2.2: Integral Theorems from Hydrostatic Equilibrium

a. Limits on State Variables

Following Chandrasekhar,¹ we wish to define a quantity \(I_{\sigma, \nu}(\mathrm{r})\) which is effectively the σth moment of the mass distribution further weighted by \(\mathrm{r}^{-\nu}\). Specifically \[I_{\sigma, \nu}(r) \equiv \frac{G}{4 \pi} \int_0^r \frac{[M(r)]^\sigma}{r^\nu} d M(r)\label{2.2.1}\]

There are quite a variety of physical quantities which can be related to \(I_{\sigma, \nu}\). For example, \[4 \pi I_{11}=G \int_0^r M(r) \rho 4 \pi r^2 \frac{d r}{r}\label{2.2.2}\]

is just the absolute value of the total gravitational energy of the star.

We can use this integral quantity to place limits on physical quantities of interest if we replace \(\rho\) by \(<\rho>\) as defined by equation \ref{2.1.2}. Since \[r^\nu=\left[\frac{M(r)}{\frac{4}{3} \pi\langle\rho\rangle(r)}\right]^{\nu / 3}\label{2.2.3}\]

we may rewrite \(I_{\sigma,\nu}\) as \[I_{\sigma, \nu}(r)=\frac{G}{4 \pi} \int_0^r[M(r)]^{\sigma-\nu / 3}\left(\frac{4 \pi}{3}\right)^{\nu / 3}\langle\rho\rangle(r)^{\nu / 3} d M(r)\label{2.2.4}\]

Now since our assumption of the monotonicity of \(\rho\) requires \(\rho_{\mathrm{c}} \geq<\rho>(\mathrm{r}) \geq\rho(\mathrm{r})\), we can obtain an inequality to set limits on \(I_{\sigma, \nu}\). Namely, \[\mathrm{\frac{G}{4 \pi}\left[\frac{4 \pi}{3}\right]^{\nu / 3} \rho_c^{\nu / 3} \frac{M^{\sigma+1-\nu / 3}(r)}{(\sigma+1-\nu / 3)} \geq I_{\sigma, \nu}(r) \geq \frac{G}{4 \pi}\left[\frac{4 \pi}{3}\right]^{\nu / 3}<\rho>^{\nu / 3}(r) \frac{M^{\sigma+1-\nu / 3}(r)}{(\sigma+1-\nu / 3)}}\label{2.2.5}\]

Now let us relate \(<\boldsymbol{P}>\), \(<\boldsymbol{T}>\), and \(<\boldsymbol{g}>\) to \(I_{\sigma, \nu}\), where these quantities are defined as \[\begin{aligned}
\langle P\rangle & \equiv \int_0^M P(r) \frac{d M(r)}{M} \\
\langle T\rangle & \equiv \int_0^M T(r) \frac{d M(r)}{M} \\
\langle g\rangle & \equiv \int_0^M g(r) \frac{d M(r)}{M}
\end{aligned}\label{2.2.6}\]

Making use of the result that the surface pressure and temperature are effectively zero compared to their internal values, we can eliminate the temperature by using the ideal-gas law, integrate the first two members of equations \ref{2.2.6} by parts and eliminate the pressure gradient by utilizing hydrostatic equilibrium. We obtain \[\begin{aligned}
& \langle P\rangle=\frac{I_{2,4}(R)}{M} \\
& \langle T\rangle=\frac{4 \pi \mu m_h}{3 k} \frac{I_{1,1}(R)}{M} \\
& \langle g\rangle=\frac{4 \pi I_{1,2}(R)}{M}
\end{aligned}\label{2.2.7}\]

The last of these expressions comes immediately from the definition of g. Applying the inequality [(equation \ref{2.2.5}], we can immediately obtain lower limits for these quantities of \[\begin{aligned}
& \langle P\rangle \geq \frac{3 G M^2}{20 \pi R^4}=5.4 \times 10^8\left(\frac{M}{M_{\odot}}\right)^2\left(\frac{R_{\odot}}{R}\right)^4 \quad \mathrm{~atm} \\
& \langle T\rangle \geq \frac{G \mu m_h M}{5 k R}=4.61 \times 10^6 \mu\left(\frac{M}{M_{\odot}}\right)\left(\frac{R_{\odot}}{R}\right) \quad \mathrm{K} \\
& \langle g\rangle \geq \frac{3 G M}{4 R^2}=2.05 \times 10^4\left(\frac{M}{M_{\odot}}\right)\left(\frac{R_{\odot}}{R}\right)^2 \quad \mathrm{~cm} / \mathrm{s}^2
\end{aligned}\label{2.2.8}\]

