PHYS-638-07f: Problem set #0 Solutions. 1. Angles, magnitudes, inverse square law: a. How far from the Earth would the Sun have to be moved so that its apparent angular diameter would be 1 arc second? (Express your answer in cm, solar radii, and AU.) First convert arcsec to radians: 1 arcsec= π/(180 60 60)= 4.8 10 6 radians. Then D = 2R α = 2R 4.8 10 = 4.1 6 105 R = 2.9 10 16 cm. (1) Another way to look at it is to recall the actual angular diameter of the Sun is about 0.5 o = 1800 arcsec. Thus to get to 1 arcsec, the Sun would have to move away by a factor 1800, or to 1800 AU. b. How far away would a Frisbee of diameter 30 cm have to be to subtend the same angle? By above, D = 30 4.8 10 6 = 6.3 107 cm = 63 km. (2) c. At the distance you calculated in (a), what would the Sun s flux here on Earth be (i.e. what would the solar constant be)? In class we noted that the solar constant is F = 1.4 10 6 erg/s/cm 2. If we now move the Sun 1800 times further away, then by the inverse-square law, the flux (which is what we mean by the solar constant) would decrease by a factor 1/1800 2. Thus F,new = 1.4 106 1800 2 = 0.43erg/s/cm 2 (3) d. What would the Sun s apparent magnitude be? (Use m = -26.7 for the actual Sun, the one that s at 1 AU.) The difference in apparent magnitude between two stars is just 2.5 times the log of the ratio of the flux. Remembering that a lower flux gives a larger magnitude (i.e. dimmer stars have bigger m), we have m,new = m + 2.5 log(f /F,new) = 26.7 + 2.5 log(1800 2 ) = 10.4. (4)
2 Remembering that the brightest stars are around magnitude zero, we see that the sun would still be a very bright star, about 10,000 times brighter than the brightest actual star! (Since m=-10 is 10 magnitudes brighter than m=0, and each difference of 5 in magnitude represents a factor 100 in brightness, so 100 2 =10,000). 2. Galaxies: distance, magnitude, and solid angle: a. What is the apparent magnitude of a galaxy that contains 10 11 stars identical to the Sun (i.e., assume its luminosity is equal to 10 11 ) if it s at a distance of 50 million parsecs? Using eqn. (14) in DocOnotes1-stars.pdf, the apparent magnitude m of an object with luminosity at a distance D is given by m = 4.8 2.5 log(/ )+5 log(d/10 pc) = 4.8 2.5 11+5 6 log(5) = +10.8, (5) with the latter equality plugging in the above numbers to give m +10.8. b. What is the flux of this galaxy here on the Earth, in cgs units? F = 4πD 2 = 10 11 4 10 33 erg/s 4π (5 10 7 3 10 18 ) 2 cm 2 = 1.4 10 9 erg/cm 2 /s (6) c. If the galaxy is circular in shape, as seen from the Earth, and has a diameter of 50,000 pc, what is its apparent angular diameter? α = s D = 50 103 pc 50 10 6 pc = 10 3 rad = 206 arcsec = 3.4 arcmin (7) d. What solid angle does it subtend (in steradians and in square arc seconds)? Ω = (π/4)206 2 arcsec 2 = 3.2 10 4 arcsec 2 (8) Ω = 4π ( α ) ( ) 2 10 3 2 = 4π = 3 10 7 ster (9) 2π rad 2π e. How does the galaxy s surface brightness (energy/time/area/solid angle) compare to the Sun s (express this as a ratio)?
