An Introduction to Modern Astrophysics · Problem 10.4
The Sun — Carroll & Ostlie
Physics — Astrophysics · coulomb-barrier-maxwell-boltzmann
Problem
(a) Taking into consideration the Maxwell--Boltzmann velocity distribution, what temperature would be required for two protons to collide if quantum mechanical tunneling is neglected? Assume that nuclei having velocities ten times the root-mean-square (rms) value for the Maxwell--Boltzmann distribution can overcome the Coulomb barrier. Compare your answer with the estimated central temperature of the Sun. (b) Using Eq. (8.1), calculate the ratio of the number of protons having velocities ten times the rms value to those moving at the rms velocity. (c) Assuming (incorrectly) that the Sun is pure hydrogen, estimate the number of hydrogen nuclei in the Sun. Could there be enough protons moving with a speed ten times the rms value to account for the Sun's luminosity?Intuitive explanation
The Maxwell–Boltzmann distribution tells us that most particles cluster near the rms velocity, but the tail extends to much higher speeds. The key insight is that the Coulomb barrier between two protons is so high that classical physics requires an unreasonably hot temperature—but the exponential tail of the distribution contains a tiny fraction of particles moving fast enough to collide. By comparing the required temperature to the Sun's actual core temperature, you'll see why quantum tunneling must be the real mechanism allowing fusion to occur.
Intuitive explanation
The Maxwell–Boltzmann distribution tells us that most particles cluster near the rms velocity, but the tail extends to much higher speeds. The key insight is that the Coulomb barrier between two protons is so high that classical physics requires an unreasonably hot temperature—but the exponential tail of the distribution contains a tiny fraction of particles moving fast enough to collide. By comparing the required temperature to the Sun's actual core temperature, you'll see why quantum tunneling must be the real mechanism allowing fusion to occur.