Midterm Exam 2 Solutions - Solar System Astronomy | ASTR 1030, Exams of Astronomy

Download Midterm Exam 2 Solutions - Solar System Astronomy | ASTR 1030 and more Exams Astronomy in PDF only on Docsity! SOLUTIONS ASTR 1030 - Solar System Astronomy Midterm 2 Formulae PLEASE: • Do not begin until all the midterms are handed out and the instructor announces the start. • Print your name clearly. Answers will be on the web: http://lasp.colorado.edu/~ergun/ASTR1030/. 18 42 40 Acceleration, Velocity, Distance, and Time: Momentum:. Force: Angular Momentum: Ratios:. Energy and Power: Mass and Energy: Force of Gravity: Acceleration from Gravity: Orbital Speed: Kepler’s Third Law: Newton’s Form of Kepler’s 3rd Law: Escape velocity.: Surface Area: Radioactive decay: v x2 x1– t2 t– 1 --------------- dx dt ----- a v2 v1– t2 t1– -------------- dv dt -----= =⎝ ⎠ ⎛ ⎞;= = p mv= F ma= I mvr m2π f rotr 2 = = Object1 ModelSize Object1 ActualSize --------------------------------------------- Object2 ModelSize Object2 ActualSize ---------------------------------------------= Power Energ y Time⁄= E mc 2 = FGravity GM1M2 r 2⁄= aGravity GMBig r 2⁄= G 6.67 10 11–× m3 kg s2⋅⁄= v 2πa p --------= p 2 years( ) a3 AU( )= p 2 4π2 GMBig ---------------a 3 = vescape 2GM RPlanet ---------------= A 4πR2= M Mo⁄ 1 2⁄( ) t τ1 2⁄⁄ = Sidereal Day (applies only if revolution and rotation have the same sense, use -1 otherwise):. Synodic Periods: Light Wave: Photons: Wien’s law: Stefan-Boltzmann Law Doppler Shift: Small angle formula: Diffraction Limit: Radius of a Star’s Wobble: Center of Mass: NDays yr⁄ Sidereal NDays yr⁄ Solar 1+= TDay Sidereal TDay Solar NDays yr⁄ Solar NDays yr⁄ Sidereal -------------------- ⎝ ⎠ ⎜ ⎟ ⎛ ⎞ = psynodic 1 yr psidereal psidereal 1 yr– --------------------------------⎝ ⎠ ⎛ ⎞= f λ× c= E hf hc λ ----= h 6.626 10 34–× J s⋅= = 1eV 1.6 10 19– J×= λ peak 2.9 10 6× T ------------------- Knm o( )= P A --- σT4 σ 5.7 10 8–× W m 2 K 4 ------------= = v c Δλ λo ------= Δλ λshifted λrest–= α s 2πd -------- 360°×= θa b α 2.5 105× λ dt ---⎝ ⎠ ⎛ ⎞ arcsec= aS vS p 2π⁄= aPMP aSMS=ASTR 1030, Midterm 2, October 24, 2008. Page: 1 SOLUTIONS Part 1: Short Answer (Give a short answer to each question). (1) (4 pts) Name four major differences between Terrestrial planets and Jovian planets. Size (Jovian are larger) Location (Jovian are outside the frost line) Density (Jovian are less dense) (a) _______________________________________________ (b) ______________________________________________ Composition (Jovian are gas) Rings (Jovian) Moons (Jovian have more) Temperature (Jovian are cooler) (c) _______________________________________________ (d) ______________________________________________ (2) (2 pts) List two ways that telescopes can increase sensitivity to faint objects. Larger apertures gather more light. Exposure time can be increased to many hours. (a) _______________________________________________ (b) ______________________________________________ (3) (1 pt) What type of light is emitted from the human body as thermal radiation (310 oK)? Infrared _______________________________________________ (4) (1 pt) What type of light is the most intense emission from large stars such as Rigel (20,000 oK)? Ultraviolet _______________________________________________ (5) (1 pt) What type of light is most commonly used for communications on Earth? Radio _______________________________________________ (6) (1 pt) What type of light has the highest energy? Gamma Rays _______________________________________________ (7) (4 pts) List the four processes that shape the surfaces of the terrestrial planets Volcanism Impact Cratering (a) _______________________________________________ (b) ______________________________________________ Erosion Tectonics (c) _______________________________________________ (d) ______________________________________________ (8) (4 pts) List the four classes of materials that are believed to be in the Nebula that formed our solar system. Abundance: 0.2% (a) Metals (b) Rock (c) Ices (d) Gas Density: 7 gm/cm3 Abundance: 0.4% Density: 3 gm/cm3 Abundance: 1.4% Density: 1 gm/cm3 Abundance: 98% Density: < 1 gm/cm3 _________________ ________________ ________________ _________________ASTR 