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Unformatted text preview: Astronomy 210 Fall 2010 Homework Set #2: Solutions 1. A sketch is useful. Scales are exaggerated to make the angular difference more apparent. θ shell opp θ conj r 1 The distances to the stars are approximately r opp = r shell- 1 AU = 9 AU (distance when the stars are in opposition to the sun, and nearest to the earth) and r conj = r shell + 1 AU = 11 AU (distance when the stars are in conjunction with the sun, farthest from the earth). If the angular separation when in opposition is θ opp = 2 deg , then the stars’ separation on the celestial sphere is s ≈ r opp θ opp , using the small angle approximation. In the same approximation, the angle between the stars at conjunction is θ conj ≈ s r conj = r opp r conj θ opp = 1 . 64 ◦ (1) Thus the angular difference is ∆ θ = θ opp- θ conj ≈ parenleftBigg 1- r opp r conj parenrightBigg θ conj = 0 . 36 ◦ (2) This certainly observable by the human eye–recall that the angular diameters of the sun and moon are 0 . 5 ◦ , so about the same as those very easily observable scales. In the context of the heliocentric model, this means that if the celestial sphere is even 10 times larger than the Earth-Sun distance, it would still lead to an observable “morphing” of the constellations. Since such behavior is not observed, the upshot is that the heliocentric moded requires that the celestial sphere is very much farther than 10 AU. Prior to Galileo’s observations, then, this would have been a big strike against the heliocentric model, as it demands that there is such a discrepancy in scale between the Sun and planetary distances and the celestial distances. Of course, this nevertheless turns out to be correct, but given what was known at the time, one could rightly have criticized the heliocentrial model for its excesses of scale (or perhaps marvelled at how large it demanded the Universe to be!)....
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This note was uploaded on 10/06/2011 for the course ASTRO 210 taught by Professor Fields during the Spring '08 term at University of Illinois, Urbana Champaign.
- Spring '08