Heat alone is three orders of magnitude short ── it burns anyway, because there is a way through, up on the exponent
Everyone knows the Sun runs on fusion. Put numbers in and you get stuck immediately ── the Sun's centre is about three orders of magnitude too cold to fuse anything. In 1929 Atkinson and Houtermans noticed the contradiction, read Gamow's alpha-decay paper from the year before (Episode 1), and produced the answer: you don't have to go over. Tunnel through. And the story does not end there. Heat says "higher energy is rarer"; tunneling says "higher energy is easier" ── two opposing exponentials collide and squeeze the reaction into a single narrow band of energy. This "Gamow peak" is what decides why stars can burn quietly for ten billion years.
| Quantity | Value |
|---|---|
| central temperature | 1.5×10⁷ K |
| measured in energy, \(k_BT\) | 1.3 keV |
| Coulomb energy needed to bring two protons within 1 fm | 1.4 MeV |
| ratio | about 1100× |
A plain ratio, in the style of Episode 1 of "Temperature That Clicks." To get over thermally, the Boltzmann factor (Episode 2 there) gives
$$e^{-1.4\,\mathrm{MeV}/1.3\,\mathrm{keV}}=e^{-1083}\approx 10^{-470}$$There are about \(10^{57}\) protons in the Sun, so at most \(10^{114}\) pairings. \(10^{-470}\) means it never happens once in the whole history of the universe. Classically, the Sun does not shine.
"If it's too cold, make it hotter" ── but the Sun's central temperature is fixed by its mass and gravity; you cannot move it. In the 1920s this was a genuine crisis. Eddington was convinced the Sun could only be powered by fusion, and the temperature was plainly insufficient. When told to go and find a hotter star, he is said to have replied, "go and find a hotter place."
Apply Episode 1's tool directly. The wall is the Coulomb potential \(V(r)=Z_1Z_2e^2/4\pi\varepsilon_0 r\). Put it into \(S_E=\int\sqrt{2m(V-E)}\,dr\) and integrate, and something remarkably clean comes out.
\(E_G\) is the Gamow energy, fixed entirely by who the partners are. For two protons it is \(E_G=493\) keV (with \(\alpha=1/137\) the fine-structure constant and \(m_r\) the reduced mass).
The shape is the point ── the larger \(E\), the smaller the exponent, so the easier it is to get through. But it sits under a square root, so the effect is gentle.
Try the typical energy at the Sun's centre, \(E=k_BT=1.3\) keV:
$$\sqrt{493/1.3}=19.5\qquad\Longrightarrow\qquad P\approx e^{-19.5}\approx 3\times10^{-9}$$\(10^{-470}\) has become \(10^{-9}\). An improvement of 461 orders of magnitude. That happens often enough among the Sun's \(10^{57}\) protons. Episode 1's lesson ── change what is on the exponent and the answer changes by orders ── is what lights the stars.
Here is the real subject of this episode. A reaction needs two things at once.
| What is required | Probability | Higher energy means |
|---|---|---|
| a proton that fast exists | \(e^{-E/k_BT}\) | worse (rarer) |
| it gets close and tunnels | \(e^{-\sqrt{E_G/E}}\) | better (easier to get through) |
Opposite directions. Multiply them and a sharp peak appears at neither extreme but in between.
Differentiate the exponent of \(\exp\!\left(-\dfrac{E}{k_BT}-\sqrt{\dfrac{E_G}{E}}\right)\) with respect to \(E\) and set it to zero:
$$E_0=\left(\frac{E_G\,(k_BT)^2}{4}\right)^{1/3}$$At the Sun's centre (\(k_BT=1.29\) keV, \(E_G=493\) keV),
$$E_0=\left(\frac{493\times1.29^2}{4}\right)^{1/3}\ \mathrm{keV}\approx \mathbf{5.9\ keV}$$So ── fusion in the Sun happens neither at the typical proton energy (1.3 keV) nor at the barrier (1400 keV), but in between, at 5.9 keV. The peak is only about 6 keV wide. Out of not \(10^{23}\) but \(10^{57}\) protons, only the fraction that lands in this narrow window keeps the star shining.
The figure shows the three curves just described. Blue falling to the right is "a proton that fast exists"; red rising to the right is "it tunnels"; and their product is the orange peak. The vertical axis is logarithmic and spans more than 30 orders of magnitude.
