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Foundations of Astronomy ASTR/PHYS 2500
Exam 2 *
Physical Constants Gravitational constant Elementary charge Vacuum permittivity Electron Volt Speed of light in vacuum Planck constant Reduced Planck constant Boltzmann constant Stefan-­‐Boltzmann constant Proton mass Electron mass Atomic mass unit Mass of an iron atom Astronomical Constants
G e ε0 eV c h h/2π k σSB mp me u or amu mFe -­‐11
6.673 x 10 -­‐19
1.602 x 10 -­‐12
8.854 x 10 -­‐19
1.602 x 10 8
2.998 x 10 -­‐34
6.626 x 10 -­‐34
1.055 x 10 -­‐23
1.381 x 10 -­‐8
5.670 x 10 -­‐27
1.673 x 10 -­‐31
9.109 x 10 -­‐27
1.6605 x 10 -­‐26
9.2731 x 10 3
-­‐1 -­‐2
m kg s C 2 -­‐1
-­‐1
C J m J -­‐1
m s J s J s -­‐1
J K -­‐1
-­‐2 -­‐4
J s m K kg kg kg kg * Mass of Earth Mass of Sun Mass of Moon Equatorial radius of Earth Mean radius of Earth Equatorial radius of Sun Equatorial radius of Moon Mean density of Earth Mean density of Sun Mean density of Moon Jupiter radius Jupiter mass Luminosity of Sun Effective temperature of Sun Light-­‐year Astronomical Unit Parsec Length of solar day Length of sidereal day Length of a year (Julian) 24
5.974 x 10 30
1.989 x 10 22
7.36 x 10 6378 6371 8
6.955 x 10 1737 5515 1408 3346 7 6.99x10
317.8 26
3.839 x 10 5778 12
9.461 x 10 8
1.496 x 10 13
3.086 x 10 24 hours = 86400 seconds 23hours 56 minutes = 86160 s 8
3.15576 x 10 kg kg kg km km m km -­‐3
kg m -­‐3
kg m -­‐3 kg m
m Mean Earth masses W K km km km s ASTR/PHYS 2500 Exam 2 ii
1a. Clouds in the Earth’s atmosphere have high albedo (A) compared to the average value (Aavg) of the Earth. What is the effect of increasing the cloud cover on the equilibrium temperature of our planet? 1b. The Sun’s rate of spin about its axis of rotation is decreasing in time. What is responsible for carrying off angular momentum and hence reducing the Sun’s spin? 1c. Briefly describe the RV method for planet detection. 1d. What condition is met when an astronomical body is in hydrostatic equilibrium? 1e. The apparent magnitude of the Deneb, Vega and Altair, the stars that make up the Summer Triangle, are mV = 1.25 (Deneb), 0.03 (Vega), and 0.77 (Altair). Which star appears the brightest in the night sky? 1f. What is the “color index” of a star, and what information does is give about the star? 1g. Why is the phenomenon of quantum tunneling important in the production of fusion energy in the Sun? ASTR/PHYS 2500 Exam 2 iii
source: nasa.gov. 2. The above plot is a variant of an HR diagram. (The luminosity scale is given in solar units and assume that the absolute magnitude is in the V-­‐band.) Use it, or numerical estimates from it, for the following exercises. 2a. Mark and label the Sun (spectral type=G2V, MV = 4.83, B-­‐V=+0.66) and the star Vega (spectral type: A0V, MV = 0.5, B-­‐V=0.0) in the HR diagram. [Assume the tick marks on the lower horizontal axis define the range of each letter class (OBGA…).] 2b. Antares A (MV = -­‐5.3) has narrow absorption lines compared to the Sun, and is redder (higher B-­‐V = 1.83, compared to the Sun). Estimate the luminosity class (name and/or number) of Antares A, and give a brief rationale for your answer. 2c. This past spring, Prof. Zheng Zheng (UofU) discovered a hypervelocity star, LAMOST-­‐HVS1, cruising with a speed of 477 km/s outward from the center of our Galaxy. If the spectral type of the star is B5V, and the apparent magnitude is mV = 13.0 mag, estimate that this star is from us. [Hint: give your answer in kpc, where 1 kpc =1000 pc.] Source: Zheng, Z. et al. 2014, ApJ Letters, 785, 23. d (kpc): ___________ ASTR/PHYS 2500 Exam 2 iv
3a. Kepler-­‐12 is a Sun-­‐like star with of mass M = 1.2 M!. It has a planet, Kepler-­‐12b, that transits every 4.4 days, causing the starlight to dim by a factor of 1.7% over a period of 4.1 hr. Kepler’s third law gives the planet’s semimajor axis as a = 0.056 AU. From this information, estimate the radius of this planet, r, and the radius of the star, R. [Assume a uniformly bright disk for the star, and a circular orbit for the planet at 90o inclination. Please use the units shown for you answer.] r (RJupiter) = _____________ 7
RJupiter= 6.99x10 m R (RSolar) = _____________ 8
RSolar= 6.96x10 m Source: kepler.nasa.gov [Note: RV studies provide the mass of both the planet and star. The planet turns out to be only 0.43 times Jupiter’s mass. Explaining the planet’s radius, given its mass, is challenging!] 3b. The supergiant in the η Carinae binary has a mass M = 120 M! and luminosity L = 5x106 L!. Estimate the star’s lifetime τ under the assumptions that all of its mass is initially in the form of hydrogen, and that 20% of it (the amount in its core) burns to helium at a constant rate. Assume that 4.20x10-­‐12 J are released per conversion of 4 protons into helium (4He). τHe (yr) = _____________ [Could life as we know it evolve on a planetary system around such a star?] 3c. The energy per H atom (i.e. proton) released in fusion to 4He is 1.04x10-­‐12 J. Suppose that in such a massive star as the supergiant in (3b), the protons fused to iron (56Fe). If the mass of an iron nucleus is mFe = 9.29x10-­‐26 kg = 55.52 mp, estimate energy per proton released as 56 protons fusion to 56Fe. [Optional: What is the lifetime of the star in this case compare with (3b)?] Energy per proton (J): _________________ [Optional: τFe (yr) = _________________]