= Solution
A <planetary magnetic field> is evidence about electrical conductivity, fluid motion and thermal evolution, but its presence alone does not identify a present <planetary dynamo>. One must separate internal global fields, crustal <remanent magnetization> and currents driven by the <solar wind>. The following discussion uses the observational picture available by 2009, including the early MESSENGER flybys and the Voyager, Galileo and Cassini results.
Outside a planet's internal electric currents, the internal component is approximately a potential <magnetic field>. Its <spherical harmonic> of degree $\ell$ decreases as $r^{-(\ell+2)}$. Measurements at different positions and altitudes can therefore distinguish a deep, low-degree source from localized crustal anomalies and external magnetospheric currents. Repeated observations, radio emissions, aurorae and particle measurements supply complementary evidence for a <magnetosphere>, while slow secular variation and records of past reversals test the internal field's evolution.
For <Earth>, ground observatories and satellite measurements reveal a predominantly dipolar field of several tens of microtesla at the surface, tilted relative to the rotation axis and changing with time. Magnetic minerals in rocks record a much older field and repeated <geomagnetic reversals>. These observations, together with seismic evidence for a liquid outer core, support an operating <planetary dynamo> in conducting iron alloy. Core temperatures exceed the relevant <Curie temperatures>, so ordinary permanent <remanent magnetization> cannot explain the main field. Thermal <convection>, cooling and <compositional convection> associated with inner-core growth supply buoyancy; the <Coriolis force> organizes the motion. The observed nonaxisymmetric components and time dependence are consistent with a dynamic source, rather than a fixed bar magnet.
Mariner 10 and the first two MESSENGER flybys established a weak but coherent intrinsic field at <Mercury>, with roughly dipolar structure closely aligned with its rotation axis. It is much weaker than Earth's surface field, but its global character favors a deep source over an explanation solely by localized crustal patches. A <planetary dynamo> in a fluid part of Mercury's large iron-rich core is the natural interpretation, supported by evidence for a molten core component. Light elements can depress the alloy's freezing temperature, and thermal or compositional buoyancy can maintain motion. Slow rotation is not enough to rule out <dynamo action>. The limited pre-orbital measurements do not uniquely determine the detailed core structure or the mechanism responsible for the field's weakness.
For <Venus>, spacecraft including Pioneer Venus found no appreciable internally generated global field. Nevertheless its ionosphere interacts with the <solar wind>, producing magnetic draping, a bow shock and an <induced magnetosphere>. Those external signatures are not evidence for an internal <planetary dynamo>. Venus is comparable to Earth in size and may possess a substantial conducting core; the absence of a detectable dynamo therefore need not mean that all of its core is solid. Insufficient core heat loss or stable stratification can prevent the required <convection>. Its slow rotation can affect flow organization, but rotation rate by itself is not a sufficient explanation of the contrast with Earth or Mercury.
For <Mars>, Mars Global Surveyor found strong, spatially localized crustal anomalies, especially in ancient southern terrain, rather than an appreciable present global dipole. This is <remanent magnetization>, indicating that an ancient field magnetized the crust while it formed and cooled. The distribution of anomalies and weakly magnetized major impact basins constrain when that ancient <planetary dynamo> operated; crustal magnetization is a record of past activity, not a present global dynamo. Modern interaction with the <solar wind> also produces an <induced magnetosphere>. Cooling and changes in core heat extraction can explain the cessation of dynamo-driving <convection>; the observations alone do not prove complete solidification of the core.
For <Jupiter>, radio emissions, its intense radiation belts and measurements by Pioneer, Voyager and Galileo demonstrate a strong global intrinsic field and an enormous <magnetosphere>. The field is broadly dipolar but tilted and contains higher multipoles. It is generated in a deep conducting fluid rather than by a permanent solid magnet: the principal conducting region is <metallic hydrogen> in the high-pressure interior. Cooling and gravitational energy release maintain <convection>, while rapid rotation organizes the flow. Plasma supplied by Io and currents in the <magnetosphere> strongly modify the observed external field, but cannot account for the internally sourced planetary component. Jupiter's large magnetic moment also reflects its large size; a moment ratio should not be confused with the ratio of local surface field strengths.
