hw1 - Physics 160: Stellar Astrophysics Homework #1...

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Unformatted text preview: Physics 160: Stellar Astrophysics Homework #1 Due Friday September 30th at 5pm Reading: Carroll & Ostlie sections 1.3, 3.1-3.6 Problems: (120 points + 20 bonus points) (1) [15 pts] The two brightest stars in Orion, α Orionis (aka Betelguese) and β Orionis (aka Rigel) are located at opposite corners of the constellation Orion, at coordinates: α Ori: 05h 55m 10.3s, +07° 24′ 25.4″ β Ori: 05h 14m 32.2s, −08° 12′ 05.9″ (a) [10 pts] What is the angular separation of these two stars, and hence the angular size of the Orion constellation? Give your answer in arcseconds, arcminutes and degrees. (b) [5 pts] Both stars are located roughly 200 pc away. Using the small angle formula, what is the approximate physical separation of the two stars in parsecs? (2) [20 pts] Ground-based parallax measurements rely on the Earth’s orbit to provide the baseline for angular measurements. What if we performed parallax measurements from Jupiter instead, which has an orbital semi-major axis of 5.2 AU? (a) [5 pts] If the parallax angle for a star at 10 pc is 0.1” as measured from the Earth, what would the parallax angle be for the same star from Jupiter? (b) [5 pts] If the best measurement equipment available can measure an angle of 0.001”, what is the maximum distance we can measure by parallax from (i) Earth and (ii) Jupiter? (c) [5 pts] To measure the full parallax angle, we need to make measurements for at least ½ of an orbit, or 6 months for Earth. How long would we have to wait to make a parallax measurement from Jupiter? (Don’t just look this up – use Kepler’s Laws!) (d) [5 pts] Based on these considerations, would a parallax program from Jupiter make sense? (3) [15 pts] The Hubble Space Telescope (HST) has an angular resolution of ≈0.05 arcseconds. The table below lists the radii of various types of stars and planets. What is the maximum distance (in the units specified) for which each object could be resolved by HST? Rigel (blue supergiant) 25 R Maximum distance object can be resolved pc Betelgeuse (red supergiant) 800 R pc 1 R pc 61 Cygni (red dwarf) 0.2 R pc Jupiter (giant planet) 0.1 R AU Earth (terrestrial planet) 0.01 R AU Object The Sun (yellow dwarf) Radius (4) [30 pts] It is common to estimate the distance of a star from its spectral type and an assumed absolute magnitude for that spectral type. However, if the star is an unresolved binary, this distance could be underestimated. (a) [5 pts] Assume star ∂ Makeyupus has an identical spectral type as the Sun (absolute V magnitude = 4.83) and an apparent visual magnitude of V = 2.42 How far away would we estimate the star to be? (b) [10 pts] It is discovered that ∂ Makeyupus is a binary. What are the true apparent magnitudes of both components if (i) the two components have the same brightness (ii) one component is half as bright? (hint: use the definition for apparent magnitude in equation 3.4) (c) [10 pts] Based on part (b), estimate the true distance to ∂ Makeyupus for both of the cases above, assuming the brighter component is a Sun-like star. (d) [5 pts] The binary components are separated from each other by 5 AU. Can they be resolved with the Hubble Space Telescope in either of the two cases above? (5) [15 pts] The star Acrux (α Crucis) is located 100 pc away, has a surface temperature of 28,000 K and a luminosity of 25,000 L. (a) [2.5 pts] What is its radius (in solar radii)? (b) [2.5 pts] What is its distance modulus? (c) [2.5 pts] What is its absolute bolometric magnitude? (d) [2.5 pts] What is its apparent bolometric magnitude? (e) [5 pts] What is the radiant flux at the surface of the Earth? How does this compare to the Sun’s radiant flux at the surface of the Earth? (in both cases, ignore scattering and absorption from our atmosphere) (6) [20 pts BONUS] A Cepheid is a pulsating star that undergoes periodic variations in its luminosity. There is a direct relationship between the period (P) of the variation and the average luminosity of the star (equation 14.2 in C&O): log10 L/L = 1.15 log10 Pdays + 2.47 (a) [10 pts] Expand this relation to derive an expression that relates the variability period of a Cepheid to its average radius and effective temperature. Leave the expression in terms of the solar units. (b) [10 pts] ∂ Cephei is a Cepheid with an average Teff = 6200 K and variability period of 5.4 days. What is its average radius in units of solar radii? Would this star be classified as a dwarf or giant star? (7) [25 pts] The human body has an average temperature of 310 K (98.6ºF) and surface area of 1.4 m2. Assume the skin is a perfect radiator, and that you are standing in a room with a constant temperature of 293 K (68ºF). (a) [5 pts] What is the peak wavelength λmax of your thermal spectrum? What part of the electromagnetic spectrum is this found? (b) [5 pts] How much energy per second (in Watts) do you radiate in the form of blackbody radiation? (c) [5 pts] How much energy per second (in Watts) do you absorb from the environment? (assume your skin is a perfect absorber of thermal radiation) (d) [5 pts] Based on b) and c), what is your net energy loss per second from just radiation? (e) [5 pts] How many food calories does this correspond to in a day? (Note: 1 food calorie = 4.184 kJ) Is this an efficient form of weight loss? ...
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