Entropy Maximization, Cutoff Distribution, and Finite Stellar Masses

Table 2

Salient features of King-like lowered isothermal models 𝑓^Low (cf. (9)) and our MPD distribution f (cf. (23)).

Feature	𝑓^Low	𝑓
Physical motivation before equilibrium	As a general stellar cluster evolves tides are generated	As a general stellar cluster evolves entropy is produced
Theoretical basis at equilibrium	Tidal force of the galaxy	Entropy maximization
Equation obeyed	Simple, approximate, and nonunique in (9)	Transcendental, exact, and unique in (23)
Role of Planck constant	Not needed since the treatment is classical Newtonian.	Needed explicitly to find the cell degeneracy 𝑔𝑗, the index 𝐽 of (27), and the huge cutoff number K.
Upper energy cutoff in the spectrum	Imposed by hand at 𝜀=0 in (9).	Predicted by theory to occur at 𝜀𝐾<0 in (19).
Functional form	For example, 𝑒^−𝛽𝜀−1 in which the subtraction term 1 is constant.	𝑓^𝐵−𝜃/𝑔 in which the subtraction term 𝜃/𝑔 is strongly energy-dependent.
Mass and radius of the cluster	Finite	Finite
Number density profile near the edge	𝑛(𝑟)∝King(1/𝑟−1/𝑟𝑡)^2	𝑛(𝑟)∝MPD(1/𝑟−1/𝑟𝑡)^5/2
Quality of fit to 𝑛^1/2 and 𝑛^2/5	Good in Figure 4	Good in Figure 3