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Stars/Radiative dynamo

A radiative dynamo is "a dynamo taking place in the radiative layers" of a star.

It is a theoretical construction to explain the magnetohydrodynamic properties of plasma occurring in the outer atmospheric layers of astronomical objects including stars. As such it is a part of theoretical stellar science and theoretical astrophysics.

Antidynamos
An "antidynamo theorem is one of several results that restrict the type of magnetic fields that may be produced by dynamo action."

No "axisymmetric magnetic field can be maintained through a self-sustaining dynamo action by an axially symmetric current."

A "dipole, an axisymmetric magnetic field. These magnetic fields are self-sustained through fluid motion in the Sun or planets, with the necessary non symmetry for the planets deriving from the Coriolis force caused by their rapid rotation, and one cause of non-symmetry for the Sun being its differential rotation."

Successful "dynamos do not possess a high degree of symmetry."

Disc dynamos
A disk generator "is a DC electrical generator comprising an electrically conductive disc or cylinder rotating in a plane perpendicular to a uniform static magnetic field. A potential difference is created between the center of the disc and the rim (or ends of the cylinder), the electrical polarity depending on the direction of rotation and the orientation of the field."

Large "research generators can produce hundreds of volts, and some systems have multiple generators in series to produce an even larger voltage. They are unusual in that they can source tremendous electric current, some more than a million amperes, because the homopolar generator can be made to have very low internal resistance."

Then, in reverse, more than a million amperes as a current between the rim of a disc and the center creates a potential difference and rotates an electrically conductive disc in a plane perpendicular to form a uniform magnetic field.

"Since cosmical clouds of ionized gas are generally magnetized, their motion produces induced electric fields [..] For example the motion of the magnetized interplanetary plasma produces electric fields that are essential for the production of aurora and magnetic storms".

The "rotation of a conductor in a magnetic field produces an electric field in the system at rest. This phenomenon is well known from laboratory experiments and is usually called 'homopolar ' or 'unipolar' induction."

Helioseismology
Helioseismology has shown that "[at] the tachocline [within the Sun,] the rotation abruptly changes to solid body rotation in the solar radiation zone. "

The Sun is a stellar example where a radiative dynamo is not occurring within its radiative zone.

Magnetic dynamos
"An alternative to the radiative–dynamo model is that the magnetic field originates in the material that formed the star. If the protostellar cloud which forms a star is weakly magnetic, conservation of magnetic energy would result in a very strong main–sequence field. We call these fossil fields [...] In order for the fossil field model to work, the field must be able to survive the collapse of the protostellar cloud during the star formation process. The fossil field argument also relies on a stable field configuration being reached that would avoid destruction on main–sequence lifetimes. Certain stable configurations have been found [...] and simulations have suggested that arbitrary field configurations do relax to these stable states [...] However, simple field configurations are still subject to the same instabilities as the fields [...] generated by dynamo action, in particular the Tayler instability [...] the fossil field model predicts field evolution similar to that of the dynamo model [...] the fossil field strength has to be several orders of magnitude larger than the initial field in the case of a magnetic dynamo in order to reproduce the same final field. [...] a significant fraction of flux could survive [from the pre-main sequence] but only if the magnetic diffusivity was sufficiently low."

Planetary sciences
"According to dynamo theory, the [Earth's magnetic] field is generated within the molten outer core region where heat creates convection motions of conducting materials, generating electric currents. These in turn produce the Earth's magnetic field. The convection movements in the core are chaotic; the magnetic poles drift and periodically change alignment. This causes field reversals at irregular intervals averaging a few times every million years. The most recent reversal occurred approximately 700,000 years ago. "

Minerals
"Radioactive potassium [...] appears also to be a substantial source of heat in the Earth's core"

"Radioactive potassium, uranium and thorium are thought to be the three main sources of heat in the Earth's interior, aside from that generated by the formation of the planet. Together, the heat keeps the mantle actively churning and the core generating a protective magnetic field."

Much "less potassium [occurs] in the Earth's crust and mantle than [is] expected based on the composition of rocky meteors that supposedly formed the Earth. If, as some have proposed, the missing potassium resides in the Earth's iron core, how did an element as light as potassium get there, especially since iron and potassium don't mix?"

At "the high pressures and temperatures in the Earth's interior, potassium can form an alloy with iron never before observed. During the planet's formation, this potassium-iron alloy could have sunk to the core, depleting potassium in the overlying mantle and crust and providing a radioactive potassium heat source in addition to that supplied by uranium and thorium in the core."

The "new alloy [is created] by squeezing iron and potassium between the tips of two diamonds [a diamond anvil] to temperatures and pressures characteristic of 600-700 kilometers below the surface - 2,500 degrees Celsius and nearly 4 million pounds per square inch, or a quarter of a million times atmospheric pressure."

"Our new findings indicate that the core may contain as much as 1,200 parts per million potassium -just over one tenth of one percent."

"This amount may seem small, and is comparable to the concentration of radioactive potassium naturally present in bananas. Combined over the entire mass of the Earth's core, however, it can be enough to provide one-fifth of the heat given off by the Earth."

"With one experiment, Lee and Jeanloz demonstrated that potassium may be an important heat source for the geodynamo, provided a way out of some troublesome aspects of the core's thermal evolution, and further demonstrated that modern computational mineral physics not only complements experimental work, but that it can provide guidance to fruitful experimental explorations,"

"More experiments need to be done to show that iron can actually pull potassium away from the silicate rocks that dominate in the Earth's mantle."

"They proved it would be possible to dissolve potassium into liquid iron."

"Modelers need heat, so this is one source, because the radiogenic isotope of potassium can produce heat and that can help power convection in the core and drive the magnetic field. They proved it could go in. What's important is how much is pulled out of the silicate. There's still work to be done."

"If a significant amount of potassium does reside in the Earth's core, this would clear up a lingering question - why the ratio of potassium to uranium in stony meteorites (chondrites), which presumably coalesced to form the Earth, is eight times greater than the observed ratio in the Earth's crust. Though some geologists have asserted that the missing potassium resides in the core, there was no mechanism by which it could have reached the core. Other elements like oxygen and carbon form compounds or alloys with iron and presumably were dragged down by iron as it sank to the core. But at normal temperature and pressure, potassium does not associate with iron."

"Early in Earth's history, the interior temperature and pressure would not have been high enough to make this alloy."

"But as more and more meteorites piled on, the pressure and temperature would have increased to the point where this alloy could form."

"The Earth is thought to have formed from the collision of many rocky asteroids, perhaps hundreds of kilometers in diameter, in the early solar system. As the proto-Earth gradually bulked up, continuing asteroid collisions and gravitational collapse kept the planet molten. Heavier elements - in particular iron - would have sunk to the core in 10 to 100 million years' time, carrying with it other elements that bind to iron."

"Gradually, however, the Earth would have cooled off and become a dead rocky globe with a cold iron ball at the core if not for the continued release of heat by the decay of radioactive elements like potassium-40, uranium-238 and thorium-232, which have half-lives of 1.25 billion, 4 billion and 14 billion years, respectively. About one in every thousand potassium atoms is radioactive."

"The heat generated in the core turns the iron into a convecting dynamo that maintains a magnetic field strong enough to shield the planet from the solar wind. This heat leaks out into the mantle, causing convection in the rock that moves crustal plates and fuels volcanoes."

Pure "iron and pure potassium [combined] in a diamond anvil cell [that] squeezed the small sample to 26 gigapascals of pressure while heating the sample with a laser above 2,500 Kelvin (4,000 degrees Fahrenheit), which is above the melting points of both potassium and iron. [Repeat] six times in the high-intensity X-ray beams of two different accelerators - Lawrence Berkeley National Laboratory's Advanced Light Source and the Stanford Synchrotron Radiation Laboratory - to obtain X-ray diffraction images of the samples' internal structure. The images confirmed that potassium and iron had mixed evenly to form an alloy, much as iron and carbon mix to form steel alloy."

"In the theoretical magma ocean of a proto-Earth, the pressure at a depth of 400-1,000 kilometers (270-670 miles) would be between 15 and 35 gigapascals and the temperature would be 2,200-3,000 Kelvin."

"At these temperatures and pressures, the underlying physics changes and the electron density shifts, making potassium look more like iron."

"At high pressure, the periodic table looks totally different."

"The work by Lee and Jeanloz provides the first proof that potassium is indeed miscible in iron at high pressures and, perhaps as significantly, it further vindicates the computational physics that underlies the original prediction."

"If it can be further demonstrated that potassium would enter iron in significant amounts in the presence of silicate minerals, conditions representative of likely core formation processes, then potassium could provide the extra heat needed to explain why the Earth's inner core hasn't frozen to as large a size as the thermal history of the core suggests it should."

Dynamo theory
Def. any conversion of mechanical energy into electrical energy and associated magnetic fields is called a dynamo.

Def. "a dynamo taking place in the radiative layers" of a star, or other astronomical object, is called a radiative dynamo, or stellar radiative dynamo.

"[M]otions resulting from [a linear magnetohydrodynamic] instability act as a dynamo to sustain the magnetic field." "Supersonic flows are initially generated by the Balbus-Hawley magnetic shear instability."

