2026 National Astronomy Competition · 85 т.

Условие

Note that the provided Answer Sheet for this question is only for subtasks (c)(vi) and (d)(v). Do all other work on blank pages.

In this question, we do a detailed dive into several elements of neutron star dynamics and energy losses. While understanding the internal nature and makeup of a neutron star requires quantum chromodynamics and general relativity, other properties, such as rotation rate, are far more elementary to model. Over time, a neutron star’s rotation rate slowly decreases due to factors such as energy loss and mass accretion. By measuring changes in a neutron star’s rotation rate, astronomers can make inferences about the physics surrounding a neutron star and its environment. Neglect more complex behavior (such as QED/QCD/GR corrections), instead assuming Newtonian mechanics holds unless otherwise stated.

Let $\Omega(t)$ be the star’s angular velocity around its axis as a function of time. In several scenarios, we can observe a proportionality relation of the form $\dot\Omega\propto-\Omega^n$, where $n$ is the neutron star’s braking index. (Throughout this problem, dots denote time derivatives, i.e. $df/dt=\dot f$ and $d^2f/dt^2=\ddot f$.)

(a) (10 points) We will begin by examining the basic behavior of the time evolution of $\Omega$.

(i) (5 points) Express $n$ in terms of $\Omega$, $\dot\Omega$, and $\ddot\Omega$.

(ii) (3 points) If $n>1$, then we can deduce that $\Omega(t)\propto t^{-\alpha}$ for some $\alpha$. Compute $\alpha$ in terms of $n$.

(iii) (2 points) What type of curve does $\Omega(t)$ versus $t$ trace if $n=1$?

This relation allows astronomers to measure $n$ for a given star. We will now examine several mechanisms for rotational slowdown and the subsequent values for $n$, treat each of these mechanisms as independent of each other unless otherwise stated. (In other words, do not combine mechanisms until part (d)(v).)

(b) (8 points) First, we will look at a primitive model for mass accretion. Let $M$ be the neutron star’s mass and $R$ be its radius. Assume the rate of mass accretion $\dot M$ is constant and that the infalling matter has no initial angular momentum relative to the rotation axis. Additionally, assume (slightly unrealistically) that the neutron star is a uniformly dense sphere of matter.

(i) (2 points) Write down the expression for the angular momentum of the neutron star $L$ in terms of $M$, $R$, and $\Omega$, up to a numerical constant. Use the fact that $L$ stays constant for problems (ii) and (iii) of this subpart.

(ii) (3 points) Now, assume that as the star accretes mass, the density $\rho$ stays constant over time. Calculate the value of $n$.

(iii) (3 points) Frontier QCD simulations currently appear to imply that, in contrast to the above, $R$ stays relatively constant as $M$ changes. Treating this new relationship as exact, calculate the new value of $n$.

(c) (40 points) Next, we will examine gravitational wave emission from the neutron star. To leading order, gravitational wave emission is caused by oscillations in the mass quadrupole moment, which is the deviation from spherical symmetry when projected onto an axis, of the neutron star. The general formula for the traceless quadrupole moment terms is

where $\mathbf r$ is the displacement from the center of the ellipsoid, $i,j\in\{x,y,z\}$ are coordinate axes, $r_i$ are projections of $\mathbf r$ onto these axes (so $r_x=x$), and $\delta_{ij}=1$ for $i=j$ with $\delta_{ij}=0$ otherwise. Such integrals are over all space unless otherwise specified. (Formally, the quadrupole moment is a 3 by 3 matrix $Q$ with 9 terms $Q_{ij}$, but you will not need any knowledge of linear algebra for this problem.)

We model the neutron star as a rotating ellipsoid (which deviates very slightly from a perfect sphere).

We establish a fixed $xyz$ coordinate system for reference, which the neutron star rotates at angular speed $\Omega$ relative to. The neutron star has body symmetry axes $x'$, $y'$, and $z'=z$, as per the diagram. By the definition of body symmetry axes,

In other words, the moments of inertia around the body axes $x'$, $y'$, and $z'$ are $I_x$, $I_y$, and $I_z$, respectively, and all “cross” terms between different axes vanish. (Let $I_x>I_y$.) Some elementary trigonometry yields

We will now compute the time evolution of $Q_{ij}$. Assume $\Omega$ is constant until part (v).

(i) (12 points) Using the above coordinate transformations and inertia integrals, calculate all terms $Q_{ij}$, expressing your answer in terms of the body axis moments of inertia and trigonometric functions of $\Omega t$. For example,

since both integrals vanish as stated.

