Physical characteristics of a polytrope index 5 with finite radius

In astrophysics the polytropic law with index n is commonly used as a means of imposing a simple and ordered physical structure on a gaseous (or smoothed discrete) system. It is pointed out that the three well known classical analytical E type solutions of the Lane-Emden equation for n = 0, 1, and 5 all have some constraining physical features. Matched composite analytical solutions provide a means of extending the range of available E type analytical solutions of the Lane-Emden equations which can be used in applications. As an illustration the physical characteristics of an alternative complete polytropic model for index n = 5 which has finite mass, radius, and central condensation have been established. Buchdahl (1978) has pointed out that the polytrope of index 5 with infinite radius has in fact finite, not infinite, potential energy. A result calculated for this finite radius model is clearly in accord with Buchdahl's conclusions which are based on an explicit expression that was derived for the gravitational potential energy of the model.