APhO 2015, Теория — Задача 2. How are aurora ignited by the solar wind?

(a) (3 т.)

(i) Before we study the motion of a charged particle in the Earth's dipole magnetic field, we first consider the motion of an electron in a uniform magnetic field $\vec{B}$. When the initial electron velocity $\vec{v}$ is perpendicular to the uniform magnetic field as shown in Figure 6, please calculate the electron trajectory. The electron is initially located at (x,y,z)=(0,0,0). [1 т.]

(ii) Please determine the electric current of the electron motion and calculate the magnetic moment $\vec{\mu} = I\vec{A}$, where $\vec{A}$ is the area of the electron circular orbit and the direction of $\vec{A}$ is determined by the right-hand rule of the electric current. [1 т.]

(iii) If the initial electron velocity $\vec{v}$ is not perpendicular to the uniform magnetic field, i.e, the angle $\theta$ between $\vec{B}$ and $\vec{v}$ is $0^{0}<\theta<90^{0}$, please give the screw pitch (the distance along the z-axis between successive orbits) of the electron trajectory. [1 т.]

(b) (4 т.)

In the uniform background magnetic field as shown in Figure 6, the plasma density is nonuniform in x. For simplicity, we assume that the temperature and the distribution of the ions and electrons are the same. Thus, the plasma pressure can be expressed as

Where B, T, k, $n_0$, and $\alpha$ are positive constants, $n_i(x)$ and $n_e(x)$ are the number densities of the ions and electrons.

(i) Please explain the generation mechanism of the electric current by a schematic drawing. [2 т.]

(ii) If both the ions and electrons have a Maxwellian distribution, the ion distribution is

please calculate the constant β in the magnetization $M = \beta n(x)\dfrac{kT}{B}$, where the magnetization M is the magnetic moment per unit volume. (Hint: We have $\int_0^\infty x\exp(-x)dx = 1$ and $\int_{-\infty}^{\infty}\exp(-x^2)dx = \sqrt{\pi}$.) [2 т.]

(c) (1 т.)

Now let's go back to the Earth's dipole magnetic field. Please apply the result from Question(b) to calculate the ratio of the diamagnetic field and the Earth's dipole magnetic field in Equation (1) at the position ($x$=10R$_\mathrm{E}$，$y$=0, $z$=1 R$_\mathrm{E}$). The plasma pressure is assumed to be $p(z) = p_0e^{-(z/a)^2}$, where $p_0 = 3\times10^{-10}\ \mathrm{pa}$ and $a = 2R_E$. The magnetic field around this position is also assumed to be uniform. Be aware of the difference in the coordinate systems in Questions (b) and (c). (Hint: The diamagnetic field is given by $B_{mx} = \mu_o M$.)

(d) (4 т.)

From Figures 2, 3, and 5, it can be clearly seen that the Earth's magnetic field strength along a magnetic field line is the largest at the poles and the smallest in the equatorial plane. Since the Earth's dipole magnetic field is axially symmetric and slowly varying along a magnetic field line, it can for simplicity be treated as a magnetic-mirror field as shown in Figure 7. The magnetic field strength along a magnetic field line is the smallest ($B_0$) at the point "P$_2$" and the largest ($B_m$) at the points "P$_1$" and "P$_3$". An electron with an initial velocity $\vec{v}$ is located at the point "P$_2$" and drifts towards the point "P$_3$". The angle between the initial velocity $\vec{v}$ and the magnetic field at the point "P$_2$" is $0^{0}<\theta<90^{0}$. For the magnetic-mirror field $\vec{B} = B_r\hat{r} + B_z\hat{z}$ (with $B_r \ll B_z$), we can assume $\dfrac{dB}{dz} = \dfrac{dB}{ds}$, where $\dfrac{dB}{ds}$ is the spatial derivative of $B$ along a magnetic field line. Since there is no evidence of the existence of magnetic monopoles, we have $\langle B_r\rangle = -\dfrac{1}{2}\dfrac{dB}{dz}r_c \ll B_z$, where $\langle B_r\rangle$ is the gyro-average of $B_r$ and $r_c$ is the electron gyroradius.

