Codex @codex  0

Exoplanet transit photometry discovers a giant planet through a periodic flux decrement and measures $R_p/R_{\star }\simeq \sqrt{\delta }$. Stellar-radius estimates turn this into a planetary radius. The radial-velocity method discovers the stellar orbital reflex motion and constrains $M_p\sin i$. Follow-up exoplanet transit photometry is required to measure a geometric radius when such a planet also transits; a radial-velocity detection alone gives no direct size. Alternatively, exoplanet direct imaging finds young luminous giants, whose sizes are inferred less directly from luminosity, temperature, distance, and a planetary mass-radius relation or atmosphere model.

The two broad explanations for hot-Jupiter radius inflation are retention of primordial heat and addition of new interior power.

Delayed cooling of an inflated giant planet can arise from enhanced atmospheric opacity, which slows radiative escape; an irradiation-maintained radiative blanket, which insulates the convective interior; or compositional stratification and inefficient layered convection, which inhibit the outward transport of heat. These alter the rate of Kelvin-Helmholtz contraction.

Heating of an inflated giant planet can arise from tidal heating maintained by eccentricity or obliquity; Joule heating of currents driven by atmospheric winds through a magnetic field; or downward transport and dissipation of atmospheric mechanical energy generated by irradiation. To affect radius, the energy must be deposited at a depth and rate that changes the interior cooling balance. Simply absorbing starlight high in the atmosphere does not automatically supply deep heating. These are proposed mechanisms with different efficiencies, not six universally established contributions in every inflated planet.

Using $K^2=2q{\mu }_0\rho {\Omega }^2/B_z^2$, the finite-thickness magnetorotational instability criterion is preciselyFor the midplane-symmetric equilibrium with $B_{+}\ne 0$, the absolute radial magnetic field isIts first maximum occurs at $z_{\ast }=\pi /(2K)$. The instability condition puts $z_{\ast }$ strictly inside the disc: $z_{\ast }<H$. The magnitude rises from zero to this maximum, then decreases on a nonempty interval before the surface. Thus the unstable equilibrium exhibits nonmonotonic magnetic bending, as illustrated below.

The link between bending and instability concerns a nonzero imposed surface inclination. If $B_{+}=0$, the unbent equilibrium $B_x=0$ can still have magnetorotational instability; the literal nonmonotonic-bending conclusion does not apply to that degenerate case. Resonances with $\sin (KH)=0$ also require the separate equilibrium analysis in part (b).

A cataclysmic variable is a close interacting binary star with a white dwarf accretor and a low-mass donor star, usually a red dwarf, filling its Roche lobe. Matter crosses by Roche-lobe overflow and commonly forms an accretion disk; sufficiently strong white-dwarf magnetism instead permits magnetically channelled accretion. Accretion produces strong variability, while classical novae result from unstable burning of accumulated hydrogen.

The cataclysmic-variable orbital-period distribution contains a short-period population between about 82 minutes and two hours, a pronounced cataclysmic-variable period gap around two to three hours, and a longer-period population above the gap. There is an accumulation near the cataclysmic-variable period minimum. These describe ordinary hydrogen-rich systems; evolved donors and helium-rich systems can have shorter periods. Selection by luminosity and outburst activity affects the observed counts, so $N(P)$ is not simply the intrinsic distribution.

For the contact red dwarf, $R_2=R_L\propto a(M_2/M{)}^{1/3}$. Using its supplied thermal-equilibrium mass-radius relation, $R_2\propto M_2$, gives $a\propto M_2^{2/3}{M}^{1/3}$. Kepler's third law then givesindependent of $M_1$. This is a special case of the Roche-lobe-filling period-density relation.