A plasma with local magnetohydrodynamic instabilities creates mechanical turbulence, motion, or shear (a dynamo) which in turn generates or sustains the local magnetic field.

When this magnetohydrodynamic dynamo occurs between or within radiative layers, a radiative dynamo is operating.

"There are three requisites for a dynamo to [occur and subsequently] operate:"

 * An electrically conductive fluid medium [such as a plasma or liquid iron]
 * [local magnetohydrodynamic instabilities]
 * An ... energy source to [create the local magnetohydrodynamic instabilities and] to drive [mechanical turbulence, motion, or shear] within the fluid.

Hydromagnetic dynamos
Magnetic induction may be represented by

 * $$ \nabla \cdot \mathbf{B}=0.$$

Conservation of mass is represented by

 * $$ \nabla \cdot \mathbf{u} = 0.$$

Conservation of momentum is given by the Navier-Stokes equation:

 * $$ \frac{D\mathbf{u}}{Dt} = -\nabla p + \nu \nabla^2 \mathbf{u} + \rho'\mathbf{g} + 2\mathbf{\Omega} \times \mathbf{u} + \mathbf{\Omega} \times \mathbf{\Omega} \times \mathbf{R} + \mathbf{J} \times \mathbf{B}, $$

where $$\nu$$ is the kinematic viscosity, $$\rho'$$ is the density perturbation that provides buoyancy (for thermal convection $$\rho'=\alpha\Delta T$$, $$\Omega$$ is the rotation rate of the Earth, and $$\mathbf{J}$$ is the electrical current density.

For heat a transport equation is

 * $$ \frac{\partial T}{\partial t} = \kappa \nabla^2 T +\epsilon $$

"where T is temperature, $$\kappa=k/\rho c_p$$ is the thermal diffusivity with k thermal conductivity, $$c_p$$ heat capacity, and $$\rho$$ density, and $$\epsilon$$ is an optional heat source."

"Often the pressure is the dynamic pressure, with the hydrostatic pressure and centripetal potential removed. These equations are then non-dimensionalized, introducing the non-dimensional parameters"

 * $$Ra=\frac{g\alpha T D^3}{\nu \kappa}, E=\frac{\nu}{\Omega D^2} , Pr=\frac{\nu}{\kappa} , Pm=\frac{\nu}{\eta}, $$

"where Ra is the Rayleigh number, E the Ekman number, Pr and Pm the Prandtl and magnetic Prandtl number. Magnetic field scaling is often in Elsasser number units $$B=(\rho \Omega/\sigma)^{1/2}$$."

Entities
"The proliferation of models has [...] created [...] different ways [to model] the core, [normalize] equations, [define] dimensionless parameters, [choose] boundary conditions, and [select] energy sources."

The "major topics [are]"
 * 1) inset and evolution of convection,
 * 2) character of the magnetic field generated, and
 * 3) comparison with the observed geomagnetic field.

"Although there are large differences in the way that the simulations are defined, the magnetic fields that they generate have some surprising similarities. The fields are dominated by the axial dipole. In some models they are most strongly generated in shear layers near the upper and lower boundaries and near the tangent cylinder, an imaginary surface touching the inner core on its equator. Convection rolls occur within which a type of the α effect distorts the toroidal field lines to create poloidal magnetic field."

Kinematic dynamo theory: "the boundary conditions defining the energy flow (e.g., an inhomogeneous heat flux or distribution of buoyancy sources) are very influential [...] They change the frequency and the mode of magnetic polarity reversals as well as the ratio in strengths of the dipole and nondipole moments."

Polarity "reversals reminiscent of the paleomagnetically observed field reversals have already been simulated by some of the models [as have other] features such as drift of the field, its secular variation, and statistical properties of Gauss coefficients".

Radiative zones
White dwarfs whose primary spectral classification is DA have hydrogen-dominated atmospheres. They make up the majority (approximately 80%) of all observed white dwarfs. .

DA spectral type, having only hydrogen absorption lines in its spectrum, white dwarf material is initially plasma—a fluid composed of nuclei and electrons. "Helium is unquestionably absent from the atmospheres of ... DA stars, and [there is a] low metal abundance".

"In a DA star the "radiative layer ... lies above the convective zone."

Only a small number of white dwarfs have been examined for fields, and it has been estimated that at least 10% of white dwarfs have fields in excess of 1 million gauss (100 T).

Coronal radiative layers
"[F]or radiative losses of the solar corona, it is meant the energy flux irradiated from the external atmosphere of the Sun (traditionally divided into chromosphere, transition region and corona), and, in particular, the processes of production of the radiation coming from the solar corona and transition region, where the plasma is optically-thin. On the contrary, in the chromosphere, where the temperature decreases from the photospheric value of 6000 K to the minimum of 4400 K, the optical depth is about 1, and the radiation is thermal."

"The energy flux irradiated from the corona changes in active regions, in the quiet Sun and in coronal holes; actually, part of the energy is irradiated outwards, but approximatively the same amount of the energy flux is conducted back towards the chromosphere, through the steep transition region. In active regions the energy flux is about 107 erg cm−2sec−1, in the quiet Sun it is roughly 8 [x] 105 - 106 erg cm−2sec−1, and in coronal holes 5 [x] 105 - 8 [x] 105 erg cm−2sec−1, including the losses due to the solar wind. The required power is a small fraction of the total flux irradiated from the Sun, but this energy is enough to maintain the plasma at the temperature of million degrees, since the density is very low and the processes of radiation are different from those occurring in the photosphere".

Whether local magnetohydrodynamic instabilities are generating a dynamo or not, these outer layers are radiative and some form of radiative dynamo may be operating.

"Many coronal heating theories have been proposed, but two theories have remained as the most likely candidates, wave heating and magnetic reconnection (or nanoflares). Through most of the past 50 years, neither theory has been able to account for the extreme coronal temperatures."

Convective dynamos
"[T]he solar cycle, generally considered as the classical case of a convective dynamo process, is probably not driven by convective turbulence at all."

Radiative α–Ω dynamos
"Models of rotationally–driven dynamos in stellar radiative zones have suggested that magnetohydrodynamic transport of angular momentum and chemical composition can dominate over the otherwise purely hydrodynamic processes."

A "number of magnetic O and B stars have been discovered [...]. Combined with this, a number of chemically peculiar A and B stars (known as Ap and Bp stars respectively) with surface field strengths up to 20kG have been identified [...]. These large–scale fields tend to have simple geometries and there is debate over whether they arise from fossil fields present during a star’s formation [...] or from a rotationally–driven dynamo operating in the radiative zone of the star [...]."

"In low–mass stars, where the outer region is convective, magnetic fields are expected to be formed in a strong shear layer at the base of the convection zone and then transported to the surface by convection and magnetic buoyancy [...]. In radiative zones there is no strong bulk motion to redistribute magnetic energy. In most dynamo models, magnetic flux is redistributed by magnetorotational turbulence [...]. This turbulence is also responsible for driving the generation of large–scale magnetic flux. This is the α-effect [...] which applies to both poloidal and toroidal components, although in rotating systems shear is generally more effective at producing toroidal field from the poloidal component and so the α–effect is needed for the poloidal field only. The toroidal field is instead maintained by the conversion of poloidal field into toroidal field by differential rotation. This is commonly referred to as an α–Ω dynamo"

Magnetic "fields [may be able to] produce turbulent instabilities which dominate the transport of angular momentum. [...] The evolution of the angular momentum distribution and magnetic field strength have a significant effect on the final fate of a star and its ejecta. Apart from causing chemical mixing, sufficiently strong magnetic fields are expected to cause magnetic braking that results in the rapid spin down of rotating magnetic stars"

The radiative α–Ω dynamo is "a magnetic model where the poloidal and toroidal components are evolved via advection–diffusion equations derived from the induction equation. [...] The magnetic field and angular momentum evolution are coupled by turbulent diffusivities, magnetic stresses and conversion of poloidal field into toroidal field by differential rotation. The dynamo is completed by regeneration of magnetic flux by a simple α–Ω dynamo."

The "magnetic turbulence from the Tayler–instability [redistributes] angular momentum in radiative zones [... By solving] for the magnetic field and hence the Alfvén velocity independently [the associated turbulent diffusion coefficients are also derived] instead of treating [them] as a function of the rotation rate. [... The] magnetic diffusivity [is] η = Prm Dmag where Prm is the turbulent magnetic Prandtl number."

The "dynamo efficiency is given by"

 * $$ \alpha = \gamma \frac{r \omega_A \Omega q}{N},$$

where N is the relevant buoyancy frequency, ω2 is the Alfvén frequency, γ is an efficiency parameter, q = ∂(log Ω)/∂(log r), and Ω(r) is the differential rotation as a function of radius.

"Strongly magnetic intermediate–mass stars typically have rotation rates much slower than other stars in their parent population (Mathys 2004). If the Alfvén radius, the radius at which the magnetic energy density is the same as the kinetic energy density in the stellar wind, is larger than the stellar radius then magnetic braking allows additional angular momentum to be carried away by the stellar wind."

