirror. b. As the inverse square of the diameter of the lens or mirror. c. As the square of the diameter of the lens or mirror. d. To the fourth power of the diameter of the lens or mirror. e. With the square of the area of the lens or mirror. Larger telescopes collect more light. The amount varies with the area of the primary optical element. 21. If a wave's frequency doubles, its wavelength a. Is halved. b. Is also doubled c. Is unchanged, because the speed of light, c, is a constant. d. Is now four times (4×) longer. e. Becomes sixteen times (16×) longer. Frequency and wavelength are inversely proportional – the opposite effect happens. 22. If new moon fell on March 2nd, what is the Moon's phase on March 14th? a. Waxing crescent b. First quarter c. Waxing gibbous technically correct answer d. Full worth one-half point e. Waning crescent The lunar cycle is 29 days. At about 14 days Moon is full, so 12 days later is between ¼ and full. 23. A star with a declination of +60.0 degrees will be a. East of the vernal equinox. b. West of the vernal equinox c. North of the vernal equinox d. South of the vernal equinox. The position of the Sun at equinox is on the celestial equator. Angles toward the North are plus, and toward South are minus. 24. The observed spectral lines of a star are all shifted towards the red end of the spectrum. Which of the following statements is true? a. This is an example of the photoelectric effect. b. This is an example of the Doppler effect. c. This is explained by Kirchhoff’s second law of spectroscopy. d. The star is not rotating. e. The star has a radial velocity towards us. Red and blue shifts are caused by the Doppler effect. A red shift mean motion away from us. 25. A planet whose semi-major axis from the Sun is 3 AU (astronomical units) would have an orbital period of how many Earth-years? a. 3 27 b. 3 c. d. 9 e. 81 Kepler’s law is P2 = a3, so when a = 3 then a3 = 27 = P2. Thus P = √27. Complete the following statements by filling in the blank (1 point each) 26. __Kepler’s_ three laws of planetary motion allow us to predict the future motion of planets. 27. The apparent annual path the Sun takes through the sky is called the __ecliptic__. 28. The energy of a photon of electromagnetic radiation depends on its __frequency__. 29. According to Wein's law, the wavelength of the peak energy emitted by a blackbody will be __decreased or halved__ if the temperature of the blackbody is doubled. This space was left blank for you to work out a solution. 30. The mean distance between the Earth and Sun is called the _astronomical_ _unit_. 31. A mirror must be _curved or concave_ in shape to reflect the light back to a focus. 32. According to Newton's three laws, the planets orbit the Sun due to __the force of gravity__. 33. Over the course of the year, the Sun's noon altitude changes by a total of __47__ degrees. The Earth’s axis of rotation for spinning is 23.5º away from the Sun. This makes the ecliptic “tilted” by 23.5º North – to – South from the celestial equator. As a result, the Sun moves from an extreme Northward position that is 23.5º North of the equator to an extreme Southward position that is 23.5º South of the equator (or −23.5º in declination). That is a total change in altitude (declination) of 47º. 34. Stars that appear blue or white in color are __hotter__ than our yellow Sun. 35. An FM station broadcasts at a frequency of 100 MHz. The wavelength is __three__ meters. Also accepted: 0.003 kilometers (Note: a Megahertz (MHz) is one million Hertz.) The speed of light is c = 300,000,000 m/s. Radio waves are light waves. Using c = λf, you divide both sides by frequency to solve for wavelength. c/f = λ. (300,000,000 m/s)/(100,000,000 1/s) = λ Note: one Hertz is equivalent to one 1/s. 3m=λ Short essay and problem-solving questions (2 points each) 36. State the relationships between the frequency, wavelength, and amount of energy carried in a photon, for an electromagnetic wave. The wave speed is equal to the wave frequency times the wavelength. Therefore the frequency is inversely proportional to the wavelength; the frequency of a light wave is equal to the speed of light divided by the wavelength. If wavelength gets longer frequency gets smaller and if wavelength gets shorter frequency gets bigger. Likewise, wavelength is inversely proportional to frequency. If frequency gets bigger wavelength gets shorter and if frequency gets smaller wavelength gets longer. The energy carried in one photon is directly proportional to the frequency of the wave. A small frequency wave has low energy photons and a high frequency wave has high energy photons. Likewise, high energy waves have short wavelengths and low energy waves have long wavelength. 37. Give three reasons the largest telescopes are all reflectors and not refractors. • Large mirrors only need one perfect optical surface, while lenses always need two perfect optical surfaces. • Large mirrors can be supported from behind, which keeps their shape stable. Large lenses must be supported on the side, and can bend, warp, or break under their own weight. • Also, because large mirrors can be supported from behind and can be made of thin and flexible materials, the shape of the optical surface can be deformed to compensate for disturbances in Earth’s atmosphere. This is called adaptive optics. Lenses cannot be deformed adaptively like this. • It is extremely difficult and expensive to cast a giant, perfect optical glass for a large lens. It is relatively easier and less expensive to coat a perfectly molded surface with a thin reflecting film and create a giant mirror. • Lenses suffer from chromatic refraction – different colors refract by different amounts and the only solution is an extremely long focal length. Mirrors do not suffer from chromatic refraction. • A large mirror can be constructed of many “tiles” of smaller mirrors that together give a curved surface to focus light to a point. You cannot construct a refracting lens out of “tiles” of small lenses. 