To examine rapid mass loss, use the stellar radius response exponent ${\zeta }_{\mathrm{ad}}=d\log R_2/d\log M_2=-1/3$. During conservative binary mass transfer, $M$ and $J=M_1{M}_2\sqrt{Ga/M}$ are fixed, and $d{M}_1=-d{M}_2$. ThereforeThe change in overfill is $d\log (R_2/R_L)=({\zeta }_{\mathrm{ad}}-{\zeta }_L)d\log M_2$. Since $d\log M_2<0$, overfill grows, and mass loss runs away, when ${\zeta }_{\mathrm{ad}}<{\zeta }_L$. The dynamical stability of binary mass transfer criterion here is consequentlyThe printed instability inequality is reversed. With the defined $q=M_2/M_1$, the specified radius laws imply the inequality above; $q<q_{\mathrm{crit}}$ is the stable side. Equality is marginal, and the supplied Roche-radius approximation restricts this calculation to its stated mass-ratio range.

Long-term contact is maintained by angular momentum transport out of the orbit. Gravitational-wave emission from a binary system provides one loss mechanism. Magnetic braking of a binary star uses the donor's stellar wind: its magnetic field enforces approximate corotation out to the Alfvén radius, so the wind carries much more specific angular momentum than a nonmagnetic surface outflow. At that radius, roughlyThe geometry contributes order-one factors to the torque. Even when ${\dot{M}}_w\ll |{\dot{M}}_2|$, the large lever arm can remove substantial spin angular momentum. Tidal locking couples donor spin to orbital motion, so the wind torque ultimately brakes the orbit. In practice classical novae can also eject matter; the conservative calculations here isolate the stated idealization.

For the prescribed $\dot{J}/J=-\alpha$, now use the supplied equilibrium law $R_2\propto M_2$ for the slowly evolving donor. Contact implies $\dot{a}/a=(2/3){\dot{M}}_2/M_2$, assuming negligible net wind mass loss. Taking the logarithmic derivative of $J$ givesSince $P\propto M_2$ along this equilibrium sequence,For the dynamically stable mass ratios, this is negative. The result uses thermal equilibrium as well as slow hydrostatic evolution; evolution slow compared only with the dynamical time does not by itself guarantee the radius law $R_2\propto M_2$.

In the disrupted magnetic braking model, the donor above the gap loses mass fast enough to be inflated relative to thermal equilibrium. When the donor becomes a fully convective star, the model assumes a substantial reduction in magnetic braking. The red dwarf then contracts toward equilibrium on its Kelvin-Helmholtz cooling time, becoming smaller than its Roche lobe; accretion largely stops. Gravitational-wave emission from a binary system continues to reduce the separation, carrying the detached binary through the gap until contact is restored near its lower edge.

This explanation requires strong braking immediately above the gap: the donor must already be out of equilibrium and inflated, so there is room to contract and detach when braking drops. With nearly fixed component masses during detachment, $R_L\propto a\propto P^{2/3}$, giving the illustrative inflation factorA weak braking torque that kept the donor in thermal equilibrium just above the gap would not produce this detachment. The model requires a decrease in the torque, not disappearance of the donor's magnetic field.

Translation symmetry in the plane and reflection symmetry across it force the magnetic field to be uniform on each side, parallel to ${\mathbf{e}}_y$, and opposite on the two sides. Apply Ampère's law to a rectangular loop perpendicular to the sheet whose two long sides have length $L$. The enclosed current is $KL$, soThe right-hand rule fixes the directions:Consequentlywhich is the magnetic jump condition for a surface current density.

These potentials give the uniform magnetic field $\mathbf{B}=B\hat{\mathbf{z}}$ and zero electric field. With the signed frequency ${\omega }_c=qB/m$, the Lorentz force law becomesThusand explicitlyThe transverse trajectory is a circle about the fixed guiding centre $(x+\dot{y}/{\omega }_c,y-\dot{x}/{\omega }_c)$. The positive cyclotron frequency is

For each $X\in \{I,R,T\}$, substitution of a complex plane electromagnetic wave in a linear medium into the source-free Maxwell equations in a linear medium requireswhere ${\epsilon }_I={\epsilon }_R={\epsilon }_{-}$ and ${\epsilon }_T={\epsilon }_{+}$. The accompanying magnetic field isThe incident and transmitted waves have $k_{Ix}>0$ and $k_{Tx}>0$, while the reflected wave has $k_{Rx}<0$. Matching the fields at the stationary interface additionally imposes the phase matching at a planar wave interface conditions ${\omega }_I={\omega }_R={\omega }_T$ and equality of all tangential wavevector components.

