The Miyamoto-Nagai disc potential is given by (see Lecture notes): GM PM (R, 2) = VR+...

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The Miyamoto-Nagai disc potential is given by (see Lecture notes): GM PM (R, 2) = VR²+ (a + √b² +2²)² This is an example of a 'potential density pair' because the above potential is an analytic solution to the Poisson equation in cylindrical coordinates (R. 2) [V²(R. 2) = 4лGp(R. 2)]. The above potential is produced by the following density distribution: (b²M aR² + (a +3√/z² + b²)(a + √z² + b²)² [R² + (a + √√2² +6²)²] 5/2 (22 +6²)3/2 which gives a reasonable approximation to real Galactic discs while remaining analytic. PM (R, 2) = (BM) (22) (23) 1. Prove that the density profile in equation 23 has a gravitational potential given by equation 22. 2. Work out the force due to the Miyamoto-Nagai disc potential. Implement this in your code. 3. Calculate how the orbits change in this potential as you move from an infinitely thin disc (with b→ 0) to a spherical galaxy (with a → 0). The Miyamoto-Nagai disc potential is given by (see Lecture notes): GM PM (R, 2) = VR²+ (a + √b² +2²)² This is an example of a 'potential density pair' because the above potential is an analytic solution to the Poisson equation in cylindrical coordinates (R. 2) [V²(R. 2) = 4лGp(R. 2)]. The above potential is produced by the following density distribution: (b²M aR² + (a +3√/z² + b²)(a + √z² + b²)² [R² + (a + √√2² +6²)²] 5/2 (22 +6²)3/2 which gives a reasonable approximation to real Galactic discs while remaining analytic. PM (R, 2) = (BM) (22) (23) 1. Prove that the density profile in equation 23 has a gravitational potential given by equation 22. 2. Work out the force due to the Miyamoto-Nagai disc potential. Implement this in your code. 3. Calculate how the orbits change in this potential as you move from an infinitely thin disc (with b→ 0) to a spherical galaxy (with a → 0).

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