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HW7_prob1_orbit6-4potential

# HW7_prob1_orbit6-4potential - Problem Orbits with a 6-4...

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Problem: Orbits with a 6-4 central potential (a) The equation of motion for the one-dimensional radial problem is μ ·· r = f ef ( r ) = dV dr + l 2 μr 3 , (1) with the 6-4 potential is de fi ned as V ( r ) = k ³ a r ´ 6 ³ a r ´ 4 ¸ , (2) where k is a constant that determines the strength of the potential, a is a constant with dimensions of length, l is the constant angular momentum, μr 2 · θ , and μ is the reduced mass of the two particle system. This equation can be written explicitly as m ·· r = f ef ( r ) = k a 6 ³ a r ´ 7 4 ³ a r ´ 5 ¸ + l 2 μr 3 = dV 0 dr , (3) where V 0 is the e ff ective potential for this problem, V 0 = V ( r ) + l 2 2 μr 2 . (4) To discuss the qualitative nature of the motion, we need to make a plot of V 0 versus r . It will be convenient to introduce a scaled radial coordinate x = r/a . This allows us to write e V ( x ) = V 0 k = c 2 x 2 + x 6 x 4 , (5) where c is the positive constant c = l 2 μka 2 . (6) We can make convenient choices for the value of c by investigating when circular orbits are possible. The condition to be satis fi ed is that the e ff ective force should vanish, f ef ( r c ) = k a " 6 μ a r c 7 4 μ a r c 5 # + l 2 μr 3 c = 0 , (7) where r c is the radius of the circular orbit. Using Eq.(6) it follows that 6 x 7 c 4 x 5 c + cx 3 c = 0 , (8) where x c = r c /a . After multiplying by x 7 c , Eq.(8) becomes a quadratic equation for the variable x 2 c , cx 4 c 4 x 2 c + 6 = 0 , (9) 1

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HW7_prob1_orbit6-4potential - Problem Orbits with a 6-4...

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