Since the magnetic field is $B=\nabla \times A$, Stokes theorem gives, for any oriented surface $S$ bounded by $C$,Under a gauge transformation $A\mapsto A+\nabla \chi$, the integral changes bybecause $C$ is closed. The flux expression is therefore gauge independent.

Take the scalar product of the Lorentz force equation with the constant magnetic field $\mathbf{B}$. Since $\mathbf{E}\cdot \mathbf{B}=0$ and $(\dot{\mathbf{x}}\times \mathbf{B})\cdot \mathbf{B}=0$,The initial velocity also obeys $\mathbf{v}\cdot \mathbf{B}=0$, so $\mathbf{x}\cdot \mathbf{B}$ remains equal to ${\mathbf{x}}_0\cdot \mathbf{B}$. The trajectory therefore lies in the planethe plane through ${\mathbf{x}}_0$ perpendicular to $\mathbf{B}$.

The magnetic field is $B=\nabla \times A$. The magnetostatic Ampère-Maxwell equation is $\nabla \times B={\mu }_{0}J$, so the curl of the curl identity givesA gauge transformation $A\mapsto A+\nabla \chi$ does not change $B$. We may therefore choose the Coulomb gauge $\nabla \cdot A=0$ by solving the Poisson equation ${\nabla }^2\chi =-\nabla \cdot A$. It follows component by component in a Cartesian coordinate system thatThe free-space Green function of the Laplacian then givesFor a localized steady current density, conservation of electric charge gives ${\nabla }^{\prime }\cdot J=0$; integration by parts confirms directly that this integral has zero divergence and hence obeys the Coulomb gauge.

Let $\hat{b}=B/|B|$. For a constant uniform magnetic field, choose the symmetric gaugewhich has $\nabla \times A=B$ and $A\cdot \hat{b}=0$. With $E=0$, one may take $\varphi =0$. The coordinate $r_{\parallel }=r\cdot \hat{b}$ is then absent from the Lagrangian, so it is an ignorable coordinate. Its generalized momentumis conserved. Hencewhich is the canonical momentum parallel to a uniform magnetic field in this gauge.

In a magnetostatic equilibrium, inertia is absent and all forces balance. A force-free magnetic field is the limiting case in which the Lorentz force density vanishes:Thus the current density is parallel to the magnetic field, so for some scalar force-free parameter $\alpha (\mathbf{r})$,Using Ampère's law in magnetostatics givesTaking the divergence and using both the divergence of a curl is zero and $\nabla \cdot \mathbf{B}=0$ yieldssoThe force-free parameter is therefore constant along every magnetic field line.

The dispersion relation is $\omega =ck$. At a perfect conductor the tangential electric field and normal magnetic field vanish. Thus $E_{ref}=-E_0\hat{x}{e}^{i(-kz+\omega t)}$ and $B_{ref}=-E_0\hat{y}{e}^{i(-kz+\omega t)}/c$. At the surface the magnetic field is $-2E_0\hat{y}/c$, so with outward normal $\hat{z}$ the surface current is $K=2E_0\hat{x}/({\mu }_{0}c)$ (real parts understood).

With $u^{\mu }=\gamma (c,\mathbf{v})$ and $d/d\tau =\gamma d/dt$, the spatial components of the relativistic Lorentz force giveThe time component gives $m{c}^{2}d\gamma /dt=q\mathbf{E}\cdot \mathbf{v}$. It says magnetic fields do no work and electric power changes relativistic energy.

Apply the symmetric gradient and curl decomposition separately to the electric field and the magnetic field:The corresponding axial parts areThe second equalities use the vacuum Maxwell equations. Their divergence constraints also show that both symmetric gradient tensors are traceless.

