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Near the polar axis of a magnetic dipole field, $B\propto r^{-3}$. Conservation of magnetic flux in a flux tube consequently gives $A(r)\propto r^3$. A dipole magnetic-field line intersecting a sphere of radius $R_{\star }$ at small polar angle ${\theta }_{\star }$ obeysThis area is for one polar tube. The approximation requires ${\theta }_{\star }^{2}r/R_{\star }\ll 1$; the full magnetic dipole field does not remain a narrow polar tube at arbitrarily large radius.

In magnetically channelled accretion, a strong magnetic field guides an accretion flow along flux tubes. The component of the Lorentz force density parallel to the magnetic field vanishes, so the longitudinal flow can be treated hydrodynamically while the field controls the transverse geometry. A magnetic dipole field near a neutron star channels matter toward small polar caps.

In the narrow polar flux tube, the magnetic field lines are almost radial and the transverse width is much smaller than the radial scale. The Lorentz force density has no component along the magnetic field, so the longitudinal magnetically channelled accretion is hydrodynamic. Let $v(r)>0$ denote inward speed. Mass conservation, the Euler equations for an inviscid fluid, and the polytropic equation of state giveUsing the adiabatic sound speed $c^2=\gamma K{\rho }^{\gamma -1}$ and $A\propto r^3$,The Bernoulli equation iswhere the right-hand side comes from matching to a nearly stationary reservoir with negligible gravitational potential. The fixed magnetic field provides transverse confinement; neglecting magnetic forces in this one-dimensional equation concerns their longitudinal component.

A regular sonic point must make both sides of the flow equation vanish:Substituting into the Bernoulli equation yieldsFor $\gamma >1$, positive finite sound speed therefore requires the critical adiabatic index for dipolar accretionTo check that this gives real regular crossings, differentiate the flow equation at the sonic point and put $x=r_s{v}_s^{\prime }/c_s$. The transonic accretion in a power-law tube calculation givesThe minus sign gives the transonic branch whose Mach number increases inward. At $\gamma =7/5$ the positive-energy reservoir cannot match a finite sonic point; for larger $\gamma$ the required $c_s^2$ is negative. A physical surface-crossing solution also requires $r_s>R_{\star }$ and validity of the narrow flux tube approximation up to the sonic region.

The dipole magnetic-field line geometry gives the angular radius at the sonic point as ${\theta }_s\simeq {\theta }_{\star }\sqrt{r_s/R_{\star }}$. The sonic region must remain a narrow flux tube for the one-dimensional transonic branch equations to apply. Thus a necessary small-angle constraint isThis also makes the transverse sound-crossing time $r_s{\theta }_s/c_s$ small compared with the radial flow time $r_s/c_s$. The restriction must hold at any larger matching radius as well. To match to a reservoir with negligible gravitational potential within the same narrow dipolar approximation, one needs an overlap regionHence the stronger useful sufficient scaling is ${\theta }_{\star }^2\ll R_{\star }{c}_0^2/(G{M}_{\star })$. The small-angle dipolar model should not be extrapolated to literal infinity, where its boundary magnetic field line turns toward the equatorial plane.

The dipolar flux-tube area has $A(r)=A_{\star }(r/R_{\star }{)}^3$. Applying transonic accretion in a power-law tube with $n=3$ and a polytropic equation of state givesHere $c_0$ and ${\rho }_0$ are the reservoir sound speed and mass density, and $c_s$ is the sound speed at the sonic point. The mass accretion rate isFor two equal polar caps of surface angular radius ${\theta }_{\star }\ll 1$, $A_{\star }\simeq 2\pi R_{\star }^2{\theta }_{\star }^2$. One loaded cap gives half the total rate. The finite-radius transonic branch exists for $1<\gamma <7/5$, must cross outside the star ($r_s>R_{\star }$), and requires ${\theta }_{\star }^2{r}_s/R_{\star }\ll 1$ for the sonic region to remain a narrow flux tube.