"Owing to the strong magnetically–induced turbulence, the toroidal field behaves roughly as BΦ ∝ r−3 and the poloidal field behaves as A ∝ r−2 so both are much stronger towards the core than at the surface of the star [...]. The toroidal field falls to zero within a very narrow region near the surface of the star to meet the boundary conditions. The strength of the toroidal field predicted is around nine orders of magnitude larger than the poloidal field. This is because the Ω–effect, the conversion of poloidal field into toroidal field by differential rotation, is much stronger than the α–effect which regenerates the poloidal field."

The "surface value of the field [is taken] to be the strength of the toroidal field just below the boundary layer. [Taking] the poloidal field [as the surface field means] a larger value of γ to produce a stronger field. In this case the toroidal field is around six orders of magnitude larger than the poloidal field. So a surface poloidal field of 103 G would correspond to a toroidal field of 109 G just below the surface. The fields then increase by several orders of magnitude towards the core. Not only do these field strengths seem unreasonably energetic but also the magnetic stresses result in cores that are spinning near or above break–up velocity. However, spectropolarimetric observations have concluded that the large–scale structure of the external magnetic fields of massive stars are largely dipolar so there must be some mechanism for converting the toroidal field into poloidal field at the surface. It is likely that the stellar wind stretches the field lines in the radial direction, changing the toroidal field to a radial geometry as material is ejected from the stellar surface".

"The transition between a strong [...] field and no field is sharpest in rapid rotators. This transition is caused by the interaction between hydrodynamic and magnetic turbulence. If [the kinetic Prandtl number] exceeds [the magnetic Prandtl number] for a sufficiently large region of the radiative envelope, the magnetic field decays exponentially and cannot be sustained by the dynamo."

Tayler-Spruit dynamos
The "[Tayler–Spruit dynamo mechanism (Spruit 2002)] asserts that pinch–type instabilities (Tayler 1973; Spruit 1999) arise in toroidal fields that drive magnetic turbulence that enforces solid–body rotation. The growth of instabilities is controlled by magnetic diffusion which ultimately determines the equilibrium strength of the field."

"For stars more massive than around 15M⊙ the Kelvin–Helmholtz turbulence dominates over the magnetic turbulence and a stable field cannot be sustained by the dynamo."

Differential rotations
"Both the core and the radiative zone dynamo models involve a significant amount of differential rotation for the generation of a large-scale toroidal field." But, "the buoyant rise time [for a magnetic field generated by a core dynamo] from the core can become much longer than the age of [OBA type stars] for weakly magnetized flux-tubes".

"Magnetic fields can be created in stably stratified (non-convective) layers in a differentially rotating star. A magnetic instability in the toroidal field (wound up by differential rotation) replaces the role of convection in closing the field amplification loop."

At right is a diagram of the internal rotation in the Sun, showing differential rotation in the outer convective region and almost uniform rotation in the central radiative region. The transition between these regions is called the tachocline.

"Until the advent of helioseismology, the study of wave oscillations in the Sun, very little was known about the internal rotation of the Sun. The differential profile of the surface was thought to extend into the solar interior as rotating cylinders of constant angular momentum. Through helioseismology this is now known not to be the case and the rotation profile of the Sun has been found. On the surface the Sun rotates slowly at the poles and quickly at the equator. This profile extends on roughly radial lines through the solar convection zone to the interior. At the tachocline the rotation abruptly changes to solid body rotation in the solar radiation zone. "

Electromagnetics
Albert "Einstein believed that there might be an asymmetry between the charges of the electron and proton so that the Earth's magnetic field would be produced by the entire Earth."

"In the case of the Earth, the magnetic field is induced and constantly maintained by the convection of liquid iron in the outer core. A requirement for the induction of field is a rotating fluid. Rotation in the outer core is supplied by the Coriolis effect caused by the rotation of the Earth. The Coriolis force tends to organize fluid motions and electric currents into columns [...] aligned with the rotation axis. Induction or creation of magnetic field is described by the induction equation:

 * $$\frac{\partial \mathbf{B}}{\partial t} = \eta \nabla^2 \mathbf{B} + \nabla \times (\mathbf{u} \times \mathbf{B}),$$

where u is a velocity, B is the magnetic field, t is time, and $$\eta=1/\sigma\mu$$ is the magnetic diffusivity with $$\sigma$$ electrical conductivity and $$\mu$$ permeability. The ratio of the second term on the right hand side to the first term gives the Magnetic Reynolds number, a dimensionless ratio of advection of a magnetic field to diffusion."

Magnetohydrodynamic dynamos
A "magnetohydrodynamic dynamo in a rapidly rotating spherical shell [is modeled with] changing electrical resistivity. When resistivity is sufficiently small, total magnetic energy can grow more than ten times larger than total kinetic energy of convection motion which is driven by an unlimited external energy source. When resistivity is relatively large and magnetic energy is comparable or smaller than kinetic energy, the convection motion maintains its well‐organized structure. [...] when resistivity is small and magnetic energy becomes larger than kinetic energy, the well‐organized convection motion is highly irregular. The magnetic field is organized in two ways. One is the concentration of component parallel to the rotation axis and the other is the concentration of perpendicular component. The parallel component tends to be confined inside anticyclonic columnar convection cells, while the perpendicular component is confined outside convection cells."

X-rays
"A "saturation" limit in stellar activity marked by a maximum in X-ray surface flux is observed in the very rapidly rotating stars such as the young Pleiades dK stars and the very active RS CVn systems [... This] activity saturation cannot in itself explain why the active stars do not appear to show much long-term variability because such stars as the Hyades dwarfs [...] are not at the saturation limit. The lack of substantial long-term variability must set in at activity levels below the saturation level, or, in evolutionary terms, it must persist well beyond the point at which a young, rapidly rotating star spins down to below its saturated state."

Visuals
A shell dynamo is a dynamo of near-surface circulation (the shell) with the resulting shear-induced conversion of mechanical energy into electrical energy and associated magnetic fields. For example, meridional flow that is poleward near the surface of a photosphere is complemented with an equatorward super-flow deeper in the photosphere as a countercurrent.

Regarding "the stability of the dynamical behaviour of axisymmetric α2ω dynamo models in rotating spherical shells as well as spheres [...] the spherical dynamo models are more stable in the following senses:
 * 1) [minimize] chaotic behaviour and
 * 2) are robust with respect to changes in the functional form of α. [Yet]
 * 3) are capable of producing chaotic behaviour for certain ranges of parameter values and
 * 4) possess, in the combined "space" of parameters and boundary conditions, regions of complicated behaviours, [...] regimes in which small changes in either the dynamo parameters or the boundary conditions can drastically change the qualitative behaviour of the model."

For an axisymmetric mean field dynamo, the "standard mean field dynamo equation [...] is of the form"

 * $$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{u} \times \mathbf{B} + \alpha \mathbf{B}) - \nabla \times (\eta_t \nabla \times \mathbf{B}),$$

where u is the mean velocity, $$\mathbf{B}$$ is the mean magnetic field, t is time. "The quantities α (giving rise to the α effect) and ηt (the turbulent magnetic diffusivity) appear in the process of parameterization of the second order correlations 〈u' x B'〉 between the fluctuations u' and B' by"

 * $$\langle \mathbf{u}' \times \mathbf{B}' \rangle = \alpha \mathbf{B} - \eta_t \nabla \times \mathbf{B}.$$

A "functional form for α [may be] given by"

 * $$\alpha = \frac{\alpha_0 cos(\theta)}{1 + \mathbf{B}^2},$$

where "the exact functional (and in general precise tensorial) forms of α, and in principle also of ηt, are complicated and not well understood in the solar and stellar settings."

The images at the top right of this lecture show the magnetic field lines of the poloidal field Bp and contours of the toroidal field Bt for a solution showing temporal chaos in an axisymmetric spherical shell dynamo.

Shell dynamo models "for the solar convection zone with positive α - effect in the northern hemisphere [include] a meridional circulation which is directed equatorward at the bottom and poleward at the top of the convection zone [may have two] different rotation patterns"
 * 1) a simple variation of the rotation rate with depth and
 * 2) the rotation law as derived by helioseismology.

"Dynamos in differentially rotating stars differ from those in stars that rotate rigidly because rotational shear generates a strong toroidal field and enforces an axisymmetric field geometry."

"Helioseismology gives us detailed information about the internal rotation profiles apart from the well-known surface phenomenon of the equatorial acceleration of δΩ ≃ 0.06 per day. One finds super-rotation beneath the equator and sub-rotation beneath the poles. Close to the equatorial plane, the rotation rate is essentially constant on cylindrical surfaces, while close to the poles the surfaces of isorotation are rather disk-shaped."

The "differential rotation [and] the meridional flow [...] influence the mean-field dynamo [...] This influence can be expected to be just a modification if its characteristic time-scale τdrift exceeds the (half-)cycle time τcyc of about 11 yr." Bold added.

If "the drift is poleward at the bottom of the convection zone, the dynamo might fail to maintain a solar-type magnetic cycle. On the other hand, an equatorward directed meridional flow can produce the observed solar-type butterfly diagram even in the case that a circulation-free dynamo would produce an antisolar-type butterfly diagram. [The] phase relation between the toroidal and the radial field components [is] negative in the solar photosphere [...] It is almost impossible to explain this observation by virtue of an αΩ-dynamo and a rotation law with positive shear [This] situation is changed if meridional circulation is taken into account.".

