ISEGORIA / MATH ENCYCLOPEDIA
Stellar astrophysics: how stars shine and die
A star is a ball of gas held up against its own weight by pressure, and almost everything about it follows from that balance. Plot brightness against surface temperature and the stars fall on a diagonal band where mass alone sets colour, size, luminosity and lifespan. Inside, one equation fixes the run of density, pressure and temperature from the centre to the surface. When the fuel is gone, quantum mechanics takes over: a ball of electrons that cannot be squeezed closer holds up a white dwarf, and past a maximum mass of about 1.4 suns even that fails.
Before you begin: Thermodynamics and nuclear physics
Predict, manipulate, then test yourself: every experiment ends with practice questions that are generated fresh at three levels. Graphs illustrate the mathematics; they do not replace a proof.
1. The Hertzsprung–Russell diagram
Two numbers say most of what a star is: how much light it gives out (its luminosity L) and how hot its surface is (its effective temperature T). Plot one against the other and the stars do not scatter. Most fall on a diagonal band, the main sequence, where they fuse hydrogen in their cores and where mass alone fixes everything else: heavier stars are hotter, brighter, larger and shorter-lived. The Stefan–Boltzmann law makes every point of the diagram a radius, so giants and white dwarfs separate themselves at once. Drag the gold point along the main sequence and watch the star beside the Sun change colour and size, while its spectrum slides to shorter wavelengths with Wien’s law. Switch to “any star” to put a point anywhere and read off the radius it demands.
Worked example. A main-sequence star of 10 solar masses has \(L=1.4\,M^{3.5}=4400\,L_\odot\) and \(R=M^{0.6}=4.0\,R_\odot\), so \(T_{\rm eff}=5772\,{\rm K}\,(L/R^2)^{1/4}=23{,}600\) K, a blue-white B star whose spectrum peaks at \(\lambda_{\max}=2.898\times10^{-3}/23600=123\) nm in the ultraviolet. Its life is about \(10\ {\rm Gyr}\times10/4400=23\) Myr, four hundred times shorter than the Sun’s although it has ten times the fuel.
Watch out. The mass–luminosity and mass–radius relations drawn here are empirical fits (\(L\propto M^{2.3}\), \(M^{4}\), \(M^{3.5}\) and \(M\) in four mass ranges, \(R\propto M^{0.9}\) below one solar mass and \(M^{0.6}\) above), good to a few tens of percent: real stars scatter with composition and age, and every star drifts up and to the right of its starting point as it ages. The catalogue stars are approximate. Colours are those of a black body, which real stellar spectra only roughly follow, and the lifetime assumes 10% of the hydrogen is burned at 0.7% efficiency.
Why does a star of ten solar masses live hundreds of times less long as the Sun, when it has only ten times the fuel?
Because it burns the fuel far faster: its luminosity is about 4400 times the Sun’s (\(L\propto M^{3.5}\)), so it spends its 10 times larger supply 440 times faster, and lasts about 1/440 as long, roughly 23 million years.
2. Hydrostatic equilibrium and the polytrope
Why does a star neither collapse nor blow apart? At every radius the pressure gradient must carry the weight of the gas above it, \(dP/dr=-Gm\rho/r^2\). If the pressure follows a power of the density, \(P=K\rho^{1+1/n}\), the equations of structure fold into one dimensionless equation, the Lane–Emden equation, and its solution θ(ξ) fixes the whole interior: density, pressure, temperature and enclosed mass. The index n measures how soft the gas is. The star is cut open so that you can see the core. A larger n packs the same mass into a smaller, denser centre, and the central temperature follows from the virial scaling \(T_c\propto\mu M/R\), which is why a star that contracts heats up. Drag the probe along the equator and read the local conditions.
Worked example. The Sun as an \(n=3\) polytrope (\(M=1\,M_\odot\), \(R=1\,R_\odot\), \(\mu=0.6\)): \(\xi_1=6.897\) and \(\rho_c/\bar\rho=54.18\), so \(\rho_c=54.18\times1.41=76\) g/cm³. The central pressure is \(P_c=11.05\,GM^2/R^4=1.2\times10^{16}\) Pa and the central temperature \(T_c=11.8\) MK. Half of the mass lies inside \(r=0.283\,R\), and the gravitational energy is \(|\Omega|=1.5\,GM^2/R\).
Watch out. A polytrope is a model of the structure, not a calculation of it. The real Sun has \(\rho_c\approx150\) g/cm³ and \(T_c\approx15.7\) MK, because its composition and opacity change with radius, so no single n describes it. Here the gas is taken to be ideal with a constant mean molecular weight μ (0.6 for ionised solar material, 0.5 for pure hydrogen, 4/3 for pure helium), so that \(T\propto P/\rho=\theta\); radiation pressure and degeneracy are ignored. For n ≥ 5 the solution does not reach zero density at a finite radius, so n stops at 4.5.
What happens to the ratio of central to mean density as n increases from 0?
It rises without bound: 1 for n = 0 (a uniform ball), 5.99 for n = 3/2, 54.2 for n = 3, because the same mass is packed ever more tightly towards the centre.
3. Degeneracy pressure and the Chandrasekhar limit
A white dwarf is a star that has run out of fuel and is held up by something other than heat: electrons squeezed so close together that the Pauli principle forces them into ever higher momentum states, at any temperature. This degeneracy pressure depends on density alone. While the electrons are slow, \(P\propto\rho^{5/3}\), and a heavier dwarf is a smaller one. Squeeze further and the electrons approach the speed of light, the pressure softens to \(\rho^{4/3}\), and there is a largest mass that can be supported: the Chandrasekhar limit, about 1.46 solar masses for carbon and oxygen, fixed only by fundamental constants and the number of nucleons per electron. Drag the point along the mass–radius curve to watch the dwarf shrink beside the Earth, and its density profile move from the n = 3/2 polytrope towards the n = 3 one.
Worked example. With \(\mu_e=2\) and a central density of \(10^6\) g/cm³, the electron momentum is \(x_c=p_F/m_ec=0.80\), only mildly relativistic. Integrating hydrostatic equilibrium outwards gives \(M=0.40\,M_\odot\) and \(R=1.72\,R_\oplus=11{,}000\) km. At \(3\times10^7\) g/cm³ the mass has reached \(1.0\,M_\odot\) and the radius has shrunk to \(0.88\,R_\oplus\), about that of Sirius B. The limit is \(M_{\rm Ch}=5.83/4=1.46\,M_\odot\) for carbon–oxygen, and \(1.26\,M_\odot\) for iron (\(\mu_e=2.15\)).
Watch out. This is the cold, non-rotating, Newtonian star of Chandrasekhar (1935) with non-interacting electrons. Real white dwarfs are slightly larger because they are warm (the observed Sirius B is about 5% above the curve, and the point is drawn from approximate catalogue values), and Coulomb attraction, general relativity and inverse beta decay (which begin to matter near 10⁹ to 10¹⁰ g/cm³) make real stars unstable before the curve reaches M_Ch. The right-hand end of the curve is therefore the idealised limit, not a state a real star reaches. Pure hydrogen (μₑ = 1) is a thought experiment.
Why does adding mass to a white dwarf make it smaller?
More mass means more weight to carry, so the central pressure must rise. Degeneracy pressure rises only with density (\(P\propto\rho^{5/3}\)), so the star must be squeezed denser, and since the mass is larger the radius ends up smaller: \(R\propto M^{-1/3}\).