For <Saturn>, Pioneer, Voyager and Cassini measurements likewise show an intrinsic field and a <magnetosphere>, with a dipole strikingly close to the rotation axis. A <planetary dynamo> in conducting <metallic hydrogen> is supported by the same broad interior argument as for Jupiter. Interior cooling and differentiation can provide power, including energy associated with helium separation. The near alignment does not invalidate the <Cowling anti-dynamo theorem>: that theorem excludes an exactly axisymmetric isolated dynamo field, not a predominantly axisymmetric exterior with undetected nonaxisymmetric structure in its source region. A stably stratified conducting layer above the active region is one possible way to attenuate the observable nonaxisymmetric components; it is a model, not a uniquely established consequence of the observations.
For <Uranus>, Voyager 2 in 1986 observed an intrinsic field with a dipole tilted by about $59^\circ$ to the rotation axis and significantly displaced from the centre. Higher multipoles are substantial, so a centred dipole is an inadequate description. The field and associated <magnetosphere> indicate an active <planetary dynamo>, despite the planet's small intrinsic heat flux. That low luminosity constrains its thermal evolution and may indicate inhibited deep mixing; it does not demonstrate an absence of all conducting-fluid motion. A conducting, volatile-rich fluid shell is a plausible source region. The term <ice giant> describes bulk composition, not a claim that the dynamo region is ordinary frozen ice.
For <Neptune>, Voyager 2 in 1989 also found a highly non-dipolar, displaced intrinsic field, with a dipole tilt near $47^\circ$. Its field, radio emissions and <magnetosphere> imply another active <planetary dynamo>. Its larger intrinsic heat flux than Uranus supplies evidence for substantial continuing interior energy transport. Electrically conducting water-ammonia-rich mixtures at high pressure, with ionic or otherwise mobile charges, can supply the conducting region without requiring the <metallic hydrogen> layer characteristic of <gas giants>. Numerical dynamos in convecting shells surrounding more stably stratified interiors can produce strongly nonaxisymmetric, multipolar fields resembling those of the two <ice giants>. Such models show that their unusual field geometry is compatible with <dynamo action>, but field measurements do not uniquely identify the shell's composition or thickness.
The maintenance mechanism is governed by the <resistive induction equation>,
$$
\partial_t\mathbf B=\nabla\times(\mathbf u\times\mathbf B)+\eta\nabla^2\mathbf B,
\qquad
\operatorname{Rm}=\frac{UL}{\eta},\qquad
\eta=\frac1{\mu_0\sigma_e}.
$$
A conducting fluid needs a sufficiently large <magnetic Reynolds number> and a flow geometry that regenerates magnetic flux rather than merely amplifying one component temporarily. Three-dimensional stretching and twisting can couple poloidal and toroidal fields; strictly planar regeneration and a strictly axisymmetric magnetic field are restricted by the relevant <anti-dynamo theorems>. Rotation influences this regeneration through the <Coriolis force>, but does not provide an inexhaustible power source merely by being present.
Without regeneration, the characteristic <magnetic diffusion> time is $\tau_\eta\sim L^2/\eta$. For a terrestrial core scale of order $10^6$ metres and diffusivity of order one square metre per second, this is of order $10^4$–$10^5$ years, far shorter than geological ages. Long-lived global fields therefore require continual energy input. A schematic <magnetic energy> balance is
$$
\frac{d}{dt}\int\frac{B^2}{2\mu_0}\,dV
=-\int\mathbf u\cdot(\mathbf J\times\mathbf B)\,dV
-\int\frac{J^2}{\sigma_e}\,dV,
$$
when boundary energy fluxes vanish. Mechanical work against the <Lorentz force density> supplies the energy lost as <Ohmic heating>. Thermal cooling, <latent heat> release, <compositional convection> and differentiation supply that mechanical power. Magnetic feedback eventually changes the flow and saturates the field. Strong conductivity without an adequate energy source is not sufficient for a dynamo.
\b[The 2009 observations support active global dynamos in Mercury, Earth, Jupiter, Saturn, Uranus and Neptune; Venus has no detected appreciable intrinsic dynamo field, and Mars preserves evidence of an extinct earlier dynamo in its crust.] Induced fields and surviving remanence explain why planets without a current global dynamo can still display magnetic phenomena. The diversity of field geometries and strengths constrains, rather than uniquely solves, the diversity of planetary conducting regions and their convective histories.
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