1030, Midterm 2, October 24, 2008. Page: 2 SOLUTIONS (14) If two objects are the same size but one object is 3 times hotter than the other object, the hotter object emits (a) 3 times more energy. (b) 81 times more energy. (c) 9 times more energy. (d) 12 times more energy. (e) None of the above. (15) Which of the following does NOT occur in a collapsing cloud. (a) It spins up due to conservation of angular momentum. (b) Gravitational energy is converted to thermal energy so it heats up. (c) Collisions cause the particles to lie mostly in the same plane. (d) The heavy metals and rocks go to the inner part of the protoplanetary disks. (e) Most of the mass goes to the center and forms a star. (16) How do asteroids differ from comets? (a) Asteroids and comets are both made of rocky and icy material, but asteroids are smaller in size than comets. (b) Asteroids are rocky bodies and are less dense than the comets, which are made of icy material. (c) Asteroids are made of icy material and are less dense than the comets, which are rockier. (d) Asteroids are made of icy material and are denser than the comets, which are more rocky. (e) Asteroids are rocky bodies and are denser than the comets, which are made of icy material. (17) What do we mean by the diffraction limit of a telescope? (a) It describes the maximum exposure time for images captured with the telescope. (b) It is the maximum size to which any telescope can be built. (c) It is the best angular resolution the telescope could achieve with perfect optical quality and in the absence of atmospheric distortion. (d) It describes the farthest distance to which the telescope can see. (e) It is the size of the primary mirror of the telescope. (18) Which of the following could not be determined by an observation that uses only spectroscopy? (a) The speed at which a distant galaxy is moving away from us. (b) The chemical composition of a distant star. (c) The rotation rate of a distant star. (d) The size of a distant galaxy. (e) The surface temperature of a distant star. (19) What causes stars to twinkle? (a) It is intrinsic to the stars - their brightness varies as they expand and contract. (b) Variations in the absorption of the atmosphere. (c) Variable absorption by interstellar gas along the line of sight to the star. (d) The inability of the human eye to see faint objects. (e) Bending of light rays by turbulent layers in the atmosphere. (20) Why did the solar nebula heat up as it collapsed? (a) Collisions among planetesimals generated friction and heat. (b) Nuclear fusion occurring in the core of the protosun produced energy that heated the nebula. (c) As the cloud shrank, its gravitational potential energy was converted to kinetic energy and then into ther- mal energy. (d) Radiation from other nearby stars that had formed earlier heated the nebula. (e) The shock wave from a nearby supernova heated the gas. (21) When an electron in an atom goes from a higher energy state to a lower energy state, the atom (a) Emits a photon of a specific frequency. (b) Can emit a photon of any frequency. (c) Absorbs a photon of a specific frequency. (d) Absorbs several photons of a specific frequency. (e) Can absorb a photon of any frequency.ASTR 1030, Midterm 2, October 24, 2008. Page: 5 SOLUTIONS Part 3: Analytical Problems Instructions: Solve the following problems. Express your answers in MKS or units specified in the problem and with the minimum number of significant figures needed to properly answer the question. The answer by itself will not get full credit. You must show how you came to the answer. Analytical problems are scored as marked. (1) (5 pts) Escape velocity (a) (1 pt) Suppose the Earth (RE = 6.38x10 6 m, ME = 5.97x10 24 kg) were twice the size in radius (eight times the vol- ume) with the same density. Calculate the mass of this “big” Earth. (b) (3 pts) What is the escape velocity from this planet? Answer in km/s. (c) (1 pt) Our best chemical rockets could accelerate to roughly 15 km/s. Could we have a robust space program with chemical rockets if we happened to be an a “big” Earth? (2) (10 pts) Imaging Stars (a) (4 pts) It has long been a goal to be able to image the disk of a star. Let’s examine if this is