L04 and L05 Polytropes

L04 and L05 Polytropes#

Week 2, Thursday and Week 3 Tuesday

Material covered and references#

Derivation of the Lane-Emden equation

So far, we only have two equations for stellar structure: mass continuity, and hydrostatic equilibrium. However, we have 3 unknowns: \(M_r(r)\), \(P(r)\), and \(\rho(r)\).

We need to find a relation between two of these quantities. The ideal gas law is a relation between \(P\) and \(\rho\) (called an “equation of state”, but it introduced another unknown \(T(r)\). To find \(T(r)\) we will have to talk about energy transport (later on).

Micro-objective:

  • I can explain why we need to introduce an “equation of state” to be able to solve the structure of a star.

  • I can explain why using the “ideal gas law” is not the final solution, and how using a “polytropic” equation of state can be used right now as an approximation.

But for know, we can use some useful polytropic equation of state where the pressure only depends on the density to a certain power such that

\[P(r) = K\rho(r)^{\frac{n+1}{n}}\]

The first step is to set ourselves up to be able to solve this equation (we will solve for the density). This means:

  1. Combine the hydrostatic equilibrium equation and the continuity equation to eliminate \(M_r\).

  2. We assume that our solution for the density will have this form:

\[\rho(r)=\rho_o\theta(r)^n,\]

where theta is a unit-less function that describes the variation of the density as a function of radius.

  1. We can then transform our equations of structure into the well-known Lane-Emden differential equation for polytropes, the solution of which gives \(\theta(\epsilon)\), where \(\epsilon\) is unit-less and related to \(r\).

Micro-objectives:

  • I can identify all of the terms in the polytropic equation of state.

  • I can explain the meaning of the terms in our solution ‘shape’.

  • I can explain where the \(\alpha\) constant we use to normalize the radial coordinate \(r\) comes from.

In the textbooks:

  • The best one in my opinion is in the Leblanc: Sec 5.4, until eq. 5.93

  • But you can also look in McDo: Chap 9 until eq 9.10, or Hanson Sec. 7.2.1 until eq. 7.26

  • Kip has the derivation in Chap 19 until eq 19.10, but uses a nomenclature different than the other books (\(w(z)\) instead of \(\theta(\epsilon)\))

The second step is to analyze the solution, so that we can go from our mathematical \(\theta(\epsilon)\) back to \(\rho(r)\).

  1. Once we obtain the solution for \(\theta(\epsilon)\) for a certain n index, we can use it to determine characteristics of the polytrope. We can find the value of \(\epsilon\) at the surface: it is the value of \(\epsilon\) where \(\theta\) (and therefore the density!) is null! We called this location \(\epsilon_1\).

  2. From this, we found that we are able to then transform \(\theta(\epsilon)\) into \(rho/\rho_o\) versus \(r/R\star\). We compared this with the real value of \(rho/\rho_o\) inside of our Sun.

Micro-objectives:

  • In a graph of \(\theta(\epsilon)\) versus \(\epsilon\), I can identify the center and surface of a star.

  • I can transform a graph of \(\theta(\epsilon)\) versus \(\epsilon\) into a graph of \(\rho/\rho_o\) versus \(r/R_\star\).

In the textbooks:

  • Hanson has the best description of the properties of polytropes.

Table 7.1 also gives values for epsilon_1 and theta’(epsilon_1) for various n.

The third and final step was to go back to \(\rho(r)\) (and also \(P(r)\) and \(M_r(r)\)) in real units.

We found 3 equations that relate \(M_\star\), \(R_\star\), \(P_o\), \(\rho_o\), and \(K\). As we only have 3 independent equations, this means we if define two of these quantities, we can find the other 3.

For example, you can make equations for \(P_o\), \(\rho_o\), and \(K\) that only depends on \(M_\star\) and \(R_\star\).

Another example would be for stars with degenerate matter (we’ll talk about this later in the course). In this case \(K\) is known from statistical mechanics – therefore if we pick the mass of the star, the radius, central pressure, and central density are all known.

Micro-objectives:

  • I can explain how the expressions for \(R_\star\), \(P_o\), and \(M_\star\) create a system of 3 equations and 5 unknowns.

  • I can use polytropic models to compare with the Sun’s density profile shape and central density value.

In the textbooks:

Same as above

Python skills:

  • Practice adding curves to graph, and use the astropy units and constant packages

  • Learn about functions

  • Learn about loops

Supplemental problems suggestions:#

  • Leblanc ex. 5.4 shows that a n=0 polytrope represent the constant density case.

  • Leblanc Prob. 5.10