"For sufficiently small eddy magnetic diffusivity [...] the meridional flow [is] very powerful to change the properties of α2Ω-dynamos working in the convection zone rather than in the solar tachocline. For positive but uniform ∂Ω/∂r [...] the migration of the toroidal magnetic activity belts is strongly correlated with the amplitude of the circulation. If the circulation is equatorward at the bottom of the convection zone and its amplitude is sufficient then it can indeed turn a poleward drift into an equatorward drift [...] The resulting cycle times are always between 10 and 100 years [...] Another striking property of the circulation-dominated models is that they produce the observed opposite signs of the magnetic field components".

If "the real internal rotation law from helioseismology is applied. The large negative slope ∂Ω/∂r in the polar regions unavoidably produces strong toroidal field belts at high latitudes. For these models stationary solutions are found much more frequently than those with cyclic behavior. An equatorward migration of the toroidal field belts (ca. 1 m/s at the bottom of the convective zone, [...] is only achieved in a very narrow range of flow amplitudes. That solution shows the correct cycle time and also the negativity of [the toroidal and poloidal fields]."

Oranges
"The Hyades dwarfs [...] do possess radiative cores, and based on the solar analogy, are presumably capable of generating solar-like large-scale fields."

Reds
Small-scale "magnetic fields can be generated in the solar convection zone, for example, by a turbulently driven dynamo. This "turbulent field" does not require rotation, although the generation rate increases with increasing rotation. [The] total energy stored in the turbulent field could be higher than that in the large-scale field. [... Low-mass] stars, which under conventioal dynamo theory are probably unable to generate a large-scale field due to the absence of the radiative core, should only have turbulent fields. The turbulent field theory (or the "distributive" dynamo [...]) is also particularly appealing since it might explain two important observational clues:"
 * 1) "the apparent lack of a change in coronal heating efficiency going from stars which have radiative cores to the fully convective M dwarfs [...]; and
 * 2) an absence of long-term stellar cyclic X-ray variability by more than a factor of ~2 in all of the Hyades late-type dwarfs (including those with radiative cores) uncovered in the study of Einstein and ROSAT observations [...]."

"Further support for the turbulently driven dynamo comes from the very recent ROSAT study of M dwarfs [...], the modeling of which suggests that the coronal geometry for low-mass dwarfs is dominated by relative compact loop configurations, and that the emission contribution of structures with large-scale dipolar or quadrapolar geometry is negligible."

"The ROSAT, EUVE, and Ca II observations could all be explained if turbulent magnetic activity dominated over any large-scale field activity at the rotation rates typical of active dwarfs."

The "most active M dwarfs should not exhibit cyclic activity."

The "very low mass, fully convective stars do not have radiative cores, and the large-scale field dynamo does not operate. The magnetic activity of these stars is generated by a turbulently driven dynamo process [...] More massive stars with radiative cores generate solar-like large-scale magnetic fields through the operation of an αω type shell dynamo. [...] they also generate small-scale magnetic fields through the operation of a turbulently driven dynamo. In stars with radiative cores which have relatively high rotation rates, such as the fairly young Hyades dwarfs [...] the turbulent dynamo dominates, and well-defined activity cycles are not observed. As stars evolve and spin down from young, rapid rotators, their magnetic activity changes from a regime in which the turbulent dynamo dominates to one characterized by a solar-like large-scale field shell dynamo."

Liquid objects
"Tidal forces between celestial orbiting bodies causes friction that heats up the interiors of these orbiting bodies. This is known as tidal heating, and it helps create the liquid interior criteria, providing that this interior is conductive, that is required to produce a dynamo."

Nitrogens
There "exists a class of stars that are slowly rotating (v < 60 km s−1) but exhibit significant nitrogen enrichment. It was suggested that these stars are, or once were, magnetic stars. [...] those stars [...] with nitrogen enrichment 6.8 < log10[N/H] < 7.1 and 0 < v/km s−1 < 150 cannot easily be categorized into either group of stars. They may be low–mass, fast rotators that have been partially spun down by magnetic braking, low–mass stars that are born with slow rotation or high–mass stars that are born with slow rotation."

"The [VLT–FLAMES] survey [of massive stars] observed two distinct populations of stars. The first shows increasing nitrogen enrichment with rotation rate, the second is a class of slow–rotating stars that exhibit unusually high nitrogen abundances compared to the rest of the population. This distribution of stars is well reproduced by the magnetic model."

Calciums
The "evidence from long-term Ca II emission core monitoring [...] shows that smooth solar-like cyclic variability is not generally observed in young (less than 1 Gyr) active dwarfs [...]. On shorter timescales, [...] no differences in EUV luminosity more than a factor of 2 in a sample of active stars when comparing EUVE survey fluxes to those derived from the ROSAT Wide Field Camera survey performed 2 yr earlier."

Sun
In the model shown at right the Sun and regions around it are labeled.

The Sun which is a rotating body may become a magnet due to a dynamo.

"The core of the Sun is considered to extend from the center to about 0.2 to 0.25 solar radius. It is the hottest part of the Sun and of the Solar System. It has a density of up to 150 g/cm³ (150 times the density of liquid water) and a temperature of close to 15,000,000 kelvin [15 MK] ... The core is made of hot, dense gas in the plasmic state. The core, inside 0.24 solar radius, generates 99% of the fusion power of the Sun." It is in the core region that solar neutrinos may be produced.

"The radiation zone or radiative zone is a layer of a star's interior where energy is primarily transported toward the exterior by means of radiative diffusion, rather than by convection. Energy travels through the radiation zone in the form of electromagnetic radiation as photons. Within the Sun, the radiation zone is located in the intermediate zone between the solar core at .2 of the Sun's radius and the outer convection zone at .71 of the Sun's radius. "

"Matter in a radiation zone is so dense that photons can travel only a short distance before they are absorbed or scattered by another particle, gradually shifting to longer wavelength as they do so. For this reason, it takes an average of 171,000 years for gamma rays from the core of the Sun to leave the radiation zone. Over this range, the temperature of the plasma drops from 15 million K near the core down to 1.5 million K at the base of the convection zone. "

"Within a radiative zone, the temperature gradient—the change in temperature (T) as a function of radius (r)—is given by:

 * $$\frac{\text{d}T(r)}{\text{d}r}\ =\ -\frac{3 \kappa(r) \rho(r) L(r)}{(4 \pi r^2)(16 \sigma) T^3(r)}$$

where κ(r) is the opacity, ρ(r) is the matter density, L(r) is the luminosity, and σ is the Stefan–Boltzmann constant. Hence the opacity (κ) and radiation flux (L) within a given layer of a star are important factors in determining how effective radiative diffusion is at transporting energy. A high opacity or high luminosity can cause a high temperature gradient, which results from a slow flow of energy. Those layers where convection is more effective than radiative diffusion at transporting energy, thereby creating a lower temperature gradient, will become convection zones. "

"The convection zone of a star is the range of radii in which energy is transported primarily by convection. ... Stellar convection consists of mass movement of plasma within the star which usually forms a circular convection current with the heated plasma ascending and the cooled plasma descending." This is the granular zone in the outer layer of a star.

"The solar dynamo is the physical process that generates the Sun's magnetic field. The Sun is permeated by an overall dipole magnetic field, as are many other celestial bodies such as the Earth. The dipole field is produced by a circular electric current flowing deep within the star, following Ampère's law. The current is produced by shear (stretching of material) between different parts of the Sun that rotate at different rates, and the fact that the Sun itself is a very good electrical conductor (and therefore governed by the laws of magnetohydrodynamics)."

"The tachocline ... is a thin layer of the solar interior, straddling the convection zone and the radiative interior. It is widely believed that a toroidal magnetic field of at least 105 G permeates this layer ... The tachocline naturally divides into two sublayers: an inner "radiative" layer and an outer "overshoot" layer. By current estimates, the radiative layer is twice as thick as the overshoot layer."

The "radiative" layer of the tachocline may be a source for a radiative dynamo.

Mercury
"This plot [at right] shows the measured magnitude of the magnetic field of Mercury as MESSENGER executed its first flyby of that planet. MESSENGER's Magnetometer (MAG) provided definitive identification of all boundaries of the Mercury magnetosphere system, consistent with the observations made with the Fast Imaging Plasma Spectrometer (FIPS) on the Energetic Particle and Plasma Spectrometer (EPPS) instrument, and revealed a much more quiescent system than was seen during the first Mariner 10 flyby. This state of the system was also consistent with the absence of energetic particles as documented by the Energetic Particle Spectrometer (EPS) portion of MESSENGER's EPPS instrument. Mercury lacks radiations belts similar to the Van Allen belts at the Earth discovered by James Van Allen with a simple particle experiment on Explorer I launched 50 years ago."

Mercury, despite its small size, has a magnetic field [see image and plot at right], because it has a conductive liquid core created by its iron composition and friction resulting from its highly elliptical orbit.

"Mercury’s core, already suspected to occupy a greater fraction of the planet's interior than do the cores of Earth, Venus, or Mars, is even larger than anticipated."

The "elevation ranges on Mercury are much smaller than on Mars or the Moon and documents evidence that there have been large-scale changes to Mercury’s topography since the earliest phases of the planet’s geological history."