Move the temperature slider and the peak shifts while its height changes violently. This is the core of stellar physics ── raise the temperature by 10% and the reaction rate jumps by tens of percent. That steep temperature dependence is what automatically settles a star at one temperature (too hot, it expands and cools; too cool, it contracts and heats).
An unexpected fact. Per unit volume, the Sun's centre is remarkably dim.
| Heat source | power per unit volume |
|---|---|
| the Sun's centre (its brightest part) | about 280 W/m³ |
| the Sun, averaged | about 0.27 W/m³ |
| a human body | about 1000 W/m³ |
| a compost heap | a few hundred W/m³ |
Per unit volume, you generate more heat than the centre of the Sun. The Sun is bright because it is enormous. And that dimness is exactly why the Sun lasts ten billion years ── being hard to tunnel through is not a defect but a lifetime. Without the barrier, all the hydrogen in the universe would have turned to helium in the first few minutes, and there would be no stars, no planets and no living things.
The height of the Gamow peak goes as \(\exp(-3E_0/k_BT)\). Insert \(E_0\propto T^{2/3}\) and the reaction rate becomes a power of temperature ──
| Reaction | E_G | ν at the Sun's centre | Meaning |
|---|---|---|---|
| pp chain (p + p) | 493 keV | about 4 | gentle. This is the Sun |
| CNO cycle (p + ¹²C) | 32.9 MeV | about 20 | ferocious. Heavy stars run on this |
A large \(Z\) sends \(E_G\) soaring, pushes the peak to higher energy, and steepens the temperature dependence. That stars a little heavier than the Sun switch to CNO and become short-lived is a direct property of this exponential. A star's whole life is decided by the number on the exponent.
Established: the Sun's central temperature and density and \(k_BT\approx1.3\) keV; that the Coulomb barrier is of order an MeV and classically insurmountable; the Gamow factor \(\exp(-\sqrt{E_G/E})\) and the Gamow energy \(E_G=2m_rc^2(\pi\alpha Z_1Z_2)^2\) (493 keV for p–p); the Gamow peak \(E_0=(E_G(k_BT)^2/4)^{1/3}\approx5.9\) keV and its width; the temperature dependence \(\varepsilon\propto T^{\nu}\) with \(\nu\approx4\) for pp and \(\approx20\) for CNO; the application of tunneling to stellar fusion by Atkinson and Houtermans (1929); and the Sun's power per unit volume. All standard stellar physics.
Simplifications: (1) The real rate is written \(\sigma(E)=\dfrac{S(E)}{E}e^{-\sqrt{E_G/E}}\), with the nuclear physics packed into the astrophysical S-factor \(S(E)\). The body treats \(S(E)\) as essentially constant (a good approximation for non-resonant reactions, but it fails when there is a resonance). (2) \(p+p\) involves the weak interaction, so the rate is not set by ease of tunneling alone (as noted in the body). (3) In stellar plasma the surrounding electrons screen the barrier slightly (a few percent correction in the Sun; that screening becomes the protagonist in bonus ①). (4) The derivation of the Gamow peak is a saddle-point approximation and deviates slightly from a Gaussian. (5) The figure idealises a single reaction, non-degenerate and non-relativistic. (6) There was once a "solar neutrino problem" in which only a third of the predicted neutrinos were observed; the cause was neutrino oscillation, not the stellar model, and it was resolved in 2001–02 (the picture in the body is unaffected).
\(k_BT\) at the Sun's centre is 1.3 keV; the proton–proton Coulomb barrier is 1.4 MeV ── a factor of 1100 short. The classical probability is \(e^{-1083}\approx10^{-470}\), a number that never occurs once in the history of the universe. Apply Episode 1's tunneling and it becomes \(e^{-\sqrt{E_G/E}}=e^{-19.5}\approx10^{-9}\) ── an improvement of 461 orders. The Sun shines because it tunnels.
More interesting still, heat (higher energy is worse) and tunneling (higher energy is better) are opposing exponentials. Their product peaks sharply in between, and fusion in the Sun happens neither at the average nor at the barrier but in a narrow window at 5.9 keV (the Gamow peak).
And the steepness of that exponential decides a star's life. Rates run from \(T^4\) (pp) to \(T^{20}\) (CNO), so heavy stars burn ferociously and die young. That the Sun is dimmer per unit volume than a human body and still lasts ten billion years is thanks to being hard to tunnel through.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen the temperature slider moves the Gamow peak and changes its height by orders of magnitude. The CNO button switches the partner to carbon so you can see how far the peak gets pushed out. "See the answer" opens each solution.