Poisson Equation 1 2 Potential energy: work done against gravitational forces to assemble a distribution of mass ρ(x) The potential-energy tensor Assuming a distribution of mass is already in place, and produces a potential Φ(x), the work done to add (from infinity) a further small amount of mass δm is δw = δmφ(x). If we add an small increment of density the potential is: The potential energy of a system can be obtained by integrating the know potential and/or densities. There are two useful ways: The potential-energy tensor is symmetric. For a flattened body (say along axis 3) W33 will be typically smaller than the other two components because x 3-x3 it typically smaller than the other two components. 3 4
Spherical systems Are rare in nature, but from their study one can set up formalism and derive useful quantities. Simple potentials Point mass Circular frequency Homogeneous sphere The density is constant, M(r) = 4/3!r 3 ρ Escape velocity Circular speed increases with radius (as happens in central region of spiral galaxies) 5 6 Period: Inverse of angular frequency: Plummer model Density about constant at small r, and falling to 0 at large r / 2 A test particle released at rest at radius r has an equation of motion: Poisson eq. in radial coordinates Which is an harmonic oscillator with period Hence the test particle will reach the center in 1/4 T, irrespective of the initial position r, in a time / 2 Dynamical time: 7 8
Profiles that matches the observed profiles of surface brightness Navarro, Frenk & White (NFW) profile R = observed radial distance r = real radial distance Many systems (including galaxies) can be fit by a double power law. A useful formalism is provided by the models given by: R z r often very complicated to solve. A very important case is given by the NFW model, with α=1 and β=3. Simulations show that haloes of dark matter particles follow this distribution Many elliptical galaxies follow a R 1/4 profile: A very similar profile is given by the Hubble-Reynolds law A simple form of the luminosity density The two remaining free parameters (ρ0 and a) are strongly correlated, so that these models are members of a 1 free parameter family. This parameter is r200, the distance at which the mean density is 200xcritical density ρc, or the mass inside r200, M = 200 ρc4/3!r 3 200 Concentration c = r200/a can be integrated and gives a surface brightness profile similar to the Hubble Reynolds, especially at large R: assuming that the luminosity density traces the mass density, one can compute the potential. 9 10 Assuming that the luminosity density traces the mass density, one can compute the potential using the Mass-to-Light Ratio. Axisymmetric potentials Kuzmin-Toomre potential Is produced by an infinitesimally thin disk with surface density: In spherical systems one can use the two Newton s theorems: 1) The gravitational field inside a spherical shell is null; hence Miyamoto-Nagai potential 2) The gravitational field outside a spheric shell is equal to the g.f. of a point source located in its center with the same mass; For a=0, it is the Plummer potential; for b=0 it is the K-T potential; with a and b different from 0 it a combination of both. 11 12
Miyamoto-Nagai potential b/a=0.2 Both Kuzmin and Miyamoto-Nagai potential have finite mass, and circular velocity becomes Keplerian at large R. Instead, we know that circ. velo. is about constant. Logarithmic potential b/a=1.0 b/a=5 13 14 15 16
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