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Stars and Stellar Processes

Mike Guidry

Chapter 8

Summary of Stellar Equations - all with Video Answers

Educators

Chapter Questions

01:19

Problem 1

Express the basic equations of stellar structure (4.4), (4.10), (7.41), (Z.43), and (7.34) in Lagrangian form.

Manik Pulyani
Manik Pulyani
Numerade Educator
00:03

Problem 2

Apply pure scaling (dimensional) arguments to the equation for hydrostatic equilibrium to obtain directly the Lane-Emden result
$$
M \propto R^{(3-n) /(1-n)}
$$
relating the mass $M$ and radius $R$ for a star with a polytropic equation of state
$$
P=K \rho^{1+1 / n} \equiv K \rho^\gamma,
$$
where $K$ is a constant, $P$ is the pressure, $\rho$ is the mass density, and $n$ is the polytropic index.

John Johnson
John Johnson
Numerade Educator
01:06

Problem 3

Use Eqs. (8.7) and (8.8) to demonstrate that the mass for a LaneEmden solution is given by Eq. (8.12).***

Raj Bala
Raj Bala
Numerade Educator
00:03

Problem 4

(a) Show that Eq. (8.5),
$$
\frac{1}{r^2} \frac{d}{d r}\left(\frac{r^2}{\rho} \frac{d P}{d r}\right)=-4 \pi G \rho,
$$
with appropriate boundary conditions is equivalent to the usual formulation of the hydrostatic equilibrium problem for Newtonian gravity.
(b) Prove that Eq. (8.8) with the assumption ( $\underline{8.6)}$ is equivalent to this result and therefore describes hydrostatic equilibrium for a star governed by a polytropic equation of state.
(c) Show formally that the parameter $a$ defined in Eq. (8.7) has the dimension of length.
(d) Show that the coefficient $K$ in Eq. (8.6) and the parameter $a$ in Eq. (8.7) can be written as
$$
K=\frac{P_{\mathrm{c}}}{\rho_{\mathrm{c}}^{(n+1) / n}} \quad a=\sqrt{\frac{(n+1) P_{\mathrm{c}}}{4 \pi G \rho_{\mathrm{c}}^2}},
$$
respectively. Thus, for a given polytropic index $n$ the parameters $K$ and $a$ are determined completely by the central
pressure $P_{\mathrm{c}}$ and the central density $\rho_{\mathrm{c}}$.

John Johnson
John Johnson
Numerade Educator
04:33

Problem 5

Using the form of the Lane-Emden equation given in footnote 3 after Eq. (8.8), find an analytical solution for $n=0$ and evaluate the corresponding Lane-Emden constants $\xi_1$ and $\xi_1^2\left|\theta^{\prime}\left(\xi_1\right)\right|$.

Keshav Singh
Keshav Singh
Numerade Educator
04:15

Problem 6

Show that for the Lane-Emden solution the central density $\rho_{\mathrm{c}}$ can be expressed as
$$
\rho_{\mathrm{c}}=-\left(\frac{\xi_1}{3 \theta^{\prime}\left(\xi_1\right)}\right) \frac{M}{\frac{4}{3} \pi R^3},
$$
where $\theta^{\prime}\left(\xi_1\right) \equiv(d \theta / d \xi)_{\xi_1}$.

M Hassan Anwar
M Hassan Anwar
Numerade Educator
01:46

Problem 7

Verify the polytropic curves in Figs. 8.1 and $\underline{8.2}$, and the entries for $\xi_1$ in Table 8.2, by integrating the Lane-Emden equation ( $\left.\underline{8.8}\right)$ numerically. Hint: There are various ways to do this but the simplest is to use a combined numerical/graphics package like MatLab, Maple, or Mathematica to integrate and visualize results.

Gregory Higby
Gregory Higby
Numerade Educator
02:05

Problem 8

Prove that the relationship $(\underline{8.13})$ between the mass and radius for a Lane-Emden solution follows from Eqs. (8.10) and (8.12).

James Kiss
James Kiss
Numerade Educator
10:04

Problem 9

Use Eqs. $(\underline{8.13}),(\underline{8.6})$, and $(\underline{8.11)}$ to show that for a Lane-Emden solution with polytropic index $n$ the relationship between the central density and central pressure is given by Eq. (8.14).***
1 This is just a generalization of the standard example from introductory physics of estimating a time to travel some distance $x$ as $t=x / x=x / v$, where $v$ is the average velocity.
2 Stars also burn heavier fuels after they leave the main sequence and we can associate additional nuclear burning timescales with each fuel. But these timescales typically are small compared with that for hydrogen burning on the main sequence. Thus the time to consume available hydrogen on the main sequence is a rough approximation to the total nuclear burning timescale for a star.
3 Often the Lane-Emden equation is stated in the equivalent form
$$
\frac{1}{\xi^2} \frac{d}{d \xi}\left(\xi^2 \frac{d \theta}{d \xi}\right)=-\theta^n
$$
This is more compact but not as useful computationally as the form (?8).

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator