2 Simple analytical estimates of the relaxation time

The relaxation time for a single star can be defined as the time necessary for two-body encounters to change its velocity, or energy, by an amount of the same order as the initial velocity, or energy, i.e. the time in which the memory of the initial values is lost. Thus for the velocity we have

$$T_{\rm relax} = T_{\rm cross} \frac {\upsilon^2}{\Delta\upsilon_\perp^2},$$ (1)

where $T_{\rm cross}$ is the crossing time of the system. Following e.g. Binney & Tremaine (1987) we can obtain a simple order-of-magnitude estimate of the relaxation time. Let us focus on the motion of a single star assuming that it is moving on a straight line with a constant velocity v and that the remaining stars have equal mass m and are distributed uniformly in space. We first consider that the star passes a single perturber star on its rectilinear trajectory. Then the total change of its velocity component perpendicular to the trajectory is

$$\vert \delta\upsilon_\perp \vert \approx {{ 2 G m } \over {b \upsilon}},$$ (2)

where G is the gravitational constant and b is the impact parameter, i.e. the minimum distance between the two stars if there was no gravitational attraction. To find the total change, due to all the particles, we integrate over all encounters and find

$$\Delta\upsilon_\perp^2 = \int_{b_{\rm min}}^{b_{\rm max}} \de... ...upsilon^2} \ln \left(\frac {b_{\rm max}} {b_{\rm min}}\right),$$ (3)

where N is the number of stars, m is their mass, R is some characteristic radius of the system and $b_{\rm min}$ and $b_{\rm max}$ are the minimum and maximum values for the impact parameter. Using the same approximations and including a softening $\epsilon$, Huang et al. (1993) find

 

$$\Delta\upsilon_\perp^2 \frac{ 4 G^2 m^2 N}{R^2 \upsilon^2} \left[ \frac{\epsilon^2}{b_{\rm max}^2 + \epsilon^2} -\frac{\epsi... ...+\ln\frac{b_{\rm max}^2 + \epsilon^2} {b_{\rm min}^2 + \epsilon^2} \right]\cdot$$ (4)

A somewhat more elaborate derivation following the calculation of the diffusion coefficients (e.g. Chandrasekhar 1942; Spitzer & Hart 1971; Spitzer 1987; Binney & Tremaine 1987) gives

$$T_{\rm relax} = {{F \upsilon^3} \over {G^2 m \rho~\ln ( {b_{\rm max}} / {b_{\rm min}})}},$$ (5)

where $\rho$ is the density and F is a constant, roughly equal to 0.34. The quantity $\ln ( {b_{\rm max}} / {b_{\rm min}})$ is often denoted by $\ln \Lambda$ and referred to as the Coulomb logarithm.

The appropriate values of $b_{\rm min}$ and $b_{\rm max}$ in the above equations have been discussed at length in the literature. Chandrasekhar (1942) argued that $b_{\rm min}$ is the value of b for which the angular deflection of the star is equal to  ${1 \over 2} \pi$. The value of $b_{\rm max}$has been subject to considerable controversy. Chandrasekhar (1942), Kandrup (1980) and Smith (1992) have opted for a $b_{\rm max}$ of the order of the mean inter-particle distance, while others (e.g. Spitzer & Hart 1971; Farouki & Salpeter 1982; Spitzer 1987) used for $b_{\rm max}$ a characteristic radius of the system. The numerical simulations of Farouki & Salpeter (1994) argue in favour of the latter. This is further corroborated by the results of Theis (1998).

Using the estimates $b_{\rm min} = \frac {G m} {\upsilon^2}$, $b_{\rm max}=R$ and assuming virial equilibrium, so that we can use for the velocity the estimate $\upsilon^2 \approx \frac {G N m} {R}$, we find (Binney & Tremaine 1987)

$$T_{\rm relax}={{N}\over{8~\ln N}}T_{\rm cross}.$$ (6)

Similarly for the case with softening setting $b_{\rm min} \ll \epsilon$, $b_{\rm max}=R$ and estimating the velocity by assuming virial equilibrium we find

 

$$T_{\rm relax} {{{\rm\upsilon}^4 R^2}\over{G^2 M^2}}~ {{N}\over {4\left[\ln(R^2/\epsilon^2)-1\right]}}T_{\rm cross}$$ =

$${{{\rm\upsilon}^4 R^2}\over{G^2 M^2}}~ {{N}\over {8\ln(R/\epsilon)}}T_{\rm cross}$$ = (7)

$${{N}\over{8\ln(R/\epsilon)}}T_{\rm cross}.$$ =

Equation (7) differs by a factor of 2 from that of Huang et al. (1993), the reason being that our definition of the relaxation time is based on the velocities, while that of Huang et al. (1993) is based on the energies.

It is clear that, for the number of particles used in present-day simulations of collisionless systems and the appropriate values of the softening, N is considerably larger than $R/\epsilon$. Since, however, only the logarithms of these quantities enter in Eqs. (6) and (7), the differences in the estimates of the relaxation times differ, for commonly used values of N, by less than a factor of 2. Equation (7) is more appropriate, since it includes the softening. Often a coefficient g is introduced in the Coulomb logarithm, i.e. $\Lambda~=~g N$. Giersz & Heggie (1994) estimated that the most appropriate value of g is 0.11. They also compiled in their Table 2 the values given by several other authors. They are all between 0.11 and 0.4. Independent of what is chosen for the Coulomb logarithm, equations such as (6) or (7) argue that even for a moderately low number of particles, of the order of say a few thousands, the relaxation time is comfortably high, of the order of, or higher than, 40 crossing times.