IPhO 2023 — Задача 1. Neutron Stars (10 points)

Theory — IPhO 2023 Tokyo Japan — Q2 — English (Official) — Neutron Stars (10 points) · 10 т.

Условие

We discuss the stability of large nuclei and estimate the mass of neutron stars theoretically and experimentally.

Part A. Mass and stability of nuclei (2.5 points)

The rest-energy of a nucleus $m(Z,N)c^2$ consisting of $Z$ protons and $N$ neutrons is smaller than the sum of rest-energies of protons and neutrons, hereafter called nucleons, by the binding energy $B(Z,N)$, where $c$ is the speed of light in vacuum. Ignoring minor corrections, we can approximate the binding energy consisting of the volume term with $a_V$, the surface term with $a_S$, the Coulomb energy term with $a_C$, and the symmetry energy term with $a_{\mathrm{sym}}$ in the following way.

$m(Z,N)c^2 = Am_N c^2 - B(Z,N), \qquad B(Z,N) = a_V A - a_S A^{2/3} - a_C \frac{Z^2}{A^{1/3}} - a_{\mathrm{sym}} \frac{(N-Z)^2}{A}, \qquad (1)$

where $A = Z + N$ is the mass number and $m_N$ is the nucleon mass. In the calculation, use $a_V \approx 15.8\ \mathrm{MeV}$, $a_S \approx 17.8\ \mathrm{MeV}$, $a_C \approx 0.711\ \mathrm{MeV}$, and $a_{\mathrm{sym}} \approx 23.7\ \mathrm{MeV}$ ($\mathrm{MeV} = 10^6$ electron volts).

Part B. Neutron star as a gigantic nucleus (1.5 points)

For large nuclei with a large enough mass number $A > A_c$ with a threshold $A_c$, these nuclei stay stable against nuclear fission because of the sufficiently large binding energy due to gravity.

Part C. Neutron star in a binary system (6.0 points)

Some neutron stars are pulsars regularly emitting electromagnetic waves, which we call "light" for simplicity here, at a constant period. Neutron stars often make binary systems with a White Dwarf. Let us consider the star configuration shown in Fig. 1, where a light pulse from a neutron star N to the Earth E passes near a White Dwarf W of the binary system. Measuring these pulses influenced by the star's gravity leads to an accurate estimation of the mass of W as explained below, resulting in the estimation of the mass of N.

A.1 Under the condition of $Z = N$, determine $A$ for maximizing the binding energy per nucleon, $B/A$. [0,9 т.]

A.2 Under the condition of fixed $A$, the atomic number of the most stable nucleus $Z^*$ is determined by maximizing $B(Z, A-Z)$. For $A = 197$, calculate $Z^*$ using Eq. (1). [0,9 т.]

A.3 A nucleus having large $A$ breaks up into lighter nuclei through fission in order to minimize the total rest-mass energy. For simplicity, we consider one of multiple ways to break a nucleus with $(Z, N)$ into two equal nuclei, which occurs when the following energy relation holds,

$m(Z,N)c^2 > 2m(Z/2, N/2)c^2.$

When this relation is written as

$Z^2/A > C_{\mathrm{fission}} \frac{a_S}{a_C},$

obtain $C_{\mathrm{fission}}$ up to two significant digits. [0,7 т.]

B.1 We assume that $N = A$ and $Z = 0$ is realized for sufficiently large $A$ and Eq. (1) continues to hold with the addition of the gravitational binding energy. The binding energy due to gravity is

$B_{\mathrm{grav}} = \frac{3}{5} \frac{GM^2}{R},$

where $M = m_N A$ and $R = R_0 A^{1/3}$ with $R_0 \approx 1.1 \times 10^{-15}\ \mathrm{m} = 1.1\ \mathrm{fm}$ are the mass and the radius of the nucleus, respectively. For $B_{\mathrm{grav}} = a_{\mathrm{grav}} A^{5/3}$, obtain $a_{\mathrm{grav}}$ in the MeV unit up to the first significant digit. Then, ignoring the surface term, estimate $A_c$ up to the first significant digit. In the calculation, use $m_N c^2 \simeq 939\ \mathrm{MeV}$ and $G = \hbar c / M_P^2$ where $M_P c^2 \simeq 1.22 \times 10^{22}\ \mathrm{MeV}$ and $\hbar c \simeq 197\ \mathrm{MeV} \cdot \mathrm{fm}$. [1,5 т.]

C.1 As shown in the figure below, under the constant gravitational acceleration $g$ we place two levels I and II with the height difference $\Delta h (> 0)$. Set the identical clocks at I, II, and $F$, the free-falling system, denoted by clock-I, clock-II, and clock-F, respectively.

