L-shell

Last updated May 29, 2023

The L-shell, L-value, or McIlwain L-parameter (after Carl E. McIlwain) is a parameter describing a particular set of planetary magnetic field lines. Colloquially, L-value often describes the set of magnetic field lines which cross the Earth's magnetic equator at a number of Earth-radii equal to the L-value. For example, $L=2$ describes the set of the Earth's magnetic field lines which cross the Earth's magnetic equator two earth radii from the center of the Earth. L-shell parameters can also describe the magnetic fields of other planets. In such cases, the parameter is renormalized for that planet's radius and magnetic field model.^[1]

Although L-value is formally defined in terms of the Earth's true instantaneous magnetic field (or a high-order model like IGRF), it is often used to give a general picture of magnetic phenomena near the Earth, in which case it can be approximated using the dipole model of the Earth's magnetic field.

Charged particle motions in a dipole field

The motions of low-energy charged particles in the Earth's magnetic field (or in any nearly-dipolar magnetic field) can be usefully described in terms of McIlwain's (B,L) coordinates, the first of which, B is just the magnitude (or length) of the magnetic field vector.^[2] This description is most valuable when the gyroradius of the charged particle orbit is small compared to the spatial scale for changes in the field. Then a charged particle will basically follow a helical path orbiting the local field line. In a local coordinate system {x,y,z} where z is along the field, the transverse motion will be nearly a circle, orbiting the "guiding center", that is the center of the orbit or the local B line, with the gyroradius and frequency characteristic of cyclotron motion for the field strength, while the simultaneous motion along z will be at nearly uniform velocity, since the component of the Lorentz force along the field line is zero.

At the next level of approximation, as the particle orbits and moves along the field line, along which the field changes slowly, the radius of the orbit changes so as to keep the magnetic flux enclosed by the orbit constant. Since the Lorentz force is strictly perpendicular to the velocity, it cannot change the energy of a charged particle moving in it. Thus the particle's kinetic energy remains constant. Then so also must its speed be constant. Then it can be shown that the particle's velocity parallel to the local field must decrease if the field is increasing along its z motion, and increase if the field decreases, while the components of the velocity transverse to the field increase or decrease so as to keep the magnitude of the total velocity constant. Conservation of energy prevents the transverse velocity from increasing without limit, and eventually the longitudinal component of the velocity becomes zero, while the pitch angle, of the particle with respect to the field line, becomes 90°. Then the longitudinal motion is stopped and reversed, and the particle is reflected back towards regions of weaker field, the guiding center now retracing its previous motion along the field line, with the particle's transverse velocity decreasing and its longitudinal velocity increasing.^[3]

In the (approximately) dipole field of the Earth, the magnitude of the field is greatest near the magnetic poles, and least near the magnetic Equator. Thus after the particle crosses the Equator, it will again encounter regions of increasing field, until it once again stops at the magnetic mirror point, on the opposite side of the Equator. The result is that, as the particle orbits its guiding center on the field line, it bounces back and forth between the north mirror point and the south mirror point, remaining approximately on the same field line. The particle is therefore endlessly trapped, and cannot escape from the region of the Earth. Particles with too-small pitch angles may strike the top of the atmosphere if they are not mirrored before their field line reaches too close to the Earth, in which case they will eventually be scattered by atoms in the air, lose energy, and be lost from the belts.^[4]

However, for particles which mirror at safe altitudes, (in yet a further level of approximation) the fact that the field generally increases towards the center of the Earth means that the curvature on the side of the orbit nearest the Earth is somewhat greater than on the opposite side, so that the orbit has a slightly non-circular, with a (prolate) cycloidal shape, and the guiding center slowly moves perpendicular both to the field line and to the radial direction. The guiding center of the cyclotron orbit, instead of moving exactly along the field line, therefore drifts slowly east or west (depending on the sign of the charge of the particle), and the local field line connecting the two mirror points at any moment, slowly sweeps out a surface connecting them as it moves in longitude. Eventually the particle will drift entirely around the Earth, and the surface will be closed upon itself. These drift surfaces, nested like the skin of an onion, are the surfaces of constant L in the McIlwain coordinate system. They apply not only for a perfect dipole field, but also for fields that are approximately dipolar. For a given particle, as long as only the Lorentz force is involved, B and L remain constant and particles can be trapped indefinitely. Use of (B,L) coordinates provides us with a way of mapping the real, non-dipolar terrestrial or planetary field into coordinates that behave essentially like those of a perfect dipole. The L parameter is traditionally labeled in Earth-radii, of the point where the shell crosses the magnetic Equator, of the equivalent dipole. B is measured in gauss.