38. The speed of light is 300,000 km/s. How far away is a spacecraft if a radio signal it sends out takes 10 minutes to reach Earth? (hint: there are 60 seconds in one minute) Speed is equal to distance traveled divided by the time of travel. The speed and time are given; solve for the distance. Note the speed is given in units of kilometers per second. To make progress, first convert the time in minutes into time in seconds: 10 min × (60 s/1 min) = 600 s Since speed = distance / time, distance = speed × time Distance = (300,000 km/s)(600 s) Distance = 180,000,000 km That is one hundred eighty million kilometers. 39. Explain why Earth has seasons. For two points extra credit, also explain why seasons always fall in the same calendar months every year. Seasons result from reoccurring changes in the average temperature at Earth’s surface, in the area where you live. Earth experiences this reoccurring climate change at latitudes above and below about 25° from the equator because the axis along which Earth spins is not aligned vertically with respect to the direction in which Earth orbits the Sun. Since the spin axis is tilted, the number of hours that a land mass is exposed to sunlight changes throughout the year, as the direction which Earth’s spin axis points is either more toward or more away from the direction to the Sun. Your area has less hours of sunlight when the geographic pole closest to you points away from the Sun. This relative orientation also results in the sunlight you receive being spread out over a larger area. The combination of less concentrated sunlight plus a shorter daily exposure to sunlight results in less total energy being absorbed by your land mass. As a direct result, your land-mass is not as warm. The average temperature is colder. You experience winter. At this same time, land masses the opposite pole have more hours of sunlight in a day and the relative orientation of Earth, spin-axis, and Sun results in received sunlight being concentrated over a smaller area. The combination of longer exposure and more concentrated sunlight results in more total energy being absorbed and higher average temperature. The opposite pole experiences summer. In between these annual extremes are the equinoxes, when the spin axis is parallel to the direction toward the Sun and both poles have equal hours of sunlight during their day. Extra credit: Humans have charted our calendar based on the cycle of Earth’s motion around the Sun. One calendar year is the time it takes Earth to return to the same relative position with respect to the Sun. Due to the way Earth orbits and spins, our spin axis (pole direction) precesses, and over the course of thousands of years the direction that North points, relative to the stars in our sky, changes. If our calendar were based on the position of Earth with respect to the stars (a sidereal year), then the seasons would slowly move forward through the calendar as Earth precesses. Eventually spring would come in September and summer in December, in the Northern hemisphere. However, since our calendar follows the solar year, we always find ourselves in the same relative orientation with respect to the Sun at the same calendar dates from year to year. As a consequence, we observe the constellations to slowly move forward through the calendar. Orion is a winter constellation presently, for the Northern hemisphere. In about 7,000 years Orion will be a summer constellation. 40. The star Veritas is observed to have a blue-white light of peak wavelength 450 nm, and the star Erasmus is observed to have a deep reddish light of peak wavelength 720 nm. a. Which star is hotter? Veritas is hotter. Hotter objects emit energy at shorter wavelengths. Blue and white include shorter wavelengths than red alone. b. How many times greater is the temperature of the hotter star compared to the colder star? To solve we apply Wien’s law: peak wavelength is inversely proportional to temperature. Let Veritas have wavelength λV and temperature TV, and let Erasmus have wavelength λE and temperature TE. Applying Wien’s law: (λE/ λV) = (TV/ TE) (720 nm)/(450 nm) = (TV/ TE) 1.6 = (TV/ TE) Veritas is 1.6 times hotter than Erasmus. c. Hotter blackbodies also emit more power into radiant energy. How many times more total energy per second does the hotter star radiate compared to the colder star? (hint: use the x2 button two times to raise a number to the fourth power). To solve we apply Stefan’s law: power of radiant energy emitted is proportional to temperature to the fourth power. Power emitted by Veritas = (TV)4 Power emitted by Erasmus = (TE)4 (PV/PE) = (TV)4/(TE)4 From the previous part, we know TV = 1.6TE, so (PV/PE) = (1.6 TE)4/(TE)4 (PV/PE) = (1.6)4 = 6.55 Veritas, at 1.6 times hotter, also emits about 6.6 times more radiant energy (light). Challenge problem (worth three points extra credit): Explain how astronomers can measure the mass of our Sun. Newton’s law of gravitation provides a mathematical model we can use to find the Sun’s mass. Newton’s law states that the size of the gravitational force between two objects varies directly with the product of the objects masses and inversely with the square of their distance apart. By measuring how objects fall to the Earth, we can calculate the mass of the Earth. From Kepler’s laws of planetary motion, we can calculate the distance between Earth and Sun (semi-major axis). When we know the size of the force of gravity of Sun on Earth, we can solve Newton’s law to calculate the Sun’s mass. Using Newton’s second law of motion, we figure out the size of the force of gravity from the motion of Earth. If you are interested, the Sun’s mass is M = 4π2a3/GP2 (a and P from Kepler’s laws) to within 3%.