We assume that an observer sits with clock-F, and initially $F$ is placed at the same height as that of clock-I and its velocity is zero. Since the clocks are identical, they register equal time intervals, $\Delta \tau_F = \Delta \tau_{\mathrm{I}}$. Then, we let $F$ fall freely, and work in the frame of $F$, which is considered to be inertial. In this frame, clock-II passes by clock-F with velocity $v$, so that the time dilation of clock-II can be determined by the Lorentz-transformation. When time $\Delta \tau_{\mathrm{I}}$ elapses on clock-F, time $\Delta \tau_{\mathrm{II}}$ elapses on clock-II. Determine $\Delta \tau_{\mathrm{II}}$ in terms of $\Delta \tau_{\mathrm{I}}$ up to the first order in $\Delta \phi / c^2$, where $\Delta \phi = g \Delta h$ is a difference of the gravitational potential, i.e., the gravitational potential energy per unit mass. [1 т.]

C.2 Under the gravitational potential $\phi$, time delays change the effective speed of light, $c_{\mathrm{eff}}$, observed at the infinity, though the local speed of light is $c$. When $\phi(r=\infty) = 0$, $c_{\mathrm{eff}}$ can be given up to the first order in $\phi/c^2$ as

$c_{\mathrm{eff}} \approx \left(1 + \frac{2\phi}{c^2}\right) c$

including the effect of space distortion, which was not featured in C.1. We note that the light path can be approximated as a straight line. As shown in Fig. 1 (a), we take the $x$-axis along the light path from the neutron star N to the Earth E and place $x = 0$ at the point where the White Dwarf W is the closest to the light path. Let $x_N\ (< 0)$ be the $x$-coordinate of N, $x_E\ (> 0)$ be that of E, and $d$ be the distance between W and the light path. Estimate the changes of the arrival time $\Delta t$ of the light from N to E caused by the White Dwarf with mass $M_{\mathrm{WD}}$ and evaluate the answer in a simple form disregarding higher order terms of the following small quantities: $d/|x_N| \ll 1$, $d/x_E \ll 1$, and $GM_{\mathrm{WD}}/(c^2 d) \ll 1$. If necessary, use the following formula.

$\int \frac{dx}{\sqrt{x^2+d^2}} = \frac{1}{2} \log \left( \frac{\sqrt{x^2+d^2}+x}{\sqrt{x^2+d^2}-x} \right) + C. \qquad (\log\ \mathrm{is\ the\ natural\ logarithm})$ [1,8 т.]

C.3 As shown below, in a binary star system N and W are assumed to be moving in circular orbits with zero eccentricity around the center of mass $G$ on the orbit plane. Let $\varepsilon$ be the orbital inclination angle measured from the orbit plane to the line directed toward E from $G$, and let $L$ be the length between N and W and $M_{\mathrm{WD}}$ be the mass of the White Dwarf. In the following, we assume $\varepsilon \ll 1$.

We observe light pulses from N on E far away from N. The light path to E varies with time depending on the configuration of N and W. The delay in the time interval of arriving pulses on E has maximum value $\Delta t_{\mathrm{max}}$ for $x_N \simeq -L$ and the minimum value $\Delta t_{\mathrm{min}}$ for $x_N \simeq L$ (see Fig. 1 (b) for the configuration). Calculate $\Delta t_{\mathrm{max}} - \Delta t_{\mathrm{min}}$ in a simple form disregarding higher order terms of small quantities as done in C.2. We note that the delays due to gravity from stellar objects other than W are assumed to cancel out in $\Delta t_{\mathrm{max}} - \Delta t_{\mathrm{min}}$. [1,8 т.]

C.4 The below figure shows the observed time delays as a function of the orbital phase $\varphi$ for the binary star system with $L \approx 6 \times 10^6\ \mathrm{km}$ and $\cos \varepsilon \approx 0.99989$. Estimate $M_{\mathrm{WD}}$ in terms of the solar mass $M_{\odot}$ and show the results for $M_{\mathrm{WD}}/M_{\odot}$ up to the first significant digit. Here the approximate relation, $GM_{\odot}/c^3 \approx 5\ \mu\mathrm{s}$, can be used. [0,8 т.]

C.5 In the binary system of neutron stars, two stars release energy and angular momentum by emitting gravitational waves and eventually collide to merge. For simplicity, let us consider only a circular motion with the radius $R$ and the angular velocity $\omega$ and then $\omega = \chi R^p$ holds with the constant $\chi$ depending on neither $\omega$ nor $R$ if relativistic effects are ignored. Determine the value for $p$. [0,4 т.]

C.6 The amplitude of the emitted gravitational wave from the binary system in C.5 is proportional to $R^2 \omega^2$. Figure below qualitatively shows four different temporal profiles of the observed gravitational waves before the two-star collision. Select the most appropriate profile from (a) to (d). [0,2 т.]

Решение

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Оригинал в Архива: IPhO_2023_Q2.pdf · официални решения: IPhO_2023_S2.pdf