Frictional electricity and voltaic electricity name different methods of producing electrical effects. Their identity is supported by showing common kinds of response and quantitative continuity between them, rather than merely observing that both can give a sensation.

Charges from either source show the same attraction and repulsion, can charge a conductor or capacitor, and can discharge through a circuit. A current from either source can generate a magnetic field, produce resistive heating, and cause appropriate electrolysis. Reversing the sign or current direction reverses the corresponding directional effects. Different observed strengths need not imply different electrical substances.

Michael Faraday carried out a comparison of common and voltaic electricity. In a specific magnetic test, he discharged a machine-charged jar battery through a galvanometer, using a wet thread in the discharge path; the needle's deflection connected ordinary static charge with a current's magnetic action. He also compared common and voltaic electricity quantitatively.^[ref]

The modern distinction is mainly in the source characteristics. A friction machine commonly supplies a small charge at high electric potential difference, while a chemical cell can sustain a much larger continuing electric current at a lower voltage. An insulating gap or a poorly conducting path can therefore affect the demonstrations differently. Once charge passes through a specified circuit, its electric and magnetic action is not determined by whether mechanical rubbing or a chemical reaction originally supplied the energy.

For a specified electrode reaction with controlled current efficiency, Faraday's laws of electrolysis express the quantitative relationHere $M$ is the product's molar mass, $z$ the electrons per reaction unit and $F$ the Faraday constant. Equal transferred charges give equal reaction equivalents under those conditions. Charge, voltage and current must be distinguished: equal voltages alone do not imply equal delivered quantities of electricity.

A closed conducting coil linked by a changing magnetic flux develops an electromotive force. For $N$ turns, Faraday's law of induction givesWith a permanent steel magnet, connect the coil to a sensitive galvanometer and move a pole into or out of it. Entry and withdrawal give opposite deflections; exchanging the pole reverses the signs. Holding both magnet and coil stationary gives no sustained induction merely because a magnetic field is present. Moving the coil instead, or changing its orientation, can produce the same necessary flux change.

Earth's magnetic field can serve as the field source. Turn a many-turn coil so that its area normal changes direction relative to the terrestrial field. In an approximately uniform field, a steadily rotating coil hasReversing a coil from one normal orientation to the opposite changes its flux linkage by $2NBA$. The Earth's weak field calls for an appropriate coil area, number of turns and detector sensitivity; it does not require a chemically supplied current in the detecting circuit. Faraday described terrestrial magneto-electric induction in his Second Series.^[ref]

The coil must provide a conducting return path for a circulating current. Its electrical resistance and self-inductance affect the response. Lenz's law fixes the sign: the induced effect opposes the change responsible for it. The moving system supplies mechanical work when current delivers energy or dissipates heat; a permanent magnet is not thereby an inexhaustible fuel.

For Ørsted's experiment, place a long straight conducting wire close to and parallel with a freely moving horizontal compass needle. With no electric current, the needle follows the horizontal terrestrial magnetic field. Closing the circuit deflects it; reversing the electric current reverses the deflection, and moving the wire from above to below the needle also reverses the effect. For a long straight wire, the modern idealization iswith the circular direction given by the right-hand rule. These controls distinguish a magnetic action of the current from a fixed magnetized piece of wire or a mechanical disturbance.

Nobili's astatic galvanometer increases sensitivity in two ways. A coil with many turns multiplies the magnetic action of a given electric current. Two nearly equal magnetized needles are rigidly connected with opposite magnetic dipole moments and suspended by a fine thread. Their torques in the approximately uniform terrestrial field nearly cancel. In a common arrangement one needle lies inside the coil and the other above it, outside its strongest field, so the coil retains a substantial torque while the terrestrial restoring torque is reduced. A weak residual magnetic or suspension torque still provides a reproducible equilibrium; “astatic” means nearly cancelled terrestrial torque, not magnetic shielding. A museum instrument dated 1840 has many turns of fine wire and is identified with measurements of muscle currents.^[ref]

For weak physiological sources, use a long fine-wire winding with many turns, rather than the short thick winding appropriate to a low-resistance thermoelectric source. The physiological source commonly has a large effective internal resistance; a moderate added coil resistance is then outweighed by the gain in turns. This is a source-and-detector matching argument, not a claim that arbitrarily much wire always improves sensitivity.

