# Neutron Star Models

## Spherical Stars

Compact objects => GR

3 equations from GR

$$G_{ab}=8\pi T_{ab}$$

• Metric of a star (2 factors)
• Energy momentum tensor ($\rho$, $p$)
• Equation of state

### Metric of a spherical star

Line element of a static spherical object, $$ds^2 = - e^{-2\Phi} dt^2 + e^{2\lambda} dr^2 + r^2 d\Omega^2$$ in which $d\Omega^2 := d\theta^2 + \sin^2\theta d\phi^2$.

### Energy momentum tensor

In the perfect fluid model, $T_{ab}$ $$T_{ab} = (\rho + P)U_a U_b + P g_{ab}$$

### Oppenheimer-Volkoff equation etc

\begin{align} \frac{d m(r)}{dr} & = 4\pi \rho(r) r^2 \\ \frac{d\Phi(r)}{dr} & = - \frac{m(r)+4\pi P(r) r^3}{r\left( r-2m(r) \right)} \\ \frac{dP(r)}{dr} & = -(\rho + P(r)) \frac{m(r) + 4\pi P(r) r^3}{r\left( r-2 m(r) \right)} \label{eq:OVEqn} \end{align}

Four variables

Fourth eq n: EoS

Initial conditions

1. $m(0)=0$
2. $e^{-2\Phi(R)} = 1-\frac{2M}{R}$
3. $p(0)$

##### Notes

• $m(r)$ is the mass function including gravitational energy.
• $\rho$ doesn't include the gravitational energy.
• Given $p(0)$ and EoS, the system is determined.

## Ideal Gas Model

$M_m = 0.7 M_{\odot}$, $\rho_c=5\times 10^{15}\mathrm{g\cdot cm^{-3}}$, $R=9.6\mathrm{km}$

### Comments on $M - \rho$ graph

• Max mass? ( + Gravitational Energy )
• Turning points? ( Left: Stable; Right: Unstable | Chap 5 & 6)
• Stability? ( Mode stability: $\xi^i=\xi^i e^{i\omega t}$ | Euler equation )
• Minimum Mass? ( No minimum? $\beta$ decay. )

## Modification of Ideal Neutron Gas Model

### Neutrons, protons, electrons

Non-interacting neutrons, protons, and electrons.

• Neutrons occur: $\rho \geq 1.2\times 10^7\mathrm{g\cdot cm^{-3}}$
• Maximum of $\frac{n_{\mathrm{neutron}}}{n_{\mathrm{proton}}}$: $\rho = 7.8\times 10^{11}\mathrm{g\cdot cm^{-3}}$
• $$\lim_{\rho\rightarrow \infty} \frac{n_{\mathrm{neutron}}}{n_{\mathrm{proton}}} = 8$$
• $$\lim_{\rho \rightarrow\infty} w := \lim_{\rho \rightarrow\infty} \frac{P}{\rho} = \frac{1}{3}$$

Results:
$M_m = 0.72 M_\odot$, $\rho_c = 5.8\times 10^{15}\mathrm{g\cdot cm^{-3}}$, $R = 8.8\mathrm{km}$;
$M_{\text{min}} \approx 0.03 M_\odot$, $\rho_c \approx 7.8\times 10^{11} \mathrm{g\cdot cm^{-3}}$, $R = 48 \mathrm{km}$
( Why miminum? )

### Harrison-Wheeler equation of state

$M_{\text{min}} \approx 0.18 M_\odot$, $\rho_c=2.6\times 10^{13}\mathrm{g\cdot cm^{-3}}$, $R=300\mathrm{km}$

## Realistic Theoretical Models

Maximum Mass
• $\pi$: $1.5 M_\odot$
• R: $1.6M_\odot$
• BJ: $1.9M_\odot$
• TNI: $2.0 M_\odot$
• TI: $2.0M_\odot$
• MF: $2.7M_\odot$

The existence of maximum mass allows us to rule out models with only the knowledge of mass of observed neutron stars.

## Neutron Star Mass: Observations

### X-Ray Binaries

#### Configuration of the system

Line of sight; Inclination;

Measure:
• Period
• Orbital

#### Theory

$$v_1 = \omega a_1 \sin i \label{variationBinary}$$

We can measure $P = 2\pi/\omega$ and $v_1$.

\begin{align} \frac{G(M_1+M_2)}{a^3} & = \omega^2 \\ a_1+a_2 & = a \\ M_1 a_1 - M_2 a_2 & = 0 \end{align}

Move observable quantities to the right.

\begin{align} f_1(M_1,M_1+M_2, i) & := \frac{M_2^3 \sin^3i}{(M_1+M_2)^2} \nonumber\\ &= v_1^3/G\omega \end{align}

mass function

3 unknown variables:
• $M_1$;
• $M_2$;
• $\sin i$

3 constraints:
• $f_X(M_O,M_O+M_X,i)$ and $f_O(M_X,M_O+M_X,i)$;
• $\sin i$ from magical techniques (eclipse duration and optical curve)

$$M_X = \frac{f_X q(1+q)^2}{\sin^3i}, \qquad q := \frac{M_X}{M_O} = \left(\frac{f_O}{f_X}\right)^{1/3}$$

## Neutron Star Mass: Observation

### Only One Mass Function

The companion of PSR 1913+16 has not been directly observed.

