= Solution
In <stellar homology>, write $r=Rx$, $m=Mf(x)$ and $\rho=\rho_c h(x)$, with common dimensionless profiles. Integrating <hydrostatic equilibrium> from the surface, where the <pressure> is negligible, gives
$$
P_c=\frac{GM\rho_c}{R}\int_0^1\frac{f(x)h(x)}{x^2}\,dx.
$$
The profile integral is common to the sequence. Similarly, <mass conservation> gives $M=4\pi\rho_cR^3\int_0^1x^2h(x)dx$. These prove the <homology scaling of central stellar pressure>:
$$
\rho_c\propto\frac{M}{R^3},\qquad P_c\propto\frac{GM\rho_c}{R}\propto\frac{GM^2}{R^4}.
$$
The <ideal gas law> consequently gives $T_c\propto\mu M/R$, with the gravitational and gas constants absorbed into the coefficient.
The <radiative diffusion in a star> equation has a characteristic gradient $T_c/R$. Hence $L\propto RT_c^4/(\kappa_c\rho_c)$. Using <Kramers' opacity law>, and comparing fixed-composition homologous models, this becomes
$$
L_{\rm rad}\propto\frac{RT_c^{15/2}}{\kappa_0\rho_c^2}
\propto\kappa_0^{-1}\mu^{15/2}M^{11/2}R^{-1/2}.
$$
The volume integral of the <stellar energy-generation rate> $\epsilon=\epsilon_0\rho T^\nu$ instead gives
$$
L_{\rm nuc}\propto M\epsilon_0\rho_cT_c^\nu
\propto\epsilon_0\mu^\nu M^{\nu+2}R^{-\nu-3}.
$$
The dimensionless integrals behind these relations do not depend on $M$ along a homologous sequence. In <stellar thermal equilibrium> the two <luminosities> agree, proving <Kramers homology with a variable nuclear exponent>:
$$
R^{\nu+5/2}\propto\epsilon_0\kappa_0\mu^{\nu-15/2}M^{\nu-7/2}.
$$
For the specified <proton–proton chain> exponent $\nu=7/2$, the mass factor cancels. Thus
$$
\boxed{R=\text{constant},\qquad L\propto M^{11/2}.}
$$
The model captures two important features of the <lower main sequence>: <hydrogen burning> primarily uses the <proton–proton chain>, and gas rather than <radiation pressure> supplies the principal support. A radiative interior and a <Kramers' opacity law> approximation are useful near the Sun. It is a simplified local model: the Sun has a convective envelope, the coolest <main sequence> stars become fully convective, and real <opacities> and burning exponents vary. In particular a literally constant radius is not a realistic law over the entire <lower main sequence>.
On this <proton–proton chain> branch $T_c\propto M$, because $R$ and <mean molecular weight> are fixed. Normalizing to the Sun gives the upper mass estimate
$$
\boxed{M_*\simeq\frac{2\times10^7}{1.5\times10^7}M_\odot=\frac43M_\odot.}
$$
For the specified <CNO cycle> exponent $\nu=23/2$, the homology relation instead reads $R^{14}\propto M^8$. Substituting into the radiative <luminosity> relation gives
$$
\boxed{R\propto M^{4/7},\qquad L\propto M^{73/14},\qquad T_c\propto M^{3/7}.}
$$
This branch retains the deliberately idealized fully radiative assumptions, even though real <CNO cycle> cores are often convective.
For a <Hertzsprung-Russell diagram>, use the <Stefan–Boltzmann law>, $L=4\pi R^2\sigma T_{\rm eff}^4$. On the <proton–proton chain> branch $T_{\rm eff}\propto M^{11/8}$, so $L\propto T_{\rm eff}^4$. On the <CNO cycle> branch $T_{\rm eff}\propto M^{57/56}$, giving $L\propto T_{\rm eff}^{292/57}$. Thus the upper branch rises slightly more steeply toward the hot, luminous upper left in a logarithmic <Hertzsprung-Russell diagram>. The sketch below matches the approximate branches continuously at their respective burning crossovers; the true crossover has both heating terms and is smooth.
The <proton–proton chain> begins with reactions between <hydrogen> nuclei, so its specific heating scales as $X_H^2\rho T^{7/2}$ in this approximation. The catalytic <CNO cycle> involves <hydrogen> and CNO nuclei, and scales as $X_HZ_{\rm CNO}\rho T^{23/2}$. Here $X_H$ is the <hydrogen mass fraction> and $Z_{\rm CNO}$ the <CNO mass fraction>, not the total abundance of every metal. Consequently
$$
\frac{\epsilon_{\rm CNO}}{\epsilon_{pp}}\propto\frac{Z_{\rm CNO}}{X_H}T^8.
$$
Keeping $X_H$ fixed while dividing $Z_{\rm CNO}$ by 256 gives the <CNO crossover temperature>
$$
\boxed{T_{*,\rm poor}=256^{1/8}(2\times10^7\,\mathrm K)=4\times10^7\,\mathrm K,\qquad
M_{*,\rm poor}\simeq\frac83M_\odot.}
$$
The mass estimate uses the unchanged <proton–proton chain> branch below crossover, together with the stipulated unchanged <opacity> and <mean molecular weight>.
The low-CNO <proton–proton chain> sequence therefore lies on the same locus, but extends to twice the transition mass. Beyond its later crossover the low-CNO <CNO cycle> sequence has the same logarithmic slope $292/57$ but a different normalization. At a fixed mass where both sequences are CNO-dominated, lowering the nuclear coefficient by 256 gives
$$
\frac{R_{\rm poor}}{R_{\rm solar}}=256^{-1/14}=2^{-4/7},\qquad
\frac{L_{\rm poor}}{L_{\rm solar}}=2^{2/7},\qquad
\frac{T_{{\rm eff},\rm poor}}{T_{{\rm eff},\rm solar}}=2^{5/14}.
$$
Thus those low-CNO models are smaller, hotter and modestly more luminous at the same mass. At the same <effective temperature>, their CNO branch is lower in <luminosity> than the solar-composition CNO branch; these comparisons hold different quantities fixed.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-42-homology-sequences.png]
{title=Radiative homology sequences for solar CNO abundance and a CNO abundance reduced by 256, with their different burning crossovers}
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