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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