“From Mercury’s extraordinarily dynamic magnetosphere and exosphere to the unexpectedly volatile-rich composition of its surface and interior, our inner planetary neighbor is now seen to be very different from what we imagined just a few years ago."

"MESSENGER’s radio tracking has allowed the scientific team to develop the first precise model of Mercury’s gravity field which, when combined with topographic data and the planet’s spin state, sheds light on the planet’s internal structure, the thickness of its crust, the size and state of its core, and its tectonic and thermal history."

"Mercury’s core occupies a large fraction of the planet, about 85% of the planetary radius, even larger than previous estimates. Because of the planet’s small size, at one time many scientists thought the interior should have cooled to the point that the core would be solid. However, subtle dynamical motions measured from Earth-based radar, combined with MESSENGER’s newly measured parameters of the gravity field and the characteristics of Mercury’s internal magnetic field that signify an active core dynamo, indicate that the planet’s core is at least partially liquid."

"Mercury’s core is different from any other planetary core in the Solar System. Earth has a metallic, liquid outer core sitting above a solid inner core. Mercury appears to have a solid silicate crust and mantle overlying a solid, iron sulfide outer core layer, a deeper liquid core layer, and possibly a solid inner core. These results have implications for how Mercury’s magnetic field is generated and for understanding how the planet evolved thermally."

"Energetic and magnetostrophic balance arguments show that a dynamo source for Mercury's observed magnetic field is problematic if one expects an Earth-like partitioning of toroidal and poloidal fields."

But, a thin shell dynamo model is consistent with the observed weak magnetic field.

From "the ratio of the dipole field at the core-mantle boundary to the toroidal field in the core for various shell thicknesses and Rayleigh numbers[...] some thin shell dynamos can produce magnetic fields with Mercury-like dipolar field intensities. In these dynamos, the toroidal field is produced more efficiently through differential rotation than the poloidal field is produced through upwellings interacting with the toroidal field. The poloidal field is also dominated by smaller-scale structure which was not observable by the Mariner 10 mission, compared to the dipole field."

Venus
"Venus and the Earth have similar radii and estimated bulk compositions, and both possess an iron core that is at least partially liquid. However, despite these similarities, Venus lacks an appreciable dipolar magnetic field."

This "absence is due to Venus’s also lacking plate tectonics for the past 0.5 b.y. (1 b.y.=109 yr). The generation of a global magnetic field requires core convection, which in turn requires extraction of heat from the core into the overlying mantle. Plate tectonics cools the Earth’s mantle; on the basis of elastic thickness estimates and convection models, [...] the mantle temperature on Venus is currently increasing. This heating will reduce the heat flux out of the core to zero over ~1 b.y., halting core convection and magnetic field generation. If plate tectonics was operating on Venus prior to ca. 0.5 Ga, a magnetic field may also have existed. On Earth, the geodynamo may be a consequence of plate tectonics; this connection between near-surface processes and core magnetism may also be relevant to the generation of magnetic fields on Mars, Mercury and Ganymede."

The lack of an appreciable Earth-like dipolar magnetic field "cannot be explained by the planet's slow rotation".

In "the absence of plate tectonics, the mantle on Venus cannot cool rapidly enough to drive core convection and a geodynamo."

"Planetary magnetic fields are produced by motion in a conductor, usually the planet’s iron core. Such motion may be due to either thermal convection or compositional convection, driven by core solidification".

"The maximum heat flux that can be extracted from the core without thermal convection is given by"

 * $$F_c = k \alpha g T/C_p,$$

"where k and α are the thermal conductivity and expansivity, g is the acceleration due to gravity, T is the core temperature, and Cp is the specific hear capacity. [...] Fc is in the range 11-30 mW·m-2. Thermal convection will cease if the heat being extracted from the core is less than Fc; in the absence of core solidification, the geodynamo will halt. Compositional convection may continue [...], but will certainly halt if the heat flux out of the core drops to zero or below (i.e., the core starts heating up). The rate at which the core loses heat is controlled by the temperature difference between core and mantle and, thus, on the rate at which the mantle is cooling".

Earth
The illustration at right is of the dynamo mechanism that creates the Earth's magnetic field: convection currents of magma in the Earth's outer core, driven by heat flow from the inner core, organized into rolls by the Coriolis force, creates circulating electric currents, which generate the magnetic field.

As "the result of radioactive heating and chemical differentiation, the Earth's outer core is in a state of turbulent convection. This sets up a process that is a bit like a naturally occurring electrical generator, where the convective kinetic energy is converted to electrical and magnetic energy. Basically, the motion of the electrically conducting iron in the presence of the Earth's magnetic field induces electric currents. Those electric currents generate their own magnetic field, and as the result of this internal feedback, the process is self-sustaining so long as there is an energy source sufficient to maintain convection."

The Earth's "magnetic field resulted from electric currents induced in the fluid outer core of the Earth."

The Earth is magnetic and a dynamo may be generating the field.

"The use of more realistic parameters in numerical geodynamo simulations tends to generate less Earth-like magnetic fields. This paradox could be resolved by considering uniform heat flux instead of uniform temperature at the core's surface."

"Electrical currents produced by motions in the Earth's fluid outer core are thought to be responsible for the planet's magnetic field."

"The Earth's main magnetic field is thought to be generated by motions in the planet's fluid outer core, which lead to an effect similar to that of a dynamo. Recent high-resolution numerical simulations produce only a non-dipolar or a dipolar but comparatively weak magnetic field unlike that of the Earth. Older models that did generate a strong, Earth-like field needed to use unrealistically high viscosities for the core fluid. Common to most of the models is the assumption of a laterally uniform core-surface temperature."

A "low-viscosity geodynamo model [used] to evaluate the effect of a different and more realistic boundary condition-a uniform heat flux at the surface of the core-on the simulation of an Earth-like magnetic field [shows] that when the surface temperature is laterally uniform, only a weak magnetic field is generated because planetary-scale fluid circulations are suppressed. In contrast, a laterally uniform heat flux at the core's surface leads to large-scale convective flows, and a comparatively strong dipole-type magnetic field."

The "dipole, which comprises much of the Earth's magnetic field and is misaligned along the rotation axis by 11.3 degrees, was caused by permanent magnetization of the materials in the earth. This means that dynamo theory was originally used to explain the Sun's magnetic field in its relationship with that of the Earth."

Moon
"The "geodynamo" that generates Earth's magnetic field is powered by heat from the inner core, which drives complex fluid motions in the molten iron of the outer core. But the moon is too small to support that type of dynamo."

"This is a very different way of powering a dynamo that involves physical stirring, like stirring a bowl with a giant spoon."

"Early in its history, the moon orbited the Earth at a much closer distance than it does today, and it continues to gradually recede from the Earth. At close distances, tidal interactions between the Earth and the moon caused the moon's mantle to rotate slightly differently than the core. This differential motion of the mantle relative to the core stirred the liquid core, creating fluid motions that, in theory, could give rise to a magnetic dynamo."

"The moon wobbles a bit as it spins--that's called precession--but the core is liquid, and it doesn't do exactly the same precession. So the mantle is moving back and forth across the core, and that stirs up the core."

A "lunar dynamo could have operated in this way for at least a billion years. Eventually, however, it would have stopped working as the moon got farther away from the Earth."

"The further out the moon moves, the slower the stirring, and at a certain point the lunar dynamo shuts off."

"Rocks can become magnetized from the shock of an impact, a mechanism some scientists have proposed to explain the magnetization of lunar samples. But recent paleomagnetic analyses of moon rocks, as well as orbital measurements of the magnetization of the lunar crust, suggest that there was a strong, long-lived magnetic field on the moon early in its history."

"One of the nice things about our model is that it explains how a lunar dynamo could have lasted for a billion years."

"It also makes predictions about how the strength of the field should have changed over the years, and that's potentially testable with enough paleomagnetic observations."

"Only certain types of fluid motions give rise to magnetic dynamos."

"We calculated the power that's available to drive the dynamo and the magnetic field strengths that could be generated. But we really need the dynamo experts to take this model to the next level of detail and see if it works."

Mars
"Mars once underwent plate tectonics, slow movement of the planet's crust, like the present-day Earth. A new map of Mars' magnetic field [at the right] made by the Mars Global Surveyor spacecraft reveals a world whose history was shaped by great crustal plates being pulled apart or smashed together."

Initial "observations [in 1999], also done with the Mars Global Surveyor’s magnetometer, covered only one region in the Southern Hemisphere. The data was taken while the spacecraft performed an aerobraking maneuver, and so came from differing heights above the crust."

"This high resolution magnetic field map, the first of its kind, covers the entire surface of Mars. The new map is based on four years of data taken in a constant orbit. Each region on the surface has been sampled many times."

“The more measurements we obtain, the more accuracy, and spatial resolution, we achieve."

"This map lends support to and expands on the 1999 results."

“Where the earlier data showed a "striping" of the magnetic field in one region, the new map finds striping elsewhere. More importantly, the new map shows evidence of features, transform faults, that are a "tell-tale" of plate tectonics on Earth."

On "Mars the direction of the magnetic field changes dramatically from place to place."