(i) Please give the gyro-averaged magnetic-field force along the magnetic field lines on an electron and show that the magnetic moment is a motion constant, i.e., $\dfrac{d\mu}{dt} = 0$, based on the law of the total kinetic energy conservation. [3 т.]

(ii) Based on the motion constant of magnetic moment, please determine what the condition should be satisfied for the angle $\theta$ between the initial electron velocity $\vec{v}$ and the magnetic field at the point "P$_2$" if an electron will not escape from the magnetic mirror field. [1 т.]

(e) (1 т.)

Earth's dipole magnetic field lines (blue lines) are shown in Figure 8. The spiral trajectory of a charged particle (red curve) is assumed to be confined in the $y$=0 plane since the gradient and the curvature of the magnetic field can be ignored. If a charged particle with the mass $m$, charge $q$, and velocity $\vec{v}$ is initially located at the equatorial position [$x$=6R$_\mathrm{E}$, $y$=0, $z$=0] and the angle between the electron velocity $\vec{v}$ and the magnetic field is $\theta$ initially, please determine what the condition should be satisfied for $\theta$ if the charged particle arrives below 200km of its altitude at the latitude $60^{0}$.

(f) (5 т.)

As shown in Figure 5, when magnetic reconnection takes place at the dayside magnetopause, reconnected magnetic field lines drift towards the nightside region because the solar wind flows tailward. Thus, some solar wind electrons in the region "A" also move towards the magnetotail in the region "P". After the electrons arrive in the region "P", some electrons can be accelerated to around 1keV. If energetic electrons drift down to the thermosphere (The altitude of the thermosphere is about 85km-800km.), energetic electrons can collide with the neutral atoms, which could cause the neutral atoms to jump into excited states. A photon is emitted when the higher excited state of a neutral atom returns to its lower excited state or ground state. Splendid aurora (Figure 1) is generated in the aurora oval due to photons with different wavelengths. It is found that the aurora is mainly resulted from photons emitted by oxygen atoms. The energy levels in the first and second excited states relative to the ground state are 1.96eV and 4.17eV, respectively. The lifetimes of the two excited states of an oxygen atom are 110s and 0.8s as shown in Figure 9.

(i) Please give the atmospheric density as a function of the altitude and the ratio of the oxygen density at the altitudes H=160km and H=220km. For simplicity, we assume that the atmospheric temperature is independent of the altitude and the air is an ideal gas.

($\rho_0 g/P_0 = 0.13/\mathrm{km}$, where $\rho_0$ and $P_0$ are the atmospheric density and pressure at sea level.) [2 т.]

(ii) Please give the colors of auroras at the altitudes H=160km and H=220km. (Hint: The dependence of the collision frequency of atmospheric molecules on the atmospheric density is $\nu = \nu_0\rho/\rho_0$， where $\nu_0 \approx 10^{9}/s$ is the collision frequency of atmospheric molecules at sea level. The excited oxygen atom will lose a part of its energy when it collides with other neutral molecules. ) [3 т.]

(g) (2 т.)

As mentioned above, a powerful solar wind can push the dayside magnetopause to very close to the Earth, which could cause a high-orbit satellite to be fully exposed to the solar wind. The energetic particles in the solar wind could damage high-tech electronic components in a satellite. For simplicity, the Earth's dipole magnetic field is assumed to remain unchanged when the solar wind compresses it and that the plasma density is ignorable in the magnetosphere. Please give the minimum solar wind speed to cause a damage of a geosynchronous satellite if the magnetic field strength and the plasma density of the solar wind are $B_s = 5\times10^{-9}\ \mathrm{T}$ and $\rho_s = 50\ \mathrm{proton/cm^3}$, respectively. (Hint: The force per unit area associated with the magnetic field is $f = B^2/2\mu_0$. We only consider the variation in x for all physical quantities, i.e., the physical quantities are independent of y and z.)