Planetary Ionospheres and Magnetospheres

Abstract

We begin this chapter with a detailed review of the ionization and recombination processes in a planetary ionosphere, as exemplified by the ionosphere we know the most about, that of the Earth. We then extend the discussion to the ionospheres of Mars and Venus, and examine atmospheric loss mechanisms. After using Maxwell’s equations to learn about electromagnetic wave propagation in an ionosphere, we review basic magnetospheric processes and look at their effects, such as particle trapping and precipitation, the ring current and other magnetospheric currents, magnetospheric convection and substorms, and magnetospheric-ionospheric coupling and aurorae. We conclude the chapter with a discussion of the magnetospheres of Mercury, Venus and Mars.

Challenges

• [11.1] In (11.26), derive λ _J  = r _x /H _x and λ _J  = (v _esc /v _th )² from the expression λ _J  = GMm/(kT _x r _x ).

• [11.2] Derive (11.93) from (11.92). [Hint: Electric current equals the amount of charge passing a given point per second; how is the current associated with a gyrating particle related to the charge of the particle and the period of the gyromotion?]

• [11.3] Derive (11.98) from (11.76) and other previous results.

• [11.4] Justify, using ${\overrightarrow{F}}_{\mathrm{B}}=q\overrightarrow{v}\times \overrightarrow{B}$, the statement preceding (11.99) that the kinetic energy of the particle is constant. [A descriptive answer is acceptable, but the description should be physically precise.]

• [11.5] Spacecraft orbiting through the trapping region of a planetary magnetosphere, such as the Earth’s Van Allen radiation belts, do not necessarily orbit in the plane of the magnetic equator; thus, they sample the magnetosphere at a range of magnetic latitudes. For uniformity of data, measured pitch angles are usually referred to the magnetic equator. If the spacecraft measures a pitch angle, α, for a particle at a location away from the magnetic equator where the magnetic field strength is B, and if on the same field line the magnetic field strength at the magnetic equator is B ₀, show that the pitch angle, α ₀, of the particle as it crosses the magnetic equator is given by

  $$ { \sin}^2{\alpha}_0=\frac{B_0}{B}{ \sin}^2\alpha $$

  [Hint: μ _B and Φ _B are constants of the motion. Express the pitch angles in terms of v _⊥ at B and v ⁰_⊥ at B ₀, and use (11.93) and (11.97).]

• [11.6] The loss cone is approximately 3° for particles on the geomagnetic equator at 6 R _E above the Earth’s surface, i.e., particles of this pitch angle mirror at ionospheric heights:

  • a. What is the mirror ratio, i.e., the ratio of the magnetic field strength at the ionosphere to that at the geomagnetic equator, along this field line? [Hint: Refer to problem 11.4. What is the pitch angle at the mirror point?]

  • b. If particles are injected with an initially isotropic distribution of pitch angles at 6 R _E on the geomagnetic equator, what fraction of these particles are precipitated?

[11.7] Equations (11.77) and (11.79) in Sect. 11.5.6 give, respectively, the $\overrightarrow{F}\times \overrightarrow{B}$ drift velocity, ${\overrightarrow{v}}_{\mathrm{D}}$, and its more commonly used form, the $\overrightarrow{E}\times \overrightarrow{B}$ drift velocity. The purpose of this problem is to derive (11.77) and (11.79).

Figure 11.45 shows a uniform magnetic field, $\overrightarrow{B}$, out of the page. In addition to any magnetic forces, all charges in this region also feel a downward force, $\overrightarrow{F}$, of external origin (e.g., gravitational or electrical) of uniform magnitude regardless of position. ($\overrightarrow{F}$ is unrelated to $\overrightarrow{B}$.)

• a. Suppose you take hold of the charge shown above and run with a constant (vector) velocity, $\overrightarrow{v}$, holding tightly onto the charge so it cannot execute gyromotion. In what direction would you have to run so that the total force (or net force) on the particle is zero? Draw a vector diagram showing and labelling $\overrightarrow{B},\overrightarrow{v}$, and all forces. Then write down the vector equation relating $\overrightarrow{F},\overrightarrow{v}$, and $\overrightarrow{B}$. (In this and the other parts, below, keep careful track of minus signs.)

  

  Fig. 11.45

• b. Evaluate the quantity $\left(\overrightarrow{v}\times \overrightarrow{B}\right)\times \overrightarrow{B}$ for this situation, using the unit vectors shown above. Express your answer in terms of a single unit vector, then rewrite your answer to express the quantity $\left(\overrightarrow{v}\times \overrightarrow{B}\right)\times \overrightarrow{B}$ in terms of $\overrightarrow{v}$.

• c. Obtain (11.77) from your answers to parts a and b.

• d. Now assume that the force, $\overrightarrow{F}$, is caused by an electric field vertically downward in the reference frame of the diagram (i.e., $\overrightarrow{F}=q\overrightarrow{E}$). In your reference frame, the force on a stationary particle is zero. It follows that, by running, you have placed yourself in a reference frame in which the electric field is zero: if you let go of the particle, it will remain at rest in your reference frame, so evidently there is no electric field.

If a particle is now released at rest in the original reference frame of the diagram, what initial velocity will you see for this particle? What subsequent motion do you expect to see for this particle? (Remember, in your reference frame there is only a magnetic field.) By extension, what subsequent motion would an observer see who is at rest in the reference frame of the diagram? What is the $\overrightarrow{E}\times \overrightarrow{B}$ drift velocity of the guiding center in the reference frame of the diagram? (Also check that the drift velocity is in the direction of $\overrightarrow{E}\times \overrightarrow{B}$.)