(ii) (7 points) For the first step of our calculations, we will need the second time derivative $\frac{d^2Q_{ij}}{dt^2}$ of the quadrupole moment. Calculate these terms and indicate which ones are nonzero. (You may find one or more of the identities $\cos2\alpha=\cos^2\alpha-\sin^2\alpha$, $\sin2\alpha=2\sin\alpha\cos\alpha$, and $\cos^2\alpha+\sin^2\alpha=1$ helpful.)

Detecting gravitational waves involves measuring the strain $h$, or the fractional change in length $\Delta L/L$ of spacetime caused by the gravitational wave. For the purposes of this problem, we ignore nuances such as viewing angle and GW polarization regimes. As such, we approximate the behavior as

where $d$ is the distance from the observer to the neutron star. (Note that there is a time offset due to the nonzero time it takes gravitational waves to reach the observer, but we ignore this, as it is not highly relevant in understanding the qualitative behavior of the system.)

(iii) (3 points) For convenience, we only look at one term of $h_{ij}$ to capture the underlying behavior. Calculate the amplitude $h_0$ and angular frequency $\Omega_{GW}$ of the oscillations in $h_{xx}$. Express your answers in terms of $I_z$, $\epsilon=\frac{I_x-I_y}{I_z}$, $\Omega$, and fundamental constants.

(iv) (6 points) According to general relativity, the radiated power is given by

where the sum runs over all 9 pairs of indices $i,j\in\{x,y,z\}$. Calculate $P$ in terms of $I_z$, $\epsilon$, $\Omega$, and fundamental constants.

(v) (6 points) The power for the gravitational waves comes purely from the rotational kinetic energy $E=\frac12I_z\Omega^2$ of the neutron star. Assuming its mass, radius, and inertia moments stay constant, calculate the value of $\dot\Omega$ in terms of $\Omega$, $I_z$, $\epsilon$, and fundamental constants. From this, extract the value of $n$.

(vi) (6 points) As the neutron star slowly loses rotational kinetic energy while keeping its inertia moments constant, oscillations of the observed gravitational wave strain $h$ change in both period and amplitude. On your Answer Sheet, sketch a graph for $h_{xx}(t)$ versus time $t$ that reflects these changes. (Exaggerate your graph so that significant visible changes in both period and amplitude occur over the course of several periods. Your graph need not be to scale.)

(d) (27 points) Finally, we will consider magnetic dipole radiation, which is electromagnetic radiation induced by the rotation of the star and subsequent movement of its magnetic field. This is the main source of energy loss in pulsar systems. Let the magnetic dipole moment be $\mu$ with an inclination $\chi$ to the rotation axis (here, the z-axis). Take $\chi\in(0,\frac\pi2)$.

Some simple geometry gives

in component form.

(i) (4 points) Calculate the second derivative $\ddot{\boldsymbol\mu}(t)$ in component form. Using this, evaluate its magnitude $|\ddot{\boldsymbol\mu}|$.

(ii) (5 points) According to the Larmor formula, the power radiated by the magnetic dipole is given by

where $\mu_0$ is the permeability of free space. As in the previous section, assume that this power comes from a change in the rotational kinetic energy $E=\frac12I_z\Omega^2$, where we again take $I_z$ as the (constant) moment of inertia around the rotation axis. Calculate $\dot\Omega$ as a function of $\Omega$ and any other relevant variables. Using this, calculate the value of $n$. (Refer to this value as $n_0$.)

(iii) (9 points) Nevertheless, neutron stars are sometimes observed to have $n$-values that significantly differ from $n_0$. We will now model this. Specifically, take your same $P=-\dot E$ expression as before, but now assume that $\mu$ and $\chi$ are no longer fixed, instead potentially having small nonzero rates of change. Using your result from part (a)(i), calculate a new value for $n$. Express your answer in terms of $\Omega$, $\dot\Omega$, $\mu$, $\dot\mu$, $\chi$, $\dot\chi$, and fundamental constants. Again, assume that the mass, radius, and composition of the neutron star stay constant.

(iv) (2 points) Assume that the neutron star’s magnetic moment decays exponentially according to

Rewrite your expression for $n$ from part (iii) in terms of $\Omega$, $\dot\Omega$, $\tau_B$, $\chi$, $\dot\chi$, and fundamental constants.

(v) (7 points) The Crab Pulsar is observed to have a long-term average braking index of $n_{obs}=2.51\pm0.01$. For each of the following, mark on your Answer Sheet whether it would cause $n$ to deviate toward $n_{obs}$, away from $n_{obs}$, or have no change, assuming a default value of $n=n_0$ (remember $\dot\Omega<0$ and $\chi\in(0,\frac\pi2)$):

(A) A gradual increase in the inclination $\chi$,

(B) A gradual decrease in the internal magnetic dipole moment $\mu$,

(C) Gradual accretion of a nearby star (use your result from part (b)(iii) of the problem).

(D) Significant gravitational wave emission (use your result from part (c) of the problem).

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