Subfield: Modern Physics

- Proton mass, $m_{p}=1.67 \cdot 10^{-27} \mathrm{~kg}$
- Neutron mass, $m_{n}=1.67 \cdot 10^{-27} \mathrm{~kg}$
- Electron mass, $m_{e}=9.11 \cdot 10^{-31} \mathrm{~kg}$
- Avogadro's constant, $N_{0}=6.02 \cdot 10^{23} \mathrm{~mol}^{-1}$
- Universal gas constant, $R=8.31 \mathrm{~J} /(\mathrm{mol} \cdot \mathrm{K})$
- Boltzmann's constant, $k_{B}=1.38 \cdot 10^{-23} \mathrm{~J} / \mathrm{K}$
- Electron charge magnitude, $e=1.60 \cdot 10^{-19} \mathrm{C}$
- 1 electron volt, $1 \mathrm{eV}=1.60 \cdot 10^{-19} \mathrm{~J}$
- Speed of light, $c=3.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s}$
- Universal Gravitational constant,

$$
G=6.67 \cdot 10^{-11}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{kg}^{2}
$$

- Solar Mass

$$
M_{\odot}=1.988 \cdot 10^{30} \mathrm{~kg}
$$

- Acceleration due to gravity, $g=9.8 \mathrm{~m} / \mathrm{s}^{2}$
- 1 unified atomic mass unit,

$$
1 \mathrm{u}=1.66 \cdot 10^{-27} \mathrm{~kg}=931 \mathrm{MeV} / \mathrm{c}^{2}
$$

- Planck's constant,

$$
h=6.63 \cdot 10^{-34} \mathrm{~J} \cdot \mathrm{s}=4.41 \cdot 10^{-15} \mathrm{eV} \cdot \mathrm{s}
$$

- Permittivity of free space,

$$
\epsilon_{0}=8.85 \cdot 10^{-12} \mathrm{C}^{2} /\left(\mathrm{N} \cdot \mathrm{m}^{2}\right)
$$

- Coulomb's law constant,

$$
k=\frac{1}{4 \pi \epsilon_{0}}=8.99 \cdot 10^{9}\left(\mathrm{~N} \cdot \mathrm{m}^{2}\right) / \mathrm{C}^{2}
$$

- Permeability of free space,

$$
\mu_{0}=4 \pi \cdot 10^{-7} \mathrm{~T} \cdot \mathrm{m} / \mathrm{A}
$$

- Magnetic constant,

$$
\frac{\mu_{0}}{4 \pi}=1 \cdot 10^{-7}(\mathrm{~T} \cdot \mathrm{m}) / \mathrm{A}
$$

- 1 atmospheric pressure,

$$
1 \mathrm{~atm}=1.01 \cdot 10^{5} \mathrm{~N} / \mathrm{m}^{2}=1.01 \cdot 10^{5} \mathrm{~Pa}
$$

- Wien's displacement constant, $b=2.9$. $10^{-3} \mathrm{~m} \cdot \mathrm{K}$
- Stefan-Boltzmann constant,

$$
\sigma=5.67 \cdot 10^{-8} \mathrm{~W} / \mathrm{m}^{2} / \mathrm{K}^{4}
$$

Question: A stable star of radius $R$ has a mass density profile $\rho(r)=\alpha(1-r / R)$. Here, "stable" means that the star doesn't collapse under its own gravity. If the internal pressure at the core is provided solely by the radiation of photons, calculate the temperature at the core. Assume the star is a perfect black body and treat photons as a classical ideal gas. Use $R=7 \times 10^{5} \mathrm{~km}$ and $\alpha=3 \mathrm{~g} / \mathrm{cm}^{3}$. Round your answer to the nearest kilokelvin. We treat photons as a classical gas here to neglect any relativistic effects.

Solution:
For a star that doesn't collapse in on itself, there must be some source of pressure $p(r)$ that balances out the pressure due to gravity. Define the function:
$$
m(r)=\int_{0}^{r} 4 \pi r^{2} \rho(r) d r
$$
inside the star. The gravitational pressure on an infinitesimal shell of radius $r$ and thickness $d r$ is given by:
$$
d p_{g}=-\frac{G m(r) \rho(r) 4 \pi r^{2} d r}{r^{2} 4 \pi r^{2}}=-\frac{G m(r) \rho(r)}{r^{2}} d r
$$
Hence, the pressure source must provide a pressure gradient $\frac{d p}{d r}$ given by:
$$
\frac{d p}{d r}=\frac{G m(r) \rho(r)}{r^{2}}
$$
If $\rho(r)=\alpha(1-r / R)$, then $m(r)=\frac{1}{3} \pi \alpha r^{3}(3 r / R-4)$. We need the pressure gradient:
$$
\frac{d p}{d r}=\frac{1}{3} \pi G \alpha^{2} r(3 r / R-4)(r / R-1)
$$
We integrate this equation from $r=0$ to $R$ with the obvious boundary condition $p(R)=0$. We find that $p(0)=\frac{5}{36} \pi G \alpha^{2} R^{2}$. From the Stefan-Boltzmann law, along with elementary kinetic theory, the pressure due to local radiation is given by $\frac{4 \sigma}{3 c} T_{c}^{4}$. Plugging in values gives $T_{c}=26718 \mathrm{kK}$

Final answer: $26718$ kK