Polytropes¶

The Polytopic Relation¶

We will assume that the entropy \(S\) is constant in our polytope. When this is the case, we can write down:

\[ P = K \rho^\gamma \]

Different \(\gamma\) will approximate different stellar opacities. \(K\) is a constant at all radii. We will plug these in (in gory detail) to our equations.

Recall:

Hydrostatic Equilibrium

\[ \frac{dP}{dr} = -\rho \frac{GM}{r^2} \]

Mass Continuity

\[ \frac{dM_r}{dr} = 4\pi r^2 \rho \]

We will now play some mathematical tricks. Going to HE:

\[ \frac{r^2}{\rho} \frac{dP}{dr} = -G M_r \]

Differentiate both sides:

\[ \frac{d}{dr}\left(\frac{r^2}{\rho}\frac{dP}{dr}\right) = - 4\pi G r^2 \rho \]

We have two unknowns in this equation, and so we invoke the polytope:

\[ P = K \rho^\gamma = K \rho^{1+1/n} \]

where \(n\) is the polytopic index.

A side note¶

We have done this before! A relativistic, Fermi gas (Neutron stars, WDs), we have \(n=3\) and \(\gamma=\frac43\). Similarly, non-relativistic, Fermi gas has \(n=\frac32\) and \(\gamma =\frac53\) (fully convective stars, giant planets).

Back to Polytropes¶

Let’s define dimensionless variables:

\[ \boxed{\rho = \rho_c \theta^n} \]

\[ \boxed{P = P_c \theta^{n+1}} \]

\[ \boxed{r = \alpha \xi} \]

Here, \(\rho_c\) is the central density, \(P_c\) is the central pressure, \(\alpha\) is some length constant, and \(\xi\) is a dimensionless radius-like quantity. Note that:

\[ \boxed{\alpha^2 = \frac{K(n+1) \rho_c^{\frac{1-n}{n}}}{4\pi G}} \]

We can thus re-write our polytrope differential equations in these dimensionless units, giving us the Lane-Emden equation:

\[ \frac{1}{r^2}\frac{d}{dr}\left(\frac{r^2}{\rho}\frac{dP}{dr}\right) = -4\pi G\rho \rightarrow \boxed{\frac{1}{\xi^2}\frac{d}{d\xi}\left(\xi^2 \frac{d\theta}{d\xi}\right) = - \theta^{n}} \]

We will solve this for \(\theta(\xi)\), and thus we need boundary conditions to solve this. What are these boundary conditions?

Boundary Conditions¶

\(\theta\vert_{\xi=0} = 1\). (central pressure is the central pressure)
\(\frac{d\theta}{d\xi}\vert_{\xi=0} = 0\). (continuity across the core)

Solving The Equation¶

There are only three analytic solutions to this equation.

n = 0

\[ \theta(\xi) = 1 - \frac{\xi^2}{6} \]

where \(\xi_{max} = \sqrt{6}\).

Here, we assume that the pressure is not related to the density – an incompressible fluid! We find this where \(\rho = \rho_c\), or things like Earth’s interior.

n = 1

\[ \theta(\xi) = \frac{\sin\xi}{\xi} \]

where \(\xi_{max}=\pi\).

n = 5

\[ \theta(\xi) = \left(1+\frac{\xi^2}{3}\right)^{-1/2} \]

where \(\xi_{max} = \infty\).

Any time that \(n>5\), \(\xi_{max} = \infty\). Stars tend to live in between \(n=1\) and \(n=5\).

Numerical Solutions¶

A solution for \(n=1.5\) is a pretty good approximation for a fully convective star. The Sun is most closely approximated by \(n=3\) polytopes.

For a given \(n\), we need two constants:

\(K\) and \(\rho_c\).

If we are given those two things, polytopes allow us to give you back \(M\), \(R\), \(\rho(r)\), \(L(r)\), \(T(r)\), etc.

Other Useful Polytope Information¶

Here are some useful relations coming from the Lane-Emden equation.

\[ R = \alpha \xi_{max} = \left(\frac{K(n+1)}{4\pi G}\right)^{1/2} \rho_c^{\frac{1-n}{2n}}\xi_{max} \]

\[ M = \int_0^{R} 4\pi r^2 \rho(r) \, \mathrm{d}r = 4\pi \alpha^3 \rho_c \int_0^{\xi_{max}} \xi^2 \theta^n \mathrm{d}\xi = 4\pi \left(\frac{K(n+1)}{4\pi G}\right)^{3/2} \rho_c^{\frac{3-n}{2n}} \left(-\xi^2 \frac{d\theta}{d\xi}\right)\Bigg\vert_{\xi_{max}} \]

We can combine the two equations (with something like \(R^{(3-n)/n} \cdot M^\frac{n-1}{n}\)):

\[ R^{\frac{3-n}{n}} M^{\frac{n-1}{n}} = \underbrace{\frac{K}{GN_n}}_\text{const} \rightarrow \boxed{M \propto R^{\frac{n-3}{n-1}}} \]

Let’s examine our two reference points.

1. \(n=1.5\) Polytrope: A Fully Convective Star

\[ RM^{1/3} = \frac{K}{0.42 G} \]

2. \(n=3\) Polytrope: The Sun

\[ M = \frac{K}{0.36 G} \]

We can thus get other quantities, like the central density:

\[ \rho_c = \frac{3M}{4\pi R^3} a_n \text{ where } a_n\equiv \frac{-\xi}{3\left(\frac{d\theta}{d\xi}\vert_{\xi=\xi_{max}}\right)} \]

The central pressure:

\[ P_c = \frac{4\pi G \rho_c^2 \alpha^2}{n+1} = c_n \frac{GM^2}{R^4} \]

And, lastly, \(T_c\), for an ideal gas:

\[ T_c = \frac{\mu}{\mathcal{R}}\frac{P_c}{\rho_c} = b_n \frac{\mu}{\mathcal{R}}\frac{GM}{R} \]

note that \(a_n\), \(b_n\), and \(c_n\) are constants where tables exist!

An Example¶

Can we write \(L\) and \(T_{eff}\) for a polytrope?

We should get:

\[ T_{eff} \propto M^{1/7} R^{1/49} \]

(weakly radius dependendent) for a modified H- opacity. If we do luminosity, we find:

\[ L = M^{4/7} R^{102/49} \]

Plugging in with numbers, we find, for a fully convective star::

\[ T_{eff} = 2400 \text{ K } \left(\frac{M}{M_\odot}\right)^{1/7} \left(\frac{R}{R_\odot}\right)^{1/49} \]

Stellar Astrophysics

Polytropes

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