Quantum Theory of White Dwarf Stars

1. Quantum Theoryof White Dwarf Stars IV

2. Topics • The Sun • Hydrostatic Equilibrium • The Sun’s Central Pressure • White Dwarfs

3. The Sun Power Output of Sun • 3.826 x 1026 Watts Unit of Power • 1 Watt = 1 Joule/second

4. The Sun is a Fusion Reactor 1H + 1H → 2H + e+ + n e+ + e- → g + g 2H + 1H → 3He + g 3He + 3He → 4He + 1H + 1H 1 p+p collision in 1022 leads to fusion 4H →4He Deuterium creation 3He creation 4He creation

5. 4 1H → 4He Mass of Proton (i.e., 1H nucleus) 1.007276 amu Mass of a-particle (i.e., 4He nucleus) 4.001506 amu where 1 Atomic Mass Unit (amu) is 10-3 kg of Carbon (126C) / NA = 1.66 x 10-27 kg NA = 6.022 x 1023 is Avogadro’s Number

6. 4 1H → 4He Mass Deficit 4 x 1.007276 amu = 4.029104 amu 4 1H - 4.001506 amu 4He = 0.0276 amu That is, 0.0276 / 4.0291 = 0.007 of the mass of a proton is converted to photons and neutrinos.

7. How Long will the Sun Shine? Available Hydrogen Fuel • 0.1 of the Sun’s mass is hot enough to fuse hydrogen to helium • 0.007 of that mass is converted fuel = (0.1) x (0.007) x (2 x 1030) kg = 1.4 x 1027 kg that is, about 233 times the Earth’s mass

8. How Long will the Sun Shine? Available Energy E = mc2 = (1.4 x 1027 kg) x (3 x 108 m/s)2 = 1.26 x 1044 Joules

9. How Long will the Sun Shine? Estimated Lifetime 1.26 x 1044 J / (4 x 1026 J/s x 3.15 x 107/s/yr) 10 billion years

10. The Sun’ Old Age In about 5 billion years, the Sun will, after passing through a red giant phase, become a hot Earth-sized star called a white dwarf A white dwarf star is a super-dense object whose existence is direct consequence of relativity and the quantum nature of the world.

11. Helix Nebula 450 ly

12. Hydrostatic Equilibrium A star is in hydrostatic equilibrium if at any radius r the net outward force, due to the pressure, balances the inward force of gravity Pressure exterior to shell Pressure due to gravity Pressure interior to shell

13. Hydrostatic Equilibrium - II Consider a thin shell of gas of thickness Dr. Its mass is Dm = (A Dr) r(r), where A is the shell’s area and r(r) is the shell’s density Pressure exterior to shell Pressure due to gravity Pressure interior to shell

14. Hydrostatic Equilibrium – III The gravitational force on the shell is where m(r) is the mass up to radius r

15. Hydrostatic Equilibrium – IV The pressure difference Dp between the interior and exterior of the shell must equal the pressure due to gravity F / A In the limit of infinitely thin shells we have

16. Pressure at Sun’s Core Equation of Hydrostatic Equilibrium The pressure p(0) in the core is the sum of the pressure differences between all shells, from r = R at the surface to r = 0 at the center Problem: Compute core pressure assuming a constant density

17. Pressure at Sun’s Core - II For a constant density r(r) = d, the central pressure is Using M = 2 x 1030 kg, R = 7 x 108 m for the Sun we find p(0) = 1.3 x 1014 Pa (1 Pascal = 1 N/m2) Note: pressure at Earth’s surface ~ 105 Pa

18. The Standard Solar Model Central density r(0) = 1.5 x 105 kg / m3 Central Pressure p(0) = 2.3 x 1016 Pa Approximate Profile Bahcall – Pinsonneault http://www.sns.ias.edu/~jnb

19. Challenge Problem Compute, by numerical integration, the pressure in the Sun’s core assuming the density profile Compare your result with the value p(0) = 2.3 x 1016 Pa

20. Pressure at Core of White Dwarf A white dwarf star is so dense that the gravitational compression forces within it must be enormous. But since these stars are stable these forces must be balanced by an equally enormous outward pressure. What provides this pressure? The electrons, which form a relativistic quantum gas within the core!

21. Dx White Dwarf – II According to the uncertainty principle, Dp Dx = h the momentum of an object, confined to a region of linear dimension Dx, will fluctuate by an amount Dp = h / Dx in that dimension

22. Dx White Dwarf – II We assume that each electron occupies, on average, a volume Dx3 and they are as tightly packed as possible. In this case, the electron density, that is, the number of electrons per unit volume is n = 1 / Dx3

23. Dx White Dwarf – III The momentum fluctuations Dp can therefore be written as Dp = h / Dx = h n1/3 The electron pressure P = F / Dx2, where F, the force, is the rate of change of momentum: F = Dp / Dt Therefore, the pressure is P = Dp / (Dt Dx2)

24. Dx White Dwarf – IV The pressure is P = Dp / (Dt Dx2). We now assume that the electrons are moving at the speed of light, c We can then write P = Dp c / (cDt Dx2) = Dp c / Dx3 = h c n4/3(using, n = 1 / Dx3, Dp = h n1/3)

25. White Dwarf – V Since the star is electrically neutral, the number of electrons equals the number of protons N = M / mp where mp is the proton massand M is the mass of the Sun. Therefore, the electron density is roughly n = N / (4/3 p R3)

26. White Dwarf – V We assume that the electron pressure hcn 4/3 is sufficient to balance the pressure due to gravity. Since the pressure needed, at the core, is roughly GM2 / R4, we can write the equilibrium condition as hc n4/3 = GM2 / R4

27. White Dwarf – VI The equilibrium condition yields the remarkable prediction that a relativistic electron gas can support a mass no greater than about M ~ N3 mp where N = MPlanck / mp = 1.3 x 1019 and MPlanck = √(hc / G) (The Planck mass) = 2.18 x 10-8 kg Problem: Complete calculation

28. White Dwarf – VII We have found that the pressure of a relativistic electron gas can support a maximum mass of about 1.8 x Sun’s mass! A more accurate calculation gives 1.4 x Sun’s mass, a result first worked out in the 1920s by the Indian physicist Chandrasekhar and now known as the Chandrasekhar limit

29. Summary • The Uncertainty Principle • Because of the uncertainty principle, the smaller the region to which an object is confined the greater the fluctuations in its momentum • Chandrasekhar Limit • The fluctuating momentum leads to a pressure, which, for electrons, is sufficient to prevent a mass of up to 1.4 x Sun’s mass from collapsing under its own gravity. This is a beautiful example of a macroscopic consequence of the quantum nature of the world