The Parker solution family, v(r), is expressed as the contour of H(r,v), the total specific energy (kinetic energy, thermal enthalpy, and gravitational potential per unit mass), . The parameters with subscription 0 stand for values taken at a reference point, generally the critical (trans-sonic) point. The left plot is a bird-eye-view of H(r,v) drawn in x-y-z space; x axis is for the heliocentric distance (radius), y is for the flow speed, and z and the colors are for the value of H that is offset and then rescaled with logarithm for display purpose. A movie (2.4 Mbyte) provides 360-degree view.

   There are two key parameters to determine the system; the specific heat ratio (γ) and the typical temperature (T₀). The index γ characterizes the relationship between density and pressure (or temperature), taking over the equation of state. It is frequently chosen to be 1 or numbers slightly greater than 1, so that the polytrope approximation will mimic near-isothermal high-heat conduction case (the specific heat ratio for the monatomic or proton or electron must be 5/3). Here it is set at 1.05. Given the solar mass (and gravity constant), the typical temperature is another key parameter generally defined at the critical trans-sonic point. 1 MK is often chosen, for this number yields reasonably good consequences at wide range of regions from solar surface, corona to interplanetary space.
   The plot above contains all possible Parker solution types. Among them,the solar wind of the trans-sonic outward flow is expressed by the special one running, in the drawing above, from left-bottom (sub-sonic) to critical point (trans-sonic) to the upper-right (super-sonic); the critical point is at the x-point dividing yellow and blue areas, and the integration constant H₀ at the right hand side of the total enthalpy equation above has to be zero. Another solution line that passes the critical point but runs from upper-left (super-sonic near the Sun) to the x-point to bottom-right (sub-sonic regime at region distant from the Sun) is of few meanings for the solar wind, though, there are lots of meanings for other systems like accretion disc.