# Flat rotation curves and gravitational potential

I have been reading about spiral galaxies rotation curves, and I have a question I would like to clarify.

For example, many of them have flat rotation curves after some characteristic distance $r>r_{c}$ from the center. If this is so, then, when one computes the gravitational potential by solving:

$$v^{2}=r\dfrac{d\phi}{dr},$$

it means that at large distances, where $v(r)$ is almost constant, $\phi(r)$ increases as a logarithm of the distance.

If one keeps putting massive particles in the outer parts of the galaxy; one has two options:

1. The velocity is almost constant in a finite distance range and then, almost at the edge of the galaxy, starts to decrease.
2. The velocity keeps being constant till $r\to+\infty.$

If 1) is true, then why it is necessary dark matter to embbed the whole galaxy in a halo, since at the edge of the galaxy everything becomes keplerian again?

If 2) is true, then, since a classical massive particle can't tunnel the potential barrier created by $\phi(r)\sim \ln(r/r_{c})$, doesn't it mean that there's a radial cutoff $r_{\Lambda}$ for every particle such that it can't move beyond that orbit of radius $r_{\Lambda}$? Does't it mean that the galaxy is a self-bound and finite object?

• You're considering a specific type of halo, for example, NFW or Burkert. They give $v(r)\sim 1/r$ for large values of $r.$ But NFW faces many problems and Burkert is phenomenological, so there's no physics behind. then, 2) doesn't seem that crazy – user115376 Nov 11 '16 at 19:25
• Extreme-energy cosmic rays have energies exceeding $5\times10^{19} eV$: this energy is still finite. Charged particle will radiate. The problem in 2) arise when neutral high energy particles: photons+neutrinos. Photons, no problem, the effective potential, only accounts for gravitational lensing. Neutrinos, two cases. a) Stick to Standard Model and consider massless. The same as photons. b) Go to BSM and consider the mass. They interact weakly and non-electromagnetically, doesn't matter if they could reach "infinite energies" at the end you won't see them. This is not a contradiction. – user115376 Nov 12 '16 at 9:14