Equation for L in a Dipole Magnetic Field

In a centered dipole magnetic field model, the path along a given L shell can be described as^[5]

r=L\cos ^{2}\lambda

where $r$ is the radial distance (in planetary radii) to a point on the line, $\lambda$ is its geomagnetic latitude, and $L$ is the L-shell of interest.

L-shells on Earth

For the Earth, L-shells uniquely define regions of particular geophysical interest. Certain physical phenomena occur in the ionosphere and magnetosphere at characteristic L-shells. For instance, auroral light displays are most common around L=6, can reach L=4 during moderate disturbances, and during the most severe geomagnetic storms, may approach L=2. The Van Allen radiation belts roughly correspond to L=1.5–2.5, and L=4–6. The plasmapause is typically around L=5.

L-shells on Jupiter

The Jovian magnetic field is the strongest planetary field in the solar system. Its magnetic field traps electrons with energies greater than 500 MeV ^[6] The characteristic L-shells are L=6, where electron distribution undergoes a marked hardening (increase of energy), and L=20-50, where the electron energy decreases to the VHF regime and the magnetosphere eventually gives way to the solar wind. Because Jupiter's trapped electrons contain so much energy, they more easily diffuse across L-shells than trapped electrons in Earth's magnetic field. One consequence of this is a more continuous and smoothly-varying radio-spectrum emitted by trapped electrons in gyro-resonance.

Related Research Articles

Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles give rise to a magnetic field, which acts on other currents and magnetic moments. Magnetism is one aspect of the combined phenomena of electromagnetism. The most familiar effects occur in ferromagnetic materials, which are strongly attracted by magnetic fields and can be magnetized to become permanent magnets, producing magnetic fields themselves. Demagnetizing a magnet is also possible. Only a few substances are ferromagnetic; the most common ones are iron, cobalt, and nickel and their alloys. The rare-earth metals neodymium and samarium are less common examples. The prefix ferro- refers to iron because permanent magnetism was first observed in lodestone, a form of natural iron ore called magnetite, Fe₃O₄.

<span class="mw-page-title-main">Magnetopause</span> Abrupt boundary between a magnetosphere and the surrounding plasma

The magnetopause is the abrupt boundary between a magnetosphere and the surrounding plasma. For planetary science, the magnetopause is the boundary between the planet's magnetic field and the solar wind. The location of the magnetopause is determined by the balance between the pressure of the dynamic planetary magnetic field and the dynamic pressure of the solar wind. As the solar wind pressure increases and decreases, the magnetopause moves inward and outward in response. Waves along the magnetopause move in the direction of the solar wind flow in response to small-scale variations in the solar wind pressure and to Kelvin–Helmholtz instability.

<span class="mw-page-title-main">Paramagnetism</span> Weak, attractive magnetism possessed by most elements and some compounds

Paramagnetism is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, diamagnetic materials are repelled by magnetic fields and form induced magnetic fields in the direction opposite to that of the applied magnetic field. Paramagnetic materials include most chemical elements and some compounds; they have a relative magnetic permeability slightly greater than 1 and hence are attracted to magnetic fields. The magnetic moment induced by the applied field is linear in the field strength and rather weak. It typically requires a sensitive analytical balance to detect the effect and modern measurements on paramagnetic materials are often conducted with a SQUID magnetometer.

The solar wind is a stream of charged particles released from the upper atmosphere of the Sun, called the corona. This plasma mostly consists of electrons, protons and alpha particles with kinetic energy between 0.5 and 10 keV. The composition of the solar wind plasma also includes a mixture of materials found in the solar plasma: trace amounts of heavy ions and atomic nuclei such as C, N, O, Ne, Mg, Si, S, and Fe. There are also rarer traces of some other nuclei and isotopes such as P, Ti, Cr, ⁵⁴Fe and ⁵⁶Fe, and ⁵⁸Ni, ⁶⁰Ni, and ⁶²Ni. Superimposed with the solar-wind plasma is the interplanetary magnetic field. The solar wind varies in density, temperature and speed over time and over solar latitude and longitude. Its particles can escape the Sun's gravity because of their high energy resulting from the high temperature of the corona, which in turn is a result of the coronal magnetic field. The boundary separating the corona from the solar wind is called the Alfvén surface.