We need other equations:
1. Periastron advance: involves each mass and distance of the system
• Doppler effet:
• First order: doppler effect due to line of sight motion
• Second order: time dilation due to velocity ($v^2$)
• Gravitational

Solve out mass.

## Neutron Star Mass: Observation

### Observation Results

#### Observation Results

1. We have models with max mass larger than all observations. HW, ideal neutron gas
2. All neutron stars mass should be $M_{\text{Cha}}$ in advanced stellar evolution theory.

Shaded region: $1.2 \lt M/M_\odot \lt 1.6$

$M \geq 2.10 \pm 0.28 M_\odot$, $R \geq 13.8 \pm 1.8 km$

## Maximum Mass Upper limit of all neutron stars.

### Nauenberg & Chapline (1973)

#### ¶   Assumptioins

1. GR
2. $dP/d\rho\geq 0$
3. $dP/d\rho \leq c^2$: causality
4. Density along radius is continuous

#### ¶   Procedure

1. Pick a density $\rho_0$ (matching density)
2. Find the path that maximized the mass. $P=P_0+c^2(\rho-\rho_0)$
3. Solve OV equation using HW for $\rho\lt \rho_0$ and $P=P_0+c^2(\rho-\rho_0)$ for $\rho\gt\rho_0$

#### ¶   Results

$$M_{\text{max}} \approx 3.2 M_{\odot}$$

### Nauenberg & Chapline

\begin{align} \frac{d\rho}{dn}& = \frac{\rho+P/c^2}{n}\\ M & = \frac{4}{3}\pi \rho R^3 \\ A & = 2\pi n\left( \frac{3}{8\pi\rho} \right)^{3/2} (\chi - \sin\chi\cos\chi) \\ \sin\chi & = \left( \frac{8\pi \rho}{3} \right)^{1/2} R \\ \left| \frac{\partial M}{\partial \chi} \right|_A & = 0 \qquad \text{Equilibrium} \end{align}

5 equations

$\rho$, $P$, $n$, $R$, $A$, $\chi$

We can find $P/\rho$ as a function of $\chi$.

$$\frac{P}{\rho} = \xi(\chi), \qquad \xi(\chi) = \frac{6\cos\chi}{9\cos\chi - \frac{2\sin^3\chi}{ \chi-\sin\chi\cos\chi } }-1$$

### Nauenberg & Chapline

$$\xi(\chi) = \frac{6\cos\chi}{9\cos\chi - \frac{2\sin^3\chi}{ \chi-\sin\chi\cos\chi } }-1$$

### Nauenberg & Chapline

#### Stability

\begin{align} & \left\vert \frac{\partial^2 M}{\partial\chi^2} \right\vert_A \gt 0 \\ \Rightarrow & \Gamma \gt \Gamma_c(\chi) \end{align} with \begin{align} \Gamma & := \frac{\partial \ln P}{\partial \ln n} = \left( 1+\frac{\rho}{P} \right)\frac{dP}{d\rho} \\ \Gamma_c(\chi) & = (\xi+1)\left( 1+ \frac{3\xi+1}{2}\left( \frac{\xi+1}{6\xi}\tan^2\chi -1 \right) \right) \end{align}

### Nauenberg & Chapline

$$\xi\leq 0.364 \Rightarrow \chi\leq 1.12 \Rightarrow \frac{M}{R} \leq 0.405$$

### Nauenberg & Chapline

#### Causality: $dP/d\rho\leq c^2$

Maximum mass when $\Gamma=\Gamma_c(\chi_c)$: $$\frac{\xi_c+1}{\xi_c}v^2 = \Gamma_c$$

$$\rho_c\geq \frac{\rho_0-P_0/v^2}{1-(\xi_c+1)/\Gamma_c}$$

$$M\leq \frac{1}{2} \sqrt{\frac{3}{8\pi}} \sqrt{ \frac{1-(\xi_c+1)/\Gamma_c}{\rho_0-P_0/v^2} } \sin^3\chi_c$$

### Nauenberg & Chapline

Use BBP EoS to find largest allowed value of $\rho_0-P_0/v^2$.
($\rho_0 = 5\times 10^{14}\mathrm{g\cdot cm^{-3}}$)
Then set $v=1$.

$$M_{\text{max}} \approx 3.6 M_\odot$$

## Backup: Ideal Gas Model

### Equation of State

\begin{align} P & = \frac{m_ic^2}{\lambda_i^3}\phi_i(x) \\ \rho_0 & = \sum_i n_i m_i \end{align}

### Symbols

\begin{align*}\lambda_i & =\hbar/(m_i c) \\ x & := p_F/(m_i c) \\ \phi_i(x) & := \frac{g}{6\pi^2}\int_0^x\frac{x^4 dx}{(1+x)^{1/2}} \end{align*}

#### Equation of State: Backup

##### Number Density

\begin{align} n_i & = \frac{g}{h^3} \int_0^{p_F} 4\pi p^2 dp \nonumber \\ & = \frac{4g\pi}{3h^2} p_F^2 \nonumber \\ & = \frac{1}{3\pi^2\lambda_i} \end{align}

##### Energy density

Baryons are much heavier than electrons so the energy density is almost dominated by baryons.