Isothermal cooling flows

Details of the solution are derived for a steady, self-similar, comoving, isothermal cooling flow. The distribution of gas phases can be expressed in terms of the mass how function, (M) over dot (r, T), which gives the mass per unit time of gas hotter than temperature T flowing into a sphere of radius r. Self-similarity allows this to be separated as (M) over dot (r, T) = (M) over dot (r)g(T), where (M) over dot (r) is the usual mass flow rate and g(T) is a dimensionless function expressing the distribution of the phases. It is shown that, for (M) over dot (r) proportional to r(n), g(T) = exp [-3 eta/3-eta integral(0)(T) dT'/T'/(T(m)/T')(2)Lambda(T')/Lambda(T(m))-1], where T(m) is the maximum temperature of the hot gas. In the units used here, the corresponding solution for the differential emission measure from within a sphere of radius r is de/dT = 5/2 (M) over dot g(T)/Lambda(T)-Lambda(Tm)(T/Tm)(2), where (M) over dot = (M) over dot (r) = is the total mass flow rate into the sphere.