3 Assuming both the galaxy and the sun emit isotropically from their surface, then the ratio of surface brightness is just given by the ratio of their surface flux, I gal I = gal/4πr 2 gal /4πR 2 ( ) = 10 11 7 10 10 2 = 2.2 10 14. (10) 5 10 4 3 10 18 3. Equilibrium Temperature of Earth: a. Assuming Earth is a blackbody, use the known luminosity and distance of the Sun to estimate Earth s average equilibrium surface temperature if the solar energy it intercepts is radiated to space according the Stefan-Boltzman law. Compare this to the temperature on a moderate spring day in Delaware. Equating the solar flux intercepted by the Earth s cross-sectional area πr 2 e to the total blackbody emission over its surface area 4πR 2 e, we find Thus T e = σte 4 4πRe 2 = πre 2 4πa 2 e (11) ( ) 1/4 = 281 K = 8 C (12) 16πa 2 e σ On moderate spring day in Delaware, temperature is a bit higher, ca. 20 C (68 F). But given the approxmiations, this is pretty close to the characteristic temperature computed for simple blackbody! b. According to http://en.wikipedia.org/wiki/earth, Earth s albedo, meaning the fraction of received light that is absorbed is only ca. 0.367, with the rest (63.3% ) being refected by, e.g. clouds, snow, etc., without contributing any heat to Earth. So now redo the calculation in (a) reducing the solar input energy by this amount. If only a fraction 0.367 of Sun s luminosity is actually absorbed by Earth, then the equilibrium temperature should be reduced by a factor 0.367 1/4 = 0.78, reducing the above equilibrium temperature now to T e = 219 K = -54 C. c. Which result seems more reasonable? Briefly discuss what other physics might be important to include to understand the actual surface temperature of Earth. This apparently more realistic model thus seems to give a temperature that is much lower than the typical temperature of the actual Earth. The key piece of physics missing is the greenhouse effect, which blocks the re-radiation of solar energy, forcing the surface of the Earth to be warmer
4 than it would otherwise be, much like a blanket at night keeps our skin at a higher temperature than it would otherwise be. Bottom line: the greenhouse effect and the albedo effect roughly cancel, making the simple blackbody temperature in part (a) come out about right! 4. Parallax of Mars: In 1672, an international effort was made to measure the parallax angle of Mars at opposition, when it was on the opposite side of the Earth from the Sun, and thus closest to Earth. a. Consider two observers at the same longitude but one at latitude of 45 degrees North and the other at 45 degrees South. Work out the physical separation s between the observers given the radius of Earth is R E 6400 km. Viewed from the center of the Earth, the two observers at ±45 o are separated by 90 o, thus forming a right angle. The radius lines to each observer thus form the two lengths of an isosceles triangle with the observers separated by the base, with length s = 2R e = 8.8 10 3 km. (13) b. If the parallax angle measured is 25 arcsec, what is the distance to Mars? Give your answer in both km and AU. D = s α = 8.8 10 3 km 25arcsec/206000 (arcsec/radian) = 7.3 107 km = 0.48 au. (14) 5. Brightness of the full moon: a. Given the Sun s radius R and distance to Earth a e, compute the Sun s solid angle on the sky, both in steradians, and as a fraction of the full sky s 4π steradians. Ω /2π = 1 dµ = 1 µ = 1 1 (R /a e ) 2 (1/2)(R /a e ) 2 1.1 10 5 (15) µ Ω 6.8 10 5 ster (16) Ω /4π 5.5 10 6 (17)
5 b. By coincidence, the moon covers about this same fraction. Use this fact together with the moon s albedo of 0.07 (meaning the moon reflects back 7% of the sunlight that hits its surface) to estimate how much dimmer the full moon appears in optical light compared to sunlight on Earth. Since the solar flux declines as (R /a e ) 2, then a perfectly reflecting moon would have its surface brightness reduced by this fraction. With an albedo of 0.07, this surface brightness would be reduced by that extra factor. Finally, since the solid angles are the same, the ratio of surface brightness is also the ratio of apparent brightness. f moon f = 0.07 ( R a e ) 2 = 1.6 10 6 (18) c. Given that the Sun s apparent magnitude is m = 26.7, use your answer for part (c) to estimate the apparent magnitude of the full moon in optical light. m moon = 26.7 2.5 log(f moon /F ) = 12.2 (19) d. Assume that the sunlight that is not reflected is re-radiated as if the moon were a blackbody. Compute the average equilibrium temperature of the moon. Surface flux of the moon s thermal emission is given by: F em = σt 4 moon = 0.93πR2 moon 4πa 2 e 4πR 2 moon = 0.93 4πR2 σt 4 16πa 2 e = 0.93R2 σt 4 4a 2 e (20) T moon = 0.93 0.25 R 2a e T 0.048T 273 K (21) e. Estimate the peak wavelength at which the moon emits this blackbody radiation, comparing it to the peak wavelength of the Sun s spectrum. What part of the electromagnetic spectrum does this correspond to? λ max,moon = λ max, T T moon 500 nm 0.048 10, 400 nm (22) This is in the infrared.
