Homework 1 Solutions

9010097 2571 note if you used the small angle formula


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Unformatted text preview: use of the dot product between vectors with spherical coordinates φ, θ: cos Γ = (cos θ1 cos φ1, cos θ1 sin φ1, sin θ1) (cos θ2 cos φ2, cos θ2 sin φ2, sin θ2) = cos θ1 cos θ2 cos (φ1 ­φ2) + sin θ1 sin θ2 = 0.9010097 =&gt; Γ = 25.71º Note: if you used the small angle formula, you would have been off by a bit: Γ ≈ (Δθ2 + cosθΔϕ2)1/2 = ([12.44º]2 + cos(55.53º)[40.95]2)1/2 = 33.23º This formula really should only be applied when Γ &lt; 0.1º (b) [10 pts] Alkaid is 31 pc away, Dubhe is 38 pc away. How far apart are these two stars in physical distance in units of pc? Let’s project the vectors pointing to Alkaid and Dubhe onto a polar coordinate map separated by the angle you calculated above, with Alkaid aligned with the x Cartesian axis: The Cartesian coordinates are then: ALK = [31,0], DUB = 38 [cos 25.71º, sin 25.71º] = [34.24, 16.49] Separation vector = DUB – ALK =...
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