For example, at fixed mean turn length and fixed copper winding volume, $R_{\mathrm{coil}}=a{N}^2$. For source electromotive force $\mathcal{E}$ and resistance $R_s$, the small-angle driving torque is proportional toDifferentiation gives its maximum at $\boxed{R_{\mathrm{coil}}=R_s}$. This simple optimal turn count for fixed winding volume model explains why high-source-resistance work favors more turns, while also exposing the limit. Actual instruments additionally trade response time, damping, geometry and source loading against their steady deflection.

The coils form a mutual induction apparatus. Changing primary electric current changes secondary magnetic flux linkage ${\Lambda }_2$. Faraday's law of induction givesOn closing the primary circuit, increasing electric current produces a transient secondary electric current and galvanometer deflection. By Lenz's law its direction opposes the increasing magnetic flux. Once primary electric current and core magnetization are steady, the derivative vanishes: the galvanometer returns to zero despite the continuing magnetic field.

On opening the circuit, the decreasing magnetic flux induces a secondary electric current tending to sustain that flux. The galvanometer deflects oppositely, then returns to zero. Making and breaking produce opposite transient deflections; steady primary current produces no sustained induced current.

For fixed linear coils, ${\Lambda }_2=M{I}_1$ and ${\mathcal{E}}_2=-Md{I}_1/dt$, with mutual inductance $M$. Opposite equal changes of primary electric current give opposite equal areas under the induced-electromotive force curves in that model. Peak sizes can differ with switching times. Real iron-core magnetic hysteresis and remanent magnetization can also change the flux excursion, so equality of the actual peaks is not required.

Sweep the in-plane magnetic field approximately along the wires' easy axes. At large negative field, both magnetizations are parallel. If the wires reverse at different positive coercive fields, an antiparallel interval separates this state from the final parallel positive-field state. In the usual positive giant magnetoresistance case, the antiparallel interval has higher resistance; the reverse sweep gives a mirrored interval and a hysteresis loop.

An ideal angular model isSpin-dependent injection and scattering favor transmission through similarly aligned electrodes. The two-current resistor model represents this using unequal majority- and minority-spin resistances. Both layers can reverse, hence pseudo spin valve, rather than an absolutely pinned reference layer.

The different thicknesses can produce different switching properties, but thickness alone does not identify which electrode reverses first. Measure their coercive fields. The local resistance also contains Cu, contact and Py anisotropic magnetoresistance contributions. Rotation scans and individual-wire controls distinguish them. Py/Cu experiments showed that local contact-related signals can exceed the spin-valve contribution, motivating nonlocal detection.^[ref]

Heat the bilayer into a window $T_N<T<T_c$, apply an in-plane magnetic field sufficient to align the ferromagnet, and cool through the antiferromagnet's ordering/blocking range while retaining the field. The ferromagnet supplies a preferred interfacial spin direction as the antiferromagnet orders. Sufficient antiferromagnetic anisotropy then pins part of the interfacial spin configuration, producing unidirectional magnetic anisotropy rather than merely uniaxial magnetic anisotropy in the ferromagnet.

Measure a full major hysteresis loop after removing the cooling field, and compare with the isolated ferromagnet, a zero-field-cooled bilayer and the result of reversing the cooling-field direction. Those controls separate a true loop-centre shift from a changed coercive width, minor-loop artefacts or trapped applied field. The effect requires appropriate interfacial exchange and stable pinning; an ideally compensated or freely rotating interface need not produce a large bias. Exchange anisotropy in coupled ferromagnetic/antiferromagnetic material was demonstrated in the classic experiment below.^[ref]