6 f. For a filter that is centered broadly on this wavelength, what is difference in the apparent magnitude of the moon compared to the optical magnitude computed in part (c). m = 2.5 log[f moon,vis /f moon,ir ] = 2.5 log[0.07/(.93/4)] 1.3 (23) 6. The star Dschubba (δ Sco) has a parallax p = 8 mas: Assuming it is a spherical blackbody with radius R = 7.5 R and surface temperature T = 28, 000 K, compute Dshuba s a. uminosity, in erg/s, and ; By Stefan-Boltzmann law: = 4πR 2 σt 4 = 1.2 10 38 erg/s = 30, 000 (24) b. Absolute magnitude: By definition of absolute magnitude normalized to the sun s value M sun = +4.8, we have M = 4.8 2.5 log(30, 000) = 6.4. (25) c. Apparent magnitude; A parallax of α = 8 mas 0.008 arcsec implies a distance of D = 1/0.008 = 125 pc. Thus the apparent magnitude is d. Distance modulus; m = M + 5 log(d/10pc) = 6.4 + 5 log(12.5) = 0.9. (26) m M = 5 log(12.5) = 5.5 (27) e. Radiant flux at the star s surface, in CGS, and relative to surface flux of the Sun; F(R) = 4πR = 30, 000 4 1033 erg/s 2 4π(7.5 7 10 10 ) 2 cm = 3.4 2 1013 erg/cm 2 /s. (28) Relative to the flux at sun s surface, we have F(R) F (R ) = / (R/R ) = 30000 = 533. (29) 2 (7.5) 2
7 f. Radiant flux at the Earth s surface, and the ratio of this to the solar irradiance; F = 4πD = 30, 000 4 1033 erg/s 2 4π(1253 10 18 ) 2 cm = 6.8 2 10 5 erg/cm 2 /s. (30) Relative to the solar flux at earth s orbit, we have g. Peak wavelength λ max. By Wien s displacement law, F F = / (D/au) 2 = 30000 (125 206, 000) 2 = 4.5 10 11. (31) λ max = 2.9 106 nm K T = 2.9 106 nm 100nm. (32) 2.8 104 7. Absorption by coal dust. Imagine you re a coal miner working under a 1000-watt lamp that is at distance of 10 meter away from your work location. a. Assuming the lamp emits its light isotropicallly, what is the flux of light on your workspace, in watt/m 2? (Assume the coal mine walls are perfect absorbers, i.e with zero albedo). Since the walls absorb all indirect light, the local flux is just due to the direct light, for which the flux decreases following the inverse-square law: F(r) = 4πr = 1000 2 4π 10 = 0.8 2 watt/m2. (33) b. Now suppose there is a cave-in that fills the mine with black, spherical coal dust particles of diameter 0.1 mm, and with a uniform number density of 20 particles per cubic centimeter. What is the optical depth between you and the lamp? The cross section of dust of diameter d = 0.1 mm is σ = (π/4)d 2 = 7.8 5 cm 2. The optical depth is given in terms of number density n = 20/cm 3, cross section σ, and distance D = 1000 cm, τ = nσ D = 20 7.8 5 10 3 = 1.56. (34) c. What is now the flux on your workspace?
8 Applying the exponential attenuation by optical depth, we find F abs = F(r)e τ = 0.8 e 1.56 = 0.17 watt/m 2. (35) d. What distance had this flux value before the cave in? Solving the inverse-square law for distance, we find r = 4πF = 10 3 4π0.17 = 21.8 m. (36) 8. Absorption by interstellar dust. Imagine you re an astronomer observing a star with luminosity 1000 that is at a distance of 10 parsec. a. What is the flux of light you observe? Give your answer in both watt/m 2 and /parsec 2. Again, the flux decreases following the inverse-square law: F(r) = 4πr = 1000 2 4π 10 = 0.8 /parsec 2 = 0.8 2 4 26 (3 16 ) 2 = 3.5 10 7 watt/m 2. (37) b. Now assume the space between you and the star contains spherical dust particles of diameter 1 micron and number density of 6000 particles per cubic kilometer. Assuming the dust absorbs or scatters light in proportion to its geometric cross-section, what is the optical depth between you and the star? Again, the cross section of dust of diameter d = 1 µm is σ = (π/4)d 2 = 7.8 9 cm 2. The optical depth is given in terms of number density n = 6000/km 3 = 6 12 cm 3, cross section σ, and distance D = 10 pc = 3 19 cm, c. What is now the flux you observe? τ = nσ D = 6 12 7.8 9 3 19 = 1.4. (38) Again, applying the exponential attenuation of optical depth, we find F obs = F(r)e τ = 3.5 10 7 e 1.4 = 8.6 10 8 watt/m 2. (39) d. If you knew the star s luminosity but didn t know about the dust, what distance would you infer for the star based on your observed flux??
9 Again, inverting the inverse-square law to infer a distance in terms of the known luminosity and the observed flux, you infer 10 r = = 3 4 26 = 6 17 m = 20pc. (40) 4πF obs 4π8.6 8 e. What is the change in the apparent magitude m of this star resulting from the dust absorption. Using the definition of apparent magnitude, we have m = 2.5 log(f/f obs ) = 2.5 log(e 1.4 ) = 2.5 1.4 log(e) = 1.5. (41) f. Finally, generalize your analysis to derive an expression for how the extinction magnitude A, defined as the change in apparent magnitude m due to absorption, depends on the absorption optical depth τ to any star. Again, applying the magnitude definition, A m = 2.5 log(e τ ) = 2.5τ log(e) = 1.08τ. (42)