Astro 210 Homework 9 Solutions - Astronomy 210 Spring 2011...

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Unformatted text preview: Astronomy 210 Spring 2011 Homework Set #9: Solutions 1. How far can the eye see? (a) [5 points] By definition, the difference between any two magnitudes is related to the loga- rithm of the ratio of fluxes: m 2 − m 1 = − 5 2 log 10 F 2 F 1 (1) In this problem we are interested in the difference between an object’s apparent and absolute magnitude, i.e., m − M . The apparent magnitude measures the observed flux of the object at its actual distance, i.e., L/ 4 πd 2 . On the other hand, the absolute magnitude measured the flux the object would have at distance d = 10 pc: L/ 4 πd 2 . Using these expression and eq. (1), m − M = − 5 2 log 10 F 2 F 1 = − 5 2 log 10 L/ 4 πd 2 L/ 4 πd 2 = 5log 10 d d (2) Indeed, this is just the distance modulus μ ≡ m − M found in class, and so it would be fine to just take this expression as the starting point. Solving eq. (2) for distance, we have d = d 10 μ/ 5 = d 10 ( m − M ) / 5 = 10 1+( m − M ) / 5 pc (3) This holds for any flux (i.e., magnitude) m . If we are observing at the limit of detection, then m = m lim and the maximum observed distance is d lim = 10 1+( m lim − M ) / 5 pc (4) which was to be shewn. (b) [5 points] We repeatedly use eq. (4), with m lim = 6 but for different values of M : i. The sun-like star with M ⊙ = 4 . 83 can be seen by the naked eye out to d lim = 17 pc (5) ii. The most luminous star in James Kaler’s table of the brightest stars in the sky is Aludra, with absolute magnitude M = − 7 . 5. Stars of this luminosity can be see out to d lim = 5012 pc (6) iii. The least luminous nearby stars is DEN 0255-4700, with absolute magnitude M = 24 . 44. Stars of this luminosity can be see out to d lim = 2 . 1 × 10 − 3 pc (7) The nearest star to us is just over 1 pc away, and that this is the typical spacing between neighboring stars in our Galaxy. Thus even the closest low luminosity stars are too far away to be seen by the naked eye. Solar-luminosity stars can be seen out to moderate distances and volumes, while high-luminosity stars can be see over enormous distances and volumes....
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Astro 210 Homework 9 Solutions - Astronomy 210 Spring 2011...

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