Similar "stripes in the crustal magnetic field on Earth. Stripes form whenever two plates are being pushed apart by molten rock coming up from the mantle, such as along the Mid-Atlantic Ridge. As the plate spreads and cools, it becomes magnetized in the direction of the Earth’s strong global field. Since Earth’s global field changes direction a few times every million years, on average, a flow that cools in one period will be magnetized in a different direction than a later flow. As the new crust is pushed out and away from the ridge, stripes of alternating magnetic fields aligned with the ridge axis develop. Transform faults, identified by “shifts” in the magnetic pattern, occur only in association with spreading centers."

"Plate tectonics provides a unifying framework to explain several Martian features. First, there is the magnetic pattern itself. Second, the Tharsis volcanoes lie along a straight line. These formations could have formed from the motion of a crustal plate over a fixed “hotspot” in the mantle below, just as the Hawaiian islands on Earth are thought to have formed. Third, the Valles Marineris, a large canyon six times as long as the Grand Canyon and eight times as deep, looks just like a rift formed on Earth by a plate being pulled apart. Even more, it is oriented just as one would expect from plate motions implied by the magnetic map."

Plate "tectonics does give us a consistent explanation of some of the most prominent features on Mars.”

Jupiter
"The interior of Jupiter is the seat of a strong dynamo that produces a surface magnetic field in the equatorial region with an intensity of ~ 4 Gauss. This strong magnetic field and Jupiter’s fast rotation (rotation period ~ 9 h 55 min) create a unique magnetosphere in the solar system which is known for its immense size (average subsolar magnetopause distance 45-100 RJ where 1 RJ = 71492 km is the radius of Jupiter) and fast rotation [...]. Jupiter’s magnetosphere differs from most other magnetospheres in the fact that it derives much of its plasma internally from Jupiter’s moon Io. The heavy plasma, consisting principally of various charge states of S and O, inflates the magnetosphere from the combined actions of centrifugal force and thermal pressure."

In "the absence of an internal heavy plasma, the dipole field would balance the average dynamic pressure of the solar wind (0.08 nPa) at a distance of ~ 42 RJ in the subsolar region [...] the observed average magnetopause location of ~ 75 RJ [...] The heavy plasma is also responsible for generating an azimuthal current exceeding 160 MA in the equatorial region of Jupiter’s magnetosphere where it is confined to a thin current sheet (half thickness ~ 2 RJ in the dawn sector)."

"The energization of plasma by various electrical fields as it diffuses inwards is responsible for the creation of radiation belts in the inner magnetosphere of Jupiter. It is believed that the radial diffusion is driven by the ionospheric dynamo fields produced by winds in Jupiter’s atmosphere"

"In situ and remote observations of Io and its surroundings from Voyager showed that Io is the main source of plasma in Jupiter’s magnetosphere [...] "

"It is estimated that upward of 6 × 1029 amu/s (~ 1 ton/s) of plasma mass is added to the magnetosphere by Io. The picked-up plasma consists mostly of various charged states of S and O and populates a torus region extending from a radial distance of ~ 5.2 RJ to ~ 10 RJ."

"The next most important source of plasma in Jupiter’s magnetosphere is the solar wind whose source strength can be estimated by a consideration of the solar wind mass flux incident on Jupiter’s magnetopause and the fractional amount that makes it into the magnetosphere (< 1%). Such a calculation suggests that the solar wind source strength is < 100 kg/s (Hill et al. 1983) considerably lower than the Io source. Nevertheless, the number density of protons (as opposed to the mass density) may be comparable to the iogenic plasma number density in the middle and outer magnetospheres where the solar wind may be able to gain access to the magnetosphere."

"The escape of ions (mainly H+ and H2+ ) from the ionosphere of Jupiter provides the next significant source of plasma in Jupiter’s magnetosphere. The ionospheric plasma escapes along field lines when the gravity of Jupiter is not able to contain the hot plasma (~ 10 eV and above). The escape however is not uniform and depends on the local photoelectron density, the temperature variations of the ionosphere with the solar zenith angle, other factors such as the auroral precipitation of ions and electrons and the ionospheric heating from Pedersen currents. In situ measurements show that in Io’s torus, protons contribute to less than 20% of total ion number density and constitute < 1% of mass suggesting that the ionosphere is not a major source of plasma in Jupiter’s magnetosphere. [The] ionospheric source strength [is] in the range of ~ 20 kg/s."

The "surface sputtering of the three icy satellites by jovian plasma provides the last significant source of plasma in Jupiter’s magnetosphere. Because the icy moons lack extended atmospheres and the fluxes of the incident plasma are low at the locations of these moons, the total pickup of plasma from these satellites is estimated to be less than 20 kg/s based on the plasma sputtering rates provided".

"Other minor constituents found in the torus [...] were Na+ (with an abundance of < 5%) and molecular ions SO+ and SO2+ (both with abundances of < 1% of the total). The average mass of a torus ion is ~ 20 and the average fractional charge on an ion is ~ 1.2 [...]. The bulk velocity of the plasma was found to be ~ 75 km/s, close to the corotational value."

Io
Io has enough tidal heating to liquify its inner core, even if the moon is not conductive enough to support a dynamo.

The "orbital and gravitational relationships between Io, its sister moons Europa and Ganymede, and Jupiter cause massive, rapid flexing of its rocky crust. These tidal flexures generate tremendous heat within Io’s interior, which is released through the many surface volcanoes observed."

“Io has no impact craters; it is the only object in the Solar System where we have not seen any impact craters, testifying to Io’s very active volcanic resurfacing.”

"Io is extremely active, with literally hundreds of volcanic sources on its surface. Interestingly, although Io is so volcanically active, more than 25 times more volcanically active than Earth, most of the long-term surface changes resulting from volcanism are restricted to less than 15 percent of the surface, mostly in the form of changes in lava flow fields or within paterae."

“Our mapping has determined that most of the active hot spots occur in paterae, which cover less than 3 percent of Io’s surface. Lava flow fields cover approximately 28 percent of the surface, but contain only 31 percent of hot spots.”

“Understanding the geographical distribution of these features and hot spots, as identified through this map, are enabling better models of Io’s interior processes to be developed.”

Enceladus
With "a diameter only slightly more than 300 miles, Enceladus just doesn’t have the bulk needed for its interior to stay warm enough to maintain liquid water underground."

"With temperatures around 324 degrees below zero Fahrenheit, the surface of Enceladus is indeed frozen. However, in 2005 NASA's Cassini spacecraft discovered a giant plume of water gushing from cracks in the surface over the moon's south pole, indicating that there was a reservoir of water beneath the ice. Analysis of the plume by Cassini revealed that the water is salty, indicating the reservoir is large, perhaps even a global subsurface ocean. Scientists estimate from the Cassini data that the south polar heating is equivalent to a continuous release of about 13 billion watts of energy."

"To explain this mysterious warmth, some scientists invoke radiation coupled with tidal heating. As it formed, Enceladus (like all solar system objects) incorporated matter from the cloud of gas and dust left over from our sun’s formation. In the outer solar system, as Enceladus formed it grew as ice and rock coalesced. If Enceladus was able to gather greater amounts of rock, which contained radioactive elements, enough heat could have been generated by the decay of the radioactive elements in its interior to melt the body."

"Enceladus' orbit around Saturn is slightly oval-shaped. As it travels around Saturn, Enceladus moves closer in and then farther away. When Enceladus is closer to Saturn, it feels a stronger gravitational pull from the planet than when it is farther away. Like gently squeezing a rubber ball slightly deforms its shape, the fluctuating gravitational tug on Enceladus causes it to flex slightly. The flexing, called gravitational tidal forcing, generates heat from friction deep within Enceladus."

"The gravitational tides also produce stress that cracks the surface ice in certain regions, like the south pole, and may be reworking those cracks daily. Tidal stress can pull these cracks open and closed while shearing them back and forth. As they open and close, the sides of the south polar cracks move as much as a few feet, and they slide against each other by up to a few feet as well. This movement also generates friction, which (like vigorously rubbing your hands together) releases extra heat at the surface at locations that should be predictable with our understanding of tidal stress."

"To test the tidal heating theory, scientists with the Cassini team created a map of the gravitational tidal stress on the moon's icy crust and compared it to a map of the warm zones created using Cassini's composite infrared spectrometer instrument (CIRS). Assuming the greatest stress is where the most friction occurs, and therefore where the most heat is released, areas with the most stress should overlap the warmest zones on the CIRS map."

"However, they don't exactly match."

"For example, in the fissure called the Damascus Sulcus, the area experiencing the greatest amount of shearing is about 50 kilometers (about 31 miles) from the zone of greatest heat."

"Enceladus' wobble, technically called "libration," is barely noticeable."

"Cassini observations have ruled out a wobble greater than about 2 degrees with respect to Enceladus' uniform rotation rate."

A "computer simulation [...] made maps of the surface stress on Enceladus for various wobbles, and found a range where the areas of greatest stress line up better with the observed warmest zones."

"Depending on whether the wobble moves with or against the movement of Saturn in Enceladus' sky, a wobble ranging from 2 degrees down to 0.75 degrees produces the best fit to the observed warmest zones,"

"The wobble also helps with the heating conundrum by generating about five times more heat in Enceladus’ interior than tidal stress alone, and the extra heat makes it likely that Enceladus' ocean could be long-lived, according to Hurford. This is significant in the search for life, because life requires a stable environment to develop."