A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field. A permanent magnet's magnetic field pulls on ferromagnetic materials such as iron, and attracts or repels other magnets. In addition, a nonuniform magnetic field exerts minuscule forces on "nonmagnetic" materials by three other magnetic effects: paramagnetism, diamagnetism, and antiferromagnetism, although these forces are usually so small they can only be detected by laboratory equipment. Magnetic fields surround magnetized materials, and are created by electric currents such as those used in electromagnets, and by electric fields varying in time. Since both strength and direction of a magnetic field may vary with location, it is described mathematically by a function assigning a vector to each point of space, called a vector field.

An aurora , also commonly known as polar lights or northern lights, is a natural light display in Earth's sky, predominantly seen in high-latitude regions. Auroras display dynamic patterns of brilliant lights that appear as curtains, rays, spirals, or dynamic flickers covering the entire sky.

A Van Allen radiation belt is a zone of energetic charged particles, most of which originate from the solar wind, that are captured by and held around a planet by that planet's magnetosphere. Earth has two such belts, and sometimes others may be temporarily created. The belts are named after James Van Allen, who is credited with their discovery. Earth's two main belts extend from an altitude of about 640 to 58,000 km above the surface, in which region radiation levels vary. Most of the particles that form the belts are thought to come from solar wind and other particles by cosmic rays. By trapping the solar wind, the magnetic field deflects those energetic particles and protects the atmosphere from destruction.

The South Atlantic Anomaly (SAA) is an area where Earth's inner Van Allen radiation belt comes closest to Earth's surface, dipping down to an altitude of 200 kilometres (120 mi). This leads to an increased flux of energetic particles in this region and exposes orbiting satellites to higher-than-usual levels of ionizing radiation.

Earth's magnetic field, also known as the geomagnetic field, is the magnetic field that extends from Earth's interior out into space, where it interacts with the solar wind, a stream of charged particles emanating from the Sun. The magnetic field is generated by electric currents due to the motion of convection currents of a mixture of molten iron and nickel in Earth's outer core: these convection currents are caused by heat escaping from the core, a natural process called a geodynamo.

In physics, the dynamo theory proposes a mechanism by which a celestial body such as Earth or a star generates a magnetic field. The dynamo theory describes the process through which a rotating, convecting, and electrically conducting fluid can maintain a magnetic field over astronomical time scales. A dynamo is thought to be the source of the Earth's magnetic field and the magnetic fields of Mercury and the Jovian planets.

In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current, permanent magnets, elementary particles, various molecules, and many astronomical objects.

A levitated dipole is a type of nuclear fusion reactor design using a superconducting torus which is magnetically levitated inside the reactor chamber. The name refers to the magnetic dipole that forms within the reaction chamber, similar to Earth's or Jupiter's magnetospheres. It is believed that such an apparatus could contain plasma more efficiently than other fusion reactor designs. The concept of the levitated dipole as a fusion reactor was first theorized by Akira Hasegawa in 1987.

<span class="mw-page-title-main">Guiding center</span>

In physics, the motion of an electrically charged particle such as an electron or ion in a plasma in a magnetic field can be treated as the superposition of a relatively fast circular motion around a point called the guiding center and a relatively slow drift of this point. The drift speeds may differ for various species depending on their charge states, masses, or temperatures, possibly resulting in electric currents or chemical separation.

In atomic physics, the electron magnetic moment, or more specifically the electron magnetic dipole moment, is the magnetic moment of an electron resulting from its intrinsic properties of spin and electric charge. The value of the electron magnetic moment is −9.2847647043(28)×10⁻²⁴ J⋅T⁻¹. In units of the Bohr magneton (μ_B) it is −1.00115965218128(18) μ_B, a value that was measured with a relative accuracy of 1.7×10⁻¹³.

In quantum physics, the spin–orbit interaction is a relativistic interaction of a particle's spin with its motion inside a potential. A key example of this phenomenon is the spin–orbit interaction leading to shifts in an electron's atomic energy levels, due to electromagnetic interaction between the electron's magnetic dipole, its orbital motion, and the electrostatic field of the positively charged nucleus. This phenomenon is detectable as a splitting of spectral lines, which can be thought of as a Zeeman effect product of two relativistic effects: the apparent magnetic field seen from the electron perspective and the magnetic moment of the electron associated with its intrinsic spin. A similar effect, due to the relationship between angular momentum and the strong nuclear force, occurs for protons and neutrons moving inside the nucleus, leading to a shift in their energy levels in the nucleus shell model. In the field of spintronics, spin–orbit effects for electrons in semiconductors and other materials are explored for technological applications. The spin–orbit interaction is at the origin of magnetocrystalline anisotropy and the spin Hall effect.

The ions and electrons of a plasma interacting with the Earth's magnetic field generally follow its magnetic field lines. These represent the force that a north magnetic pole would experience at any given point. Plasmas exhibit more complex second-order behaviors, studied as part of magnetohydrodynamics.

