The local approximation

There are two alternative ways of further simplification. One is the orbit average, which uses that any distribution function, being a steady state solution of the collisionless Boltzmann equation, can be expressed as a function of the constants of motion of an individual particle (Jeans' theorem). For the sake of simplicity, it is assumed that all orbits in the system are regular, as it is the case for example in a spherically symmetric potential; thus the distribution function $f$ now only depends on maximally three independent integrals of motion (strong Jeans' theorem). Let us transform the Fokker-Planck equation to a new set of variables, which comprise the constants of motion instead of the velocities $v_i$. Since in a spherically symmetric system the distribution only depends on energy and the modulus of the angular momentum vector $J$, the number of independent coordinates in this example can be reduced from six to two, and all terms in the transformed equation (8) containing derivatives to other variables than $E$ and $J$ vanish (in particular those containing derivatives to the spatial coordinates $x_i$). Integrating the remaining parts of the Fokker-Planck equation over the spatial coordinates is called orbit averaging, because in our present example (a spherical system) it would be an integration over accessible coordinate space for given $E$ and $J$ (which is a spherical shell between $r_{\rm min}(E,J)$ and $r_{\rm max}(E,J)$, the minimum and maximum radius for stars with energy $E$ and angular momentum $J$). Such volume integration is, since $f$ does not depend any more on $x_i$ carried over to the diffusion coefficients $D$, which become orbit-averaged diffusion coefficients.

Orbit-averaged Fokker-Planck models treat very well the diffusion of orbits according to the changes of their constants of motion, taking into account the potential and the orbital structure of the system in a self-consistent way. However, they are not free of any problems or approximations. They require checks and tests, for example by comparisons with other methods, like the one described in the following.

We treat relaxation like the addition of a big non-correlated number of two-body encounters. Close encounters are rare and thus we admit that each encounter produces a very small deflection angle. Thence, relaxation can be regarded as a diffusion process .

A typical two-body encounter in a large stellar system takes place in a volume whose linear dimensions are small compared to other typical radii of the system (total system dimension, or scaling radii of changes in density or velocity dispersion). Consequently, it is assumed that an encounter only changes the velocity, not the position of a particle. Thenceforth, encounters do not produce any changes ${\Delta \bf x}$, so all related terms in the Fokker-Planck equation vanish. However, the local approximation goes even further and assumes that the entire cumulative effect of all encounters on a test particle can approximately be calculated as if the particle were surrounded by a very big homogeneous system with the local distribution function (density, velocity dispersions) everywhere. We are left with a Fokker-Planck equation containing only derivatives with respect to the velocity variables, but still depending on the spatial coordinates (a local Fokker-Planck equation).

In practical astrophysical applications, the diffusion coefficients occurring in the Fokker-Planck equation are not directly calculated, containing the probability $\Psi$ for a velocity change ${\Delta \bf v}$ from an initial velocity ${\bf v}$. Since $D(\Delta v_i)$, and $D(\Delta v_i\Delta v_j)$ are of the dimension velocity (change) per time unit, and squared velocity (change) per time unit, respectively, one calculates such velocity changes in a more direct way, considering a test star moving in a homogeneous sea of field stars. Let the test star have a velocity ${\bf v}$ and consider an encounter with a field star of velocity ${\bf v}_f$. The result of the encounter (i.e. velocity changes $\Delta v_i$ of the test star) is completely determined by the impact parameter $p$ and the relative velocity at infinity $v_{\rm rel} = \vert{\bf v} - {\bf v}_f\vert$; thus by an integration of the type

$$<\!\Delta {\dot v}_i\!>_p = 2\pi\int (\Delta v_i) \,v_{\rm rel}\, n_f p dp ,$$ (10)

the rate of change of the test star velocity due to encounters with $v_{\rm rel}$, in field of stars with particle density $n_f$, averaged over all relevant impact parameters is computed. The integration is normally carried out from $p_0$ (impact parameter for $90^o$ deflection) until $R$, which is some maximum linear dimension of the system under consideration. Such integration generates in subsequent equations the so-called Coulomb logarithm $\ln \Lambda$; we will argue later that it can be well approximated by $\ln (0.11 N)$, where $N$ is the total particle number. The diffusion coefficient finally is