"The wobble is probably caused by Enceladus' uneven shape."

"Enceladus is not completely spherical, so as it moves in its orbit, the pull of Saturn's gravity generates a net torque that forces the moon to wobble."

"Enceladus' orbit is kept oval-shaped, maintaining the tidal stress, because of the gravitational tug from a neighboring larger moon Dione. Dione is farther away from Saturn than Enceladus, so it takes longer to complete its orbit. For every orbit Dione completes, Enceladus finishes two orbits, producing a regular alignment that pulls Enceladus' orbit into an oval shape."

Uranus
"The discovery of [Uranus]'s non-dipolar, non-axisymmetric magnetic [field at the right] destroyed the picture-established by Earth, Jupiter and Saturn-that planetary magnetic fields are dominated by axial dipoles."

"Planetary magnetic fields are generated by complex fluid motions in electrically conducting regions of the planets (a process known as dynamo action), and so are intimately linked to the structure and evolution of planetary interiors."

Three-dimensional "numerical dynamo simulations [...] model the dynamo source region as a convecting thin shell surrounding a stably stratified fluid interior."

This "convective-region geometry produces magnetic fields similar in morphology to [that] of Uranus [The field is] non-dipolar and non-axisymmetric, and [results] from a combination of the stable fluid's response to electromagnetic stress and the small length scales imposed by the thin shell."

The planet had "a strong planetary magnetic field of Uranus and an associated magnetosphere and fully developed bipolar magnetotail [and a] detached bow shock wave [which] was observed upstream at 23.7 Uranus radii (1 RU = 25,600 km) and the magnetopause boundary at 18.0 RU. [The] maximum magnetic field of 413 nanotesla was observed at 4.19 RU [The] planetary magnetic field is well represented by that of a dipole offset from the center of the planet by 0.3 RU. The angle between Uranus' angular momentum vector and the dipole moment vector has the surprisingly large value of 60 degrees. [The] field of Uranus may be described as that of an oblique rotator. The dipole moment of 0.23 gauss R3U, combined with the large spatial offset, leads to minimum and maximum magnetic fields on the surface of the planet of approximately 0.1-1.1 gauss. The rotation period of the magnetic field and [that] of the interior of the planet is estimated to be 17.29±0.10 [hr]."

Neptune
"The discovery of [Neptune]'s non-dipolar, non-axisymmetric magnetic [field contributes to destroying] the picture-established by Earth, Jupiter and Saturn-that planetary magnetic fields are dominated by axial dipoles."

The "convective-region geometry produces magnetic fields similar in morphology to [that of] Neptune. [The field is] non-dipolar and non-axisymmetric, and [results] from a combination of the stable fluid's response to electromagnetic stress and the small length scales imposed by the thin shell."

The "rotation axis of [Neptune] is inclined by only 29° to the orbital plane [...] The magnetic dipole axis of Neptune is tilted at an angle of 47° to the spin axis of the planet. The extrapolated near-equatorial surface field is 1.42 µT, corresponding to a magnetic moment (equatorial surface field times radius cubed) of 2.16 x 1017 Tm3 close to 27 times greater than the terrestrial magnetic moment. The quadrupole moment if Neptune is quite large and makes a greater contribution to the surface magnetic field than at any other planet. The most forward portion of the magnetopause is estimated to lie on average at about 26 Neptunian radii in front of the planet, and of the bow shock at about 34 Neptune radii."

Brown dwarfs
"Stars with masses M > 0.3 M⊙ have an outer convective zone and an interior radiative region that need not be rotating at the same rate. A poloidal magnetic field in the convective layers will be stretched and amplified into strong toroidal fields when it is dragged by convective overshoot ... into the radial shear in rotation that resides at the boundary (in and near the so-called "tachocline" ... For less massive stars and young brown dwarfs, the energy is transported throughout the star by convection; no radiative core is present. For this reason, it has been supposed that the activity and its dependence on rotation might change near the spectral type where the radiative layer disappears (about M5.5)"

Giant stars
Notation: let the symbol AGB indicate an asymptotic giant branch star with a hydrogen-exhausted core.

Notation: let the symbol E-AGB indicate an AGB star with a hydrogen-exhausted core.

For a 7 M⊙ AGB model sequence, "[o]n the E-AGB, the convective envelope appears clearly separated from the stellar core by a radiative layer ... Density and temperature drop significantly within this layer". "As evolution proceeds luminosity and radiation pressure increase ... The base of the convective envelope moves inwards into deeper and hotter parts of the interior until nuclear reactions become important ... just before the first thermal pulse ..., the radiative "buffer" layer disappears, and the convection cuts into the hydrogen-burning shell. ... high lithium abundances ... in ... oxygen rich, luminous (Mbol = -6.2... -6.8) AGB stars [are produced at the base of the convective envelope which] has a base temperature of 75 ˑ 106K, sufficient to reduce the duration of the Li-rich phase well below 104yrs".

Hypotheses

 * 1) The magnetic field of the solar surface is being generated by direct electron incidence.

title: Stars/Radiative dynamo


text:
Statistical Analysis

Purpose of Course
In this course, we will look at the properties behind the basic concepts of probability and statistics and focus on applications of statistical knowledge. We will learn how statistics and probability work together. The subject of statistics involves the study of methods for collecting, summarizing, and interpreting data. Statistics formalizes the process of making decisions—and this course is designed to help you cultivate statistic literacy so that you can use this knowledge to make better decisions. Note that this course has applications in sciences, economics, computer science, finance, psychology, sociology, criminology, and many other fields. Every day, we read articles and reports in print or online. After finishing this course, you should be comfortable asking yourself whether the articles make sense. You will be able to extract information from the articles and display that information effectively. You will also be able to understand the basics of how to draw statistical conclusions. This course will begin with descriptive statistics and the foundation of statistics. You will then learn about probability and random distributions, the latter of which enables us to work with several aspects of random events and their applications. Finally, we will examine a number of ways to investigate the relationships between various characteristics of data. By the end of this course, you should have a grasp on what statistics represent, how to use them to organize and display data, and how to test the data to make effective conclusions.

This course was originally adapted for Wikiversity from the MA121/ECON104 Course Introduction to Statistics at Saylor.org

Learning Outcomes
Upon completion of the course, you will be able to:
 * Define the meaning of descriptive statistics and statistical inference.
 * Distinguish between a population and a sample.
 * Explain the purpose of measures of location, variability, and skewness.
 * Calculate probabilities.
 * Explain the difference between how probabilities are computed for discrete and continuous random variables.
 * Recognize and understand discrete probability distribution functions, in general.
 * Identify confidence intervals for means and proportions.
 * Explain how the central limit theorem applies in inference.
 * Calculate and interpret confidence intervals for one population average and one population proportion.
 * Differentiate between Type I and Type II errors.
 * Conduct and interpret hypothesis tests.
 * Compute regression equations for data.
 * Use regression equations to make predictions.
 * Conduct and interpret ANOVA (Analysis of Variance).

Content Units
The links below will direct you to the navigation page for each unit of this course.

Unit 1: Data And Descriptive Statistics

 * Unit 1 Navigation
 * Unit 1 Content

Unit 2: Probability Topics

 * Unit 2 Navigation
 * Unit 2 Content

Unit 3: Random Variables and Distributions

 * Unit 3 Navigation
 * Unit 3 Content

Unit 4: Central Limit Theorem and Confidence Intervals

 * Unit 4 Navigation
 * Unit 4 Content

Unit 5: Hypothesis Testing

 * Unit 5 Navigation
 * Unit 5 Content

Unit 6: Correlation, Regression, and ANOVA

 * Unit 6 Navigation
 * Unit 6 Content

About the Resources in This Course
This course project draws upon three main types of resources:

The first are readings and video lectures from Barbara Illowsky and Susan Dean’s Collaborative Statistics, which is available freely under a Creative Commons Attribution 2.0 Generic (CC BY 2.0) license from the following location: http://cnx.org/content/col10522/latest/

The second type of resources in this course are lectures from Khan Academy. These lectures are available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported (CC BY-NC-SA 3.0) license. Khan Academy has many lectures available from http://www.khanacademy.org/

Finally, the above resources have been woven together and organized into a format analogous to a traditional college-level course by professional consultants that work as experts within the subject area. This process was facilitated by The Saylor Foundation. Additionally, if you have worked through all of the material contained in this project, you may be interested in taking the final exam provided by Saylor.org or completing other courses available there that are not yet on Wikiversity.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Statistical Analysis


text:
Statistical Analysis/Unit 1 Navigation

This is the main navigation page for Unit 1 of the course Introduction to Statistical Analysis, developed using openly licensed materials from Saylor.org's Introduction to Statistics. Below you will find a full description of Unit 1 in general, as well as for each subunit. Follow the links within each subunit description to access particular topics, or proceed directly to the Unit 1 Content Page.

UNIT 1: DATA AND DESCRIPTIVE STATISTICS
In today's world, we have access to large volumes of data every day. The first step of data analysis is to accurately summarize all of this data, both graphically and numerically, so that we can understand what the data is saying. We see and use data in our lives every day. To be able to use and interpret the data correctly is essential to making informed decisions. In this unit, you will learn about descriptive statistics, which is used to summarize and display data. After completing this unit, you will know what you can do once you have collected data and how to present that data.

