Question

Prove that the total gravitational potential energy can be written as ρ(z)φ(z)dr, where ρ(z) and φ(z) are the density and gravitational poi.(.mi.ial ai. locai;ion x in space, Π iptXtively Use the above formula for W to obtain the total gravitational energy of the Plummer sphere with the graviational potential: φ(r) = (r2+p)

Answer 1

To solve this problem the details calculations are given below
The gravitational force F^vector(x^vector) on particle m_s at position x^vector is due to the mass distribution p(x^vector). According to Newton's Inverse-square law, the force rho F^vector (x^vector) on the particle m_s at location x^vector due to mass rho m(x^vector) at location x^vector is rho F^vector(x) = Gm_s (x^vector - x^vector)/ Vertical-bar x^vector - x^vector Vertical-bar ^3 rho m(x^vector) = Gm_s (x^vector - x^vector)/ Vertical-bar x^vector - x^vector Vertical-bar ^3 rho (x^vector) d^3 x^vector Total force on the particle m_s is now F^vector(x^vector) = m_s g^vector(x): g(x^vector) = G integral (x^vector - x^vector)/ Vertical-bar x^vector - x^vector Vertical-bar ^3 rho (x^vector) d^3 x^vector g(x^vector) is the gravitational field the force per unit mass. Gravitational potential: p(x^vector) = -G integral p(x^vector) d^3 x^vector/ Vertical-bar x^vector - x^vector Vertical-bartherefore nabla x^vector8(1/ Vertical-bar x^vector - x^vector Vertical-bar ) = (x^vector - x^vector)/ Vertical-bar x^vector - x^vector Vertical-bar 63 therefore g^vector (x^vector) = nabla x^vector integral G p(x^vector) d rho x^vector/ Vertical-bar x^vector - x^vector Vertical-bar = -nabla psi Now assume that 'build' up the galaxy slow. We have a galaxy with a density f_p, with 0 < f <