solutions-ps04 - Physics 341 Problem Set#4 Solutions 1 We...

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Physics 341: Problem Set #4 Solutions 1. We showed in class that we can derive the total mass from a visual binary orbit, M = m 1 + m 2 = 4 π 2 d 3 ˜ α 3 GP 2 where M is the total mass, ˜ α is the angular semi-major axis in radians ( ˜ α = ˜ α 1 + ˜ α 2 ), P is the orbital period, and d is the distance to the system. Often for visual orbits of stars, we know ˜ α and P very precisely, but the distance d is much more uncertain. Let’s call the fractional uncertainty on the distance f d , meaning that we think the true distance is within d = d 0 ± f d d 0 (where d 0 is our best guess and f d 1 ). For example, if d = 100 ± 3 pc , the fractional uncertainty would be f d = 3 / 100 = 0 . 03 (or in other words, a 3% uncertainty). (a) If f d is the fractional uncertainty on the distance, what is the fractional uncertainty on the total mass f M ? Hint: All you need to know is M d 3 and then use our favorite Taylor expansion approximation that (1 ± x ) α 1 ± αx when x 1 . The fractional uncertainty in the mass f M will be related to f d in a simple way. We can substitute d = d 0 ± f d d 0 = d 0 (1 ± f d ) into our formula for the total mass: M = 4 π 2 ˜ α 3 GP 2 d 3 = 4 π 2 ˜ α 3 GP 2 d 3 0 (1 ± f d ) 3 = 4 π 2 ˜ α 3 d 3 0 GP 2 (1 ± f d ) 3 = M 0 (1 ± f d ) 3 where M 0 is our best estimate of the mass (corresponding to the distance d 0 ), M 0 = 4 π 2 ˜ α 3 d 3 0 /GP 2 . Now, because the fractional uncertainty is assumed to be small, f d 1, we can use our favorite Taylor expansion approximation, that (1 ± x ) α 1 ± αx when x 1. So in our case we have M = M 0 (1 ± f d ) 3 M 0 (1 ± 3 f d ) We define the fractional uncertainty in the mass f M just like we did for f d : M = M 0 ± f M M 0 = M 0 (1 ± f M ) Comparing our results above, we see that the fractional uncertainty in the mass is just f M = 3 f d , three times the fractional uncertainty in the distance. (b) Our best estimate for the distance to the center of the Galaxy is d = 8 . 1 ± 0 . 5 kpc . What is the fractional uncertainty in the distance, f d ? Using your result from part (a), what is the fractional uncertainty f M in our mass estimate for Sgr A*? 1
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The fractional uncertainty in the distance is just f d = 0 . 5 kpc 8 . 1 kpc = 0 . 062 = 6 . 2% As we showed above, because the mass scales as d 3 , the fractional uncertainty in the mass is about three times as large, f M = 0 . 186 or 18.6%. If you would like to check this, try calculating the mass for Sgr A*, using distances of 7.6 kpc, 8.1 kpc, and 8.6 kpc (i.e., covering the quoted uncertainty range). (c) Sirius is a visual binary with an orbital period P = 49 . 94 yr . Sirius A (the bright star) has an angular semimajor axis of ˜ α A = 2 . 419 00 , while Sirius B (the fainter star) has ˜ α B = 5 . 191 00 . The distance to the Sirius system is 2 . 64 ± 0 . 01 pc . What are the masses of Sirius A and B? What are the uncertainties in the masses?
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