For example, suppose you are interested in buying a new mobile phone with a particular type of a camera. Suppose you are not sure about the prices of any of the phones with this feature so you log on to a website that provides you with a sample data set of prices, given your requirements. Now, looking at all the prices in the sample can sometimes be confusing. A better way to compare might be to look at the median price and the variation of prices. The median and variation are two of several ways that you can describe data. You can also ask the website to graph the data so that it is easier to see what the price distribution looks like. In this unit, you will study precisely this, i.e. you will learn numerical and graphical ways to describe and display your data. You will understand the essentials of calculating common descriptive statistics for measuring center, variability, and skewness in data. You will learn to not only calculate, but to interpret these measurements and graphs.

Remember: Descriptive statistics are, as their name suggests, “descriptive.” They do not generalize beyond the data considered. Inferential statistics can be used to generalize the findings from sample data to a broader population.

Time Advisory
Time Advisory: This unit will take you approximately 10 hours to complete.
 * Subunit 1.1: 2 hours
 * Subunit 1.2: 3 hours
 * Subunit 1.3: 5 hours

Learning Outcomes
Upon completion of this this unit, you will be able to:
 * Apply various types of sampling methods to data collection.
 * Create and interpret frequency tables.
 * Display data graphically and interpret the following types of graphs: stemplots, histograms, and boxplots.
 * Recognize, describe, and calculate the measures of location of data: quartiles and percentiles.
 * Recognize, describe, and calculate the measures of the center of mean, median, and mode.
 * Recognize, describe, and calculate the following measures of the spread of data: variance, standard deviation, and range.

Subunits
Subunit one consists of three main topics:

Introduction to Statistics
This topic will introduce you to a definition of statistics, as well as some broad concepts related to data, sampling, variation, and frequency. Learning about these general topics is essential to creating a basic understanding of the practice of statistics.

 * 1.1.1: Statistics, Probability, and Key Terms
 * 1.1.2: Data and Sampling
 * 1.1.3: Variation and Frequency

Descriptive Statistics: Displaying Data
This topic will introduce you to some of the basic methods for displaying descriptive statistical data including stem plots, line graphs, bar graphs, as well as histograms and box plots.

 * 1.2.1: Stem and Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs
 * 1.2.2: Histograms
 * 1.2.3: Box Plots

Descriptive Statistics: Measures
This topic will introduce you to some of the basic measures used in statistics such as those for central tendency (Mean, Median, and Mode), as well as for the spread of data (Variance and Standard Deviation).

 * 1.3.1: Measures of the Location of the Data
 * 1.3.2: Measures of the Center of the Data
 * 1.3.3: Skewness and the Mean, Median, and Mode
 * 1.3.4: Measures of the Spread of the Data

About the Resources in This Course
This course project draws upon three main types of resources:

The first are readings and video lectures from Barbara Illowsky and Susan Dean’s Collaborative Statistics, which is available freely under a Creative Commons Attribution 2.0 Generic (CC BY 2.0) license from the following location: http://cnx.org/content/col10522/latest/

The second type of resources in this course are lectures from Khan Academy. These lectures are available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported (CC BY-NC-SA 3.0) license. Khan Academy has many lectures available from http://www.khanacademy.org/

Finally, the above resources have been woven together and organized into a format analogous to a traditional college-level course by professional consultants that work as experts within the subject area. This process was facilitated by The Saylor Foundation. Additionally, if you have worked through all of the material contained in this project, you may be interested in taking the final exam provided by Saylor.org or completing other courses available there that are not yet on Wikiversity.

title: Statistical Analysis/Unit 1 Navigation


text:
Statistical Analysis/Unit 1 Content

Statistics, Probability, and Key Terms

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 1: Sampling and Data
 * "Section 1: Sampling and Data”
 * “Section 2: Statistics”
 * “Section 3: Probability”
 * “Section 4: Key Terms”

Instructions: Please read each of the linked sections above in their entirety.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

Data and Sampling

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: "Chapter 1: Sampling and Data”
 * “Section 5: Data”
 * “Section 6: Sampling”

Instructions: Please read each of the linked sections above in their entirety.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

Variation and Frequency

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 1: Sampling and Data”
 * “|Section 7: Variation”
 * “|Section 8: Answers and Rounding Off”
 * "|Section 9: Frequency”

Instructions: Please read each of the linked sections above in their entirety.

 * Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Video Lecture 1: Sampling and Data”

Instructions: Please view the lecture to the right in its entirety.

 * Assignment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 1 Practice: Sampling and Data”

Instructions: Please solve problems 1-6 under the section titled “Key Terms” and problems 7-12 under the section titled “Discussion Questions.” Next, click on the hyperlink titled “|Homework” and solve problems 1, 3, 5, 7, 9, 11, 16, 19, 21, 25-27. Please click on the “[ Show Solution ]” link below the problem to check your solution.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

Stem and Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 2: Descriptive Statistics”
 * “Section 1: Descriptive Statistics”
 * “Section 2: Displaying Data”
 * “Section 3: Stem and Leaf Graphs (Stemplots), Line Graphs and Bar Graphs”

Instructions: Please read each of the linked sections above in their entirety.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

Histograms

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 2: Descriptive Statistics”
 * “Section 4: Histograms”

Instructions: Please read the section above in its entirety.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

Box Plots

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 2: Descriptive Statistics”
 * “Section 5: Box Plots”

Instructions: Please read the section above in its entirety.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

Measures of the Location of the Data

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 2: Descriptive Statistics”
 * “Section 6: Measures of the Location of the Data”

Instructions: Please read the section above in its entirety.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

Measures of the Center of the Data

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 2: Descriptive Statistics”
 * “Section 7: Measures of the Center of the Data”

Instructions: Please read the section above in its entirety.

 * Lectures: Khan Academy’s Statistics
 * “The Average”
 * “Sample vs. Population Mean”

Instructions: Please view the linked lecture titled “Statistics: The Average” in its entirety (12:35 minutes). This lecture provides an introduction to descriptive statistics and central tendency as well as ways to measure the average of a set using median, mean, and mode. Then, view the lecture titled “Sample vs. Population Mean” in its entirety (6:42 minutes). This lecture discusses the difference between the mean of a sample and the mean of a population.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages displayed above.

Skewness and the Mean, Median, and Mode

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 2: Descriptive Statistics”
 * “Section 8: Skewness and the Mean, Median, and Mode”

Links: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 2: Descriptive Statistics,” “Section 8: Skewness and the Mean, Median, and Mode”

Instructions: Please read the section above in its entirety.

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

Measures of the Spread of the Data

 * Readings: Barbara Illowsky and Susan Dean’s Collaborative Statistics: “Chapter 2: Descriptive Statistics”
 * “Section 9: Measures of the Spread of the Data”

Instructions: Please read the section above in its entirety.

 * Lecture: Barbara Illowsky and Susan Dean’s Collaborative Statistics
 * “Video Lecture 2: Descriptive Statistics”

Instructions: Please view the lecture to the right in its entirety.

 * Lectures: Khan Academy’s Statistics
 * “Variance of a Population”
 * “Sample Variance”
 * “Standard Deviation”
 * “Alternate Variance Formulas”

Instructions: Please view all four lectures in their entirety. First, view “Variance of a Population” (12:23 minutes). Secondly, watch the lecture titled “Sample Variance” (12:23 minutes), which explains how to estimate the variance of a population using the variance of a sample. Then, watch the video titled “Standard Deviation” (13:07 minutes), which will provide a review for what you have learned so far as well as an introduction to the standard deviation. Finally, watch the video titled “Alternate Variance Formulas” (12:17 minutes) to learn alternates for the variance of a population.

 * Assignment: Barbara Illowsky and Susan Dean’s Collaborative Statistics: Chapter 2: Descriptive Statistics
 * “Practice 1: Center of the Data”
 * “Practice 2: Spread of the Data”

Instructions: Please solve all the problems in these two sections. Next, click on the hyperlink titled “Homework” and solve problems 1, 3, 5, 7, 9, 12, 14, 15, 17, 21, 24-30. Please click on “[ Show Solution ]” link below the problem to check your solution

Terms of Use: Please respect the copyright and terms of use displayed on the webpages.

About the Resources in This Course
This course project draws upon three main types of resources:

The first are readings and video lectures from Barbara Illowsky and Susan Dean’s Collaborative Statistics, which is available freely under a Creative Commons Attribution 2.0 Generic (CC BY 2.0) license from the following location: http://cnx.org/content/col10522/latest/

The second type of resources in this course are lectures from Kahn Academy. These lectures are available under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported (CC BY-NC-SA 3.0) license. Kahn Academy has many lectures available from http://www.khanacademy.org/

Finally, the above resources have been woven together and organized into a format analogous to a traditional college-level course by professional consultants that work as experts within the subject area. This process was facilitated by The Saylor Foundation. Additionally, if you have worked through all of the material contained in this project, you may be interested in taking the final exam provided by Saylor.org or completing other courses available there that are not yet on Wikiversity.

title: Statistical Analysis/Unit 1 Content