They do not understand how that which differs with itself is in agreement: harmony consists of opposing tension, like that of the bow and the lyre.--Heraclitus
Stability of Stars
Though their deaths can be brief and dramatic, stars spend their lives in stable configurations that last for millions or even billions of years. This stability arises from a competition between the star's self gravity, which tends to cause the star to collapse under its own weight, and its internal pressure, which pushes the star outward. If the gravity force throughout the star exceeds the pressure force, the star will shrink. As the star contracts, the pressure rises, which impedes further contraction. By contrast, if the pressure force exceeds the gravity throughout the star, the star expands. This lowers the pressure and prevents further expansion. In this way, there is a natural tendency for all parts of the star to achieve a balance between gravity and pressure, and when this occurs, the star is in hydrostatic equilibrium.
Models of Stars
It is possible to model stars in hydrostatic equilibrium. In doing so, one seeks the mass density configuration that provides the balance between the gravity and the pressure. We generally model gravity with Newton's Univeral Law of Gravitation. This tells us that, for our spherical star, the weight of a parcel of the star is proportional to the mass of that bit of the star times the mass of all the material lying inside the radial location of that parcel of the star divided by the square of the distance from the parcel to the center of the star. This is exactly analogous to the fact that your weight is due to the pull of every atom inside the Earth on your body. For the pressure force, we need a rule that explains how the pressure of a parcel of gas in the star depends on that parcel's density, temperature, and composition. In general, this is a complicated law that depends on the details of the atomic physics of the gas making up the star. We call this complicated pressure law an Equation of State. Since realistic equations of state are complicated, realistic models of stars are complicated too.
Polytropes
Fortunately there is a fairly simple equation of state that works well for many situations arising in astrophysics. This is the polytropic equation of state in which the pressure P is proportional to the (n + 1)/n power of the mass density ρ, where n is a constant. We write this equation of state mathematically as P = K ρ(n + 1)/n, where K is a constant of proportionality and n is the polytropic index. A gaseous sphere in hydrostatic equilibrium whose pressure obeys the polytropic equation of state is known as a polytrope
A polytrope is described by a second order differential equation called the Lane-Emden equation:
whose solution φ gives the mass density profile (and pressure) of the sphere as a function of its scaled position ξ along its radius. The first root of the solution to this equation ξ=ξ1 determines the position where the mass density goes to zero and therefore determines the sphere's radius. n is the order of the equation, which, for physically relevant situations, varies from 0 to 5. With a solution for the Lane-Emden equation, one may substitute in a central density and mass of a real star and get a physical polytrope, that is, a good approximation to the physical structure of a real gaseous sphere in hydrostatic equilibrium, such as a star.
Closed-form solutions to the Lane-Emden equation are available for the cases n = 0, 1, and 5. In particular, for n = 0, the solution is φ(ξ) = 1 - ξ2/6, but the mass density is uniform throughout the star. For n = 1, the solution is φ(ξ) = sin(ξ)/ξ. For n = 5, the solution is φ = (1 + ξ2/3)-1/2. For this solution, the density is infinite in the center of the star. For other values of n, the Lane-Emden equation must be solved numerically.
The Polytrope Tool
The purpose of the Polytrope Tool is to allow an internet user to calculate polytropes numerically and graph and download the results. The tool requires the user to choose a polytropic index in the range 0 to 4.999999. The n = 5 case, which results in an infinite central density and an infinite radius of the star, is not implemented in the current version of the tool. Upon submission of the data, the server at Clemson University computes the relevant solution to the Lane-Emden equation and then sends the results to the user's browser. The user can then graph and download the results. The user can then choose a central density and a mass of the star and compute a physical polytrope. Again the user can graph and download the results. In this way, the Polytrope Tool is intended to be an easy-to-use means to understand some basics of stellar structure.
The best way to start using the tool is to try out the tutorials. Help links in the tool also aid the user in determining the correct input. More background and other related information is available from the Papers & Links sidebar link. We hope you enjoy the tool, and, as always, we welcome feedback.
