Physics 616

• Prof. Andrew W. Steiner
(or Andrew or "Dr. Steiner")
• Office hour: 103 South College, Thursday 11am
• Email: awsteiner@utk.edu
• Homework: Electronically as .pdf
• You may work with each other on the homework, but you must write the solution in your own words

Use down and up arrows to proceed to the next or previous slide.

Outline

• Ch. 10 in Schutz
• Some material from Shapiro and Teukolsky
• And other assorted references

General Properties of Compact Objects

• Neutron stars, white dwarfs, strange quark stars, etc.
• Speed of sound is $c_s=\left(d p/d\rho\right)^{1/2}_{S,N}$ where here $S$ refers to constant entropy (or constant entropy per particle)
• Classical speed of sound, $\sqrt{K/\rho}$, often written in terms of the bulk modulus
• Limit that $c_s > 0$ (hydrodynamic stability; similar to Rayleigh-Taylor) and $c_s < 1$ (causality)
• Of course, we know the EOS at low energy densities from experiment
• What is the ground state of matter at near zero (e.g. laboratory) density?

Constant-Density Solution

• Analytical solutions to the TOV equations are rare; most of them are unphysical
• First is $\rho=\mathrm{constant}$
• Then the enclosed mass is trivial, $m=4 \pi \rho r^3/3$ and $M=4 \pi \rho R^3/3$
• Choose $\rho$, that represents our choice in the EOS, then choose $M$, and that specifies $R$
• Then the central pressure is given in Eq. 10.52 in Schutz
• Inside the star, Eq. 10.54 gives $\Phi$, and $\Lambda$ can be computed (for any EOS) from Eq. 10.29
• Note we can go from the EOS to the metric functions, thus showing that we have completely specified the system

Buchdahl's solution

• EOS: $$\rho = 12 \left( p_{*} p \right)^{1/2} - 5 p$$
• Note that this is $\rho(p)$ instead of $p(\rho)$
• Polytropes: $p \propto \rho^{\Gamma}$ with $\Gamma=1+1/n$
• For small $p$ reduces to the $n=1$ polytrope
• Solution outlined on p. 268
• Constraints on various parameters from causality
• There is a scale invariance, if $\rho$, $p$ and $p_{*}$ are all multiplied by the same constant, the EOS is unchanged

Upper Mass Limit for White Dwarfs

• Supported by degenerate electrons, EOS is $$p = \frac{2 \pi}{3 h^3} \left(\frac{3 h^3}{8 \pi \mu m_p}\right) \rho^{4/3}$$
• What is the relationship between the pressure and the chemical potential for this EOS?
• Newtonian limit is sufficiently accurate $$\frac{dm}{dr} = 4 \pi r^2 \rho \quad \mathrm{and} \quad \frac{dp}{dr} = - \frac{m \rho}{r^2}$$
• Schutz uses dimensional analysis $$M = R^3 \bar{\rho} \quad \mathrm{and} \quad \frac{\bar{p}}{R} = \frac{\bar{\rho} M}{R^2}$$
• And then using the EOS implies $$M = \left(\frac{3 k^3}{4 \pi}\right)^{1/2} \approx 0.32~\mathrm{M}_{\odot}$$ (which is a factor of 4 too small)

Neutron Stars

• Supported by neutron degeneracy pressure and (a larger contribution from) the repulsion between nucleons due to QCD
• Created in core-collapse supernovae, thus minimum mass around $1~\mathrm{M}_{\odot}$
• No simple calculation of the maximum mass, must be larger than 2, but could be almost $3~\mathrm{M}_{\odot}$.
• Radii between 10 and 13 km
• Observations of large mass neutron stars are revolutionary
• Rotation can significantly change the structure
• Strong differential rotation can lead to toroidal neutron stars (but these are unlikely in reality)
• Magnetic fields near $10^{18-19}$ G can as well, but these are also difficult to generate in realistic simulations

Stellar Stability in Newtonian Systems

• Perform variations at constant mass and constant entropy
• For radial modes, leads to Strum-Liouville equation similar to that we saw last semester
• Modes must have real frequencies (sinusoids and not exponentials)
• Stability requires average value $\Gamma > 4/3$ ($n=3$ polytrope)

Stability in GR systems

• Perform variations at constant gravitational mass and constant entropy
• For radial modes, leads to Strum-Liouville equation similar to that we saw last semester
• Modes must have real frequencies (sinusoids and not exponentials)
• Stability requires $\Gamma > 4/3 + \varepsilon$
• Stability changes at critical points
• $dM/d\rho_c$ doesn't fully characterize stability. $D-E$ branch unstable (even though it appears stable)
• Mass-radius relations are not functions

Exotic Matter in Neutron Stars

• Hyperons $\Lambda,\Sigma^{+},\Sigma^{0},\Sigma^{-}$, Bose condensates $\pi^{-}, K^{-}$, and deconfined quarks $u,d,s$ can appear at high density.
• No charmed, bottom, or top quarks because densities are not large enough
• Transition between normal matter and exotic matter can be first or second order
• "Ehrenfest classification": First-order means discontinuity in the first derivative of the free energy, Second-order means discontinuity in the second derivative...
• "Modern": First-order means a phase transition with a latent heat
• Prototypical first-order transition is from a Maxwell construction ("Equal-area construction")
• In NSs, can lead to constant $p$ as a function of $\rho$
Wikipedia from "Pgrass"

CSS parameterization

• Simple non-trivial neutron star EOS by Han and Alford
• "Normal matter" at low densities
• First-order phase transition to exotic (e.g. quark) matter at high densities
• Indicates surface inside the star with continuous pressure but discontinuous energy density (but $\mu$ and $T$ must still be continuous in equilibrium!)
Alford and Han (2016)

CSS parameterization II

• Left panel: maximum mass, right panel: radius of maximum mass star
• With this particular EOS, configurations $D$ and $B$ are not possible, and radii must be smaller than 13 km
Alford and Han (2016)

Other Phase transitions

• Other first-order phase transitions are possible
Spinella et al. (2016)
• Pressure is not flat, but only weakly varying with $\rho$
• Require one to abandon the idea of local charge neutrality
• Transitions may involve changes in composition, addition of superfluidity/superconductivity, etc.
• Second-order transitions are also possible (e.g. between quark and hadronic matter)

Group Work

• Work towards computing a neutron star mass-radius curve using the Buchdahl EOS. Use $p_{*} = 4.7 \times 10^{-5}~\mathrm{km}^{-2}$. What mass star does this correspond to? How do the units make sense in the EOS and in the various equations in Schutz's book? You may want to check Lattimer's 2001 paper on Canvas.