STANDARDISED UNITS AND TIME SCALES . D.C. Heggie Department of Mathematics, University of Edinburgh, King's Buildings, Edinburgh EX9 3JZ, U.K. and R.D. Mathieu Center for Astrophysics, 60 Garden Street, Cambridge, Ma 02138, U.S.A. l. Units For the purpose of comparison of results obtained by different authors, it is very convenient if they share a common system of units. The following system of units seems to find quite wide (if not universal) favour. The units are such that G = 1 M = 1 E= -1/4 where G is the gravitational constant, M is the total initial mass, and E is the initial energy. The corresponding units of mass, length and time are then Um = M Ul = - GM^2/4E (1) Ut = GM^{5/2}/(-4E)^{3/2} (cf. Hénon 1972). The choice for E looks odd, but corresponds to a virial radius R (harmonic mean particle separation) equal to unity for a system in virial equilibrium. In N-body work a somewhat different, actually N-dependent, system is often used (cf. Aarseth 1972), but leads to a crossing time scale proportional to N^{-1/2}. This system is also unsuitable for galaxy simulations, where neither the number of stars nor the number of particles in the simulation is relevant to the important dynamical time scales. There are of course stellar dynamical calculations for which the system (1) is unsuitable, e.g. unbound systems or cosmological simulations. And even with regard to systems for which these units are appropriate, it is not suggested that system should be the {\sl only} system in which the results of dynamical calculations are expressed. What could be recommended 234 is that it should be {\sl one} of the systems used in all published results. In addition, however, the procedure by which quantities are to be converted into astrophysical units, e.g. parsecs, km/sec, solar masses, etc., should be stated explicitly. (Errors can easily be made in efforts to track down the definition of dimensionless variables, or quantities expressed in arbitrary units, and furthermore the repeated labour involved is a waste of time.) This is not to say that it is best to use astrophysical units in the first place; to do so involves choosing particular values for M and E, whereas many stellar dynamical calculations are formally valid for any choice of these values. Thus the unit of density in the system (1) could be quoted as 10^3 (/10^6 M_sun) (R/10pc)^{-3} M_sun pc^{-3}, where M and R are, respectively, the mass and virial radius of the astrophysical system to which the calculations are to be applied. From the observer's point of view, the applicability of theoretical results is enormously en- hanced if they are presented in a manner analogous to the presentation of obtainable data. Since the latter is usually constrained by our perspective on the universe, it is incumbent on theorists to make full use of the greater flexibility available to them in the presentation of their results. Oft-cited examples are the projection of three_dimensional density profiles onto two dimensions, and the conversion of anisotropic velocity distributions into tangential- and radial-velocity distri- butions. 2. Relaxation times For theoretical purposes one needs both local and global measures of the relaxation time scale. The choices made by Spitzer and Hart (1971) are adopted quite commonly, i.e. the local relaxation time t_rf = v_mf^3/(35.4 G^2m rho_f log10(0.4N) (2) and the half-mass relaxation time t_rh = 0.0600 M^{1/2}Rh^{3/2}/G^{1/2}m log10 (0.4N) (3) where we have given the form of trf appropriate when all stars have the same mass m, v_mf^2 is the mean square (three_dimensional) speed of the stars, rho_f is their mass density, and Rh is the radius containing half the total mass. Both choices have arbitrary aspects, and even contentious ones (the argument of the 'Coulomb logarithm'). For theoretical purposes it would be preferable, perhaps, to choose a relaxation time which simplifies the Fokker-Planck equation as much as possible. This was the basis of the old reference time introduced by Spitzer & Härm (1958), but since Spitzer evidently subsequently preferred eq.(2), we are unable to suggest any better alternative. The important point is that it is essential to state precisely what definition of relaMation time is being adopted. It is not even enough to say 'Spitzer & Hart (1971), eq.(5)' since this equation gives two definitions for trh, which agree only if a further approximation is made. It is also necessary to make clear whether natural or common logarithals are intended. These remarks are trivial, but they are made simply because confusion has arisen in the .. literature in cases where such points have not been stated explicitly. It is worth pointing out that a similar confusion exists among observers as well, indeed at a more fundamental level. One finds in the observational literature a variety of applications of different formulae for the relaxation 235 times of stellar systems. Given that relaxation times vary tremendously throughout any given stellar system (because of density gradients_, and that they are sensitive to the spectrum of stellar masses, the suitability of many of the quoted relaxation times to the problems under study also varies greatly. Two frequently quoted time scales are the central relaxation time, typically derived from models fitted to density profiles, and a mean relaxation time within, say, the half-mass radius. These two time scales generally differ greatly .and, depending on the issue being addressed, only one and quite possibly neither is the appropriate choice. In addition, one of the observationally most accessible indicators of dynamical evolution is the presence of mass segregation; clearly, in this case, the precise inclusion of a mass spectrum in the derivation of the evolutionary time scales is necessary, as well as recognition of the fact that the time scales will vary depending upon the stellar component in question. Indeed, as we begin to study the younger open clusters and Magellanic Cloud clusters, the proper treatment of the evolution times with respect to the mass spectrum becomes absolutely critical. Mathieu (1983) has discussed this point in detail with regard to the young open cluster M35; cf. also McNamara & Sekiguchi (1986). Given the existing theoretical literature it is possible to compute properly the relevant time scales for most problems (to within the limits of our understanding of relaxation processes). However, the widespread use of mean relaxation times, such as eq.(3), indicates that this is not usually done. The community of cluster observers is in need of a sort of tutorial discussion of relaxation time scales, including both a review of the basic physics involved, and a set of straightforward procedures for calculating the appropriate evolutionary time scales for a range of problems. This exercise will not only be of great value for the general community, but will pose a challenging problem for stellar dynamicists as well. Several difficult issues will have to be addressed, including the very definition of relaxation time scales in the presence of density gradients, mass spectra and binaries. In addition a detailed comparison of the analytic theory and N-body simulations remains to be done. The work of Casertano et.al. (1986) is an important step in this direction. However, the detailed study of this difficult and fascinating problem should not unduly delay the preparation of a tutorial discussion for the use of the more general community studying stellar systems. References Aarseth, S.J., 1972, in M. Lecar (ed.), Gravitational N-Body Problem, Reidel, Dordrecht, p.88 Casertano, S., Hut, P. & McMillan, S.L.W., 1986, Ap.J., in press Hénon, M., 1972, in M. Lecar (ed.), Gravitational N-Body Problem, Reidel, Dordrecht, p.406 McNamara, B. & Sekiguchi, K., 1986, A.J., in press Mathieu, R.D., 1983, Ph.D. dissertation, University of California, Berkeley Spitzer, L., Jr. & Härm, R., 1958, Ap.J., 127, 544 Spitzer, L., Jr. & Hart, M.H., 1971, Ap.J., 164, 399 . .