The pitch angle of a charged particle is the angle between the particle's velocity vector and the local magnetic field. This is a common measurement and topic when studying the magnetosphere, magnetic mirrors, biconic cusps and polywells. See Aurora and Ring current

The magnetosphere of Jupiter is the cavity created in the solar wind by Jupiter's magnetic field. Extending up to seven million kilometers in the Sun's direction and almost to the orbit of Saturn in the opposite direction, Jupiter's magnetosphere is the largest and most powerful of any planetary magnetosphere in the Solar System, and by volume the largest known continuous structure in the Solar System after the heliosphere. Wider and flatter than the Earth's magnetosphere, Jupiter's is stronger by an order of magnitude, while its magnetic moment is roughly 18,000 times larger. The existence of Jupiter's magnetic field was first inferred from observations of radio emissions at the end of the 1950s and was directly observed by the Pioneer 10 spacecraft in 1973.

The dipole model of the Earth's magnetic field is a first order approximation of the rather complex true Earth's magnetic field. Due to effects of the interplanetary magnetic field (IMF), and the solar wind, the dipole model is particularly inaccurate at high L-shells, but may be a good approximation for lower L-shells. For more precise work, or for any work at higher L-shells, a more accurate model that incorporates solar effects, such as the Tsyganenko magnetic field model, is recommended.

The impact of the solar wind onto the magnetosphere generates an electric field within the inner magnetosphere - the convection field-. Its general direction is from dawn to dusk. The co-rotating thermal plasma within the inner magnetosphere drifts orthogonal to that field and to the geomagnetic field B_o. The generation process is not yet completely understood. One possibility is viscous interaction between solar wind and the boundary layer of the magnetosphere (magnetopause). Another process may be magnetic reconnection. Finally, a hydromagnetic dynamo process in the polar regions of the inner magnetosphere may be possible. Direct measurements via satellites have given a fairly good picture of the structure of that field. A number of models of that field exists.

References

↑ Galileo - Glossary of Selected Terms. NASA Jet Propulsion Laboratory, (2003).
↑ McIlwain, Carl E. (1961), "Coordinates for Mapping the Distribution of Magnetically Trapped Particles", Journal of Geophysical Research , 66 (11): 3681–3691, Bibcode:1961JGR....66.3681M, doi:10.1029/JZ066i011p03681, hdl: 2060/20150019302 ^{[ permanent dead link ]}
↑ Introduction to Space Science, Robert C Haymes, Wiley & sons, 1971. Chapter 7, "Van Allen Radiation" and Chapter 9, "Planetary Magnetism"
↑ The Radiation Belt and Magnetosphere. W. N. Hess, Blaisdell Publishing Co 1968
↑ Walt, Martin (1994). Introduction to Geomagnetically Trapped Radiation. New York, NY: Cambridge University Press. ISBN 978-0-521-61611-9.
↑ Jupiter’s radio spectrum from 74 MHz up to 8 GHz . Imke de Pater et al. Icarus, Volume 163, Issue 2, June 2003, Pages 434-448.

Other references

Tascione, Thomas F. (1994), Introduction to the Space Environment (2nd ed.), Malabar, FL: Kreiger
Margaret Kivelson and Christopher Russell (1995), Introduction to Space Physics, New York, NY: Cambridge University Press, pp. 166–167

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[galileo_glossary-1] Galileo - Glossary of Selected Terms. NASA Jet Propulsion Laboratory, (2003).

[2] McIlwain, Carl E. (1961), "Coordinates for Mapping the Distribution of Magnetically Trapped Particles", Journal of Geophysical Research , 66 (11): 3681–3691, Bibcode:1961JGR....66.3681M, doi:10.1029/JZ066i011p03681, hdl: 2060/20150019302 ^{[ permanent dead link ]}

[3] Introduction to Space Science, Robert C Haymes, Wiley & sons, 1971. Chapter 7, "Van Allen Radiation" and Chapter 9, "Planetary Magnetism"

[4] The Radiation Belt and Magnetosphere. W. N. Hess, Blaisdell Publishing Co 1968

[walt-5] Walt, Martin (1994). Introduction to Geomagnetically Trapped Radiation. New York, NY: Cambridge University Press. ISBN 978-0-521-61611-9.

[jupiter_radio-6] Jupiter’s radio spectrum from 74 MHz up to 8 GHz . Imke de Pater et al. Icarus, Volume 163, Issue 2, June 2003, Pages 434-448.

[1]

[2]

[3]

[4]

[5]

[6]