$$D(\Delta v_i) = \int <\!\Delta {\dot v}_i\!>_p f({\bf v}_f) d^3\!{\bf v}_f,$$ (11)

where $f({\bf v}_f)$ is the velocity distribution of the field stars. In an equal mass system, $f({\bf v}_f)$ should be equal to the distribution function of the test stars occurring in the Fokker-Planck equation for self-consistency. In case of a multi-mass system, however, $f({\bf v}_f)$ could be different from the test-star distribution, if the diffusion coefficient arising from encounters between two different species of stars is to be calculated. The diffusion coefficients are (for an exact procedure see Appendix 8.A of BT87):

$$D(\Delta v_i) = 4\pi G^2 m_f \ln\Lambda \frac{\partial{}}{\partial {v_i}}h({\bf v})$$

$$D(\Delta v_iv_j) = 4\pi G^2 m_f \ln\Lambda \frac{{\partial^2}}{ \partial v_i \partial v_j} g({\bf v})$$ (12)

with the Rosenbluth potentials RosenbluthEtAl57

$$h({\bf v}) = (m+m_f) \int {f({\bf v}_f)\over\vert{\bf v}-{\bf v}_f\vert} d^3\!{\bf v}_f$$

$$g({\bf v}) = m_f \int f({\bf v}_f) \vert{\bf v}-{\bf v}_f\vert d^3\!{\bf v}_f \ .$$ (13)

With these results we can finally write down the local Fokker-Planck equation in its standard form for the Cartesian coordinate system of the $v_i$:

$$\bigg(\frac{\delta f}{\delta t} \bigg)_{\rm enc}= -4\pi G^2... ... {\partial^2 g \over\partial v_i\partial v_j} \Bigr)\Biggr]$$ (14)

Note that in RosenbluthEtAl57 the above equation is given in a covariant notation, which allows for a straightforward transformation into other curvilinear coordinate systems.

Before going ahead the question is raised, why such approximation can be reasonable, regarding the long-range gravitational force, and the impossibility to shield gravitational forces as in the case of Coulomb forces in a plasma by opposite charges. The key is that logarithmic intervals in impact parameter $p$ contribute equally to the mean square velocity change of a test particle, provided $p\gg p_0$ (see e.g. Spitzer87, chapter 2.1). Imagine that on one hand side the lower limit of impact parameters ($p_0$, the $90^o$ deflection angle impact parameter) is small compared to the mean interparticle distance $d$. Let on the other hand side $D$ be a typical radius connected with a change in density or velocity dispersions (e.g. the scale height in a disc of a galaxy), and $R$ be the maximum total dimension of the system. Just to be specific let us assume $D=100d$, and $R=100D$. In that case the volume of the spherical shell with radius between $D$ and $R$ is $10^6$ times larger than the volume of the shell defined by the radii $d$ and $D$. Nevertheless the contribution of both shells to diffusion coefficients or the relaxation time is approximately equal. This is a heuristic illustration why the local approximation is not so bad; the reason is with other words that there are a lot more encounters with particles in the outer, larger shell, but the effect is exactly compensated by the larger deflection angle for encounters happening with particles from the inner shell. If we are in the core or in the plane of a galactic disc the density would fall off further out, so the actual error will be smaller than outlined in the above example. By the same reasoning one can see, however, that the local approximation for a particle in a low-density region, which suffers from relaxation by a nearby density concentration, is prone to failure.

These rough handy examples should illustrate that under certain conditions the local approximation is not a priori bad. On the other hand, it is obvious from our above arguments, that if we are interested in relaxation effects on particles in a low-density environment, whose orbit occasionally passes distant, high-density regions, the local approximation could be completely wrong. One might think here for example of stars on radially elongated orbits in the halo of globular clusters or of stars, globular clusters, or other objects as massive black holes, on spherical orbits in the galactic halo, passing the galactic disc. In these situations an orbit-averaged treatment seems much more appropriate.