possible with today’s technology. Suppose we try to image a star that is 2x106 km in diameter (slightly larger than our Sun) that is 10 ly away. What is the angular size of this star? Answer in arcsec. (b) (3 pts) What is the resolution of our largest optical telescopes (10 m diameter)? Use 500 nm wavelength. Answer in arcsec. Can we image this star? (c) (3 pts) Suppose we used a 1 m space telescope that imaged its corona (same size) in X-rays (1 nm). What is the res- olution in arcsec? Can we image this star? Answer: (a) The mass would be eight times that of Earth: (b) The escape velocity is: (c) We could not escape with current rocket technology. In fact, it would be nearly impossible to achieve orbit. The space program would not be so robust. M 8ME 4.78 25×10 kg= = v 2GM R ----------- 2 6.67 10 11–× m3 kg s2⋅⁄( ) 4.78 25×10 kg( ) 2 6.38 6×10 m( ) -------------------------------------------------------------------------------------------- 22.4km s⁄= = = Answer: (a) Using the small-angle formula, the angle between the stars is: (b) The diffraction limit of a telescope (in arcsec) is: No, we cannot resolve this star. (c) The diffraction limit of a telescope (in arcsec) is: Yes we can image the star, but with only about 20 pixels across, the image would be very fuzzy. α sπd ----- 180°× 2 10 6 km× π 10ly( ) 9.46 1012km ly⁄×( ) -------------------------------------------------------------- 180°× 1.2 6–×10 deg 3600arcsec 1° ------------------------⎝ ⎠ ⎛ ⎞ 4.36 3–×10 arcsec= = = = αd 2.5 10 5 λ dt ---⎝ ⎠ ⎛ ⎞× arcsec 2.5 105 5 7–×10 m 10m --------------------⎝ ⎠ ⎛ ⎞× 0.0125 arcsec= = = αd 2.5 10 5 λ dt ---⎝ ⎠ ⎛ ⎞× arcsec 2.5 105 1 9–×10 m 1m --------------------⎝ ⎠ ⎛ ⎞× 2.5 4–×10 arcsec= = =ASTR 1030, Midterm 2, October 24, 2008. Page: 6 SOLUTIONS (3) (7 pts) Imaging Neptune (a) (3 pts) Neptune is about 30 AU from Earth. It’s mean radius is about 24,500 km. What is Neptune’s angular size as viewed from 30 AU away? Express your answer in degrees and arcseconds. Hint: Use the diameter to calculate angular size. (b) (1 pt) In reality, we would like to have at least 10 times better (smaller) angular resolution than calculated above to produce fuzzy image of Neptune (otherwise it appears as a dot). Divide your answer in part (a) by 10 to get the need resolution to make a fuzzy image of Neptune. (c) (3 pts) How large of a telescope is needed to produce an image of Neptune in visible light (500 nm)? Can the Hub- ble Space Telescope do it (d=2.5m)? . (4) (5 pts) Rapid Rotation (a) (3 pts) Some stars rotate so rapidly that they are quite flat- tened. The rotation speeds can reach 300 km/s. Looking at this star in Lyman-α (121.567 nm), what would the spectral shift (Δλ) on the right side of the star? Answer in nm. It it positive or negative? Would it be red-shifted or blue-shifted? (b) (2 pts) What is the spectral shift on the left side of the star? Answer in nm. Would it be red-shifted or blue-shifted? Answer: (a) Using the small-angle formula, the angular size of Neptune is: (b) One would need ~0.225 arcsec resolution to “image” Neptune. (c) The diffraction limit of a telescope (in arcsec) is: where λ is the wavelength of the light and dt is the diameter of the telescope. In this problem, we know α and λ, but not dt. Solve for dt to get: Yes, HST’s diameter is sufficient. α s 2πd -------- 360°× 2( )2.45 1× 0 4 km 2π 30AU( ) ------------------------------------- 360°× 4.96x 10× 4 km 2π 29AU( ) 1.5 108km AU⁄×( ) --------------------------------------------------------------------- 360°× 6.24 10× 4– °( ) 3600arcs 1° --------------------⎝ ⎠ ⎛ ⎞ 2.25arcsec= = = = = αd 2.5 10 5 λ dt ---⎝ ⎠ ⎛ ⎞× arcs= dt 2.5 10 5 λ α --⎝ ⎠ ⎛ ⎞× 2.5 105 500nm( ) 0.225arcsec -------------------------× arcsec( ) 0.55m= = = Star Answer: (a) Using the Doppler shift formula: The spectral shift on the right side: Moving away, the shift is positive. Red-shift. (b) The spectral shift on the left side is: Moving toward us, the shift is negative. Blue-shift. v c Δλ λo ------= Δλ v c - λo 300km s⁄ 3 5×10 km s⁄ -------------------------- 121.567nm( ) 0.122nm= = = Δλ v c - λo 300–( )km s⁄ 3 5×10 km s⁄ ---------------------------- 121.567nm( ) 0.122–( )nm= = =ASTR 1030, Midterm 2, October 24, 2008. Page: 7