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Answer to Question #71335 in Astronomy | Astrophysics for brett

Question #71335
Assume that dwarf spherical galaxy has the so-called Plummer sphere potential Φ(r) = − (r0v02)/(sqr(r2+r02)), with constants v0 = 125 km/s, and r0 = 2.5 kpc (core radius). [This is the force per unit mass of the test particle, a.k.a. specific potential. The same way we talk about the specific force, which is simply force/mass or acceleration!] The objective is to fully characterize the system based on the potential. This is always easy in spherically symmetric systems. Find the specific force field created by specific potential Φ(r). Find asymptotic behavior of F and Φ with r at r <<r0 and at r → ∞. Note: by *behavior* we mean not only the value, but also functional form, in this case a certain power law. For example, 1/(1+x3 ) behaves like ≈ 1−x3/3 for x <<1 (first 2 terms of Taylor expansion; we can give only one leading term, but here it was so easy..), and like ∼ x −3 at large radii; the function goes to zero as the 3rd power of radius.
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