Homework 4

Release Date:

2024/03/25

Due Date:

2024/04/02 23:59:59

Topic:

Planetary Magnetic Fields

Disclosure:

This homework is based on a hypothetical setting and does not represent any real mission endorsed by NASA or any other space agency.

Particle Fact Sheet

Particle

Mass (kg)

Charge (C)

Proton

1.67e-27

1.6e-19

Electron

9.11e-31

-1.6e-19

Magnetic Fields of Planets

Magnetic fields are an important aspect of planetary science. The magnetic field of a planet is generated by the motion of the liquid iron core of the planet. The magnetic field of a planet can protect the planet from the solar wind and it tells us about the evolution history of the planet. The seafloor of the Earth shows linear magnetic stripes that are parallel to the mid-oceanic ridges. They are the result of the Earth’s magnetic field flipping its polarity over geological time. Mars does not have a global magnetic field, but it has crustal magnetic fields, which are remnants of the planet’s past magnetic field when the dynamo was active. Europa, a moon of Jupiter, is believed to host an ocean beneath its icy crust due to the magnetic induction of Jupiter’s varying magnetic field.

1. (5’) Magnetic field of Callisto

Callisto is the outermost Galilean moon of Jupiter. The rocks on Callisto’s surface are billions of years old, which suggests that Callisto has not experienced any geological activity for a long time. The surface of Callisto is heavily cratered to saturation, meaning that almost every part of the surface is covered with craters. In fact, Callisto has the oldest and most heavily cratered surface in the Solar System.

Callisto is composed of a mixture of rock and ice. It is less affected by the magnetic field of Jupiter and the tidal forces of Jupiter due to its larger distance from Jupiter compared to the other Galilean moons. Measurements of the density and the moment of inertia of Callisto suggest that Callisto is an undifferentiated body, meaning that iron, rock and ice are not separated into layers. The Galileo spacecraft also did not detect an internally generate magnetic field around Callisto. Given the context above, answer the following questions:

  1. (2’) Please explain why Callisto lacks an internally generated magnetic field

  2. (3’) Why are the lacking of an internally generated magnetic field and the heavily cratered surface of Callisto consistent with each other?

2. (5’) Magnetic field of Venus

Venus does not have a global magnetic field. The lack of a global magnetic field of Venus may be due to the slow rotation of Venus or due to the current absence of a liquid iron core. However, whether Venus ever had a global magnetic field in the past is still an open question. One way to find out is to study the crustal magnetic fields of Venus. Suppose that Venus has a crustal magnetic layer that is between the surface and some depth below the surface. We know that the permanet magnetic field ceases to exist above the Curie temperature of the material, which is about 800 K. The surface temperature of Venus is about 740 K and the thermal gradient of Venus’ crust is about 15 K/km. Given the facts above, estimate the maximum depth of the crustal magnetic layer of Venus, if there is any.

3. (5’) Energy of charged particles

We use the unit eV to measure the energy of charged particles. For a non-relativistic particle with velocity \(v\), the kinetic energy of the particle is given by:

\[KE = \frac{1}{2} \times m \times v^2\]

where \(m\) is the mass of the particle. The solar wind contains a stream of charged particles and many of them are protons and electrons. Suppose that the energy of the solar wind particles is 1 keV (kilo-electron-volt).

  1. (2’) Calculate the velocity of a proton with an energy of 1 keV.

  2. (2’) Calculate the velocity of an electron with an energy of 1 keV.

Suppose that the proton hits the magnetic field of Earth and starts to gyrate around the magnetic field lines. The magnetic field of Earth is approximately 0.5 Gauss.

  1. (1’) What is the maximum radius of the gyration of the proton with a kinetic energy of 1 keV around the magnetic field lines of Earth?

4. (5’) Motion of charged particles in magnetic fields

Use the UCLA IGPP Space Physics Exercise tool to visualize charged particle motion in electric and magnetic fields. At the following website, you will find a variety of tools designed to provide direct visualization of particle motion in different electromagnetic environments. You will be using those tools to visualize particle motion for different initial and ambient conditions. Please use the following link to access the tool:

[link]

For this problem, you will submit your sketch of the particle trajectory under the listed conditions (screenshots are not accepted) and provide a brief explanation of the trajectory. Please also note that whether the particle is an ion or an electron.

  1. (1’) Sketch the particle trajectory under uniform magnetic field conditions.

  2. (2’) Sketch the particle trajectory under \(E\times B\) drift conditions.

  3. (2’) Sketch the particle trajectory under \(\nabla B\) drift conditions.