Since these theorems apply for any gas sphere in hydrostatic equilibrium where the ideal-gas law applies, we can use them for establishing the range of values to be expected in stars in general. In addition, it is possible to use the other half of the inequality to place upper limits on the values of these quantities at the center of the star.

b. β* Theorem and Effects of Radiation Pressure

We have consistently neglected radiation pressure throughout this discussion and a skeptic could validly claim that this affects the results concerning the temperature limits. However, there is an additional theorem, also due to Chandrasekhar¹ (p.73), which places limits on the effects of radiation pressure. This theorem is generally known as the β* theorem. Let us define β as the ratio of the gas pressure to the total pressure which includes the radiation pressure. The radiation pressure for a photon gas in equilibrium is just \(\mathrm{P}_{\mathrm{r}}=\mathrm{aT}^{4 / 3}\). Combining these definitions with the ideal-gas law, we can write \[\begin{aligned}
P_g & =\beta P_T=\left[\frac{3}{a}\left(\frac{k}{\mu m_h}\right)^4 \frac{1-\beta}{\beta}\right]^{1 / 3} \rho^{4 / 3} \\
P_r & =(1-\beta) P_T=\frac{a T^4}{3} \\
T & =\left[\frac{3 k(1-\beta)}{\mu a m_h \beta}\right]^{1 / 3} \rho^{1 / 3} \\
P_c & =\frac{P_{g, c}}{\beta_c}=\frac{1}{\beta_c}\left[\frac{3}{a}\left(\frac{k}{\mu m_h}\right)^4 \frac{1-\beta_c}{\beta_c}\right]^{1 / 3} \rho_c^{4 / 3}
\end{aligned}\label{2.2.9}\]

Using the integral theorems to place an upper limit on the central pressure, we get \[P_c \leq \frac{1}{2} G\left(\frac{4 \pi}{3}\right)^{1 / 3} \rho_c^{4 / 3} M^{2 / 3}\label{2.2.10}\]

Equation \ref{2.2.10}, when combined with the last of equations \ref{2.2.9} and solved for M, yields \[M \geq\left(\frac{6}{\pi}\right)^{1 / 2}\left[\frac{1-\beta_c}{\beta_c^4}\left(\frac{k}{\mu m_h}\right)^4 \frac{3}{a}\right]^{1 / 2} G^{-3 / 2}\label{2.2.11}\]

Now we define β* to be the value of β which makes Equation \ref{2.2.11} an equality, and then we obtain the standard result that \[\frac{1-\beta^*}{\left(\beta^*\right)^4} \geq \frac{1-\beta_c}{\beta_c^4}\label{2.2.12}\]

Since \((1-\beta)/\beta^4\) is a monotone increasing function of \((1-\beta)\), \[1-\beta^* \geq 1-\beta_c=\frac{P_{r, c}}{P_T}\label{2.2.13}\]

Equation \ref{2.2.11} can be solved directly for M in terms of β* and thus it places limits on the ratio of radiation pressure to total pressure for stars of a given mass. Chandrasekhar¹ (p.75) provides the brief table of values shown in Table 2.1.

Table 2.1 Stellar Mass vs. 1-β*

\(1-\beta^*\)	\(\left(M / M_{\odot}\right) \mu^2\)
0.01	0.56
0.02	0.81
0.10	2.14
0.20	3.83
0.30	6.12
0.40	9.62
0.50	15.50
0.60	26.52
0.70	50.93
0.80	122.50
0.90	520.7
1.00	\(\infty\)

As we shall see later, m is typically of the order of unity (for example \(\mu\) is \(1 / 2\) for pure hydrogen and 2 for pure iron). It is clear from Table 2.1, that by the time that radiation pressure accounts for half of the total pressure, we are dealing with a very massive star indeed. However, it is equally clear that the effects of radiation pressure must be included, and they can be expected to have a significant effect on the structure of massive stars.